els ele af ens esse
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have
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ARTES
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OF MIC
Q
158
B29
1855
V. I







ELEMENTS
OF
414
LIBRARY
OF THE
ERSITY OF MICH
NATURAL PHILOSOPHY.
BY
bisevod?
Alleem
almes
hambers
W. H. C. BARTLETT, LL. D.,
PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE UNITED STATES
MILITARY ACADEMY AT WEST POINT.
Y wall lo
1.-MECHANIC S.
THIRD EDITION.
NEW YORK:
PUBLISHED BY A. S. BARNES AND COMPANY,
NO. 51 JOHN-STREET.
CINCINNATI: H. W. DERBY AND COMPANY.
1855.

2014 MARTETE

Entered, according to Act of Congress, in the year One Thousand
Eight Hundred and Fifty,
By W. H. C. BARTLETT,
In the Clerk's Office of the District Court of the United States for the Southern District
of New York,

Reclassed 9-30-37 yu
PREFACE.
THE present volume is the first of three in which its author desires
to offer, to academies and colleges, a course of Natural Philosophy,
including Astronomy. It embraces the subject of Mechanics-the
ground-work of the whole. It is intended to be complete within
itself, and to have no necessary dependence, for the full compre-
hension of its contents, upon those which are to follow. In its
preparation, constant reference was made to the admirable labors of
M. PONCELET, and much valuable assistance was derived from the
work of M. PESCHEL.
Large type, marginal notes, tables of reference, and numerous
diagrams, often repeated, have swollen the volume beyond the limits
originally intended; but whatever of inconvenience may thence arise,
will, it is hoped, be more than compensated by the facilities which
these sources of increased size cannot fail to bring to the aid both of
the teacher and student.

3011719
iw ni sont to to
galloo bu
திருக
bagged onlor oft aligus send fotojos
gohd
A gributo
M

Dbjects of Physical Science.
Physics of Ponderable Bodies
Primary Properties of Bodies
Secondary Properties
Force
CONTENTS.
INTRODUCTION.
Page
9
12
13
14
19
PART FIRST.
MECHANICS OF SOLIDS.
Space, Time, Motion, and Force.
Action of Forces, Equilibrium, Work.
Varied Motion.....
I.
29
II.
42
III.
67
IV.
102
Forces whose Directions meet in a Point..
V.
... 123
Forces whose Directions are Parallel......

6
CONTENTS.
Page
VI.
134
Centre of Gravity of Bodies
VII.
Motion of Translation of a Body or System of Bodies
150
VIII.
ЕТИОЦ
Equilibrium of a System of Heavy Bodies..
IX.
162
Equilibrium of Several Forces, Virtual Velocities, and Motion of a Solid Body.. 167
KOIT
X.
Motion and Equilibrium of a Body about an Axis
Central Forces..
XI.
XII.
Motion of the Heavenly Bodies ....................
Pendulum....
Balistic Pendulum ...
Funicular Machine
..... 180
200
230
XIII.
241
266
XIV.
XV.
Action of Bodies resting upon each other, and upon Inclined Planes........
XVI.
271
299
Friction and Adhesion..
313
Tables of Friction without Unguents
319
Tables of Friction of Unctuous Surfaces...
322
Tables of Friction with Unguents...
324
Table of Friction on Trunnions in their Boxes.
365

CONTENTS.
7
The Wedge
XVII.
Page
345
XVIII.
Stiffness of Cordage.
379
Table of Rigidity of Cordage
382
XIX. SOPILY.
Wheel and Pulley
Wheel and Axle
XX.
387
406
Screw..
Lever
428
XXI.
440
XXII.
Atwood's Machine....
444
XXIII.
Impact of Bodies....
..... 458
PART SECOND.
MECHANICS OF FLUIDS.
I.
Introductory Remarks
476
II.
Mechanical Properties of Fluids
.. 480
III.
Work of the Power and of the Resistance..
.. 488
IV.
Pressure of Heavy Fluids...
Table of Tenacities..
493
508

8
CONTENTS.
Page
V.X
510
Equilibrium of Floating Bodies..
Specific Gravity.
VI.
MX
Table of the Densities of Water at different Temperatures.
Table of Specific Gravities..
VII.
517
522
.... 531
Compressible Fluids
536
ebA ban fool W
VIII.
XX
Air-Pump
540
IX.
Weight and Pressure of the Atmosphere.....
550
X.
Mariotte's Law..
.. 556
XI.
Law of the Pressure, Density, and Temperature..
... 559
XII.
Barometer
.... 556
Table for Finding Height by the Barometer
578
оOTа THAT
XIII.
Water-Pumps
Siphon
Motion of Fluids.
Motion of Gases and Vapors
Table of Aperture Coefficients...
Discharge of Fluid through Pipes..
XIV.
582
.. 595
XV.
598
XVI.
.. 606
618
XVII.
622

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NATURAL PHILOSOPHY.
singgro no dily bowohns
ed you aboy Jugpianos ziedT
r off
edt to Blanta bai
gra edt et borso
INTRODUCTION.
To brodt femilo
odbed inllante
THE term nature is employed to signify the assemblage Nature.
of all the bodies of the universe; it includes whatever
od
exists and is the subject of change. Of the existence of
these bodies we are rendered conscious by the impressions Bodies.
they make on our senses. Their condition is subject to a
variety of changes, whence we infer that external causes
of
200
The Loa
are in operation to produce them; and to investigate Physical science.
nature with reference to these changes and their causes,
is the object of Physical Science.
bodies.
All bodies may be distributed into three classes, viz.: Classification of
unorganized or inanimate, organized or animated, and the
heavenly bodies or primary organizations.
The unorganized or inanimate bodies, as minerals, Inanimate
water, air, form the lowest class, and are, so to speak, definite period,
bodies, no
the substratum for the others. These bodies are acted no life.
on solely by causes external to themselves; they have
no definite or periodical duration; nothing that can
properly be termed life.
organs, vitality.
The organized or animated bodies, are more or less Animated bodies,
perfect individuals, possessing organs adapted to the per-
formance of certain appropriate functions. In consequence
of an innate principle peculiar to them, known as vitality,
bodies of this class are constantly appropriating to them-
selves unorganized matter, changing its properties, and

10
NATURAL PHILOSOPHY.
and limited
duration.
deriving, by means of this process, an increase of bulk.
Reproduction, They also possess the faculty of reproduction. They
retain only for a limited time the vital principle, and,
when life is extinct, they sink into the class of inanimate
bodies. The animal and vegetable kingdoms include all
the species of this class on our earth.
Animal and
vegetable
kingdoms.
Celestial bodies;
organs-
continents, ocean,
atmosphere.
Earth existed
The celestial bodies, as the fixed stars, the sun, the
comets, planets and their secondaries, are the gigantic
individuals of the universe, endowed with an organization
on the grandest scale. Their constituent parts may be
compared to the organs possessed by bodies of the second
class; those of our earth are its continents, its ocean,
its atmosphere, which are constantly exerting a vigorous
action on each other, and bringing about changes the
most important.
The earth supports and nourishes both the vegetable
long before plants and animal world, and the researches of Geology have
and animals.
demonstrated, that there was once a time when neither
plants nor animals existed on its surface, and that prior to
the creation of either of these orders, great changes must
have taken place in its constitution. As the earth existed
thus anterior to the organized beings upon it, we may
Heavenly bodies infer that the other heavenly bodies, in like manner, were
-the support of called into being before any of the organized bodies which
animals and
vegetables.
Natural
philosophy,
probably exist upon them. Reasoning, then, by analogy
from our earth, we may venture to regard the heavenly
bodies as the primary organized forms, on whose surface
both animals and vegetables find a place and support.
Natural Philosophy, or Physics, treats of the general
external changes. properties of unorganized bodies, of the influences which
act upon them, the laws they obey, and of the external
changes which these bodies undergo without affecting
their internal constitution.
Chemistry;
Chemistry, on the contrary, treats of the individual
properties of bodies, by which, as regards their constitu-

INTRODUCTION.
11
tion, they may be distinguished one from another; it also Internal changes.
investigates the transformations which take place in the
interior of a body-transformations by which the sub-
stance of the body is altered and remodelled; and lastly,
it detects and classifies the laws by which chemical
changes are regulated.
anatomy,
Natural History, is that branch of physical science Natural History-
which treats of organized bodies; it comprises three chemistry,
divisions, the one mechanical-the anatomy and dissec- physiology.
tion of plants and animals; the second, chemical-animal
and vegetable chemistry; and the third, explanatory-
physiology.
physical.
Astronomy teaches the knowledge of the celestial Astronomy,
bodies. It is divided into Spherical and Physical astron-spherical and
omy. The former treats of the appearances, magnitudes,
distances, arrangements, and motions of the heavenly
bodies; the latter, of their constitution and physical con-
dition, their mutual influences and actions on each other,
and generally, seeks to explain the causes of the celestial
phenomena.
physical laws.
Again, one most important use of natural science, is
the application of its laws either to technical purposes- Application of
mechanics, technical chemistry, pharmacy, &c.; to the phe-
nomena of the heavenly bodies-physical astronomy; or to
the various objects which present themselves to our notice
at or near the surface of the earth-physical geography,
meteorology-and we may add geology also, a science which
has for its object to unfold the history of our planet from
its formation to the present time.
philosophy, a
Natural philosophy is a science of observation and ex- Natural
periment, for by these two modes we deduce the varied science of
information we have acquired about bodies; by the observation and
former we notice any changes that transpire in the condi-
tion or relations of any body as they spontaneously arise
experiment.

12
NATURAL PHILOSOPHY.
Apparatus;
experimental
physics.
without interference on our part; whereas, in the per-
formance of an experiment, we purposely alter the natural
arrangement of things to bring about some particular con-
dition that we desire. To accomplish this, we make use
of appliances called philosophical or chemical apparatus, the
proper use and application of which, it is the office of Ex-
perimental Physics to teach.
If we notice that in winter water becomes converted
Observation, into ice, we are said to make an observation: if, by
means of freezing mixtures or evaporation, we cause water
to freeze, we are then said to perform an experiment.
experiment.b
These experiments are next subjected to calculation,
by which are deduced what are sometimes called the laws
Laws of nature. of nature, or the rules that like causes will invariably pro-
duce like results. To express these laws with the greatest
possible brevity mathematical symbols are used. When
it is not practicable to represent them with mathematical
precision, we must be contented with inferences and
assumptions based on analogies, or with probable ex-
planations or hypotheses.
Hypotheses and
probability of
their truth.
A hypothesis gains in probability the more nearly it
accords with the ordinary course of nature, the more
numerous the experiments on which it is founded, and the
more simple the explanation it offers of the phenomena
for which it is intended to account.
PHYSICS OF PONDERABLE BODIES.
Physical
properties; the
senses.
All the senses
not equally
employed.
§ 1.-The physical properties of bodies are those ex-
ternal signs by which their existence is made evident to
our minds; the senses constitute the medium through
which this knowledge is communicated.
All our senses, however, are not equally made use of
for this purpose; we are generally guided in our decisions
by the evidence of sight and touch. Still sight alone is

INTRODUCTION13
ITAW
.
Hod
frequently incompetent, as there are bodies which cannot
be perceived by that sense, as, for example, all colorless
gases; again, some of the objects of sight are not sub-
stantial, as, the shadow, the image in a mirror, spectra
formed by the refraction of the rays of light, &c. Touch, Touch.
on the contrary, decides indubitably as to the existence of
any body.
secondary
properties of
The properties of bodies may be divided into primary Primary and
or principal, and secondary or accessory. The former, are
such as we find common to all bodies, and without which bodies.
we cannot conceive of their existing; the latter, are not
absolutely necessary to our conception of a body's ex-
istence, but become known to us by investigation and
experience.
todyle
gal tognat od
PRIMARY PROPERTIES.
§ 2. The primary properties of all bodies are extension
and impenetrability.
and thickness.
Extension is that property in consequence of which Extension;
every body occupies a certain limited space. It is the length, breadth,
condition of the mathematical idea of a body; by it, the
volume or size of the occupied space, as well as its boun-
dary, or figure, is determined. The extension of bodies is
expressed by three dimensions, length, breadth, and thick-
ness. The computations from these data, follow geometri-
cal rules.
Impenetrability is evinced in the fact, that one body Impenetrability.
cannot enter into the space occupied by another, without
previously thrusting the latter from its place.
A body then, is whatever occupies space, and possesses
extension and impenetrability. One might be led to im- Body defined.
agine that the property of impenetrability belonged only
to solids, since we see them penetrating both air and Air and water
water; but on closer observation it will be apparent that impenetrable.
this property is common to all bodies of whatever nature.

14
NATURAL PHILOSOPHY.
Experiment.
Experiment.
If a hollow cylinder into which a piston fits accurately, be
filled with water, the piston cannot be thrust into the
water, thus showing it to be impenetrable. Invert a glass
tumbler in any liquid, the air, unable to escape, will pre-
vent the liquid from occupying its place, thus proving the
impenetrability of air. The diving-bell affords a familiar
illustration of this property.
The difficulty of pouring liquor into a vessel having
only one small hole, arises from the impenetrability of the
air, as the liquid can run into the vessel only as the air
makes its escape. The following experiment will illus-
trate this fact:
In one mouth of a two-
necked bottle insert a funnel
a, and in the other a siphon b,
the longer leg of which is im-
mersed in a glass of water.
Now let water be poured into
the funnel a, and it will be
seen that in proportion as this
water descends into the vessel
is
F, the air makes its escape
through the tube b, as
proved by the ascent of the
bubbles in the water in the
tumbler.
a
F
Fig. 1.
SECONDARY PROPERTIES.
Secondary
properties.
expansibility.
The secondary properties of bodies are compressibility,
expansibility, porosity, divisibility, and elasticity.
§3.-Compressibility is that property of bodies by
Compressibility, virtue of which they may be made to occupy a smaller
space; and expansibility is that in consequence of which
they may be made to fill a larger, without in either case
altering the quantity of matter they contain.

847
15
INTRODUCTION.
Both changes are produced in all bodies, as we shall Change of
presently see, by change of temperature; many bodies temperature,
may also be reduced in bulk by pressure, percussion, &c. percussion.
pressure,
§ 4. Since all bodies admit of compression and ex-
pansion, it follows of necessity, that there must be
interstices between their minutest particles; and that
property of a body by which its constituent elements do
not completely fill the space within its exterior boun-
dary, but leaves holes or pores between them, is called Porosity.
porosity. The pores of one body are often filled with Pores filled with
some other body, and the pores of this with a third, as in
the case of a sponge containing water, and the water in its
turn, containing air, and so on till we come to the most
subtle of substances, ether, which is supposed to pervade Ether pervades
all bodies and all space.
other bodies.
all bodies and all
space.
invisible pores
In many cases the pores are visible to the naked eye; Visible and
in others they are only seen by the aid of the microscope,
and when so minute as to elude the power of this instru-
ment, their existence may be inferred from experiment.
Sponge, cork, wood, bread, &c., are bodies whose pores
are noticed by the naked eye. The human skin appears
full of them, when viewed with the magnifying glass; the
porosity of water is shown by the ascent of air bubbles
when the temperature is raised.
85.-The divisibility of bodies is that property in Divisibility.
consequence of which, by various mechanical means,
such as beating, pounding, grinding, &c., we can reducé
them to particles homogeneous to each other, and to the
entire mass; and these again to smaller, and so on.
By the aid of mathematical processes, the mind may Infinite
be led to admit the infinite divisibility of bodies, though divisibility;
their practical division, by mechanical means, is subject practical
to limitation. Many examples, however, prove that it limitation.
may be carried to an incredible extent. We are fur- Smallness of some
nished with numerous instances among natural objects,
natural objects.

16
NATURAL PHILOSOPHY.
o dat whose existence can only be detected by means of the
most acute senses, assisted by the most powerful arti-
ficial aids; the size of such objects can only be calculated
approximately.
Mechanical
subdivisions in
the arts
Mechanical subdivisions for purposes connected with
the arts are exemplified in the grinding of corn, the
pulverizing of sulphur, charcoal, and saltpetre, for the
manufacture of gunpowder; and Homoeopathy affords a
remarkable instance of the extended application of this
To property of bodies.owlod Borog
tod Borog to solod aysel Judyb
Divisibility of Some metals, particularly gold and silver, are suscep-
gold.selbod odio tible of a very great divisibility. In the common gold
lace, the silver thread of which it is composed is covered
1
with gold so attenuated, that the quantity contained in a
abering dt foot of the thread weighs less than 600 of a grain. An
Ha a bod e inch of such thread will therefore contain 72300 of a grain
boldly of gold; and if the inch be divided into 100 equal parts,
soog oldiival each of which would be distinctly visible to the eye, the
Divisibility of
dyes.
quantity of the precious metal in each of such pieces
would be 7200000 of a grain. One of these particles ex-
amined through a miscroscope of 500 times magnifying
power will appear 500 times as long, and the gold covering
it will be visible, having been divided into 3,600,000,000
parts, each of which exhibits all the characteristics of
this metal, its color, density, &c.af
Dyes are likewise susceptible of an incredible divisi-
ability. With 1 grain of blue carmine, 10 lbs. of water
may be tinged blue. These 10 lbs. of water contain about
617,000 drops. Supposing now, that 100 particles of car-
mine are required in each drop to produce a uniform tint,
it follows that this one grain of carmine has been sub-
edindivided 62 millions of times. fant to
In the spider's According to Biot, the thread by which a spider lets
thread, thread of herself down is composed of more than 5000 single
the silkworm.
ethiothreads. The single threads of the silkworm are also of
emos to seonllame an extreme fineness. Sidiboroat
In blood.
Our blood which appears like a uniform red mass, con-

INTRODUCTION.
17
sists of small red globules swimming in a transparent fluid
called serum. The diameter of one of these globules does
not exceed the 4000th part of an inch: whence it follows
that one drop of blood, such as would hang from the
point of a needle, contains at least one million of these
globules.
investigations.
But more surprising than all, is the microcosm of organ- In the Infusoria.
ized nature in the Infusoria, for more exact acquaintance
with which we are indebted to the unwearied researches
of Ehrenberg. Of these creatures, which for the most
part we can see only by the aid of the microscope, there
exist many species so small that millions piled on each
other would not equal a single grain of sand, and thou-
sands might swim at once through the eye of the finest
needle. The coats-of-mail and shells of these animalcules
exist in such prodigious quantities on our earth that,
according to Ehrenberg's investigations, pretty extensive Ehrenberg's
strata of rocks, as, for instance, the smooth slate near Bilin,
in Bohemia, consist almost entirely of them. By micro-
scopic measurements 1 cubic line of this slate contains Microscopic
about 23 millions, and 1 cubic inch about 41,000 millions measurement,
of these animals. As a cubic inch of this slate weighs 220
grains, 187 millions of these shells must go to a grain, weight.
each of which would consequently weigh about the 197
millionth part of a grain. Conceive further that each of
these animalcules, as microscopic investigations have
proved, has his limbs, entrails, &c., the possibility vanishes
of our forming the most remote conception of the dimen-
sions of these organic forms.
smelling.
In cases where our finest instruments are unable to Divisibility
render us the least aid in estimating the minuteness of detected by
bodies, or the degree of subdivision attained; in other
words, when bodies evade the perception of our sight and
touch, our olfactory nerves frequently detect the presence
of matter in the atmosphere, of which no chemical analysis
could afford us the slightest intimation.
Thus, for instance, a single grain of musk diffuses in a Instance of musk.
2

18
NATURAL PHILOSOPHY.
Oil of lavender.
Elasticity, its
measure.
Examples of
elastic bodies.
ivory.
large and airy room a powerful scent that frequently lasts
for years; and papers laid near musk will make a voyage
to the East Indies and back without losing the smell.
Imagine now, how many particles of musk must radiate
from such a body every second, in order to render the
scent perceptible in all directions, and you will be aston-
ished at their number and minuteness.
In like manner a single drop of oil of lavender evapo-
rated in a spoon over a spirit-lamp, fills a large room with
its fragrance for a length of time.
§ 6.-Elasticity is the name given to that property of
bodies, by virtue of which they resume of themselves their
figure and dimensions when these have been changed or
altered by any extraneous cause. Different bodies possess
this property in very different degrees, and retain it with
very unequal tenacity. The measure of a body's elasticity,
is the ratio obtained by dividing the capacity of resti-
tution inherent in the body, by the capacity of the cause
producing the change, both being supposed measurable.
Thus, if R denote the capacity of restitution, F that of the
extraneous cause, and e the elasticity, then will
e=
R
F
When F and R are equal, the body is said to be perfectly
elastic; when R is zero, the body is said to be non-elastic.
These limits embrace all bodies in nature, there being
none known to us which reach either extreme.
The following are a few out of a large number of
highly elastic bodies; viz., glass, tempered steel, ivory,
whalebone, &c.
Let an ivory ball fall on a marble slab smeared with
Experiment with some coloring matter. The point struck by the ball
shows a round speck which will have imprinted itself
on the surface of the ivory without its spherical form
being at all impaired.

INTRODUCTION.
19
Fluids under peculiar circumstances exhibit considera-
melted metals.
ble elasticity; this is particularly the case with melted Elasticity of some
metals, more evidently sometimes than in their solid state.
The following experiment illustrates this fact with regard
to antimony and bismuth.
and antimony.
Place a little antimony and bismuth on a piece of Melted bismuth
charcoal, so that the mass when melted shall be about
the size of a peppercorn; raise it by means of a blowpipe
to a white heat, and then turn the ball on a sheet of paper
so folded as to have a raised edge all round. As soon
as the liquid metal falls, it divides itself into many minute
globules, which hop about upon the paper and continue
visible for some time, as they cool but slowly; the points
at which they strike the paper, and their course upon
it, will be marked by black dots and lines.
The recoil of cannon-balls is owing to the elasticity Recoil of
of the iron and that of the bodies struck by them.
cannon-balls.
FORCE.
87.-Whatever tends to change the actual state of a
body, in respect to rest or motion, is called a force. If a Forces.
body, for instance, be at rest, the influence which changes
or tends to change this state to that of motion is called
force. Again, if a body be already in motion, any cause
which urges it to move faster or slower, is called force.
nature; existence
Of the actual nature of forces we are ignorant; we Ignorant of their
know of their existence only by the effects they produce, known by their
and with these we become acquainted solely through the effects on bodies.
medium of the senses. Hence, while their operations are
going on, they appear to us always in connection with
some body which, in some way or other, affects our
senses.

20
NATURAL PHILOSOPHY.
Universal forces,
attractions, and
repulsions.
Atomical action;
attraction of
gravitation.
Force of cohesion
and of
dissolution.
Inertia,
Known by
experience;
passive in
character.
§ 8.-We shall find, though not always upon super-
ficial inspection, that the approaching and receding of
bodies or of their component parts, when this takes place
apparently of their own accord, are but the results pro-
duced by the various forces that come under our notice.
In other words, that the universally operating forces are
those of attraction and of repulsion.
89.-Experience proves that these universal forces are
at work in two essentially different modes. They are
operating either in the interior of a body, amidst the
elements which compose it, or they extend their influence
through a wide range, and act upon bodies in the aggre-
gate; the former distinguished as Atomical and Molecular
action, the latter as the Attraction of gravitation.
§ 10.-Molecular forces and the force of gravitation,
often co-exist, and qualify each other's action, giving rise
to those attractions and repulsions of bodies exhibited at
their surfaces when brought into sensible contact. This
resultant action is called the force of cohesion or of
dissolution, according as it tends to unite different bodies,
or the elements of the same body, more closely, or to
separate them more widely.
§ 11.-Inertia is that principle by which a body resists
all change of its condition, in respect to rest or motion. If
a body be at rest, it will, in the act of yielding its con-
dition of rest, while under the action of any force, oppose
a resistance; so also, if a body be in motion, and be
urged to move faster or slower, it will, during the act
of changing, oppose an equal resistance for every equal
amount of change. We derive our knowledge of this
principle solely from experience; it is found to be com-
mon to all bodies; it is in its nature conservative, though
passive in character, being only exerted to preserve the
rest or particular motion which a body has, by resisting

INTRODUCTION.
21
all variation in these particulars. Whenever any force
acts upon a body, the inertia of the latter reacts, and this
action and reaction are, as we shall see in the proper Action equal to
place, equal and directly opposed to each other.
reaction.
determined by
§ 12.-Molecular action chiefly determines the forms Forms of bodies
of bodies. All bodies are regarded as collections or molecular action.
aggregates of minute elements, called atoms, and are
formed by the attractive and repulsive forces acting
upon them at immeasurably small distances.
bodies;
Several hypotheses have been proposed to explain the Constitution of
constitution of a body, and the mode of its formation. Boscovich.
The most remarkable of these was by Boscovich, about
the middle of the last century. Its great fertility in
the explanations it affords of the properties of what is
called tangible matter, and its harmony with the laws
of motion, entitle it to a much larger space than can be
found for it in a work like this. Enough may be stated,
however, to enable the attentive reader to seize its leading
features, and to appreciate its competency to explain the
phenomena of nature.
1. All matter consists of indivisible and inextended First postulate.
atoms.
2. These atoms are endowed with attractive and repul- Second postulate.
sive forces, varying both in intensity and direction by a
change of distance, so that at one distance two atoms
attract each other, and at another distance they repel.
3. This law of variation is the same in all atoms. It Third postulate.
is, therefore, mutual; for the distance of atom a from
atom b, being the same with that of b from
a,
b, b must attract a with precisely the same force.
if a attract
4. At all considerable or sensible distances, these mu- Fourth postulate.
tual forces are attractive and sensibly proportional to the
square of the distance inversely. It is the attraction
called gravitation.
5. In the small and insensible distances in which sensi-
ble contact is observed, and which do not exceed the Fifth postulate.

22
NATURAL PHILOSOPHY.
Sixth postulate.
Molecule,
particle, body.
Add inertia.
Exponential
curve.
1000th or 1500th part of an inch, there are many alterna-
tions of attraction and repulsion, according as the distance
of the atoms is changed. Consequently, there are many
situations within this narrow limit, in which two atoms
neither attract nor repel.
6. The force which is exerted between two atoms when
their distance is diminished without end, and is just
vanishing, is an insuperable repulsion, so that no force
whatever can press two atoms into mathematical contact.
Such, according to Boscovich, is the constitution of a
material atom and the whole of its constitution, and the
immediate efficient cause of all its properties.
Two or more atoms may be so situated, in respect
to position and distance, as to constitute a molecule. Two
or more molecules may constitute a particle. The par-
ticles constitute a body.
Now, if to these centres, or loci of the qualities of
what is termed matter, we attribute the property called
inertia, we have all the conditions requisite to explain,
or arrange in the order of antecedent and consequent, the
various operations of the physical world.
Boscovich represents his law of atomical action by
what may be called an exponential curve. Let the dis
Fig. 2.
G'
G
A
D'm
C
C'
CH
G
9
n
[Pn
tance of two atoms be estimated on the line CAC, A
being the situation of one of them while the other is
placed anywhere on this line. When placed at i, for
example, we may suppose that it is attracted by A, with

INTRODUCTION.
23
a certain intensity. We can represent this intensity by
the length of the line il, perpendicular to A C, and can
express the direction of the force, namely, from i to A, Attractive
because it is attractive, by placing il above the axis A Cordinates above.
Should the atom be at m, and be repelled by A, we can
express the intensity of repulsion by mn, and its direc- Repulsive
tion from m towards G by placing mn below the axis.
This may be supposed for every point on the axis, and
a curve drawn through the extremities of all the perpen-
dicular ordinates. This will be the exponential curve or
scale of force.
ordinates below.
sides of axis.
As there are supposed a great many alternations of Curve on opposite
attractions and repulsions, the curve must consist of many
branches lying on opposite sides of the axis, and must
therefore cross it at C', D', C", D", &c., and at G. All
these are supposed to be contained within a very small
fraction of an inch.
gravitation.
Beyond this distance, which terminates at G, the force Force of
is always attractive, and is called the force of gravitation,
the maximum intensity of which occurs at g, and is
expressed by the length of the ordinate G'g. Further
on, the ordinates are sensibly proportional to the square
of their distances from A, inversely. The branch G' G"
has the line A C, therefore, for its asymptote.
Within the limit A C' there is repulsion, which be-
comes infinite, when the distance from A is zero; whence
the branch C'D, has the perpendicular axis, Ay, for its
asymptote.
An atom being placed at G, and then disturbed so
as to move it in the direction towards A, will be repelled,
the ordinate of the curve being below the axis; if dis-
turbed so as to move it from A, it will be attracted, the
corresponding ordinates being above the axis. The point Position of
G is therefore a position in which the atom is neither indifference
attracted nor repelled, and to which it will tend to return
when slightly removed in either direction, and is called Limit of
the limit of gravitation.
gravitation.

24
NATURAL PHILOSOPHY.
Limits of
cohesion.
If the atom be at C', or C", &c., and be moved ever so
little towards A, it will be repelled, and when the disturb-
ing cause is removed, will fly back; if moved from A, it
Fig. 2.
G'
G"
D'm
C
c
C'
C
G 9
n
Permanent
molecule.
Positions of
indifference.
Limits of
dissolution.
Molecules of
different orders;
parucles.
Dn
will be attracted and return. Hence C', C", are positions
similar to G, and are called limits of cohesion, C' being
termed the last limit of cohesion. An atom situated at any
one of these points will, with that at A, constitute a
permanent molecule of the simplest kind.
On the contrary, if an atom be placed at D', or D", &c.,
and be then slightly disturbed in the direction either from
or towards A, the action of the atom at A will cause it to
recede still further from its first position, till it reaches a
limit of cohesion. The points D', D", &c., are also posi-
tions of indifference, in which the atom will be neither
attracted nor repelled by that at A, but they differ from
G, C', C", &c., in this, that an atom being ever so little
removed from one of them has no disposition to return
to it again; these points are called limits of dissolution.
An atom situated in one of them cannot, therefore, con-
stitute, with that at A, a permanent molecule, but the
slightest disturbance will destroy it.
It is easy to infer, from what has been said, how three,
four, &c., atoms may combine to form molecules of differ-
ent orders of complexity, and how these again may be
arranged so as by their action upon each other to form
particles. Our limits will not permit us to dwell upon
these points, but we cannot dismiss the subject without

INTRODUCTION.
25
suggesting a consequence which the reader will find of
interest when he comes to the subjects of light and heat.
We allude to those characteristics of the sun by which he Inference-light
is the main source of these principles to the inhabitants of and heat of sun.
the earth.
It results from the laws of gravitation, that every Attraction of
atom in a spherical solid body is attracted towards the spherical masses.
centre by a force directly proportional to its distance from
that point. The pressure towards the centre will, there-
fore, increase as the magnitude of the sphere increases,
and may ultimately become so great as to force the atoms
near enough to each other to bring them within the last
limits of cohesion, in which case, the mass, composed of
atoms thus urged into close proximity, becomes perfectly Production of
elastic. The magnitude of this elastic mass will be
greater in proportion as the whole sphere is greater.
Every body falling upon the sphere will, on reaching
its position at the surface, send the motion with which it
arrived towards the centre to agitate the atoms of the
elastic mass. These being once disturbed will, under the Effect of a falling
forces thus called into play, vibrate indefinitely about
their positions of rest by virtue of their inertia.
elasticity.
body.
hypothesis.
It is only necessary therefore to suppose, that the Nebular
heavenly bodies have been formed by the gravitation of
the particles of a vast nebula towards its centre, and to
adopt the hypothesis which modern discoveries have
revived and forced upon us, viz., that heat and light are Light and heat,
but the effects of vibratory motion, to account for the effects of motion.
incandescent and self-luminous character of the sun. The Incandescence
same principle furnishes an explanation of the internal
heat of our earth which, together with all the heavenly
bodies, would doubtless appear self-luminous were the
acuteness of our sense of sight increased beyond its
present limit in the same proportion that the sun exceeds
the largest of these bodies. The sun far transcends all Those of the sun
the other bodies of our system in regard to heat and light,
simply because of his vastly greater size.
and luminosity of
the sun.
greater because of
his greater size.

26
NATURAL PHILOSOPHY.
Effects of
§ 13. The molecular forces are the effective causes
which hold together the particles of bodies. Through
molecular action. them, the molecules approach to a certain distance where
they gain a position of rest with respect to each other.
The power with which the particles adhere in these
relative positions, is called, as we have seen, cohesion.
This force is measured by the resistance it offers to
mechanical separation of the parts of bodies from each
other.
Measure of
cohesion.
Three states of
aggregation.
On the degree of this force, the three states or ag-
gregate forms called solid, liquid, and gaseous depend.
These different states of matter result from certain definite
relations under which the molecular attraction and repul-
sion establish their equilibrium; there are three cases, viz.
two extremes and one mean. The first extreme is that in
which attraction predominates among the atoms; this pro-
Solid, gas, liquid. duces the solid state. In the other repulsion prevails, and
the gaseous form is the consequence. The mean obtains
when neither of these forces is in excess, and then matter
presents itself under the liquid form.
Formulæ.
External
peculiarities of
bodies; subject to
change.
Let A represent the attraction and R the repulsion,
then the three aggregate forms may be expressed by the
following formulæ :
A> R solid,
AR gas,
A = R liquid.
These three forms or conditions of matter may, for the
most part, be readily distinguished by certain external
peculiarities; there are, however, especially between solids
and liquids, so many imperceptible degrees of approxima-
tion, that it is sometimes difficult to decide where the one
form ends and the other begins. It is further an ascer-
tained fact that many bodies, (perhaps all,) as for instance
water, are capable of assuming all three forms of aggrega-
tion.

INTRODUCTION.
27
Thus, supposing that the relative intensity of the Change of
molecular forces determines these three forms of matter, it molecular action
follows from what has been said above, that this term
may vary in the same body.
The peculiar properties belonging to each of these
states will be explained when solid, liquid, and aëriform
bodies come severally under our notice.
in same body.
molecular forces
between
§ 14. The molecular forces may so act upon the atoms Action of
of dissimilar bodies as to cause a new combination or
union of their atoms. This may also produce a separation dissimilar bodies.
between the combined atoms or molecules in such manner
as to entirely change the individual properties of the
bodies. Such efforts of the molecular forces are called
chemical action; and the disposition to exert these efforts, chemical action.
on account of the peculiar state of aggregations of the
ultimate atoms of different bodies, chemical affinity.
Chemical affinity.
magnitude.
§ 15.-Beyond the last limit of gravitation, atoms Attraction of
attract each other: hence all the atoms of one body attract bodies of sensible
those of another, thus giving rise to attractions between
bodies of sensible magnitudes through sensible distances. Intensity of this
The intensities of these attractions are directly proportional attraction.
to the number of attracting atoms, and inversely as the
squares of their distances apart.
The term universal gravitation is applied to this force Universal
when it is intended to express the action of the heavenly gravitation.
bodies on each other; and that of terrestrial gravitation or Terrestrial
simply gravity, where we wish to express the action of gravity.
the earth upon the bodies forming with itself one whole.
The force is always of the same kind however, and varies
in intensity only by reason of a difference in the number Effects of this
of atoms and their distances. Its effect is always to gen-
erate motion when the bodies are free to move.
force.
Gravity, then, is a property common to all terrestrial Gravity common
bodies, since they constantly exhibit a tendency to ap-
proach the earth and its centre. In consequence of this
to all bodies. Its
consequences.

28
NATURAL PHILOSOPHY.
Mechanics,
Statics,
tendency, all bodies, unless supported, fall to the surface
of the earth, and if prevented by any other bodies from
doing so, they exert a pressure on these latter.
This is one of the most important properties of terres-
trial bodies, and the cause of many phenomena, of which
a fuller explanation will be given presently.
§ 16.-That branch of Natural Philosophy which treats
of the action of forces on bodies, is called MECHANICS.
Mechanics is usually considered under four separate
heads, viz.: Statics, which treats of the mutual destruction
of forces when applied to solid bodies; Hydrostatics, the
same when applied to fluids; Dynamics, which investigates
Hydrodynamics. the motions of solids; and Hydrodynamics, which discusses
the motions of fluids.
Hydrostatics,
Dynamics,
Mechanics of
solids,
and of fluids.
Statics and Dynamics will be treated together, under
the general head, MECHANICS OF SOLIDS, as will also
Hydrostatics and Hydrodynamics, under the head, ME-
CHANICS OF FLUIDS.

PART FIRST.
MECHANICS OF SOLIDS.
I.
SPACE, TIME, MOTION, AND FORCE.
§ 17.-Space is indefinite extension, without limit, and Space.
contains all bodies.
§ 18.-Time is any limited portion of duration.
We Time;
may conceive of a time which is longer or shorter than
a given time. Time has, therefore, magnitude, as well as has magnitude.
lines, areas, &c.
To measure a given time, it is only necessary to obtain Time measured.
equal times which succeed each other without intermission,
to call one of these equal times unity, and to express, by
a number, how often this unit is contained in the given
time. When we give to this number the particular name
of the unit, as hour, minute, second, &c., we have a com- Units of time.
plete expression for time.
instruments.
The Instruments usually employed in measuring time Time
are clocks, chronometers, and common watches, which are
too well known to need a description in a work like
this.
The smallest division of time indicated by these time-
pieces is the second, of which there are 60 in a minute,

30
NATURAL PHILOSOPHY.
chronometers.
Performance of 3600 in an hour, and 86400 in a day; and chronometers,
which are nothing more than a species of watch, have
been brought to such perfection as not to vary in their
rate a half a second in 365 days, or 31536000 seconds.
Time represented
by lines.
Rest;
absolute and
relative.
Example of
relative rest.
Motion, like rest,
is relative.
It is continuous.
Thus the number of hours, minutes, or seconds, be-
tween any two events or instants, may be estimated
with as much precision and ease as the number of yards,
feet, or inches between the extremities of any given dis-
tance.
Time may be rep-
resented by lines, by
laying off upon a
given right line AB,
the equal distances
Fig. 3.
Ag
from 0 to 1, 1 to 2, 2 to 3, &c., each one of these equal
distances representing the unit of time.
§ 19.-A body is in a state of absolute rest when it
continues in the same place or position in space. There is
perhaps no body absolutely at rest; our earth being,
without cessation, in motion about the sun, nothing con-
nected with it can be at rest. In what follows, rest must,
therefore, be considered but as a relative term. A body
is said to be at rest, when it preserves the same position
in respect to other bodies which we may regard as fixed.
A body, for example, which continues in the same place
in a boat, is said to be at rest in relation to the boat,
although the boat itself may be in motion in relation
to the banks of a river on whose surface it is floating.
§ 20.-A body is in motion when it occupies succes-
sively different positions in space. Motion, like rest, is
but relative. A body is in motion when it changes its
place in reference to those which we may regard as
fixed.
Motion is essentially continuous; that is, a body cannot
pass from one position to another without passing through

MECHANICS OF SOLIDS.
31
a series of intermediate positions; the motion of a point
describes, therefore, a continuous line.
When we speak of the path described by a body, Path of a body.
we are to understand that of a certain point connected
with the body. Thus, the path of a ball, is that of its
centre, &c.
rectilinear
§ 21. The motion of a body is curvilinear or recti- Curvilinear and
linear, according as the path described is a curve or motion.
right line.
When the motion is curvilinear, we may
consider it as taking place upon a polygon, of which Direction of a
the sides are very small and sensibly coincide with the body's motion.
curve. The prolongation of any one of these sides will
be a tangent to the curve, and will indicate the direction
of the body's motion while upon this side.
Conceive the time employed by a body to pass from Uniform motion.
one position to another, to be divided into a number
of very small and equal parts. If the portions of the
path successively described in these equal times be equal,
the motion is said to be uniform. If otherwise, the mo-
tion is said to be varied. It is accelerated when these varied motion:
elementary paths are greater and greater; retarded, when accelerated and
less and less in the order of time.
retarded.
§ 22.-Velocity is the rate of a body's motion. The Velocity;
rapidity or slowness of motion is indicated by the greater
or less length of the path described by the body, during
each of the small and equal portions of time into which
the whole time is divided. This length is taken as the
measure of the velocity when the small portion of time is its measure.
made to denote the unit of time.
The velocity is constant in uniform motion: it is vari- Constant and
able in accelerated and retarded motion.
variable.
§ 23.-In uniform motion, the small spaces described Uniform motion.
in equal consecutive portions of time being equal, it is
bvious that the space described in any given time will

32
NATURAL PHILOSOPHY.
Relation of space
to the time.
contain as many equal parts of space as there are equal
parts of time. Consequently, in uniform motion, equal
spaces will be described in equal times, whatever be the rate
of motion, and the spaces will be proportional to the times
employed in describing them.
Denote by S the length of space described during
the time T; s the length of the space described in the
small portion of time t, then, from what precedes, we have
S: T::s: t
=
3/8
t
(1).
Velocity
measured by the
space described
in any unit of
time.
Rule for finding
velocity.
a constant ratio.
§ 24.-Since in uniform motion, the spaces are propor-
tional to the times employed in describing them, the
velocity may be measured by the space described in any
time whatever, for example in a second, minute, an hour,
&c. Thus we say the velocity is 2 feet a second, or 120
feet a minute, or 7200 feet an hour, or 2 of a foot in 1
of a second, &c; all of which amounts to the same thing,
since the ratio of the space to the time is not changed.
When a body describes uniformly a certain space in
a given number of units of time, as the second, for ex-
ample, which is usually taken as the unit, the velocity is
found by dividing the whole space by the whole time, for
if we make t = one second in equation (1), s becomes the
velocity, § 22, and denoting this by V we have
Example.
S
V = 1
T
(2).
Example: The space described in 1 minute and 5
seconds or 65 being 260 feet, the space described in 18,
or the velocity, is given thus:

MECHANICS OF SOLIDS.
33
V =
Sous 260
1=
1
T65
=4f
4f.
enied doldw
bua Hate cens
Reciprocally, if the velocity be multiplied by the number
of units of time, the space will result.
giro
motion.
§ 25.-It frequently happens in practice that the ve- Periodical
locity is not constant, although the spaces described at the
end of certain equal intervals are equal. Such for instance
is the case in all periodical movements of which the dif
ferent changes are executed in the same interval of time,
although the velocity is continually varying within this
interval. The motion of a carriage and that of a pedes- Instance--
trian, are examples of this; the spaces described in
certain intervals, are often the same, while the motion is
sometimes accelerated and sometimes retarded.
carriage and
pedestrian.
and time,
represented
§ 26.-Conceive a table consisting of two vertical Relation of space
columns, in one of which are arranged the numbers ex-
pressive of the intervals of time elapsed since any given geometrically.
instant, and in the other, on the same horizontal lines,
the numbers which designate the spaces described by any
body in these intervals. Draw an indefinite right line In any kind of
OB; assume any linear dimension, as an inch, to repre-
sent the unit of time, and let the same length represent
the unit of space; with a scale of equal parts, lay off a
distance Ot representing
an interval of time given
by the table; upon a per-
pendicular to OB at the
point t, lay off a distance
te representing the dis-
tance passed over by the
body in the time Ot. Do
O
e1
Fig. 4.
es
B
motion.
the same for the other
times and corresponding
spaces of the table, and we obtain the points e, eg, eg, &c.,
3

34
NATURAL PHILOSOPHY.
In uniform
motion.
which, being united two and two by right lines, will give a
polygon. This polygon will not differ sensibly from a curve
when the intervals of time are small and differ very little
from each other. O, Ota, Ot, &c., are the abscisses,
and te, tea, to es, &c., the ordinates of this curve, of
which the origin is 0. It is obvious that by means of the
curve we may obtain, as by the table, the space de-
scribed during any given interval; so that this curve
gives the relation which connects the spaces with the
times, whatever be the nature of the motion.
Fig. 5.
e5
In uniform motion the spaces increase in the direct
ratio of the times, and the ordinates te, t2 e2, tz e3, &c.,
are therefore proportional to
the abscisses Oty, Ota, Ot tз,
&c.; hence the curve becomes
a right line. Let the axis
OB, of times, be divided into
any number of equal and
very small parts; through
the points of division draw
the ordinates or spaces, and
through the extremities of the
ordinates draw the lines e ba
by
ez
82
e3
b
44
103
33
B
3
4 5
eg by, es b4, &c., parallel to the axis of times, we shall
thus form a series of small right-angled triangles Ote,
e bae, &c., similar to the triangle Ot 4, and because
e3b4=t3 t4, we have
te: Ota: b4e4 tz ts
whence
t4 4
|
=
Ota
to ts
Kelation of spaces but be is the space s, described in the small time to t₁ = t,
to the times.
S
and te the space & described in the time Ot = T, and
the above may be written

MECHANICS OF SOLIDS.
35
S
Q|-
T
=
S
is [to
and making t = 1, s becomes the measure of the velocity
V, and we have
S
V =
T'
the same as before, equation (1).
Or, Ot may be taken as the unit of time, in which
case, te, becomes the velocity V, and we have
S
V
=
Velocity equal to
the ratio of the
space to the time.
Same for any
space and time.
In varied motion, the spaces not being proportional to Varied motion.
the times, the line Oe, e1e2, ees, &c., is not straight, and
the small spaces e ba
eg bg, &c., described in
the elementary times
t to, to to, &c., are not
equal. The velocity
must, therefore, vary
at every instant. For
the case represented
by the figure, the mo-
tion is accelerated,
because
the spaces
Fig. 6.
B
m
Accelerated
motion,
n
represented
geometrically.
be accelerated;
eg by, eg bg, &c., described in the equal elementary times,
continually increase. Now let it be supposed that at
the point es the motion ceases to be accelerated, and Motion ceases to
that it becomes uniform with the velocity which the
body had at this instant. The law of the motion after-
ward will be represented by the right line e, m, the pro- becomes
longation of es e, and since, at the instant we are
considering, the body describes a space equal to e4 b4 in
the elementary time eg b= t t it will, in virtue of
uniform.

36
NATURAL PHILOSOPHY.
Measure of the
velocity at any
instant;
its uniform motion, describe in a unit of time a space
equal to mn, obtained
by laying off from the
point es, on es b₁ pro-
duced, a distance en
equal to the unit of
But the space
time.
described in a unit of
time, at a a constant
2011
e
Fig. 6.
oved
es
33
m
rate, is the measure
of the velocity corre-
sponding to the point
B
Yes, or at the end of the time Ot. From the figure we
obtain
maitoin baru of Innoitro by: e3b:: mn: en;
bos tilginila
or making
e₁ b₁ = 8,
e3b₁ = t₁ mn = V,
V, e n = 1,
we have
equal to the ratio
of the element of
the space, to the
element of the
time.
whence
st: V: 1;
V==/
Tangent line;
will give the
velocity.
If we suppose the element of time to t sufficiently
small, the line ese will coincide with the curve to which
es m will become a tangent at the point e. This tangent
being constructed geometrically, will give, in the manner
above indicated, the velocity corresponding to the point
of the curve to which it is drawn, or the velocity at the
end of the time O tg.

MECHANICS OF SOLIDS.
37
Periodical motion, such as has
been defined in § 25, will be rep-
resented by a waved line EEE,
&c, whose undulations are regu-
larly disposed about the right
line e, eg, eg, &c., which repre-
sents the law of uniform motion.
It may be important to re-
mark that the curves which have 0
just been described, and which
connect the lengths of the spaces
E
1
Fig. 7.
t2
2
Bina dood
ybod
Molov
E
B
Geometrical
representation of
periodical
motion.
and the times, in any kind of motion, must not be con- Distinction
founded with the actual path described by the body.
between the line
giving the law of
the path
In this last, the tangent simply gives the direction of the the motion, and
motion; and to obtain the velocity, the elementary por- described by the
tion of the curve, or of the tangent line, must be divided body.
by the time during which this element is described.
§ 27.-Matter in its unorganized state, is inanimate or
inert. It cannot give itself motion, nor can it change Vids Jyndall
of itself the motion
which it may have
received. A body
at rest will forever
remain so unless dis-
turbed by something
C
82001
Fig. 8.
a
B
cannot change
extraneous to itself; or if it be in motion in any direction, Inanimate bodies
as from a to b, it will continue, after arriving at b, to move their state of rest
towards c in the prolongation of ab; for having arrived at or of motion.
b, there is no reason why it should deviate to one side
more than another. Moreover, if the body have a certain
velocity at b, it will retain this velocity unaltered, since
no reason can be assigned why it should be increased
rather than diminished in the absence of all extraneous
causes.
.02
exception
If a billiard-ball, thrown upon the table, seem to Apparent
diminish its rate of motion till it stops, it is because its explained.

38
NATURAL PHILOSOPHY.
motion is resisted by the cloth and the atmosphere. If a
body thrown vertically downward seem to increase its
velocity, it is because its weight is incessantly urging it
onward. If the direction of the motion of a stone,
thrown into the air, seem continually to change, it
is because the weight of the stone urges it incessantly
towards the surface of the earth. Experience proves that
in proportion as the obstacles to a body's motion are re-
moved, will the motion itself remain unchanged.
It results, from
what has been said,
that when a body is
Fig. 9.
put in motion and
abandoned to itself,
Consequences of its inertia will cause
inertia.
Forces; weight
and heat.
Illustration.
Forces produce
various effects.
it to move in a
straight line and
B
preserve its rate of motion unchanged. If, from any extra-
neous cause the body is made to describe a curve AB, and
this cause be removed at the point B, the inertia will
cause the body to move along the tangent BC, and to
preserve the velocity which it had at B.
8 28.-A force has been defined to be that which
changes or tends to change the state of a body in respect
to rest or motion. Weight and Heat are forces. A body
laid upon a table, or suspended from a fixed point by
means of a thread, would move under the action of its
weight, if the resistance of the table, or that of the fixed
point did not continually destroy the effort of the weight.
A body exposed to any source of heat, expands, its
particles recede from each other, and thus the state of the
body is changed.
§ 29.--Forces produce various effects according to cir-
cumstances. They sometimes leave a body at rest, by de-
stroying one another, through its intervention; sometimes

MECHANICS OF SOLIDS.
39
they change its form or break it; sometimes they impress
upon it motion, they accelerate or retard that which it has,
or change its direction; sometimes these effects are produ-
ced gradually, sometimes abruptly, but however produced
they require some definite time, and are effected by con- These effects
tinuous degrees. If a body is sometimes seen to change portions of time.
suddenly its state, either in respect to the direction or the
rate of its motion, it is because the force is so great as to
produce its effect in a time so short as to make its dura-
tion imperceptible to our senses, yet some definite portion
require definite
of time is necessary for the change. A ball fired from a A ball fired from
gun, will break through a pane of glass, a piece of board,
or a sheet of paper when freely sus-
pended, with a rapidity so great
that the parts torn away have not
time to propagate their motion to the
rest. A cannon freely suspended at
the end of a vertical cord will throw
its ball to the same point as though
it were on its carriage, which proves
Fig. 10.
a cannon.
are not.
that the piece does not move sensibly till the ball leaves Effects obvious,
its mouth, though afterward it recoils to a considerable while the times
distance. In these several cases the effects are obvious,
while the times in which they are accomplished are so
short as to elude the senses: and yet these times have
had some definite duration, since the changes, correspond-
ing to these effects, have passed in succession through
their different degrees from the beginning to the ending.
Forces which give motion to bodies are called motive Motive forces:
forces; they are accelerating when they accelerate the accelerating and
motion at each instant, and retarding when they retard it.
retarding.
of forces obtained
from experience.
§ 30.-We may form from our own experience a clear Idea of the action
idea of the mode in which forces act; when we push or
pull a body, be it free or fixed, we experience a sensation
denominated pressure, traction, or in general, effort. This
effort is analogous to that which we exert in raising a

40
NATURAL PHILOSOPHY.
pressures;
Forces are real weight, and thus forces are to us real pressures. Pressure
may be strong or it may be feeble; it therefore has magni-
tude, and may be expressed in numbers by assuming a
certain pressure as unity, which may easily be done if we
can find pressures that are equal to each other.
unit of force.
Equal forces.
by weights.
Two forces are equal when, substituted, one for the
other, in the same circumstances, they produce the same
effect, or when, being directly opposed, they destroy each
other.
Conceive a body W, suspended from the
extremity of a thread; the thread will as-
Forces measured sume a vertical direction, and an effort will
be necessary to support it; if two forces,
applied successively to the thread and in the
same manner, maintain the body at rest, these
forces are equal to each other and to the
Double, triple, weight of the body. A double, triple, &c.,
&c., force.
Fig. 11.
AT
BOW
Fig. 12.
T
force, will support two, three, &c., bodies,
similar to the first, suspended one above
another on the same thread; taking one of
these forces, that, for instance, which sup-
ports 2,5th of a cubic foot of distilled water
at the temperature of 60° Fahrenheit, and
of which the weight is called a pound, for
pound weight. unity, any force will be expressed by a
number which indicates how many pounds it will sup-
port.
Unit of force a
Forces compared
by the balance.
§ 31.-Weights are measured and compared by means
of an instrument called a balance, and of which we shall
speak hereafter. By the definition given above of equal
forces, it will be easy to find the weights of bodies what-
ever be the merits or defects of such an instrument. We
have but to require that these bodies substituted for a
certain number of standard units of weight, shall produce,
under the same circumstances, the same effect upon the
balance. Under this point of view, many devices may be

MECHANICS OF SOLIDS.
41
employed to measure the weights of bodies and conse-
quently the magnitudes of forces.
Springs, among others, in
supposing they preserve unim-
paired for a long time their
elasticity, may be, and indeed
are, used in practice, for this
purpose. Of such is the spring
balance, a sketch of which is
given in the figure. In using
this instrument, it is necessary
to determine previously the
accuracy of its divisions by
means of standard weights,
and to change the values of its
Fig. 13.
weights, os ou à usa
graduations if the elasticity of
the spring shall be found to have undergone a change
since its construction.
Use of spring
balance to
measure forces.
Verification of the
elasticity.
of gravity, small
§ 32. It is known from observation that the action Variation in force
of the force of gravity diminishes as the bodies upon within moderate
which it is exerted are elevated above the surface of limits.
the earth. The same body, therefore, which will cause by
its weight a spring to bend through a certain angle at the
surface of the sea, will cause it to bend through a less
angle when weighed at the top of a mountain, and thus
the absolute weight of the body, or magnitude of the force
which sustains it, is diminished. But this diminution for
the height of three miles does not exceed of the total
weight. Experience also shows that the weight of a body
diminishes as it approaches the equator, but for an extent
of territory equal to that of the state of New York this
variation is scarcely appreciable.
750
The directions of two plumb-lines being normal to the
surface of the earth, cannot be perfectly parallel, since Acts in parallel
they converge to a point near its centre and which is directions within
therefore distant about 4000 miles from the place of ordinary bodies.
the limits of

42
NATURAL PHILOSOPHY.
observation. These lines when separated by a distance
of 600 yards on the surface of the earth, will form with
each other an angle not to exceed 6", which is inappreci-
Force of gravity able to common instruments. It hence follows, that,
constant, and acts within ordinary limits, the force of gravity may be regarded
as constant, and acting in parallel directions.
in parallel
directions.
II.
Action of exterior
forces on bodies;
ACTION OF FORCES, EQUILIBRIUM, WORK.
§ 33.-When a force acts against a point in the surface
of a body, it exerts a pressure which crowds together the
neighboring particles; the body yields, is compressed and
its surface indented; the crowded particles make an effort,
by their molecular forces, to regain their primitive places,
and thus transmit this crowding action even to the re-
when some of the motest particles of the body. If these latter particles are
particles are fixed. fixed or prevented by obstacles from moving, the result
will be a compression and change of figure throughout the
When none of the body. If, on the contrary, these extreme particles are
particles are fixed. free they will advance, and motion will be communicated
cannot be
generated
by degrees to all the parts of the body. This internal
motion, the result of a series of compressions, proves that
a certain time is necessary for a force to produce its entire
Definite velocity effect, and the absurdity of supposing that a finite velocity
may be generated instantaneously. The same kind of
instantaneously. action will take place when the force is employed to
destroy the velocity which a body has already acquired;
it will first destroy the velocity of the molecules at and
nearest to the point of action, and then, by degrees, that
of those which are more remote in the order of distance.
Reaction equal
and contrary
to action.
§ 34.-As the molecular springs cannot be compressed
without reacting in a contrary direction, and with the

MECHANICS OF SOLIDS.
43
same effort, the agent which presses a body will experience
an equal pressure. This is usually expressed by saying
that reaction is equal and contrary to action. In pressing
the finger against a body, in pulling it with a thread, or
pushing it with a bar, we are pressed, drawn, or pushed in
a contrary direction, and with the same effort.
Fig. 14.
Two
Illustration.
weighing springs attached to the extremities of a thread or
bar, will indicate the same degree of tension, and in con-
trary directions when made to act upon each other through
the intervention of the thread or bar.
at any point in
§ 35.-In every case, the action of a force is trans- Point of
mitted through a body to the ultimate point of resistance, application, taken
by a series of equal and contrary actions and reactions line of direction.
which destroy each other, and which the molecular
springs of all bodies exert at every point of the right line,
limited by their boundaries, along which the force acts.
It is in virtue of this property of bodies, that the action of
a force may be supposed to be exerted at any point in its
line of direction.
transmit the
§36.-Bodies being more or less extensible and com- Bodies used to
pressible, a thread or bar, interposed between the power action of forces.
and resistance, will be stretched or compressed to a certain
degree, depending upon the energy with which these
forces act; but as long as the power and resistance remain
the same, the thread or bar, having attained its new
length, will cease to change. On this account, bodies,

44
NATURAL PHILOSOPHY.
and inextensible;
regarded as rigid which are usually employed to transmit the action of
forces from one point to another, may be regarded as
perfectly inextensible or rigid, especially as such bodies
are chosen and applied so as not to yield under this
action.
Inertia measured
by means of
forces;
§ 37.-We have just seen that when a force acts upon
a body to give it motion or to destroy that which it has,
the body will react or oppose a resistance equal to the
force. This resistance measures the inertia of the matter
of the body. It is obvious that for the same body, this
resistance increases with the degree of velocity imparted
or destroyed; we shall presently find that it is propor-
tional to this velocity, and that it also increases in the
action of inertia direct ratio of the quantity of
on a thread;
matter in the body. If a body,
free to move, be drawn by a
thread, the thread will stretch
and even break if the action be
too violent, and this will the
more probably happen in propor-
tion as the body is more massive.
If a body be suspended by means
under the action of a vertical cord, and a weighing
spring be interposed in the line
conduct of a
spring when
of inertia ;
of traction, the graduated scale barod od
of the spring will indicate the
weight of the body when the
latter is at rest; but if we sud-
denly elevate the upper end of
Fig. 15.
the thread, the spring will immediately bend more in
consequence of the resistance opposed by the inertia of
resistance to all the body. The motion once acquired by the body and
changes of
motion;
become uniform, the spring will resume and preserve
the degree of flexure or tension which it had when the
body was at rest. If, now, the body being in motion, the
velocity of the upper end of the thread be diminished, the

MECHANICS OF SOLIDS.
45
indicate the
spring will unbend and the scale will indicate a pressure
less than the weight of the body. The oscillations of the oscillations of a
spring may therefore serve to measure the variations in spring may
the motions of a body, and the energy of its force of changes in
inertia, which acts against or with a power exerted in the
direction of the motion, according as the velocity is in-
creased or diminished.
motion.
application, line
intensity.
§ 38. The effect of every force depends, 1st, upon its Effect of a force;
point of application; that is, the point to which it is point of
directly applied: 2d, upon the position of the line along of direction, and
which it acts or the straight line which its point of appli-
cation would describe if perfectly free: 3d, upon the
direction in which it tends to solicit its point of application
along this line, whether backward or forward: 4th, upon
its absolute intensity, measurable in pounds or any other
unit of weight.
§ 39.-Let A be the point of application of a force
which acts upon the line AB; from A, lay off upon Graphical
the direction in
which the force
representation
of a force;
acts, a distance
AP, containing as
many linear units,
say inches, as there
are pounds in the
intensity of the
A<
Fig. 16.
P
B
e
force; the force
will be fully represented. Commonly the direction of
the action is indicated by an arrow, and the intensity
of the force by some letter as P, for the sake of brevity.
Thus, we say a force P or AP, a force Q or AQ, as we
say a force of 5 pounds, a force of 8 pounds. In this way by length of line
the investigations in mechanics are reduced to those of or by symbol.
geometrical figures.

46
NATURAL PHILOSOPHY.
Equilibrium of
forces;
statical and
dynamical.
Illustration-
two men.
No case of
absolute rest.
§ 40.-When the forces applied to any body balance, or
mutually destroy each other, so as to leave the body in the
same state as before their application, these forces are said
to be in equilibrio. The equilibrium may be statical or
dynamical. In the first case, the forces finding the body at
rest, will leave it so; in the second case, the forces being
applied to the body in motion, will in no respect alter the
motion. Two men pulling with equal strength at the op-
posite ends of a cord, will be a case of statical equilibrium
if the men be at rest, and a case of dynamical equilibrium
if they be in motion.
In reality there is no case of absolute statical equi-
librium, since the earth's motion involves that of every
body connected with it, in the same way that a boat
Earth's motion. moving over the surface of the water carries every thing
on board along with it. The idea of repose is not neces-
sary to that of an equilibrium of forces, which only
requires the mutual destruction of all the forces that
act at the same instant upon a body.
Repose not
necessary to
equilibrium.
Forces in
equilibrio;
when the motion
changes.
Effect of inertia
on equilibrium
pre-
§ 41.-When a body, subjected to the action of several
extraneous forces, preserves its motion perfectly uniform,
notwithstanding these forces, these latter will, from the
definition above, be in equilibrio. If the velocity however
not in equilibrio augment or diminish, the extraneous forces will not be in
equilibrio; but if we take into account the force of inertia
of the different particles of the body, and introduce among
the extraneous forces one equal to it and capable of
venting the modification of the motion, there will again be
an equilibrium among all the extraneous forces. A horse
which draws a carriage along a road, destroys at each
instant all resistances which are opposed to his action; if
the motion is perfectly uniform, these resistances arise
only from the ground, the different frictions, &c. If the
velocity increases at each instant in consequence of an
increased effort of the horse, the inertia of the carriage
will come into action and add to the other resistances
of forces.
Illustration-
horse and
carriage.

MECHANICS OF SOLIDS.
47
above named, and the effort of the horse during this
increase of velocity, will be in equilibrio with all these
forces; if, on the contrary, the velocity diminish, the
inertia of the carriage, which tends to preserve its motion
uniform, will add its action to that of the horse to over-
come all the resistances, or to maintain the equilibrium.
an equilibrium
Thus inertia stands always ready to maintain an equi- Inertia always
librium among forces of whatever nature; and hence the ready to establish
distinction between the equilibrium of bodies and of among forces.
forces. Forces are ever in equilibrio, while bodies are not
necessarily so. If, for example, a material point be acted
action.
upon by a force, it will move in the direction of this force, Reaction equal
while the force itself is maintained in equilibrio by and contrary to
the inertia developed during the yielding of the point.
Action and reaction are equal and contrary.
§ 42.-When an equilibrium exists among several
forces, as O, P, Q, &c., one of them, as O, may be con-
sidered as preventing the effect of all the others. If, then,
we conceive a force
R, equal and directly
opposed to 0, at the
same point of appli-
cation C, this force
will destroy of it-
self the force 0, and
will therefore pro-
duce the same effect
upon the body as
the forces P, Q, &c.,
taken together. This
force R is called the
a
B
Fig. 17.
R
P
resultant of the forces, P, Q, &c., and these latter the
components of the force R.
Reciprocally, if to the resultant R of several forces P,
Q, &c., an equal force O, be immediately opposed, there
ill be an equilibrium between this force and the several
Resultant of
forces,
components of a
force.

48
NATURAL PHILOSOPHY.
Resultant and
components
defined.
Resultant of
several forces
acting along the
same line.
Mechanical work
of forces.
Resistance
overcome and
reproduced.
forces P, Q, &c.: hence, the resultant is a single force
which will produce the same effect as two or more forces;
the components are two or more forces which will produce
the same effect as a single force.
843.-When several forces act along the same straight
line and in the same direction, their joint effect will ob-
viously be the same as that of a single force equal to
their sum, which single force will be their resultant. If
some of the forces act in one direction, and others in an
opposite direction, the resultant will be a single force equal
to the excess of the sum of those which act in one direction
over the sum of those which act in the contrary direction;
and it will act in the direction of those forces which give
the greater sum, for when two unequal forces are directly
opposed, the smaller will destroy in the larger a portion
equal to itself. Three men pulling in the same direction
a cord, with efforts 10, 17, and 25 pounds, and two others
pulling in the opposite direction with efforts 12 and 19
pounds, the effect to move the cord will be the same as
though it were solicited by a single force 52-31=21
pounds, acting in the direction of the first men.
844. The most simple case of equilibrium, is that in
which two equal and opposing forces destroy each other,
and it is this to which the employment of force in the
mechanic arts is always reduced. To work, is to destroy
or overcome, in the service of the arts, resistances, such
as the force of adhesion of the molecules of bodies, the
strength of springs, the weight of bodies, their inertia,
&c., &c. To polish a body by friction, to divide it into
parts, to elevate weights, to draw a carriage along a road,
to bend a spring, to throw stones, balls, &c., &c., is to
work, to continually overcome resistances incessantly
recurring.
Mechanical work not only supposes a resistance over-
come, but a resistance reproduced along the path described

MECHANICS OF SOLIDS.
49
by the point at which the resistance is exerted, and in
the direction of this path. To take away from a body
a portion of its matter with a tool, for example, we must
not only overcome the resistance opposed by the matter
removed, but also cause the point of action of the tool to
advance in the direction of the line along which the resist-
ance incessantly recurs. The further the tool advances,
the greater will be the length of the removed portion;
on the other hand, the broader and thicker this portion,
the greater the resistance, and consequently, the greater
the effort to overcome it. The work performed, therefore, Work increases
at each instant, increases with the intensity of the effort and
the length of the path described by its point of application in described by the
the direction of the effort.
with the effort
and path
point of
application.
work when the
resistance is
§ 45.-Let us suppose a constant resistance and, there- Measure of the
fore, a constant effort which is equal and directly opposed
to it, that is, they are the same at each instant; it is constant.
obvious, from what precedes, that the work produced will
be proportioned to the length of the path described by
the point of application of the effort-double, if the path
is double, triple, if the path is triple, &c.; so that, if we
take for unity the work which consists in overcoming a
resistance over a length of 1 foot, the total work will be
measured by the number of feet passed over. But if for
another work, the constant resistance is double, triple,
&c. of what it was in the first case, for an equal length
of path, the work will be double, triple, &c. of what it
was before. If, for example, the resistance were 1 pound
in the first case, and 2, 3, 4, &c. pounds in the second, the
work for each foot of path would be 2, 3, 4, &c. times that
of 1 pound. In assuming, then, the work which consists
in overcoming a resistance of 1 pound, through a distance
of 1 foot, for the unit of work, we shall have for the
measure of the work, of which the object is to overcome
a constant resistance, the number of pounds which measures Rule.
this resistance repeated as many times as there are feet in the
4

50
NATURAL PHILOSOPHY.
Jilustration.
Equation of the
quantity of work.
Geometrical
representation of
the quantity of
work.
Work when the
resistance is
variable.
path described by the point of application of the resist-
ance.
For example, suppose a motive force employed to
draw a body on a horizontal plane; the work will be,
to overcome the resistance of the constant friction exerted
between the body and plane. Let this friction be 37.5
pounds, and the path described 64 feet, the total work
will be
37.5 x 64 = 2400 pounds,
or equal to 2400 pounds over 1 foot, or 1 pound over a
distance of 2400 feet.
In general, then, denoting by Q, the quantity of work
performed; by P the constant resistance, or its equal, the
effort necessary to overcome it; and by S, the space de-
scribed by the point of action, we shall have
Q=P. S...
linear unit,
To represent this geometri-
cally, assume any
as the inch, to represent 1
pound, and the same to repre-
sent the unit of linear length;
lay off from O on the indefi-
nite right line OB, the dis-
tance Oe, equal to the length
of path described by the point
Fig. 18.
(3).
e1 B
of action, and at e, the perpendicular e, containing as
many inches as the constant effort contains pounds; then
will the number of square inches in the rectangle Oer₁r,
express the quantity of work.
§ 46.-If the resistance, or the equal effort which de-
stroys it, instead of being the same at each instant, varies
incessantly, as is most frequently the case, the quantity
of work will not be given by the simple rule above; but,
as the effort, however variable, may, during the descrip-

MECHANICS OF SOLIDS.
51
tion of a very small portion of the path, be regarded as
constant, the corresponding portion of work will still be Elementary
measured by this constant effort into this small portion quantity of work.
of the path.
The total work, being composed of all its elements, Total quantity of
will be measured by the sum of all these elementary work.
products.
Draw the curve r, r1, T2, T3, &c., of which the abscisses
Oe, Oe, O2, Oes, &c., shall represent the spaces described
by the point of
action of the resist-
ance up to certain
given successive in-
stants of time, and
of which the ordi-
nates er, 171, C221
egr, &c., shall rep-
resent the corre-
sponding resistan-
e1e2
ces. Let ee, es e
e2 3, &c., be the
equal and very
13
3
Fig. 19.
15
86
5
ชา
76
30
6
B
Represented by
geometry.
small spaces described in successive portions of time. The
elementary portions of work during these intervals of
time, having for their measures the products of the small
spaces by the corresponding resistances, regarded as con-
stant for each one, that is, by the products
ee Xer,
e1e2 XG171
C2 C3 X C2 r21
these elementary portions of work are represented respec-
tively by the elementary areas
er se
Gr1 S2 C21
C2 12 S3 C3, &C.,
and the total work will be represented by the sum of all
these rectangles. But if we multiply suitably the points

52
NATURAL PHILOSOPHY.
an area.
of division e, eg, eg, &c., by diminishing the distances e e,
ee, eg eg, &c., it is obvious that the sum of the rectangles
will not sensibly differ from the area included by the
curve rr1 r2...", the whole path ee, described by the
point of action, and the two ordinates er and er, drawn
through its extremities.
Hence we see, that when we know from experience, the
Represented by law which connects the variable resistance with the length
of path described by its point of action, to compute the
amount of work performed, is but to construct by points,
or otherwise, the curve of this law, and to calculate the
area included by the curve, the total length of path
described and the extreme ordinates. When the unit of
length employed to construct the ordinates is the same as
that by which the length of path is measured, it is plain
that the unit of area will represent the work performed
by a unit of effort, as a pound, through a unit of length, say
a foot.
To find this area, divide the path described into an
even number of equal parts, and erect ordinates at the
points of division,
and at the extremi-
ties; number the
ordinates in the or-
der of the natural
Rule for finding numbers; add to-
the area.
gether the extreme
ordinates, increase
this sum by four
times that of the even
Oc
Fig. 20.
AY
660
B
ordinates and twice that of the uneven ordinates, and multiply
by one third of the distance between any two consecutive
ordinates.
Demonstration: To compute the area comprised by a
curve, any two of its ordinates and the axis of abscisses,
by plane geometry, it is usual to divide it into elemen-
tary areas, by drawing ordinates, as in the last figure,

MECHANICS OF SOLIDS.
53
and to regard each of the elementary figures, e 2 72 712
Fig. 21.
the rule.
e es 13 72, &c., as trapezoids; and it is obvious that the Demonstration of
error of this supposition will be less,
in proportion as the number of trape-
zoids between given limits is greater.
Take the first two trapezoids of the
preceding figure, and divide the dis-
tance e es into three equal parts, and
at the points of division, erect the or-
dinates mn, min; the area computed
from the three trapezoids em nr1,
m my ny n, my еs r3 n1, will be more ac-
curate than if computed from the two e1 2 72 71, e2 C3 73 72
The area by the three trapezoids is
e1 mem 3
em X
eri+mn
2
+m mi
mntmin
2
+ my ez
my my + ez rz
2
But by construction,
G₁ m = m m₁ = m₁ еz =
e es = & ly,
and the above may be written,
ee (er₁ + 2 m n + 2 m₁ n + (373),
but in the trapezoid m minn,
2 m n + 2 m₁n₁ = 4 er, very nearly;
whence the area becomes
(er₁ + 4 e r₂+ erg);
the area of the next two trapezoids in order, of the pre-
ceding figure, will be
eg (eg r3 + 4er4 + es rs);

54
NATURAL PHILOSOPHY.
Algebraic
and similar expressions for each succeeding pair of trape-
zoids. Taking the sum of these, and we have the whole
area bounded by the curve, its extreme ordinates, and the
axis of abscisses; or
expression of the Q=₁₁₂ [e₁₁ +42₂+2e3r3 +44 +2e5r5 +4е6r6+er,];
rule.
whence the rule.
§ 47.-When the value of the mechanical work of a
variable resistance for any distance passed over by the
Mean resistance; point of action, is found by the method just explained, if
this value be divided by the distance, the quotient will
equal to the entire be a mean resistance, or the constant effort which, exerted
work divided by through the entire path, will produce the same quantity
of work; for we have seen that for a constant resistance,
the quantity of work is measured by the product arising
from multiplying this resistance into the path described
by its point of action.
the entire path.
Examples of
mechanical
work;
that of a force
§ 48.-When a motive force is employed to bend a
spring, it will develop, at each instant, an effort which
is greater in proportion as its point of action describes,
in the direction of the effort, a greater path; an effort
bending a spring, which we have seen may be measured for each position
of the spring or point of action. The curve which gives
the law of these efforts may be constructed by the method
just given, and the area determined by the rule in § 46
will give the total mechanical work performed by the
force.
horse,
We have already taken as an example the work pro-
duced by a constant force in drawing a body over a horizon-
tal plane, and above we have taken the work which arises
from the action of a variable force in bending a spring;
of the draft of a the reasoning applied to these is applicable to all kinds
of work employed in the arts. Does a horse pull upon
the shaft of a mortar mill; a man draw water from a well;
of the effort of
2 man.

MECHANICS OF SOLIDS.
55
an artificer,
an artificer saw, plane, file, polish; a turner fashion his of the
materials in the lathe; the quantity of work performed manipulations of
is measured by the product of the effort, which is always obtained by the
equal and contrary to the resistance opposed by the same rule.
matter to the tool, into the path described by the point
of action, if the resistance is constant, or by the sum of
the partial products which measure the elementary por-
tions of work, if the resistance is variable.
observed in work.
§ 49.-In seeking to appreciate different kinds of work, Distinction to be
we must be careful not to confound that which is really
expended by the motive force, with that which is actually
effective in accomplishing an object. It is to this last that
are to be applied the foregoing considerations and measure-
ments. We shall presently examine the mode of action
of motive forces, the circumstances which modify the result
of this action, and the waste which may attend it.
inherent in
certain kinds of
§ 50. To show the complication incident to certain complication
kinds of mechanical work, take, for example, the work
of a filer: it is necessary 1st, to press upon the file to work.
make it take hold; 2d, to support continually its weight;
3d, to push it along the surface of the body; 4th, to move
it with a certain velocity back and forth, and therefore to
overcome the inertia of the file as well as that of the
matter removed. The quantity of work is the result of
these different circumstances; but this complication may
be made to disappear by separating from the result of the
work, every thing not indispensable to it, in considering
only what takes place where the metal is removed by the
file: there, we only perceive a resistance which is op-
posed to an equal and contrary effort in the direction
of the path described by the points of action of the file,
and of which the quantity of work is measured in the The work reduced
manner already described. The work of the operator before.
may be reduced to this, by supposing the file placed upon
a level surface, loaded with a given weight, and the
and measured as

56
NATURAL PHILOSOPHY.
What must be
mechanical work
of a force;
operator or motive power only employed in drawing it
uniformly in the direction of its length.
§ 51.-In general, then, we must henceforth understand
understood by by mechanical work, that which results from the simple
action of a force upon a resistance which is immediately
opposed to it, and which is continually destroyed in
causing the point of action to describe a path on the
line of direction of this resistance. The force must be
considered as a simple agent, producing an effort or
pressure measurable in pounds, and acting in a single
direction, as described in § 38; and we must be careful
not to confound, as is frequently done, the terms work
and force, with those by which we vaguely designate
all the effects, more or less complicated, arising from the
action of animate or inanimate agents upon resistances:
thus we should not speak of the force of a horse, of a man,
of a machine, without indicating the point of action of this
force, its intensity, and its direction; we should not speak
of the mechanical work of a force, without specifying the
same things of the resistance which it overcomes at each
instant, in each particular case of its application.
work of the
resistance.
Invariable
standard by
§ 52.-The most simple work, that which conveys at
once an idea of its measure, is the elevation of a weight
through a vertical height, if we omit the consideration
of inertia. The work in this case obviously increases as
the weight and vertical height increase, and is measured
the quantity of by the product of the two, agreeably to what is said in
45 and § 46; here the unit of work, is the unit of
weight raised through a unit of height.
which to estimate
work;
utility of this
standard.
The utility of this measure is its great simplicity, and
the ease it affords of estimating the pressure or effort in
pounds, and the path described by the point of action in
feet. We might, to be sure, take any other standard unit,
as, for instance, the quantity of work necessary to grind
1, 2, or 3 pounds of corn, which is the old standard of

MECHANICS OF SOLIDS.
57
millers;
millers and the proprietors of mills. But a given weight
of corn will present different degrees of resistance, accord- Standard of
ing to its quality and the kind of tool or machine
employed to grind it; so that not only is it impossible
for people generally to understand what the millers mean
by their standard, but for the millers to understand
each other. It is hence indispensable to have some objections to it.
standard which does not admit of variation, and of being
interpreted differently by different people; of such a
nature is the standard which results from the considera-
tion of the effort, and the path described by its point of
action in the direction of the effort.
comparing
It will remain to be found how many pounds of corn Means of
this unit of work is capable of grinding, how many square different
yards of boards it will saw, &c.: all this must come from standards.
careful observation and experiment. It is, above all,
essential that there shall be nothing arbitrary in the
mode of estimating the quantity of mechanical work.
53.-Different authors have given different names
to mechanical work, which should be carefully distin-
guished from the object accomplished, this latter being
but its effect.
mechanical
SMEATON calls it mechanical power; CARNOT, moment Different names
of activity; MONGE and HACHETTE, dynamic effect; Cou- given to
LOMB, NAVIER, and others, quantity of action; and this work;
last expression is now generally adopted. It will here-
after be employed, and will always signify the quantity
of work-mechanical work.
quantity of
Sometimes the mechanical work has been called quan- sometimes called
tity of motion, and sometimes living force, both of which are motion and living
but simple effects of mechanical work upon a body free force.
to move. We shall explain, in the proper place, the mean-
ing to be attached to these terms.
All work is judged of by the quantity of each par- Work judged of
ticular species of result, or useful effect, which it produces; by the useful
but we have seen that this quantiy of result is propor-
effect.

58
NATURAL PHILOSOPHY.
To express the
continued work
in numbers;
tional to the quantity of mechanical work necessary to its
production, and hence mechanical work or quantity of
action is what pays in forces.
§ 54.-When a motive force acts with a constant effort,
and its point of action moves uniformly during any con-
siderable portion of time, it will be sufficient to express
the work done in a unit of time, as a day, an hour, a
minute, or second. This will avoid the use of multiplicity
of figures in comparing the effects of different forces with
each other, while it will enable us easily to obtain the
value of the whole work, by simply multiplying the work
in the unit of time, by the number of units of time during
of time; note the which the force has acted or been working. The duration
of the work must, therefore, be noted. Thus, we say the
mechanical work of a particular horse is 120 pounds
raised through a vertical height of 3 feet in one second,
or 120 pounds raised through 180 feet in one minute,
this work being continued during 8 entire hours each
day.
work in unit
duration of the
effort.
The path
described in a
second is usually
taken:
of this.
Ordinarily, we take for the length of path, that which is
described in one second, this latter being taken as the unit
of time. But this distance, according to the definition of
uniform motion, is the measure of the velocity of the point
of action, which we have supposed constant; by this co-
incidence, the mechanical work happens to be measured
by the product of a constant effort into the velocity of its
point of action: which has misled many persons in caus-
the consequences ing them, as we shall see further on, to confound the
quantity of work or of action with the quantity of motion,
although their measures are in fact very different.
All units
arbitrary;
pound; unit of
In the same way that the unit of time is arbitrary, so
also are the units of effort or weight and distance, and
consequently the unit of work, which is always equal to
unit of effort, one the unit of effort or weight, exerted through the unit of
distance. We shall take for the unit of effort 1 pound,
and for the unit of distance 1 foot, so that the unit of
distance, one
foot:

MECHANICS OF SOLIDS.
59
work will be, as before, the effort one pound exerted through unit of work, the
a distance of one foot.
Suppose, for example, that the effort 75 pounds is
exerted through the distance 4 feet, then will
product of these;
4 x 75 = 300 units of work,
reference to time.
of which each one is equivalent to an effort of one pound and has no
exerted through a distance of one foot. This is ordinarily
expressed thus,
300lbs. f.;
and is read, 300 pounds raised through 1 foot. And this
has no reference to the time in which the work is per-
formed.
work proposed,
when time is
§ 55.-Mechanicians long felt the necessity of some Different units of
well defined unit by which to express the work performed,
or capable of being performed, by a motive force, in a considered.
given time, and several were proposed; but these ill
according among themselves, there seemed as little likeli-
hood of a general agreement in this respect as in regard
to the unit of velocity, which depends upon the units
assumed for time and space.
After the introduction of the steam-engine, the horse- Horse-power
power was proposed, and is now generally adopted as the adopted;
measuring unit. By horse-power is meant, the quantity of
work, measurable in pounds and feet, which a horse is
capable of performing in a given time; but this would
obviously be indefinite, since horses differ in strength and
endurance, were it not that some fixed value has been
agreed upon, according to the principle explained in § 51,
as the standard of horse-power. This value is the mean
of the results of a great many trials with different horses,
and is set down at 550 pounds raised through a vertical 550 lbs.
height of 1 foot in 1 second, or 33,000 pounds raised through 1 foot in
through 1 foot in 1 minute, or 1,980,000 pounds raised
1 second;

60
NATURAL PHILOSOPHY.
Example.
Error of
considering the
greatest effort
alone;
this effort may
be replaced by a
fixed obstacle;
error of
considering the
path alone.
through 1 foot in 1 hour; all of which amount to the same
thing.
When, then, we are told that à machine or engine is
of 30-horse power, or has a power equal to 30, for in-
stance, we are to understand that it will do work which
is equivalent to raising 550 x 30=16,500 pounds through
one foot in 1 second, or 33,000 x 30=990,000 pounds
through one foot in 1 minute, &c.
§ 56.--We can now appreciate the error we should
commit, if, in estimating the productive power of a motive
force or machine, we confine ourselves to the greatest
absolute effort it is capable of exerting, without regard to
the space described by its point of action; if, for ex-
ample, in estimating the productive effort of a man, we
only consider the greatest burden he is capable of sup-
porting at rest under the action of its weight; or, of a
horse, we consider alone the greatest effort, as indicated
by a spring balance, he can exert while pulling against a
fixed obstacle.
We can conclude nothing from these in respect to the
quantity of action; we must also have the path described
in a continuous manner. Simply to support a weight or
exert an effort, is not to work usefully; and this is rendered
clear from the consideration that we may in all such cases
replace the motor by an inert body, as a prop, a post,
&c.; the action and reaction being equal and contrary,
unaccompanied by any motion, there is no balance of
work either in favor of the effort or resistance.
It would be equally impossible to infer any work or
quantity of action from the path described by the point of
action, without taking into account the effort exerted at
each instant. It is obvious that a man or horse, running
at full speed, without exerting any effort except that
which he is capable of impressing upon himself, is pro-
ducing no useful effect; he overcomes no resistance ex-
ternal to himself, which it can be an object to destroy.

MECHANICS OF SOLIDS.
61
abilities
In a word, the productive effect of every motive force is Productive
measured, at each instant, by the product of the effort measured by the
into the path described in the direction of the effort; so product of the
that, if either the effort or path be zero, the quantity of
action will also be zero.
effort into the
path.
work, even on
fixed obstacles;
§ 57.-It must be remarked, however, that, since all Always some
bodies are more or less extensible and compressible, a
motive force cannot act against what are usually called
fixed obstacles, without producing a certain quantity of
action or mechanical work, such as we have defined it:
for the point to which the force is applied will yield to a
greater or less extent, and the body will be flattened or
elongated; the molecular springs will oppose a resistance;
there will be a small path described in the direction of the
force. At first the efforts of the equal and contrary re-
sistances are nothing; afterward they augment by degrees
till the effort of the power attains its maximum, and the
body its greatest change of shape; after this the action is
reduced to maintaining the body or obstacle at its state
of tension and repose, without producing henceforth any
mechanical action.
§ 58. Construct, in
the manner before de-
scribed, the curve Ori
T2...76, of which the
abscisses Oe, ese, &c.,
represent the spaces de-
scribed by the point of
action in each successive
instant of time in the
direction of the force, and
the ordinates, the cor-
responding pressures or
0
T1/
Fig. 22.
73
3
76
25
74
B
resistances opposed by the body in a contrary direction.
The quantity of work destroyed while the point of action
is describing any one of the small paths, as e, eg, is the
its value
represented
geometrically.

62
NATURAL PHILOSOPHY.
most cases be
neglected;
area of the trapezoid eer3 72, and the total quantity of
action destroyed by the molecular action of the body
during its entire change of figure, is the area comprised
by the curve, its greatest ordinate e re, which denotes the
maximum resistance, and the axis of abscisses. If, then,
it should happen that the body or obstacle is either com-
pressed or extended by any appreciable quantity as Oeg
which is the path described by the point of action, and
the greatest resistance ere should be considerable, this
quantity of work must be taken into account in certain
circumstances which will be explained.
§ 59.-But in general the bodies employed to receive
This work may in and transmit the action of forces, are selected with special
reference to their capacity to resist all change of figure;
so that when well chosen and judiciously disposed in com-
binations, the work referred to in the preceding article,
becomes so small a fraction of that developed by the force
when it produces motion, or when the space described by
the point of action is considerable in comparison with
that which measures the linear change of figure, that it
may, and indeed is in practice, neglected. It is under
this point of view only that the work developed by a
force, applied to a fixed obstacle can be said to be
nothing.
action and the
motion are at
right angles to
each other.
This work may also be neglected when the force which
develops it, acts in a direction perpendicular to the path
which the body is, by its connection with others, com-
especially when pelled to describe. The force in this case will only
compress or stretch the body uselessly, without adding to
or subtracting from the work in the direction of the
motion. A man who pushes against the side of a carriage
in a direction perpendicular to the path along which it
is moving, neither aids nor hinders the horses: and
although he actually develops a quantity of work by the
compression of the carriage, it must be totally neglected
in making an estimate of the useful effect.

MECHANICS OF SOLIDS.
63
without useful
$60. These considerations are important, as they Motive forces
prove, in general, that forces may work without produ- may work
cing any useful effect. If the different pieces, for example, effect;
which compose a machine, and which serve to transmit
motion and work, in acting upon each other, become com-
pressed or stretched, it is obvious that, even though the
point of action moves in the direction of the force, this
latter must first expend a certain quantity of work in
changing the figure of the pieces before the motion can
become regular or uniform throughout. And it may
happen that this first work of the power will be totally
lost, if the pieces, on ceasing to be compressed or stretched, the pieces
retain their altered shape: that is to say, if they be not work not being
elastic, or, more generally, if the molecular springs do not perfectly elastic.
contribute to augment the work when the effort of the
force is relaxed, as they did to diminish it when the
action began.
transmitting the
force changes.
§ 61. We also see that if the action of the force or Loss greater in
motor, or the resistance occasioned by the work, undergo proportion as the
frequent alterations, in becoming sometimes feeble and
sometimes stronger; in a word, if the pieces are often
compressed and distended, the loss of work thence arising
may bear a considerable ratio to the total work of the
power, which could not take place if the action of the
latter were constantly the same from the beginning to the
end of the work.
shocks.
62. The shock of bodies develops considerable still greater in
pressure, and produces sensible changes of figure; the the case of
quantity of action destroyed or generated will, therefore,
always be appreciable. On this account it becomes in-
dispensable, in the application of mechanics, to pay the
strictest attention to the influence of concussions which
may occur during the performance of mechanical work.
§ 63.-And hence we perceive the advantage arising Advantages of

64
64
NATURAL PHILOSOPHY.
stiff and elastic
materials
Elastic bodies
restore, in
expanding, the
work absorbed
in being
compressed.
Loss of work
are not perfectly
elastic.
from the use of very stiff and very elastic materials in the
construction of those pieces which are employed to receive
and transmit the action of forces, and to regulate the mo-
tions they produce.
§ 64. To obtain a clear idea how the molecular
springs of a body may develop or restore a certain
quantity of mechanical work, we have but to consider
what takes place at the instant when a body begins to
resume, progressively, its primitive figure after it has been
changed, and to recall what was said of the measure of the
quantity of work of a force, employed to bend a spring,
to compress or distend a body. Indeed, we have only to
estimate, in pounds, the different pressures corresponding
to each state of the body, from that of greatest compression
or distention to that of restitution, or to some intermediate
state which the body will retain of itself. If the body
resume, at last, precisely the form which it had before the
change; if, also, the pressures which correspond to the
same degree of tension-to the same shape and size of the
body, are the same, if, in a word, the body be perfectly
elastic, the quantity of work produced during the process
of restitution against a resistance opposed to it, will be
equal to that required to compress or distend it, since the
curve, which gives the law of the pressures and spaces,
will be the same in the two cases.
If, on the contrary, the body be not perfectly elastic,
when the bodies it will not return to its former figure; the pressures will
be less during the process of restitution, there will be a
loss of space described by the point of action, and, con-
sequently, less work performed than in the first change
of figure, there will be a certain quantity of action lost.
Examples of
elastic bodies;
There are scarcely any perfectly elastic bodies except
the gases and vapors, and these must be confined in a
close vessel or reservoir and acted upon by a piston.
Such contrivances, together with springs made of the most
elastic solids, serve to store up mechanical work for

MECHANICS OF SOLIDS.
65
air-gun;
future use; forces are employed to compress or bend them, their use.
in which state they are retained by mechanical contri-
vances till the work thus expended is required for other
purposes; the restraint is then removed and the work
transferred to some other body, which, in its turn, com-
municates it to something else, and so on to the ultimate
object to be attained. The balistas, catapultas, and bows Examples-
of the ancients, throwing arrows, stones, and other missiles balistas, bowe,
are examples of this; the air-gun, in which the motive
power is but a reservoir of compressed air, is well known;
and every body is familiar with the steam-engine, in
which, by the application of heat, water is expanded into
vapor whose molecular spring or elasticity is capable of
performing any amount of work, by the simple alterna-
tions of heating and cooling. No one is ignorant of steam and
the terrible effects of steam and gunpowder, when over- gunpowder.
heated, and yet, when properly managed, these agents
admit of being pent up in inert bodies or vessels, and
made to do the work not only of the lower animals, such
as horses, oxen, &c., but almost of intelligent beings. It
is by means of this principle of elasticity, that clocks
and watches, are kept in motion for days and entire
months.
mechanical
$65.-Weight also affords the means of storing up Weight as a
mechanical work, and of rendering it available when means of storing
wanted. When a motive force has elevated a body work.
through a certain height, in expending upon it a quantity
of work, measured by the product of its weight into the
height, this body, employed afterward to overcome a
resistance either directly or by means of a machine, may
restore, in its descent, precisely the same quantity of
work which had been before expended upon it. It is in
this way that motion is communicated to clocks, spits,
&c., &c.
By the action of heat, water assumes at the surface of Elevation of
the ocean the form of vapor, ascends to elevated regions
water by heat.
5

66
NATURAL PHILOSOPHY.
to break, &c., not
reproduced.
in the atmosphere, whence it is precipitated in the form
of rain, is collected into natural reservoirs, and becomes,
by its weight, a source of motion to mills, machinery, &c.
This reproduction does not obtain, however, when the
Work employed work is employed to divide, to break, to polish, to rub, to
destroy, in a word, the natural affinity of bodies. The
quantity of work thus expended is, in a mechanical point
of view, totally annihilated; it cannot be restored by the
body after it has undergone this change of state.
Portability of
and
combustibles.
Springs, like animals, and combustibles which give
springs, animals, heat, have this peculiarity, viz.: they are very portable,
and may be even used as a motive power for vehicles.
Thus carriages have been put in motion by springs at-
tached, as boats are put in motion by animals on board,
and by the vapor of heated water. But springs are
never perfect, and being subjected to the action of foreign
resistances, never restore the whole of the mechanical
work which they have received. Finally, animals, and
heat even, the primitive source of all the mechanical
work employed in the arts, require a certain expense in
nourishment and fuel which, according to the beautiful
theory of Leibig, are the same in principle. This nourish-
ment and fuel become, therefore, the representatives of a
certain amount of mechanical work, so that it is really
By impossible to create a motive force, without having pre-
viously incurred an equivalent expenditure.
Nourishment
and fuel
representatives
of mechanical
work.
achol 20
Inertia a source
of mechanical
work.
bod
§ 66. Thus far we have only examined the work of
of reproduction forces when employed to overcome the weight of bodies,
the resistance inherent to their state of aggregation or
force of affinity, their elasticity, &c. It remains to speak
of the resistance which all bodies oppose to a change of
their state in respect to motion or rest, by reason of their
inertia, of which no estimate has been made in what has
gone before, and from which it is impossible to separate
the other species of resistance in all questions affecting
quantity of work. It has already been remarked that the
To sollevel T

MECHANICS OF SOLIDS.
67
artificer must overcome the inertia of the matter ofarlanean
Lan bolalosos
which his tool is made; the draft-horse, that of the
carriage, and of the load it bears, &c. But indepen-
dently of this, it is very important to be able to estimate
the quantity of work which a body will absorb in ac-
quiring a certain degree of velocity, for this is often the
only useful object in view, as in the case of throwing
projectiles by the elastic force of gases or solids, which
gives rise to the art of balistics, employed in war. Be-
sides, it very often happens that instead of applying a
force directly to the object in view, we cause it to act
upon a free body, and subsequently, by the aid of its
inertia, concentrate the quantity of action absorbed by it
to do the work at a blow, as in the example of the pile- Examples—
ram, common hammer, &c.; the inertia of bodies is thus pile-ram and
made, like weight, elasticity, &c., to restore the quantity of hammer.
work which has been expended in subduing it; and we
now proceed to the consideration of the action of forces
employed to overcome inertia and to produce motion.
common
doldw
III.
1601
VARIED MOTION.
constant force.
8 67. We will begin with the most simple case of Varied motion;
varied motion, viz: that in which a body is pressed by a
constant force, that is to say, one which does not change
the intensity of its action, and which is equal and contrary
to the resistance opposed by the inertia in the line of
direction of the motion.
It is clear that, the pressure being the same at each
instant, the small increments or decrements of velocity
will, for the same body, also be the same; and thus the
velocity will increase or decrease with the time; in other

68
NATURAL PHILOSOPHY.
retarded.
This is
uniformly varied, elapsed since the commencement of motion.
accelerated, and called uniformly varied motion in general; which becomes
uniformly accelerated or uniformly retarded, according as
the force increases or diminishes the velocity of the
body.
Uniformly
accelerated;
§ 68. First, take the case of uniformly accelerated
motion, and recall to mind that the velocity acquired at any
instant is, § 26, measured by the space described by the
body in the unit of time succeeding this instant, if, the
force having ceased its action, the body continue to move
uniformly in virtue of its inertia; this velocity we have
seen how to calculate by means of the law which connects
the time with the spaces.
graphical
representation of
this motion.
Let O be the point
of starting. Draw the
line Ov₁ v2... V6, of which
the abscisses Ot, Ot
&c., represent the times
elapsed from the origin
or beginning of the mo-
tion, and of which the
ordinates tv, to v₂,...to V6,
represent the velocities
acquired at the end of
the times 0, 0 to,...O tg.
V1
V2
Fig. 23.
V5
V4
bs
b3
V6
Since in uniformly varied motion, the velocities tv₁
tava,... teve are proportional to the times Ot, Ota,... O te
the line Ovv2V3...V6, is a right line, which passes
through the point O from which the body takes its depar-
ture; for at this point, the velocity and time are zero
together, at the instant of starting. The distances Ot₁,
tita, tata, &c., being equal, if through the points v1, v2,
V3,...ve, lines be drawn parallel to the axis OB of times,
there will be formed a series of right-angled triangles,
Ot₁ v₁, v₁ b₂ v₂,... v; be ve, all equal to each other. The sides
tv₁, vaba, vaba,... vb, will represent the successive incre-

MECHANICS OF SOLIDS.
69
ments of velocity, which are equal and constant, by the
definition of uniformly varied motion, since the corre-
sponding intervals of time Ot, v1b2 v2 bз,.... v5 be, are
იხვი
equal.
time.
The successive intervals of time Ot, t ta, ta ta, &c., being Path represented
supposed very small, we may regard the body as moving rectangle of
uniformly during any one of them as t t or its equal velocity into
v3b4, and with the velocity tv acquired at its commence-
ment. But by virtue of uniform motion, the path
described by the body contains as many linear units as
the rectangle of the time into the velocity contains super-
ficial units, and, in this sense, the distance passed over by
the body in the time to t, will have for its measure the
product of this elementary portion of time by the velocity
tv3, or the area of the rectangle to tbv for another
interval tts, the path described will have for the measure
of its length, the area tts bv, and so on; so that the total
length of path described by the body during the time O te
will be the sum of all the partial rectangles ty to b₂ v1,
tz tz bg vq,.... tz to be vs; which sum will not differ sensibly
from the area of the triangle Oteve, when the points of
division t, t,...ts, are greatly multiplied.
From this fact, viz.: that the length of the path
described by a body in uniformly varied motion, is
represented by the area of a triangle whose base is the
time during which the motion takes place, and altitude the
velocity acquired at the end of this time, we easily deduce
several important consequences, called the laws of uni-
formly varied motion.
motion.
Since the area of the triangle Oteve, has for its measure, Laws of
the half of its base into its altitude, and as the base uniformly varied
into the altitude, or the entire rectangle, represents the
length of path described in the time Ote, with a con-
stant velocity te ve, acquired at the end of this time, it
follows,
1st. In uniformly accelerated motion, the path described First law.
at the end of any time, is half that which the body would

70
NATURAL PHILOSOPHY.
Second law.
Third law.
Formulas to
compute the
circumstances
of this motion.
describe in the same time, if it were to move uniformly with
the velocity acquired during this time.
Since the paths described during any two times, as
Ots, Ots, are represented by the triangles Ot, vg, Ots V5
respectively, and since these triangles are similar and their
areas are to each other as the squares of their homologous
sides, it also follows,
2d. In uniformly accelerated motion, the paths described
at the end of any two times, are to each other as the squares
of these times.
3d. That these paths are to each other, as the squares
of the velocities acquired at the end of the corresponding
times.
When in uniformly accelerated motion, the velocity
tvs, acquired at the end of a given time Ots, say one
second, taken as the unit of time, is given, the law of the
motion or the right line Ove, which represents it, is com-
pletely determined, and we may compute the velocity and
space which correspond to any other time.
e
Denote by and v₁, the length of path and velocity
which correspond to the first second, and by S and V, the
path and velocity corresponding to any other time, as T;
we have by the first law,
bod, and fodtreeb
Space in unit of
time;
1
=
male fina
relation of space,
time, and
velocity;
9 SVT.
S = VT.
and by the second law,
e: S: 1 X 1: TXT:: 1: T2;
whence,
space in any
time.
Forbipos
(4),
(5);
Mods At S = ex T². . . . (6);

MECHANICS OF SOLIDS.
71
14
and replacing & by its value, Eq. (4), did
S = v₁ T². .
From the third law,
e1,
or : S :: v2: V²;
whence
(7)
Space in any
time;
gribes
V² = 2 v₁S. . .
(8).
Velocity due to
any space.
By the definition of uniformly varied motion, we have,
V₁: V:: 1: T;
whence
V = v₁ T
(9).
Velocity due to
any time.
In what precedes, we have supposed the body to start The body has
from rest, so that the right line, which gives the law of the already an
motion, passes through the point of departure 0. But
if the body have already
a velocity Ovo acquired
previously, this right line
will pass through vo, the
extremity of the ordinate
which represents the ve-
locity of the body at the in-
stant from which the time is
reckoned. The velocity Ove
is called the initial velocity.
51
By drawing vot's, parallel
to OB, we see that the
velocity tv, which corre-
sponds to the time Otą, is
composed of two parts, viz.
t3 t's, and t3v3; the first is
31
Vo
it'
Fig. 24.
V5
V2
V4
B
0
tz
equal to the initial velocity Ovo, and the second to the
acquired velocity:
initial velocity.
un to sider
od to enlar
lootor

72
12
NATURAL PHILOSOPHY.
Formulas to
compute the
circumstances of
the motion;
51
velocity which the body would acquire in the time vot's,
equal to Ots, under the action of the constant force, had it
moved from the point v, with no initial velocity, as in the
preceding case; for the line vo vs gives, in reference to the
line vot's, the law of acceleration. Knowing, then, the
velocity which the force is capable of impressing upon the
body in a unit of time when moved from a state of rest,
it is easy to construct the line vo v5, in relation to vo t's or
its parallel Ots, and to deduce from it all the circumstances
of the motion.
Let it be required, for example, to find the length of
path described by the body in the time Ot. This path
will contain as many linear units as the trapezoidal area
Otv vo contains superficial units. We perceive at once,
that this length will be composed of two parts, viz.: that
described uniformly in virtue of the initial velocity Ovo,
and represented by the rectangle Ottavo, and that de-
scribed in virtue of the constant force and represented by
the triangle vot4v4. But, denoting by a the initial ve-
locity, and by T the time, we have for the measure of the
rectangle
a T,
and for the measure of the triangle, Eq. (7),
value of the
space;
value of the
velocity.
v₁ T²;
and if we denote by S the total length of path actually
described by the body, we have
S = a T + v₁T2.
T² . . (10):
and because the actual velocity at the end of any time, is
the initial velocity increased by that due to the action of
the constant force during this time, we have, Eq. (9),
V = a + v₁ T
•
(11).

MECHANICS OF SOLIDS.
73
§ 69.-If we now suppose the constant force, instead Uniformly
of increasing the initial velocity of the body, to diminish retarded motion;
it, the motion becomes uniformly retarded, and the line
Vo V4 gives the law of the motion.
By drawing vot's parallel to
Ots, we see that the velocity
v3 t3, which corresponds to the
time Ot, is nothing else than
the initial velocity Ovo dimin-
ished by the velocity t's v
which the body would acquire
under the action of the con-
stant force at the end of the
time Ots had it moved from
rest. The length of path de-
scribed is now represented by
Vo
Fig. 25.
t's to to to to
V2
the trapezoidal area Ovvo; and is equal to that which
would be uniformly described in the same time, with the initial
velocity Ovo, diminished by that which would be described in
the same time, if moved from rest under the action of the
constant force, by a motion uniformly accelerated; that is to
say, the length of path is represented by the rectangle Ots t's vo
diminished by the triangle vo v3 t3.
graphical
representation.
compute the
The equations (10) and (11), which appertain to uni- Formulas to
formly accelerated motion, become, therefore, applicable circumstances of
to uniformly retarded motion, by simply changing the this motion.
sign of the velocity generated by the constant force, and
that of the area of the triangle, which represents the path
due to the action of this force; hence,
S = aT - v₁ T².
(12),
Value of space;
V = a - v₁T
(13).
of velocity.
Let us suppose that, among other things, we desire
the time required for the force to destroy all the initial

74
NATURAL PHILOSOPHY.
print velocity; we have only to make V = 0, and equation (13)
bob becomes
bed
a — v₁ T = 0,
--
poff of
whence
Time required to
destroy all a
body's velocity.
The path
described during
the destruction of
its velocity;
is destroyed, the
body will return;
T
=
α
V1
(14);
from which we conclude that the time required for a con-
stant force to destroy all the velocity a body may have, is
equal to the quotient arising from dividing the value of
this velocity, by the velocity which the force can generate
in the body in one unit of time.
To find the length of path described by the body
during the extinction of its velocity, substitute the value
of the time above found in equation (12), and we have
S
=
a²
2 v1
. . (15);
that is to say, the space through which a body will move
during the entire destruction of its velocity by the action
of a constant force, is equal to the square of the velocity
destroyed, divided by twice the velocity which this force
can generate in the body during a unit of time.
It is important to remark, that if the force continue to
after the velocity act after having destroyed all the velocity, the body will
return along the path already described, and pass in
succession and in reverse order, as to time, through its
previous positions, at each of which it will have the same
velocity it had there before; for while the body is losing
its velocity, it may be regarded as beginning its motion.
at any point of its path with its remaining velocity or
that yet to be destroyed, which, in such case, is denoted by
and when all its velocity is destroyed, it returns from a
state of rest or begins to move backward with no initial
velocity; so that equations (4) to (9) become applicable to
a,

MECHANICS OF SOLIDS.
75
this latter motion, while equations (14) and (15) are to the
former. But from equation (8) we have
delow
oda
V = √2 v₁S,
and substituting for S its value given by equation (15)
we get
Jeads
bod
V=V
20, απ
201
a;
that is to say, the velocity V, which the body has ac-
quired in moving backward through a space S is equal to
the velocity a, with which it began to describe the same
space in its forward motion.
and have at its
previous
positions the
same velocity as
before.
§ 70.-One of the most important examples of uni- Motion of
formly accelerated motion, is that presented by the verti- falling bodies;
cal fall of heavy bodies; but, before discussing it, we will
make known some of the circumstances which accompany
and modify this motion at the surface of the earth.
We have already seen, § 32, that the force of gravity
may be considered as constant within ordinary limits.
motion;
But at the surface of our globe, all bodies are plunged into causes which
the atmosphere, and this atmosphere is itself a material modify this
body, which, by its inertia and impenetrability, opposes
with greater or less energy all kinds of motion of bodies;
this opposition is named atmospheric resistance. Experi-
ment shows us that this resistance increases as the velocity
of the body and the extent of its surface increase; thus,
in striking the air with a light flat board, the resistance
which we experience is greater in proportion as the mo- influence of
tion is more rapid, while it is scarcely sensible when the
motion is very slow; and again, the resistance will be less
if, instead of striking the air with the broad surface, we
present to it the edge of the board.
velocity and
extent of surface;

76
NATURAL PHILOSOPHY.
on the fall of
bodies;
bodies which
weigh most and
have least
surface, fall most
rapidly;
Fig. 26.
It is plain, therefore, that the presence of the air must
influence of air modify the laws of the vertical fall of bodies subjected to
the action of their weight. In permitting bodies to fall
through the air, from the same height, it is
observed that those which weigh most under
the same volume, or those which present the
least surface in the direction of the motion,
arrive soonest at the bottom; thus, a ball of
lead will fall sooner than a ball of equal
volume of common wood, and a ball of
common wood sooner than one of cork, &c.
But if made to fall in vacuo, or in a long
hollow cylinder from which the air has been
removed, experiment shows that all bodies
fall equally fast, and therefore will reach the
bottom at the same instant if they start
together. This is called the guinea and
feather experiment, from the fact that a
in vacuo all
bodies fall
equally fast;
gravity acts on
the interior and
exterior particles
guinea and feather will fall under the action of their
respective weights in vacuo, with the same velocity and,
therefore, will reach the bottom in the same time.
From this it follows, that the force of gravity acts indis-
criminately upon every particle of matter, and impresses upon
each, at every instant, the same degree of velocity in vacuo,
a fact which it is important to remember.
We may easily assure ourselves that the force of
gravity acts on the interior as well as on the exterior
of a body alike; particles of all bodies, by observing that the same body
weighs just as much by the weighing spring whether
placed in the open air, or in a close chamber; which
proves that the force of gravity acts through this chamber
envelope without undergoing any change.
distinction
between the
weight of a body
The weight of a body, is the resultant of all the actions
of the force of gravity upon its elementary particles; we
and the force of must be careful, therefore, not to confound the weight with
the force of gravity itself, which is, in fact, only the ele-
mentary force impressed upon each particle.
gravity.

MECHANICS OF SOLIDS.
77
in the air;
§ 71.-Finally, it is important to remember that the
denser bodies, such as gold, lead, iron, &c., are those Gold, lead, &c.
which, under equal volumes, or equal surfaces, will fall fall most rapidly
most rapidly in the air, because the resistance of the
latter is weaker when considered in reference to the
weight; and this resistance may become relatively so
small that we may neglect it, particularly when the fall of
the body is not very rapid.
uniformly
Galileo, an Italian philosopher, was the first to investi- the motion of
gate, experimentally, the laws which govern the motion of falling bodies
bodies falling under the action of their own weight, in accelerated.
vacuo; and he found the motion to be uniformly accelerated.
The force of gravity is, therefore, within the limits of ex-
periment, a constant accelerating force, acting with an equal
intensity at each instant whatever be the velocity ac-
quired. Atwood, an English philosopher, in resuming
the experiments of Galileo, with greatly improved means,
obtained the same results.
Laws of the
72.-Hence, when a body falls from rest through a motion of falling
certain height, in vacuo,
bodies;
1st. The velocities acquired are proportional to the first law;
times elapsed since the beginning of the motion.
2d. The total spaces passed over, or the heights of the second law;
fall, are proportional to the squares of the times elapsed.
3d. These heights are proportional to the squares of third law;
the velocities acquired at the end of each.
4th. The velocity acquired at the end of the first unit fourth law.
of time, is measured by double the height of fall passed
over during this time.
varies with the
latitude;
Although the force of gravity, may, without sensible Force of gravity
error, be regarded as constant at the same locality, it yet
varies, as we have seen, from place to place, in going
southward or northward, and cannot, therefore, generate
as much velocity in one latitude as another. From
careful experiments, made with a pendulum at different
places, it is found that the length of path described by a

78
NATURAL PHILOSOPHY.
space a body dat
will describe
under its action
in first second;
velocity it can
generate in one
second;
body in the first second of its fall from rest in vacuo, will
be given by the following formula, viz:
feet.
= 16.0904 0.04105 cos. 2.. (16),
-
in which e, is the space, and the latitude of the place.
In works on mechanics, the velocity which the force of
gravity can generate in a second of time at the surface of
the earth, is usually denoted by g; and as this velocity is
equal to twice e, Eq. (4), as given by the above equation,
we have,
feet.
g= 32.1808 0.0821 cos. 2.. (17);
hence all the circumstances of the motion of falling bodies
at any place, will be given by equations (4) to (15) after
substituting therein g for V1.
Let H represent the height, in feet, through which the
body has fallen in a given time denoted by T, and V the
velocity acquired at the bottom of this height; then, from
equations (5), (7), (8), and (9), we have
formulas which
relate to the fall
of bodies in
vacuo;
application to
examples;
H = {VT...
(18),
H
=
lg Ta..
(19),
V² = 2 g H
2g
(20),
V = g T.
(21),
in which, for all ordinary cases we may take
g= 32.1808 feet.
(22).
Suppose we are required to find the velocity acquired
and the path described at the end of 7 seconds; from
equation (21), we have
V = 32.1808 x 7 = 225.2656 feet,

MECHANICS OF SOLIDS.
79
from equation (19),
odt en wybod,
Tempo H
od I
32.1808
2
X (7)² = 788.4296 feet;
that is to say, at the end of 7 seconds, the body will
have a velocity which would carry it over a distance of results;
225.2656 feet during the 8th second, were its velocity at
the end of the seventh second to become constant, and the
space described during the seven seconds of fall, will be
788.4296 feet.
surface of the
body and height
be small;
It must be remembered that, in the atmosphere, the influence of the
body will not fall with the same velocity, on account of atmosphere;
the resistance of this medium; but from what has already in the case of
been remarked, this resistance will not have much in- metals, if the
fluence if the falling body be very dense, as iron, lead,
&c.; or if the surface of the body be small; or if the
height of fall be not great, say sixty or seventy feet. We
might, therefore, measure approximately, the height of application to
towers, depth of wells, &c., &c., by noting the time, as of towers and
indicated by a watch beating tenths or fifths of seconds, depth of wells.
required by a body to fall through the height.
If we have given the height through which a body has
fallen, it is easy to find the velocity acquired; for from
equation (20), we have
find the height
V = √2g H.
bree
Suppose a body to fall through a height of 80 feet, then
will
V = √2 x 32.1808 x 80 = 71.75 feet.
This proposition is of frequent occurrence in practical
mechanics.
The quantity V is called, the velocity due to a given
height H; and H, the height due to a given velocity V.
Velocity due to a
given height;
height due to a
given velocity.

80
NATURAL PHILOSOPHY.
A body thrown
vertically
upward;
greatest height
to which it will
ascend;
time required to
reach its greatest
height.
§ 73.-When a body, as the ball from a gun, for
example, is thrown vertically upward, its weight acts at
each instant with the same intensity to diminish by equal
degrees its primitive velocity; the motion will be uni-
formly retarded; the velocity will be totally destroyed
when the body attains a certain height, from which it will
descend, in taking successively the different degrees of
velocity which it had at the same places in its ascent, all
of which is obvious from what was said in § 69. Thus, at
the distance of 1, 5, 7, &c. feet from the place of starting,
the body will have exactly the same velocity in ascending
and descending; it will only have the direction of its
motion changed. When it returns to its point of depar-
ture, its velocity will be the same as it was at starting.
Denote by H, the greatest height the body will attain;
and V, the primitive or initial velocity; then will, equa-
tions (20) and (21),
V2
H: =
(23),
2g
V
T =
(24).
g
Example;
That is to say, the greatest height to which a body will
ascend, when thrown vertically upward, is equal to the
square of its initial velocity, divided by twice the force of
gravity; and the time of ascent will be equal to the initial
velocity, divided by the force of gravity
Let the body, for example, leave the earth with a
velocity of 150 feet a second, then will
H =
=
(150)2
2 X 32,1808
=
350.28 feet,
T=
150
32,1808
1=3
4.658 seconds.

MECHANICS OF SOLIDS.
81
This is on the supposition that the air opposes no
The body will not ascend so high in the air;
moreover, will fall with less velocity than in vacuo.
ance.
resist- effect of
and, resistance.
atmospheric
required to
body a given
velocity;
874.-We may now appreciate the quantity of work Quantity of work
or of action which the weight of a body will expend, in of the weight,
impressing upon itself a certain velocity, or in overcoming impress upon a
its inertia. Denote by W, the weight of the body, express-
ed in pounds, or, in other words, the absolute effort which
gravity exerts upon the body, and which is equal and
contrary to that necessary to support it in a given posi-
tion; this will measure the constant effort exerted upon
the body during its descent through the height H. The
quantity of work consumed during this fall will, § 45, be quantity of work
denoted by
consumed during
its fall;
W x H,
and this quantity of work will have generated in the body
the velocity V, computed by the equation
V2
2g H;
from which we have
V2
H =
2g
and multiplying both members by W,
W
WH =
= 1
g
.
(25).
velocity;
§ 75. Thus, the quantity of work developed by the work required to
weight of a body to impress a certain degree of velocity impress a given
upon itself, is equal to half the product obtained by multi-
plying the square of this velocity, by the weight of the
body, divided by the velocity g, which the force of
gravity is capable of impressing upon all bodies during
6

82
NATURAL PHILOSOPHY.
the first second of their fall. This product,
W
.V,
g
living force;
equal to double
the quantity of
action necessary
to produce it;
half the living
force lost or
the work that
overcomes the
inertia.
is what mechanicians call the living force of the body
whose weight is W. We see, therefore, that the quantity
of action expended by the weight of a body, is half the living
force impressed; or that the living force impressed, is double
the quantity of action expended by the weight.
It is to be remarked, that when a body is thrown ver-
gained, equal to tically upward with a certain velocity, the quantity of
action of the weight, which is always measured by the
product of the weight into the height to which this body
has risen, is employed, on the contrary, to destroy this
velocity, so that in the two cases of ascent and descent,
the half of the living force lost or gained, measures the
quantity of action or of work necessary to overcome the
inertia of the body, whether the object of this action be to
impress upon the body a certain velocity, or to destroy
that which it already has.
meaning of living
force;
This principle is, as we shall soon see, general, what-
ever be the motive force employed to communicate
motion to a body, and whatever be the direction of the
motion. But it is necessary first to remark upon certain
terms employed in mechanics.
§ 76.-As the expression of living force, employed to
designate the product,
W
V2,
g
not a force, but
the result of a
force's action;
may lead to error, it is proper to remark here, that it
must not be regarded as the name of any force, any more
than the name given to the product
W. H,
or the quantity of action, designates a force; it is simply

MECHANICS OF SOLIDS.
83
the result of the activity of a motive force, expressible in
pounds, which has been employed to overcome the inertia
of a body, to impress upon it a certain motion-a certain
velocity. Under this point of view, the living force is but
a dynamic effect of a force, or rather double this effect, since a dynamic effect.
W
V2 = 2 W. H.
g
work;
A body in motion, or a certain dynamic effect, may A body in motion
indeed become, in its turn, a source of work; as, for may be a cause of
example, a body thrown vertically upward is elevated
in virtue of its velocity to a certain height, as though it
were taken there by the incessant action of an animated
motor. But this is, in all respects, analogous to what
takes place when a force has developed a certain quantity
of work to bend or compress a spring; the inertia of the
matter has been brought into play in the same manner
that the molecular springs have in this latter case. This
inertia, 866, when it has been thus conquered, becomes
capable of restoring the quantity of work expended upon but cannot be a
it, as well as a compressed spring; in a word, inertia, like force any more
a spring, serves to store up a quantity of action, to body, or bent
transform it into living force, so that living force is a true
disposable quantity of action. The same may be said of
a body elevated to a certain height; this body solicited
by its weight is the source of a quantity of action, of
which we may subsequently dispose to produce a certain
amount of mechanical work. But as we cannot say that
this body, elevated to a certain height, is a force, that a
compressed spring is a force, neither can we say that a
body in motion, or that
than an elevated
spring,
W
V2
9
is a force. It is the same of men, animals in general, or animals,
of caloric, of water-courses, of wind, &c., &c.; these are but caloric, the wind,
agents of work, or motors-not simple forces.
&c.

84
NATURAL PHILOSOPHY.
Object of
mechanics as
applied to the
arts.
Hoitoat at vhod
The mass of a
body;
It is the object of mechanics, in its application to the
arts of life, to study the different transformations or
metamorphoses which the work of motors undergoes by
means of machines and implements, to compare different
quantities of work with each other, and to estimate their
value in money, or in work of this or that kind.
In short, when we speak of living force, communicated
to, or acquired by a body, it is only necessary to remem-
ber, that it relates to a real motion of the body, and is equal
to the product of the square of its velocity into its weight,
divided by the force of gravity.
877. Since the force of gravity acts indiscriminately
upon all the particles of a body, and impresses upon
them at each instant, the same degree of velocity at the
same place, the weight of a body, which is the result of
these partial actions, may give us an idea of the relative
quantity of matter it contains, or of its mass, for it is plain
that the mass must be proportional to the weight; often,
indeed, the weight is taken for the mass. But as the
intensity of the force of gravity varies from one locality to
another, and as the quantity of matter in the same body
or the mass remains absolutely the same, it is obvious that
this latter would be but ill defined by its weight. Ex-
force of gravity perience shows that the velocity impressed by the force of
gravity, in one second of time, is directly proportional to
may impress in the intensity of this force, and that therefore the ratio
proportional to
the velocity it
one second.
W
g
must remain the same for all places, since the weight is
also directly proportional to the force of gravity. Thus if
W and W', be the weights of the same body at different
places, and g and g' the intensities of the force of gravity
at those places, respectively, then will
W
g
=
W
g'

MECHANICS OF SOLIDS.
85
This invariable ratio
W
g'
is taken, in mechanics, as the
measure of the mass of a body. Denote the mass by M,
Tasidan et
then will
W
bod M:
M=
or W= Mg
dglow odi al
Mgo
(26),
Measure of the
mass of a body;
g
soned
in which W expresses the effort or pressure exerted by
the weight of the body, and g the velocity which this
weight can impress upon the body in a second of time.
Density is a term used to denote the degree of prox-
imity among the particles of a body. Its measure is the
number of particles in a unit of volume; and denoting
the volume or bulk by B, and density by D, we shall
have
гра
M= DB,
lo aptay ni omis
which, in equation (26), gives
W=D. B. g.
(26)'.
measure of the
weight.
§ 78. By substituting the value of the weight, as given
by equation (26), in the expression for the living force, we
find
W
g
V2 =
MV²;
od) to catr
Living force in
ylisso, trigim yout
van vllarney terms of the mass
and velocity;
that is to say, the living force of a body in motion, is
equal to the product of its mass into the square of its velocity.
Finally, mechanicians have agreed to call the product of
the mass of a body, as above defined, into its velocity, or
MV, the quantity of motion of the body; and this it must be quantity of
remarked is very different from the quantity of action or
of work. To understand what is meant by this new
expression, denote the quantity of motion by Q, then will
Q =
W
g
motion;
qui viiboloy to Hagu sus
nisd to hostem
=
MV
*** (27);
8

86
NATURAL PHILOSOPHY.
its meaning;
it is a pressure,
like weight;
living force
equal to the
quantity of
or, which is the same thing,
Q: W::
V: 9.
But W, is the weight of the body, and g, the velocity
which this weight can generate in this body, in one second
of time; hence Q must designate either a weight or an
equivalent effort, which can generate in the body, the
velocity V, in one second.
We see also that the living force,
MV2, or MVV = QV,
is the product of this effort, by the velocity V, or by
the path described uniformly by the body in a unit of
motion into the time in virtue of its acquired velocity.
velocity.
Use of the
denominations
mass and
quantity of
motion.
A force is
proportional to
the velocity it
when constant.
These observations show the distinction between the
quantity of motion of any body and its living force, and the
identity between this latter and double the quantity of
action.
§ 79.-It is principally to abridge and simplify the
computations and reasonings, that the denominations mass
and quantity of motion, are employed in mechanics; and
they might easily be dispensed with. But as authors
generally have used them, it becomes important to under-
stand their precise significations.
§ 80.-We have just seen that the force of gravity will
impress upon a body, during one second of time, velocities
can generate in a which are constantly proportional to its intensity, or to
given time, only the absolute weight of the body in each locality. But
this property arises only from the fact, that the weight
When the force remains constant during the fall, so that the total velocity
at the end of the fall, is proportional to the equal degrees
the small degree of velocity impressed at each instant. When the motive
force, instead of being constant, varies at each instant, it is
obvious that its intensity can no longer be measured by
is variable, it is
proportional to
of velocity
imparted at a
given instant.

MECHANICS OF SOLIDS.
87
the velocity which it impresses upon the same body
during a unit of time, and that its measure must depend
upon the small degree of velocity which it communicates
at a given instant.
the small degrees
can impress in a
By observing what takes place at the surface of the forces
earth, and in our planetary system, it is found that the proportional to
motive forces or pressures are, in fact, proportional to the small of velocity they
degrees of velocity which they impress upon the same body in
equal indefinitely small portions of time. This fact serves portion of time.
as the basis of all dynamic investigations, and must be
regarded as a general law of nature.
very small
inertia by the
impressed in a
§81.-Accordingly, let F be the measure, in pounds, Measure of the
of a force of pressure; let v be the small degree of motive force or of
velocity which it can impress upon a body at any velocity
instant or epoch, during an indefinitely small interval small time.
of time, denoted by t; also, let W be the pressure
exerted by the weight of the body at any given place,
and v' the small degree of velocity which this weight can
impress upon the body during the same short interval t.
We shall have, from the principles already established,
since F may be regarded as constant within the limited
time t
whence
F: W: v: v';
Consequences of
the above law;
W
F =
3
V.
v'
But from the first law of falling bodies
vgt: 1sec.;
whence
v = gt;

88
NATURAL PHILOSOPHY.
herefore
measure for the
W
v
v
intensity of any
motive force;
F=
X
1=
M.
(28).
g
t
That is, the intensity of any motive force, is measured by the
product of the mass into the velocity it can generate while
acting with a constant intensity, divided by the duration of the
action. Thus, when we know the small velocity v, im-
pressed in the short interval of time t, by the force F, we
may compute the value of this force, which is equal and
contrary to the resistance opposed to motion by the inertia
of the body. This resistance has been called by some the
force of inertia, and by others dynamic force. The relation
inertia exerted, given by Eq. (28), shows us that the force of inertia, which
is equal and contrary to F, is directly proportional to the
mass, and to the velocity v which this mass receives during
the elementary time t.
proportional to
the product of
mass into the
velocity
imparted;
Let F' be the measure of a second force, which acts
upon the mass M', impressing upon it in the same time
the small velocity v', then will
v'
F= M'.
t'.
relation of any
two motive forces.
which, with Eq. (28), gives
FF:: Mv M'v'.
That is to say, any two motive forces are to each other, as the
quantities of motion they can impress in the same elementary
portion of time.
§ 82.-From Eq. (28), we find
Velocity
impressed in any
short time;
v =
F.t
M

MECHANICS OF SOLIDS.
89
the force divided
from which we perceive that the degree of velocity which proportional to
a motive force impresses upon a body, during a short the intensity of
elementary portion of time, varies with the intensity of by the mass.
the force directly, and with the mass, or weight, inversely.
and contrary
83.-If now we suppose, at any instant, the force Measure of inertia
suddenly to cease to vary, and to continue to act upon the and of the equal
body with the intensity which it possessed at that instant, motive force;
the velocity will increase or diminish, proportionally to
the time, § 67, and the intensity of the force may be
measured by the definite quantity of motion which it
can impress upon the body during the first succeeding
second.
Designate by V₁ the velocity generated in the body
during the first second succeeding the instant in which
the force becomes constant, then will
V₁v: 1sec. t;
whence
Α
which, in Eq. (28), gives
=
t'
212
F = V₁M.
(29);
quantity of
and, in general, the motive force, equal and contrary to the equal to the
force of inertia, is measured, at each instant, by the quantity motion the latter
of motion it can impress during one second, if, instead of vary- can impress in a
ing, it retain unaltered the intensity it had at that instant.
When the mass becomes the unit of mass, Eq. (29)
becomes
unit of time, when
constant.
F = √₁. . . .
(30);
the force in this case is called the accelerating force, or, Accelerating
more properly, the acceleration or retardation due to the force,
force;

90
NATURAL PHILOSOPHY.
velocity
measured by the and is always measured by the velocity it is capable of im-
pressing on a unit of mass in a unit of time, acting with a
constant intensity.
impressed on a
unit of mass in
unit of time;
And from Eq. (29), which gives,
F
=
M'
. (30)',
is equal to the
motive force
divided by the
mass.
Geometrical
illustration;
it appears that the acceleration or retardation due to the
force, is, in every case, nothing more than that portion
of the entire motive force which results from dividing the
latter by the number of units in the mass acted on.
§ 84.-Trace, according to the method described for
uniformly varied motion, § 68, the curve vo V1 V2 V3, &c.,
which represents
the law of the
times and veloci-
ties; let t3 v3
and
t4v4 represent the
velocities which
correspond to the
end of the times
Ots and Ot, or
at the beginning
and end of the
very, small por-
tion of time
Fig. 27.
n
V5
V4
V3
m
V2
V1
Vo
0 to
tz t₁ = t.
B
Draw through vs the line vs b4, parallel to the axis OB
of times, and produce it till vs m=1 second; this line
will meet the ordinate t4v4, and bv4 will be the small
portion of velocity =v, impressed by the force, during
the small portion of time t. Now if, at the instant cor-
responding to the end of the time Ots, the force become
constant, it will subsequently impress upon the body equal

MECHANICS OF SOLIDS.
91
increments of velocity during the equal intervals of time t,
and the curve V3 V4 V5 will become the straight line vn,
tangent to the curve at the point v. Drawing through
m a line parallel to tv4, the portion mn will represent
the velocity V₁ impressed in one second, and the two
similar triangles, v3 b4 v4 and vg mn, will give
V4
23b4 b4 24: Vz M:
mn;
or
t:v::
1sec. :
V₁i
whence
v
V₁ = 7/7;
the value of the
velocity
impressed in one
second;
as before found.
Thus, when we know the law which connects the
velocity with the time, or the curve which represents this
law, we may, at any instant, by drawing a tangent to
the curve, determine the velocity V₁, and consequently found by the
compute the value of the intensity of the force from the tangent line;
equation,
W
F = MV₁
=
V₁;
g
the measure of
the motive force.
or, which is the same thing, the value of the equal and
contrary resistance, opposed by the inertia of the body, at
each instant during the action of the force.
§ 85.-Reciprocally, if we know the value of the in-
tensity of the force F at each instant, we deduce from it value of the
the corresponding value of
F
accelerating force,
equal to motive
force divided by
mass.
V₁ = Mi

92
NATURAL PHILOSOPHY.
Inclination of
tangent to the
curve.
or of the inclination of the tangent vn, or that of the
element of the curve of velocities to the axis OB of times.
The tangent of this inclination is given by
m n
=
= V₁
by means of this
tangent.
V3 m
and if the initial velocity Ov, be given, nothing is easier
Curve constructed than to construct the curve, of which the ordinates shall be
the successive velocities acquired under the action of the
force; since, by means of the inclinations of the tangents
or elements of the curve corresponding to each absciss of
time, those elements may be drawn one after the other,
thus forming a polygon, which will differ less and less
from the curve, in proportion as the number of values of
the force between given limits is greater.
Work necessary
velocity;
§ 86.-By the aid of what precedes, we may readily
to impress a given compute the quantity of work which must be expended
against a body, whose weight is W, by a force F, equal
and contrary to the force
of inertia, to impress upon
it a certain velocity V,
or, more generally, to
augment or diminish its
velocity by a given quan-
tity.
The quantity of work
expended
during any
small interval of time t,
has, for its measure, the
product of the intensity
of the force F, into the
0
Vo
01
Fig. 28.
V2
V3
V4
by
B
elementary portion of the path described by the body
during this time. This small path is given by the area
of the small rectangle vs to tab, whose base is the element
=
ta t=t, and whose altitude is t3 v3 V, § 67 and § 68;

MECHANICS OF SOLIDS.
93
that is to say, by the product Vt. Hence the elementary
quantity of work is
FV t,
for each instant of time, or for each small increment bv4
of velocity. But from Eq. (28) we have
F = M +;
replacing F by this value, in the preceding expression,
we have, for the elementary quantity of work,
MVv;
and it is the sum of all these partial quantities of work
which composes the total quantity of work; this sum may
be found thus:
From the point 0, as an origin, lay off the distances
Ow1, W1 W2 W2 W3, &c., to represent the different incre-
ments of velocity
elementary
quantity of work;
during the different
Fig. 29.
successive elementary
U1
portions of time t
which have elapsed
since the beginning of
motion increments
that will not be equal
in the case of a vari-
able force; then will
V2
V1
W₁
W2
Ow1, Ow2, Ows, &c.,
represent the veloci-
V₂
W
V5
geometrical
method of finding
the whole work.
205
ข
6
WB
ties of the body at the
corresponding instants: lay off these same lengths upon
the ordinates w1v1, W2 V2 W3 v3, &c., so that we shall
have
W1 V1
=
O w11
W₂ V₂ =
O was
=
Wz Vg Owg, &c.;

94
NATURAL PHILOSOPHY.
The area of a
triangle
represents the
sum of all the
products
v.
Work consumed
when the body's
motion is
accelerated;
the series of points v1, v2, v3, &c., will lie on a right line,
inclined to the axis O B, in an angle of 45°. Consider
now the velocity v3 w3.= V,
for instance, of which the in-
crement w3w
is called v.
or v3 b₁ = v4b49
The product
Vv, will here be represent-
ed by the small rectangle
vg W3 W4 b4, or by the trape-
zoid v3 W3 W4 v4, to which it
becomes sensibly equal when
the increment of velocity or
that of the time is very
small. The sum sought, of
01
V2
0 W1 W2
Fig. 29.
VT
06
V3
W3
W4 W5
We WB
all the partial products Vv, has for its measure the sum of
all the corresponding elementary trapezoids, or the area
comprised within the right line Ov7, the axis Ow, and the
ordinate wv, which latter represent the velocity acquired
from the beginning to the end of the time for which we
wish to estimate the work done by the force.
§ 87.-For example, if the body sets out from rest, and
we desire to find the sum of the products of Vv, correspond-
ing to the acquired velocity w4v4 = V', this sum being rep-
resented by the area of the triangle Ow4v4, we shall have
O w 4 X W 4 V 4
=
1 (2424) 2
=
1 V²;
hence the quantity of work corresponding to the velocity
V', and consumed by the inertia of the body whose mass
equal to half the is M, will be measured by MV, or by half the living
living force
communicated;
force communicated from the beginning of the motion, § 76.
This principle obtains, therefore, for any kind of motion,
or for a motive force different from the force of gravity.
For another velocity, w, v, V", the consumption of
work will be in like manner measured by MV, and
consequently for the interval between the positions in
which the body had the velocities V'and V", the quantity

MECHANICS OF SOLIDS.
95
of work consumed will be measured by the difference, or
MV² — — MV²,
in any interval,
equal to half the
living force at
end.
corresponding to the trapezoid ww, v, 24. But M V and work consumed
MV2 are the living forces at the beginning and end of
the interval of time during which we are considering the difference of
work of the motive force; the expression above is, there- beginning and
fore, one half the increment of living force, or half the living
force communicated in this interval; so that the principle
applies to any two instants of the body's motion, and thus
the quantity of work expended has, in every case, for its
measure, half of the living force communicated in the interval
between these two instants.
§88.-Finally, it must be remarked, that the preceding
supposes the velocity of the body to increase incessantly;
if it were otherwise, the force would be opposed to the
motion, and would be a retarding force. But the reasoning
remaining the same, would be applicable to this case, and
we should find that the quantity of work or action Work developed
developed by the resistance F, (equal and contrary to the
force of inertia now become a power,) during the time
necessary, to reduce the velocity from V' to V", would
have for its measure,
} (M V² — M V¹³),
or half the living force destroyed or lost.
when the motion
is retarded;
equal to half the
difference of
living force at
end of interval.
Thus, the diminution of the living force of a body
between any two given instants, supposes that a quantity the beginning and
of work or of action equal to the half of this diminution,
has been developed by the inertia of this body against
obstacles or resistances, as its augmentation supposes, on
the part of a power, an expenditure of work equal to the
half of this augmentation.
§ 89.-We now clearly perceive how the inertia
a body, serves to transform work into living force,
Inertia serves to
transform work
of into living force,
and into action;
and living force

96
NATURAL PHILOSOPHY.
examples in the
mechanic arts;
example of the
grist-mill;
the air-gun;
living force into work; or, to use the expressions em-
ployed, § 76, on the occasion of the vertical motion of
heavy bodies, we see that inertia will store up the work
of moters by converting it into living force, and give
this work out again when the living force comes to be
destroyed against resistances.
The mechanic arts offer a multitude of instances in
which these successive transformations take place, in
operating by means of machinery, implements, &c., &c.
The water contained in the reservoirs of grist-mills, for
example, represents a certain quantity of disposable
action, or work, which is changed into living force when
the sluice gates are opened; in its turn, this living force
acquired by the water, in virtue of its weight and descent
from the reservoir, is changed into a certain quantity of
work; this is communicated to the wheels of the mill,
and these latter transmit it to the millstones which pul-
verize the corn. The air confined in the reservoir of an
air-gun, represents the value of the mechanical work
expended by a certain moter in compressing it; on open-
ing the valve, the air acts upon the ball, impels it forward,
and converts a certain quantity of work into living force.
If this ball be thrown against a spring, or an elastic body,
the latter will be compressed in opposing a greater or less
resistance to the inertia of the former, and will finally
have destroyed all its
motion at the instant
the quantity of work,
developed by the
the action of the spring, becomes equal
ball against a
spring.
to half the living
Fig. 30.
force of the ball; the
spring being retained
by any means in its
compressed state, the
living force will be stored up as a quantity of disposable
work, so that when the restraint is removed from the

MECHANICS OF SOLIDS.
97
spring, the ball will be thrown back with a velocity such,
that the living force will be double the quantity of action or
of work, restored by the spring in unbending or expanding.
bodies resto.
the living force
impact.
§ 90.-If, then, the spring be perfectly elastic, the Perfectly
velocity communicated to the ball, will be precisely equal
to that impressed upon it by the air-gun in a contrary lost during an
direction. Thus, in the example before us, the quan-
tity of work has been alternately changed into living
force, and living force into quantity of work, without any
thing having been lost or gained. But if the spring be
not perfectly elastic, a portion of the living force im-
pressed upon the ball will be employed in destroying the
molecular force of the spring, that is to say, in producing
a permanent change in the arrangement of its particles.
always lost in the
elastic.
§ 91.-Hence, in the collision of bodies, not perfectly
elastic, there will always be a loss of quantity of work,
and this, from what has already been said, must be equal
to half the living force destroyed. Few, if any, solid Living force is
bodies are perfectly elastic, and as the vast majority are, collision of bodies
to a considerable degree, deficient in this quality, the not perfectly
quantity of work uselessly consumed by the molecular
forces will, in general, bear an appreciable ratio to that
developed by inertia during the compression; and it there-
fore follows, that if this last force, or the velocity which
occasions the collision, be considerable, there will take
place, in a very short time, a great loss in the quantity of
action; and this is why it is important, as before re-
marked, to avoid all shocks in the motion of machinery.
restored can
never exceed
creating the
§ 92. We also see, from what precedes, that it is as The work
impossible for the force of a spring to develop, in un-
bending, a living force greater than that consumed in that consumed in
bending it, as for the force of gravity, § 65, to give moter.
to a body while falling, a living force exceeding that
destroyed in it, through the same height, while rising;
indeed, the whole of the velocity will not, in general
7

98
NATURAL PHILOSOPHY.
What takes place
in periodical
motion;
be restored, and as the corresponding living force lost
in the shock, has really been employed to overcome a
certain resistance, and therefore to produce a certain
quantity of work, it is true, as before stated, that inertia
does actually perform an amount of work equivalent to
that which has been employed in putting it into action;
only it happens, that, in certain cases, a portion of this
work is diverted from the object we desire to accom-
plish, and is not, on that account, regarded as forming
a part of the useful effect, as was explained in § 50
with regard to the ordinary force of pressure.
§ 93. We have shown, by examples, how the quantity
of work or of action may be transformed alternately into
living force, and living force into quantity of action, by
means of springs and machines which store up and give
them out successively. These transformations take place,
in general, whenever the motion of a body solicited by a
motive force varies, by insensible degrees, so as sometimes
to be accelerated and sometimes retarded. This occurs,
for example, in the periodical motion spoken of in § 25,
and, in general, in all cases of forward and backward
movement, usually called alternating, and in which the
velocity becomes nothing from time to time. The motion
of the pendulum and that of the plumb-bob are evident
examples of this last kind. When the velocity of a body
increased, inertia augments, it is a sign that some portion of the motor's
action is employed to overcome the body's inertia, and to
increase its living force by double the portion thus ex-
pended, the other portion being absorbed by resistances;
if, on the contrary, the velocity of the body diminish,
inertia aids the notwithstanding the power may be exerted in the direc-
tion of the motion, a certain portion of the living force
acquired will be expended against the resistances, and will
augment the work of the motor by a quantity equal to
half the living force thus expended, and so on, according
to the number of alternations.
when the
velocity is
opposes the
force;
when the
velocity
diminishes,
force.

MECHANICS OF SOLIDS.
99
intervals between
moter is not
employed to
$94. From which we see, that when the velocity or
living force of a body oscillates between certain limits, it is
a proof that inertia has alternately absorbed and given out
portions of the moter's work. The work absorbed by Within the
inertia will be the same for all equal velocities, and for the instants of equal
interval between the instants of equal velocities there will velocities, the
be nothing lost or gained, and the power must be con-
sidered as having been entirely employed to overcome overcome inertia ;
resistances other than inertia. But, if in any interval of
time, the velocity, after having undergone alternations,
does not attain to what it was before, the half of the
difference of the living forces which correspond to the
beginning and end of this interval, measures the quantity
of work which has really been consumed or given out by
the inertia of the body. Consequently, if the body were
to set out from rest, the quantity of work consumed by its
inertia up to any instant, would be measured by half the work absorbed or
living force possessed by the body at this instant; if the given out by
velocity had increased incessantly, the inertia of the body half the living
would have opposed the motive force without intermis- force acquired or
sion; if the velocity had, during any part of the time,
diminished, the inertia would have aided the force.
$95.-All of which
may be made manifest
by means of the second
figure employed in § 86,
in observing that when
the velocity of the body
diminishes, after hav-
ing augmented during a
certain time, so will the
abscisses and ordinates
of the right line Ov7,
0
01
Fig. 31.
VT
Ve
V5
V3
W1 W2 W3
24
W5 W6
B
inertia, equal to
destroyed.
which represent this velocity; the extreme ordinates w, v7,
after receding from the point 0, while the velocity is
increasing, will, on the contrary, approach this point while
Geometrical
illustration;

100
NATURAL PHILOSOPHY.
example of a
by horses.
the velocity is diminishing, to keep the triangular area
Ow, v, constantly proportional to the quantity of work
absorbed by the inertia, or to its equal, one half the living
force. A carriage travelling at a variable rate, sometimes
carriage drawn faster, sometimes slower, offers an example of this: at
first, the horses exert a certain quantity of action to move
the carriage with a trot; then, when the velocity is dimin-
ished, by an increase of resistance, or by feebler action on
the part of the horses, the inertia of the carriage develops
against the resistances to its motion, a portion of the work
it had at first absorbed, equal to half the diminution of its
living force: and this alternation will continue till the
carriage is brought to rest, at which instant, the work
restored by the inertia will be exactly equal to the quan-
tity of work consumed, so that nothing will be lost. In
what is here said, it is understood, however, that no dimi-
nution of velocity results from opposition or holding back
of the horses, for in that case, the moter would be con-
verted into resistance.
The same
to weight as well
as to inertia.
§ 96.-The same reflections are applicable to the
reflections apply weight of a carriage in ascending and descending a hill.
The quantity of work employed in overcoming the weight
while ascending will be restored during the descent, pro-
vided the latter be not so steep as to cause the horses to
hold back, by which a quantity of work would be con-
sumed uselessly. And this consideration shows us one
of the many advantages which results from giving gentle
slopes to roads.
When a force is
employed to raise
the moter's work;
§ 97.-When a motor is employed to raise a burden
a weight, inertia through a vertical height, it takes the body from a state
retains nothing of of rest, and hence a quantity of work must be expended
to overcome its inertia. Arrived at the desired height,
the effort of the motor is relaxed to restore the body to
a state of rest, and during this diminished action, a por-
tion of the living force acquired is employed to destroy

MECHANICS OF SOLIDS.
101
an artificer's tool.
in part the effect of the body's weight, and the inertia
will finally retain nothing of what it had absorbed.
The same thing may be said of the operation of an the same is true
artificer in filing, sawing, &c., since at the end of each of the inertia of
oscillation of the tool, the velocity becomes nothing
through insensible variations. This could not be the case
if the motion were suddenly to change, or if concussions
should take place between bodies not perfectly elastic; a
portion of the living force would, in that case, be de-
stroyed, or, which is the same thing, diverted from its
intended purpose in producing a permanent change in
the arrangement of the particles of the colliding bodies.
by inertia ;
§ 98.-Finally, in order to give a fuller idea of the part Examples of the
performed by inertia in the various operations of the part performed
mechanic arts, and to demonstrate how it may serve to
explain an almost infinite variety of effects, we shall add
a few special examples to those already mentioned.
plane;
To take from a plane-stock its chisel, the carpenter the chisel of a
strikes the plane a blow on the back; a velocity is thus
suddenly impressed upon the stock which the chisel and
its wedge only partake of in part, because of their inertia
and imperfect connection with the body of the plane, and
are, therefore, left behind.
cask;
A bung is taken from a cask by striking, on either side the bung of a
of it, the stave in which it is inserted; the resistance
which the inertia of the bung opposes to the sudden mo-
tion communicated to the stave, causes the separation.
We often see a handle adjusted to a tool, as an axe or handles of tools;
hammer, by striking it on the end in the direction of its
length; the inertia of the handle and that of the tool tend
to resist the sudden motion impressed by the blow, but
the former yielding more than the latter, by reason of the
slight connection, the handle becomes inserted.
As an illustration of the agency of inertia, in trans-
forming quantity of action into living force, take the com- the common
mon sling, from which a stone may be thrown with much
sling;

102
NATURAL PHILOSOPHY.
the common
whirling top.
greater velocity than from the naked hand. Here, living
force is accumulated in the stone, by whirling it through
many accelerated turns about the hand before it is dis-
charged. The common top turns and runs along the
ground, in virtue of the living force acquired during an
accelerated unwinding of the string from the coils of
which it is thrown.
§ 99.-We would recommend to the reader, to con-
sider attentively these examples, as well as all others of
like nature which his observation and memory may
Inertia sometimes furnish. They will aid his conceptions of the manner in
which the inertia of bodies, like their weight and molecu-
lar spring, sometimes acts as a mere passive resistance,
sometimes a real and sometimes as a real motive force, according to the
circumstances.
a passive
resistance;
motive force.
It is, however, proper to remark, that the last example
is mainly concerned with the inertia of a body having a
motion of rotation, while, thus far, we have only spoken
of the living force of a body possessing a motion of transla-
tion, in which all the particles have the same velocity;
but we shall soon see, that the principles which connect
the living force with the quantity of action, are universal
and applicable to all kinds of motion.
IV.
OF FORCES, WHOSE DIRECTIONS MEET IN A
POINT.
Forces whose
directions meet
in a point;
§ 100. Thus far we have only considered the effect of
a single force, directly opposed to an equal force, viz.:
to molecular spring or elasticity, to weight, or to inertia. It
often happens that several forces are applied to a body,
in different directions, to overcome certain resistances

MECHANICS OF SOLIDS.
103
when one
effects of all the
others;
through its intervention. When a body is thus subjected
to the action of several forces, (powers, or resistances,) we forces in
say these forces are in equilibrio, when one of them equilibrio
destroys or prevents the effect which the others would prevents the
produce, if the first did not exist. The body itself is in
equilibrio, if the different forces applied to it, leave it at
rest. This last kind of equilibrium can never be abso- no absolute
lute, because all bodies connected with the earth partake
of its continual motion through space, and there is, in fact,
no rest for them. A body may, however, have relative
rest, as when it retains the same place in reference to
surrounding objects, such as mountains, houses, &c., which
we are in the habit of regarding as fixed. Thus, the idea statical and
of equilibrium is not alone related to rest, and by no equilibrium.
means excludes motion. From this results the distinction
of statical and dynamical equilibrium; the former relating
to the repose of the body, and the latter to the mutual
destruction of the forces which solicit it. Thus, a body
may be in motion while the forces acting upon it are in
equilibrio, or it may be at rest under the same circum-
stances.
equilibrium of
bodies;
dynamical
line, in same or
directions;
§ 101.-It has already been stated, § 43, that when Resultant of
several forces act along the same right line and in the several forces;
same direction, their effect will be equivalent to that of
a single force equal to their sum, and which will therefore
be their resultant. If these forces act in opposite direc- when acting
tions, and along the same straight line, their resultant will along the same
be equal to the excess of the sum of those which act in in different
one direction, over the sum of those which act in the
opposite direction, and it will act in the direction of the
greater of these sums. This is the case in which several
forces are exerted in the direction of the same cord. The
tension of the cord will be the same throughout, and it is
not possible to draw its two ends with different efforts.
The tension of a cord is the effort by which any two of tension of a cord;
its consecutive portions are urged to separate from each other,

104
NATURAL PHILOSOPHY.
the effect of
unequal forces
acting upon a
cord.
and this being the same throughout, the excess of the
sum of the forces which act in one direction over that of
those which act in the opposite direction, will be wholly
employed in overcoming the cord's inertia and giving it
motion.
§ 102.-When a body, or material point, moves from
A to B, so as to describe the rectilineal path A B, each
of the positions A and B may be projected upon the right
lines OM and
Parallelogram of ON, situated
paths;
in the same
N
plane with the
B"
F
line A B, by
drawing paral-
lels to these
lines consider-
Fig. 32.
B
A"
A
E
ed as axes, the
place A giving
the two co-or-
0
B'
M
dinates A A' and AA", and the position B the two
co-ordinates BB' and BB". The positions A' and A", on
the axes, are simultaneous with the position A; and
those of B', B", with the position B. The paths A'B'
and A" B", on the directions O M and ON, are, therefore,
described by the projections in the same time as the path
AB by the moving point. The first are called component
or relative paths in such and such directions. Prolong the
co-ordinates of the points A and B, till the parallelogram
AEBF is formed, and this principle will appear, viz.:
the rectilineal path described by a point, may always be re-
solved into two relative or component paths, in any two
resolution of any directions, and these component paths will be the sides of a
relative or
component
paths;
path into
component
paths;
parallelogram, constructed upon the path described by the
point as a diagonal, and parallel to the assumed directions.
Reciprocally, when we have the relative paths in any two
directions, the true path, called the resultant, will be that

MECHANICS OF SOLIDS.
105
diagonal of the parallelogram constructed upon the rela- composition of
tive paths which passes through their point of meeting.
the relative paths.
$103.-It has been shown, that the velocity of a body Parallelogram of
in motion, is represented by the length of path described velocities;
uniformly in any very small portion of time, assumed
as the unit of time, and that it is only in the case of
uniform motion, that the interval of time during which
the velocity is estimated, may be taken as great as we
please. The path A B, in the last figure, being described
by the body in the same time that its relative paths true and relative
A' B' and A" B" are described by its projections on
the directions O M and O N, the first may be regarded as
the point's true velocity, and the two last as its relative
velocities. Hence the true velocity of a body, is the diagonal true velocity
of a parallelogram constructed upon its two relative velocities,
estimated in any given directions whatever.
velocities;
found from
relative velocities
and the reverse.
curvilinear and
varied motion;
§ 104.-If the motion be curvilinear, the rectilineal di- Relative paths in
agonal A B can no longer represent, in general, the path
described. Nor, if the motion be varied, can its length
measure the velocity, when the time of description is con-
siderable. In such cases, conceive a given interval of time
divided into a great
number of small and
equal portions, and
Fig. 33.
T"
T
determine the relative
paths described during
each, by the projec-
tions of the moving
point on the axes.
Each pair of these
relative paths will de-
termine a parallelo-
gram, of which the
B
A
T
diagonal will be the corresponding elementary path de-
scribed by the point itself. Any one of these diagonals,
geometrical
representation;

106
NATURAL PHILOSOPHY.
construction of
the direction of
the body's
motion.
Roberval's
method of
constructing the
tangent;
results from the
law which
determines the
path.
as AB, will sensibly coincide with an element of the
curve, and its prolongation A T will be tangent to the
curvilinear path. This tangent will determine the direc-
tion of the body's motion at the instant, and may be
drawn by laying off from the projections A' and A" of
the body's place, the distances A' T" and A" T", equal
respectively to double, triple, quadruple, or any number
of times the body's relative velocities at the time, and
drawing T" T and T" T, respectively, parallel to the direc-
tions O T" and O T
§ 105.-When the law of a body's motion in two direc-
tions is known, it is always possible by the preceding
method to draw a tangent to the path described. Take,
for example, the ellipse:
this curve is generated
by fixing at two points
F and F', called the
foci, the ends of a
thread FA F', equal in
length to a given line
MM', called the trans-
verse axis, and moving
the point of a pencil A
to all positions in which
it will keep the thread
M
Fig. 34.
B
B
M
F
stretched. Since, in the motion of the describing point,
the sum of the lengths FA and A F' is always the same,
the portion FA will increase just as much as the portion
AF" will diminish, and therefore the point A tends to
describe equal relative paths, or will have equal relative
velocities, in the two directions A B and A F'. Hence,
taking upon FA produced, and upon A F', the equal
portions A B and A B', and completing the parallelogram
ABCB', the diagonal A C, passing through the position
of the point, will be a tangent line to the path described.
This method, which is due to Roberval, is very useful in

MECHANICS OF SOLIDS.
107
all cases where we know the law by which the curve is
described.
§ 106.-We have seen that any single motion may be
resolved into two others, and the reverse. This arises
from the simple fact, that a body may, in reality, be ani-
mated by two or more simultaneous velocities. To illus- Illustration of the
trate, let it be supposed that while a boat is crossing a
river, a man walks from one side of the boat to the other, velocities;
and that, starting from
the point A, for exam-
ple, he arrives at B at
the moment the boat
reaches a position such
that the point A shall
be at A', and the point
B at B'. It is plain,
that the man, though
only conscious of hav-
ing walked across the
Fig. 35.
coexistence of
simultaneous
simultaneous
motions;
boat from A to B, will, in fact, have been carried from
A to B' in reference to the surface of the river. He
will have moved, at the same time, with the velocity
which he impressed upon himself, and that impressed
upon him by the boat. This being understood, it is
easy to see that the result would be the same, if the boat example of four
first move from A to A', and afterward the man walk
across it from A' to B'; or if the boat were stationary,
while the man is crossing it from A to B, and then
were to move from B to B'. But this is not all; the
earth turns about its axis, while the boat floats along
the surface of the water, and the man walks across the
deck of the boat; add now the motion of the earth about
the sun through space, and we shall find the man ani-
mated by four simultaneous velocities, of which it is easy
to see that we shall find the resultant, in compounding,
by the rule given in § 103, first, any two, then the resul-

108
NATURAL PHILOSOPHY.
resultant of
several
simultaneous
velocities;
rule;
illustration.
Independence of
the action of
simultaneous
forces;
tant of these two with the third, and the resultant of the
three with the fourth. In fact, when a body has several
simultaneous motions, the effect is
M
Fig. 36.
"
the same as if the body had re-
ceived, one after the other, all the
motions which it possesses at the
same time. Hence, this rule,
viz.: The resultant of several simul-
taneous velocities is found by con-
structing a polygon, of which the
sides are equal and parallel to the
component velocities, and by join-
ing, with a right line, the point of
departure with the extremity of the
last side. This right line will represent the resultant required.
Thus, let the point O have the simultaneous velocities
OV, O V, OV", O V""; from the extremity V of 0 V,
draw Vm parallel, and equal to O V; from m draw m m'
parallel, and equal to O V"; from m' draw m' m" parallel,
and equal to O V", and join O with m"; the line Om" will
be the resultant velocity.
§ 107. The action of a force upon a body, whether at
rest or in motion, is always the same, and impresses upon
it the same degree of velocity. Let a body fall, for exam-
ple, under the action of its own weight, gravity will
impress upon it the same velocity in a given portion of
time, whether it set out from rest or is projected down-
ward by the action of
some other force. For
example, when a bomb-
shell is thrown into the
air, it describes a curve
under the joint action of
the living force with which
it leaves the mortar, and
the incessant action of its
る
​Fig. 37.

MECHANICS OF SOLIDS.
109
weight; and its velocity at any instant is the resultant
MR, of the velocity MQ, which it had at the begin-
ning of the very short interval of time next preceding
this instant, and the velocity MP impressed upon it by
its own weight during the same interval of time. Thus,
when two forces are applied to the same body, they im- two forces
press upon it, at each instant, and simultaneously, the impress
same degree of velocity which each would impress if the same velocity
acting alone. This degree of velocity, we have said, § 81,
is, from the general law of nature, proportional to the
intensities of the forces.
P
Fig. 38.
D
simultaneously,
as if acting
separately.
of forces;
§ 108.-Accordingly, let a material point A be acted Parallelogram
upon by the two forces Pand Q, represented in intensity and
direction by the lines A B and AC respectively. These
forces will impress simultaneously, and in their respective
directions, the same degrees of velocity Am and An, as
though each acted separately.
The resultant velocity will,
§ 107, be represented by the
diagonal Ar of the parallelo-
gram Am rn. Conceive a
force X, to act upon the point
along this diagonal, but in the
opposite direction, or from r
to A, and with such intensity
as to destroy this velocity;
no motion can take place, so
that the force X, destroying
the effect of the forces P
and Q, will maintain these
forces in equilibrio. Take, up-
on the diagonal, the distance
B
A
X
n
C
ADX, and conceive it to represent a force that acts
upon the point A, from A towards D; it will produce
the same effect as the forces P and Q, and will, therefore,
be their resultant. Now, the forces P and Q, and their

110
NATURAL PHILOSOPHY.
the resultant of
any two oblique
forces applied to
a point;
represented
a parallelogram;
resultant A D, equal in intensity to X, are proportional
to the velocities A m, An, and Ar, which they can
simultaneously produce; that is,
AC: AD:: An: Ar,
therefore DC is parallel to rn; and AD must be the diag
onal of the parallelogram constructed upon the lines A B
and AC as sides. Whence, the resultant of any two forces
applied to the same point, is represented, in magnitude and direc-
tion, by the diagonal of a parallelogram, constructed upon the
by the diagonal of lines which represent, in intensity and direction, the two forces.
It must not be forgotten that a force is, in geometrical in-
vestigations of mechanics, always represented by a portion
of its line of direction, containing as many linear units as
there are pounds in the intensity of the force. It is plain,
forces combined therefore, that forces may be combined by the same rules
as velocities; and this is confirmed by experiment. If, for
example, we attach to a cord A CB, fixed at its two ends,
a weight R= fifteen pounds, it is easy, by a balance-
spring, to measure the
efforts exerted in the
directions CA and
by the same rules
as velocities;
experimental
illustration of the
parallelogram
of forces.
C B. Laying off up-
on the vertical through
C, and from the point
C, a distance CD
equal to 15 inches,
and completing the
parallelogram by draw-
ing Da and Db paral-
lel respectively to CB
and CA, we shall find
the number of inches
in Ca and Cb to be
Fig. 39.
R
the same as the number of pounds indicated by the bal-
ances A and B.
§ 109.-By the same principle that two forces, applied

MECHANICS OF SOLIDS.
111
Fig. 40.
force into two
others;
to the same point, may, without change of effect, be re- Resolution of a
placed by a single one, may a single force be replaced by
two others, acting in given directions. Let a given force,
applied to the point 0, be
represented in direction and in-
tensity by the line Or: its com-
ponents, in any two assumed
directions, as OA and O B, are
thus found. Through the point
r, the extremity of Or, draw
rm and rn parallel, respec-
tively, to 0 B and OA; the
portions Om and On will repre-
sent the components required.
Make Om
=
A
m
P
R
Q
0
n
B
P; On Q; Or R; the angle trigonometrical
=
AOB = = rn B = 180° - rn 0.
=
Then, in the tri- relation of
angle Orn, because Om=rn = P, we shall have
resultant to its
two components;
R2 = P² + Q²+2PQ cos q,
or
R = √P² + Q² + 2 PQ cos o (31); value of resultant;
.
and because the angle Orn is equal to the angle r Om,
sin A OB, we also have, from the same
and sin rn O
-
triangle,
RQ
sin : sin r Om,
R: P: sin o
sin r On;
whence,
sin r Om =
Q sin o
R
.
(32).
its inclination to
its components.
sin r On =
P sin o
R
§ 110.-We have heretofore supposed the resistance

112
NATURAL PHILOSOPHY..
when the
resistance is not
immediately
opposed to the
force;
Fig. 41.
Quantity of work immediately opposed to the force destined to overcome
it. Let us now consider the case in which the resistance
is exerted in any line of direction other than that of the
force, and in which the point of application of the force
can only move along the line of direction of the resistance.
Let, for example,
AR represent a force
applied to the point A,
which can only move
in the direction A B
Decompose this force,
which denote by R,
into two components
P and Q-the first per-
pendicular to AB, and
the other in the direc-
tion of that line, and,
consequently, immedi-
equal to the
product of the
force into the
path, estimated in
direction of force.
P
C
Q
R
B
ately opposed to the resistance that may be overcome.
Since the point A cannot yield in a direction perpendicular
to A B, the component P can only tend to press it, without
producing any work. The component Q, is immediately
opposed to the resistance, and, if A a be the small path
described by the point of application A, the product
QX A a, will measure the elementary quantity of work
necessary to overcome the elementary quantity of resist
ance over the same path; such will be the measure of
the effective quantity of work of the force R.
Draw from the point a, ar perpendicular to A R;
Ar will obviously be the length of path described by
A in the direction of the force R, and we shall have,
from the triangles A ar and AQR, which are similar,
having a common angle A, and each a right angle,
Aa: Ar :: R: Q;
whence,
Aa x = Ar x R
C
(33);
X

MECHANICS OF SOLIDS.
113
which shows that the quantity of work of a force, not
immediately opposed to a resistance, is equal to the product
of the force into the length of path described by its point of
application, estimated in the direction of the force.
vilpolay out
sdfyns of si indt
§ 111.-When a heavy body ban vino a
is compelled to move upon the bod adi dgrodt en
curve ABC, the elementary ha
quantity of work expended by
its weight W, in causing it to
describe the elementary path
B C, is, from what has just
been shown, equal to the prod-
uct W x b'c', estimated upon
the vertical line A D'. It is
also the measure of the quan-
tity of work expended in the
direction of the curve. Add-
ing together all the elementary
quantities of work by which
the body is made to describe
D
vob quantity of work
m
Fig. 42.
tamp odt tad
211
B
C
D
doidy posci od:
of the weight of a
body, moving on
a curve;
pele yoolt
how to vijmeg
of lentigo
the whole curve, it is plain that the sum, or the whole
quantity of work expended by the weight, must be equal
to the weight multiplied into the sum of the elementary
paths b' c', which make up the whole height AD' = H;
or to W x H. This is also the measure of the quan-
tity of work performed by the component of the weight,
which acts in the direction of the motion, along the the same as that
curve. But, from § 87, the double of this last quantity
is equal to the living force of the body; that is to say, direction of
to the product
of the component
of the weight in
curve.
W
Jaigy out yd bodine
X V2;
JOLI
g
in which V denotes the velocity of the body in the direc-
tion of the curve, at the instant the work terminates;
whence
8

114
NATURAL PHILOSOPHY.
W
2 WH =
V2,
g
or
The velocity
the height, and
not on the path
described.
V² = 2g H;
that is to say, the velocity acquired by a body in moving
depends upon down a curve, under the action of its own weight, is the
same as though the body had fallen vertically through
the same height. And we see, from this investigation,
that the quantity of work which a motor must expend,
in elevating a weight along any inclined surface, is
always measured by the product of the weight of the
body, into the vertical height to which it is raised.
Elementary
quantity of work
of two forces
applied to a
point;
when the
projections of
components fall
of point of
Fig. 43.
§ 112.-It has just been shown, § 110, that the ele-
mentary quantity of work of a force, of which the point
of application is moved in a direction different from that
of the force, is measured either by the product of this
force into the length of the path described, estimated in
the direction of the force, or by the product of the real
path into that one of the two rectangular components
of the force, which acts in the direction of the motion;
and it must here be re-
marked, that this component
is nothing more nor less
than the projection of the
force on the direction of
the motion. Accordingly,
let us consider two forces,
P and Q, applied to the
point A, R their resultant,
and a A the small path de-
scribed by the point of ap-
plication. Let fall from the
Q
P
Q'
A a R'
P
points Q, P, and R, the perpendiculars QQ, R R', and
on opposite sides P P', upon A a produced; the projection of the force
P will be A P', that of Q, A Q', and that of the resul-
tant R, A R'.
application;

MECHANICS OF SOLIDS.
115
Now,
AR AP'R' P',
=
but AQ and RP, being equal and parallel, their projec-
tions A Q' and R' P' upon the same line, are equal, and
hence
AR'
=
AP'AQ'
and multiplying both members by the path A a, we have work of resultant
AR' X Aa = AP' x Aa - AQ' x Aa;
the first member is the elementary quantity of work of
the resultant R, the first term of the second member
is the elementary quantity of work of the component P,
and the last term, the elementary quantity of work of
the component Q. And it must be remarked that the
component A P' acts in the direction of the motion, while
the component 4 Q' acts in the opposite direction; so
that the effective quantity of work of these components,
which is the same as that of the components P and Q,
§ 110, is equal to the difference of the quantities of work
taken separately.
Had the motion taken
place so as to cause the
projections of the points
Q and P to fall on the
same side of the point A,
a little consideration will
show that the last equa-
tion would become
Fig. 44.
R
A
R
ARX Aa = AP' x Aa+AQ' x Aa,
equal to
difference of
work of
components;
when the
projections fall
on same side;
the work of
resultant equal to
sum of that of
and that the effective quantity of action of the compo-
nents A P' and A Q', would be the sum of the quantities components.

116
NATURAL PHILOSOPHY.
The work of
resultant equal to
the algebraic
taken separately, and the equation may be written, gen-
erally,
=
ARX Aa AP x Aa +AQ x Aa..(34).
Hence, the elementary quantity of work of the resultant
of two forces, applied to a point, is equal to the algebraic
sum of the work sum of the quantities of work of the two components.
of its
components.
Motion about a
fixed point.
When the projection of a force falls on the same side
of the point of application as the path described, and we
give the corresponding elementary quantity of work the
positive sign, then when it falls on the opposite side, the
work must have the negative sign.

Fig. 45.
§ 113.-The small space A a, may be described in
different ways.
we suppose, for example, that the
point of application A is on an axle A 0, which turns
horizontally about some point 0, taken arbitrarily in the
plane of its motion, as in the case of a bark or mortar
mill, the path A a becomes the
small are of a circle, which we
may regard as a small right line
perpendicular to A 0. From
the point a, let fall the perpen-
diculars a b, a d, and a c, upon
the directions of the forces
P, Q, and their resultant R;
then will the elementary quan-
tities of work due to these
forces be respectively P x Ab,
Qx Ad, and Rx Ae; and from § 112.
R
கள்
RX Ac = P x AbQ x Ad.
From the point 0, about which the motion takes
place, let fall the perpendiculars Op, 0q, and Or, upon
the directions of the forces P, Q, and R, respectively;
the triangles A Op and A ab are similar, since each has
a right angle, and the angle A Op, of the first, is equal

MECHANICS OF SOLIDS.
117
to the angle a Ab of the second, the sides A O and Op
being, respectively, perpendicular to the sides A a and
Ab; hence,
whence,
Ab: Aa :: Op: A0;
Ab = 0px 10;
A a
A
beligge
ம்
and, in like manner, from the similar triangles A da andstad and T
O Aq, we have
A a
AO
Ad = 0q. 10;
and from the similar triangles Aca and A Or,
A a
Ac = OrxA0;
these values, substituted in the above equation, give,
after omitting the common factors, and making Or=r,
Oq = q, and Op = P,
Rr = Pxp± Qxg... (35).
force;
The effective quantity of work which a force is capable Moment of s
of performing, while its point of application is constrained
to describe an elementary path A a, about a fixed centre
O, is called the moment of the force; the fixed point 0
is called the centre of moments; and the perpendiculars the centre of
P, 9, and r, the lever arms of the forces P, Q, and R, moments;
respectively.
The elementary quantities of work performed by the
forces P, Q, and R, during the description of the path
A a, are measured by the products Pp, Qq, and Rr,
A a
multiplied each by the constant ratio
AO
and if this
lever sms

118
NATURAL PHILOSOPHY.
the relative
measure of a
moment;
the moment of
the resultant of
two forces.
When the forces
are not applied to
the same point;
constant ratio be omitted, these products may be taken
as the relative measures of the elementary quantities of
work. Hence, the relative measure of a moment, is the
product of the intensity of the force into its lever arm; and
from Eq. (35) we see that the moment of the resultant
of two forces, applied to a point, is equal to the algebraic sum
of the moments of the components.
§ 114.-In what precedes, the two forces, P and Q,
have been supposed to be applied to the same point; if they
be applied to different points
Cand B, it is evident that we
may suppose two rigid bars,
CA and B A, to be firmly
attached to the body, and to
coincide in direction with the
given forces. These bars, if
the forces act in the same
plane, will meet at the point
A, and the latter thus becom-
ing invariably connected with
the body, may be taken as the
Fig. 46.
BP
R
shuler seas
Suillime gala
common point of application, without changing the effect
of the forces. The resultant AR will be obtained by
means of the diagonal of the parallelogram A PRQ, and
the point D, where it meets the surface, may be taken
as its point of application. If, now, the body be con-
strained to move around any point, as O, the common
point of application A, will describe the small arc of a
circle, which may be regarded as a small right line, to
be projected on the directions of the forces, as in the last
article; and the same reasoning will show us, that in this
algebraic sum of case also, the moment of the resultant is equal to the
the components. algebraic sum of the moments of the components.
the moment of
the resultant is
still equal to the
the moments of
§ 115. The relations which have just been established
between the quantities of work, and between the mo-

MECHANICS OF SOLIDS.
119
equally true,
wherever the
moments be
taken;
ments of forces and of their resultant, will always obtain These relations
wherever the point O be taken, since its selection was
entirely arbitrary; but these relations were obtained by centre of
considering the motion of the point, common to the
directions of the forces, this point being assumed as their
common point of application. To show that they are
equally true in regard to the motion of the true points of
application B, C, and D, see the last figure, we have only
to remark that the measure of the moment depends alone
upon the intensity of the force, and the length of the
perpendicular drawn from the centre of moments to its
line of direction, and is wholly independent of the posi-
tion of the point of application. The moment of the or wherever the
force P, for example, will be the same whether it be sup- application.
posed applied at A, or at the point B, where its direction
meets the surface of the body. The theorem of moments
will be true, therefore, when the forces P and Q are not
applied to the same point.
WORK OF
points of
theorem of the
quantity of work;
§ 116.-If it be shown that the quantity of work of Extension of the
a force is the same, whatever point be taken on its line
of direction as the point of application, it is obvious
that the theorem of the quantity of work, estimated
by the motion of the common point of union of two
forces and their resultant, will be equally true of all
cases in which the quantities of work of these forces are
computed in reference to the motion of their respective
points of application. Three cases may arise, according work estimated
as the body has a motion of rotation, of translation, or
of both combined.
First case. The
body and the di-
rection A P, of
the force P, being
supposed to have
a motion of rota-
al-bricol
10. uoltom
nolletame
by any point of
application on
line of direction;
Be A
edhogab lliw &
Fig. 47.
Aa'
T
BbP First-in motion
of rotation;
od at how lo
Trope
tion about the point 0, any two points, as A and B of the

120
NATURAL PHILOSOPHY.
line AP, will de- fur shot Fig. 47.
scribe arcs which
ho are proportional to
their distance, OA
and 0 B, from 0;
7
Aa
and we shall have
oilerpa to, taloy
A a
AO
Bb
=
OB
Bb
P
but the quantity of work of the force P, estimated by the
motion of its point of application supposed at A, will
have, § 113, for its measure,
-que
Aalmers
Px0pXÓA
Ο Α
or estimated by the motion of its point of application,
supposed at B, will be measured by
Bb
PX 0PX OB
Hence, the quantities of work are equal, being measured
by the product of the intensity P, the length of the per-
A a
Bb
pendicular Op, and the equal factors
and
O A'
OB
Second case. If the
Second-in
motion of
translation;
body only have a
motion of translation,
any two points of ap-
plication, as A and
B, will describe the
equal and parallel
Fig. 48.
a'
B
paths A a and Bb, which will be projected upon the
direction AP, in the equal paths A a' and Bb'; and the
quantities of work in the two cases being PX A a' and
Px Bb', are equal to each other.
Third case. Suppose the line of direction AP of the

MECHANICS OF SOLIDS.
121
motion is of
force P, to take the position A, B, in virtue of the com- Third-when the
bined motion of rotation and translation, and the points translation and of
A and B to be transferred to the positions a and b. This rotation
motion of the points A and B may be regarded as resolved
into a motion of rotation around the point 0, the centre
combined;
of a circle, tangent
to the two positions
of the line of direc-
tion, supposed in-
definitely near each
other, and of trans-
lation along the sec-
ond position of this
line. By the first,
the points A and B
are carried in the
arcs of circles to A₁
and B₁, and by the
A1
Fig. 49.
b
Bi
to atalog s
A1 A a
B BUP
oyed daid O tuode
second, from these latter positions to a and b, thus making
A a and Bb the actual paths described. Projecting these
latter paths on the primitive direction of the force by the
perpendiculars a a' and bb', we shall have for the quan-
tities of work, considered in reference to the motion of the
points A and B, PX Aa' and Px Bb', respectively.
But by projecting the points A, and B on the primitive
direction, by the perpendiculars 4, 4,' and B₁ B', we have
A a'
=
A₁'a' - AA,
Bb = B'b' - BB;
multiplying each equation by P,
od to
PX Aa PX A'a' - PX A' A,
=
Px Bb = Px Bb- Px B B.
Now Px Ai a', and P x Bi'b', are the quantities of
work, on the supposition of a simple motion of translation

122
NATURAL PHILOSOPHY.
alone, in the directio
tion A, B1, and these
have been shown,
Bold in the second case, to
be equal; whence,
A₁'a'
=
B'b'.
no matter where The products
the points of
application be
taken on the
lines of direction; and
P x AA,
sound of corol
Fig. 49.
BA
BL
AL
A14. a
B'1
BUR
PX BB,
girola noliel
Tornaltinoq buo
measure the quantities of work due to the motion of A
and B, on the supposition of a simple motion of rotation
about O, which have been shown to be equal, in the first
case; whence,
A₁ A = BB; osodi
JO bin A
and consequently,
பாம்பு
PX Aa' = Px Bb.
the work of the
Thus, the relation given in § 112, between the quantity
resultant, is equal of work of the resultant of two forces, and the total quan-
to the algebraic
sum of the
quantities of
work of the
components.
tities of work of the components, subsists in all cases,
whatever be the points of application, and whatever be
the nature of the motion.
§ 117.-Resuming Eq. (35),
Rr =
Pp Qq,
When the
or when its line
of direction
in which r, p, and q, denote the lengths of the lever arm
resultant is zero, of the resultant R and of the two components P and Q,
we see that the moment Rr, of the resultant, can only
passes through a reduce to zero when the moments of the components P
and Q are equal and have contrary signs. But the prod-
fixed point,

MECHANICS OF SOLIDS.
123
uct Rr, becomes nothing, either when R= 0, or r= 0. there will be an
In the first case, the resultant is nothing, and there will equilibrium.
be an equilibrium independently of all other considera-
tions. In the second case, the perpendicular r, which
measures the distance of the line of direction of the
resultant from the centre of moments, being nothing,
indicates that the resultant passes through the fixed point.
Again, the equality of the moments of the components,
necessarily implies an equality in the quantity of work
performed by each, and these quantities, having different
signs, destroy each other; hence, there will be an equi-
librium about a fixed point, when the resultant of the forces
which act upon the body, passes through this fixed point.
mib od grivley
V.
rottaillygs to chilog
OF FORCES WHOSE DIRECTIONS ARE PARALLEL.
8118.-It has been shown of two forces whose direc- Theorem of the
tions intersect: 1st, that the line
of direction of the resultant, will
intersect those of the compo-
nents in the same point; 2d,
that the moment of the resul-
tant is equal to the sum or dif-
ference of the moments of the
components, according as the
components tend to turn the
body upon which they act, in
the same or in opposite directions
about the centre of moments.
Now, these properties, being
entirely independent of the po-
sition of the point of meeting
O, and of its distance from the
B
bair
Fig. 50.
0
P
body or centre of moments, will not cease to be true when
quantity of work,
and of the
moments equally
true, when the
forces are
parallel.

124
NATURAL PHILOSOPHY.
add the point 0 is so far removed as to make the directions
difiegs of the forces sensibly parallel: whence we must conclude,
that the line of direction of the resultant of two parallel
forces is in the plane of the forces, is parallel to the direc-
tion of the forces, and that the moment of the resultant,
taken in reference to any point in the plane of the forces,
is equal to the sum or difference of the moments of the
components, according as they tend to turn the system in
the same or opposite directions about the centre of mo-
ments.
Resuming Eq. (31), and re-
volving the directions of the
forces P and Q about their
points of application A and B
till they become parallel, and
the forces act in the same direc-
tion, the angle will become
and we shall have
zero,
Fig. 51.
bod
R
Q
P
B
Value of
resultant when
the components
act in same
direction;
R = √P² + Q² + 2 P Q = P + Q.
Again, revolving the directions
as before, till they become par-
allel and the forces act in op-
posite directions, the angle o
will equal 180°, and Eq. (31)
reduces to
B
Fig. 52.
P
0

MECHANICS OF SOLIDS.
125
=
value of resultant
R = √P² + Q-2PQ P-Q; moto when components
act in opposite
whence we conclude, that the intensity of the resultant of directions;
two parallel components, is equal to the sum or difference of rule;
the intensities of the components according as these latter act
in the same or in opposite directions.
Now, resuming Eqs. (32), and changing the notation to
suit the first figure in § 118, we have
of neoptol odr
me, ed ai vhod
Q sin o hoitoosib otiroq
sin ROA =
R
"
CA Infog
antoelqoll
sin ROB =
P sin q
R
follemng ow
comoood
in which, if we make = 0, or 180°, we obtain
sin ROA = 0,
sin ROB = 0;0
(99)
myd,doldw
the resultant of
two parallel
that is to say, the angle which the direction of the result- the direction of
ant of two parallel forces makes with the directions of the
components, is nothing; in other words, the direction of the components, is
resultant of the parallel forces is parallel to that of the com- the components.
ponents, which is a confirmation of what we said above.
§ 119.-Passing thus to the
limits of the case in which the
directions of two forces P and
2, applied at the points A and
B of any body, meet in a
point; assume any point as K,
in the plane of the forces, and
let fall the perpendiculars Ka,
Kb. Denote by R, the in-
tensity of the resultant, sup-
posed to act along the line Re
B
b
Fig. 53.
R
PA
A
parallel to that of
The theorem of
moments true of
parallel forces.
To of Innoll
adio of
a
--K

126
NATURAL PHILOSOPHY.
then, from the principle of moments, will
RX Kc = PX Ka QX Kb;
the upper or lower sign
being taken, according as
the forces tend to turn the
body in the same or op-
posite directions about the
point K.
Replacing R by its
value PQ, the above
becomes
B
Fig. 53.
R
PA
b
-K
Relation of
resultant to its
two parallel
components;
(PQ) Kc
=
PX KaQx Kb;
which, by an obvious reduction, becomes
P (Kc - Ka)
=
Q(± Kb = Kc);
but
+
Ke-Ka = ca; Kb Kc = ±be;
whence
or
30
PX ac=Qx bc,
the distance of
P: Q: bc: ac;
that is to say, the line of direction of the resultant, divides
either component the perpendicular distance between the lines of direction.
from resultant,
proportional to of the components, into parts which are reciprocally pro-
the other
component.
portional to the forces.

MECHANICS OF SOLIDS.
127
Fig. 54.
ioslqen 20
本港 ​onil
PA
—
§ 120. Let the parallel
forces P and Q, be applied
to the points A and B. Join
A and B by a straight line,
and draw Ba' parallel to ba,
then will
24
B
RA
K
b
a
Be' = bc;
c'a' = ca;
and because C'c' is parallel to A a', the triangles Be' Cand
Ba' A, give the proportion,
Be': c'a' :: BC: CA,
whence
Jellareg ódi quot
P: Q :: BC: A 0;
that is to say, the line of di-
Fig. 55.
rection of the resultant of any
two parallel components, di-
vides the line joining their
points of application into
parts which are reciprocally
proportional to the intensities
P
Rule for position
R
of resultant;
C
A
B
of the components.
The above proportion
gives by composition,
P+Q P :: BO±AC: BC,
ed:
redw
PQ: Q:: BC AC: AC;

128
NATURAL PRILOSOPHY
relation of
resultant to either
component.
Moments of
parallel forces in
reference to an
axis;
or, replacing P Q by R, and B CAC by the whole
line BA,
RP :: AB: BC,
beilage
RQ
::
Q: AB: AC;
pall adgleda
OSI &
log oft of
vd & bas
that is to say, the resultant of two parallel components is to
either component, as the length of the straight line joining the
points of application of the components, is to the portion of this
line between the point in which it is cut by the direction of the
resultant, and the point of application of the other compo-
nent.
big 121.When two fo
§ 121.-When two forces are parallel, their moments
may not only be taken in reference to a point, but also in
reference to a right line, supposed fixed. Thus, suppose
the forces P, Q, and
their resultant R, to
act along the parallel
lines AP, BQ, and
CR, respectively.
Assume any line, as
ML, at pleasure;
conceive a plane
drawn through this
line and perpendicu-
lar to the plane of
the forces, and let
KL' be the intersec-
KM
P"
P P
Fig. 56.
conerw
-L
R
B
L
R
Q Q'
B
tion of these planes. From the point K, draw KL" per-
pendicular to the direction of the forces; then, regarding
moments referred K as the centre of moments, will
to a centre;
RX KO' = PX KA+ QX KB'.
whence
201
DA 08 9: 9+9

MECHANICS OF SOLIDS.
129
KA'
R = PX
+ Qx
KC
KB'
KO'
But from the similar triangles, KA'A, KB' B, and
KO' C, we have
KA'
KC
11
KA
KO'
KB'
KB
=
KO'
Και
which, substituted in the above equation, gives, on clear-
ing fractions,
RX KC = PX KA+ QX KB... (36).
Dividing both members by RX KC,
P KA
1 =
X
Q
+ X
KB
R KC R KC
by their
components;
From the points A, B, and C, draw the lines Aa, Bb,
and Cc, perpendicular to the line KL. Also, resolve forces replaced
each of the forces P, Q, and R, supposed applied at A, B,
C, respectively, into two components, one parallel, and the
other perpendicular, to the line KL; and let A P", BQ",
and CR" be the former, and A P', BQ', and CR', the
latter of these components.
In the similar triangles PAP', ROR', and QBQ',
we have, denoting the components A P', CR', and BQ',
by P', R', and Q', respectively,
PROR
=
P
R'
Q'
=
Ri

130
NATURAL PHILOSOPHY.
and from the similar triangles KA a, KCc, and KBb,
KA
A a
111
KC
Cc'
KB
Bb
=
KC
Oc
moments of
which values, substituted in the foregoing equation, give,
after clearing the fractions,
components
perpendicular to
the axis;
R' x Cc
=
P'x Aa + Q' x Bb.. (37).
The effective quan-
tity of work per-
formed by each of
the forces P, Q, and
R, may be replaced
by the algebraic sum
of the quantities of
work performed by
its components; but
the effective quanti-
moments of the ties of work of the
parallel
components;
moment of a
M
K
a
Fig. 56.
-L
P
A
R
B
Q
L'
P
P
R R
B
components which are parallel to the line KL, will be zero,
since the points of application are constrained to move in
planes at right angles to this fixed line, and hence the terms
in Eq. (37) will, for reasons explained in § 113, be the
measures of the relative quantities of work of the forces P,
Q, and R, being the products of the remaining components
into the perpendicular distances of their respective lines
of direction from points on the line KL.
The moment of a force in reference to a line, is the effec-
force in reference tive quantity of work which the force is capable of per-
to a fixed axis,
defined;
forming while its point of application is constrained to
describe an elementary path about this line, considered as
fixed; and its relative measure is, the product of the com-
ponent at right angles to the line, (the other being parallel

MECHANICS OF SOLIDS.
131
to it,) into the shortest distance from the fixed line to that
of the direction of the force.
The fixed line is called the axis of moments.
§ 122.-Dividing Eq. (36) by K C, we find
Ꭱ
=
P.
KA
+ Q.
KC
and substituting the values of
KB
KO'
the axis of
moments.
KA
KB
and
KC
KC'
as given on the opposite page, we find, after clearing
the fraction,
RX Cc = P x Aa + Q X
PX Aa+Q Bb;
distances of their
a line, and plane.
from which we see, that the product of the resultant of two Relation of the
parallel forces into the perpendicular distance of its point forces to the
of application from any given straight line, is equal to the points of
sum of the products of the forces into the perpendicular application from
distances of their respective points of application from their
same line. It is easy to see that the same is equally true
of any plane, since we have but to project the line joining
the points of application of the forces upon the assumed
plane, and take this projection as the axis of moments.
number of
parallel forces;
§ 123.-Now let us suppose any number of parallel
forces for instance, five. Find the resultant of any two Resultant of any
of them; compound this resultant with the third force,
and the resultant of the first three with the fourth, and
so on. The final resultant thus obtained, will be equal
in intensity to the sum of the intensities of the forces
which act in one direction, diminished by the sum of
the intensities of those which act in the opposite direction.
Its action will be in the direction of the greater sum.
And the moment of the resultant will be equal to the rule for finding;
algebraic sum of the moments of the components.
Men pulling upon parallel ropes, horses drawing upon

132
NATURAL PHILOSOPHY.
examples of
parallel forces.
The work
performed by the
resultant of
their traces attached to whipple-trees, are examples of
parallel forces.
Fig. 57.
R
P
§ 124. Suppose a body to be drawn in one direction
by any number of parallel forces P, Q, R, &c., and in the
opposite direction, by the parallel forces P', Q', R', &c.
parallel forces; If the points of the body move in
parallel lines, it is plain that the
paths described by the points of
application will be equal to each
other, and thus the quantity of
work of any force, will be given
by the product of its intensity
into the small path common to all
the forces. The total work will
be equal to the sum of the quan-
tities of work performed by the
forces P, Q, R, &c., diminished
by the sum performed by the
equal to the
algebraic sum of
the work of the
components.
P
Q
R
forces P, Q, R', &c.; that is to say, it will be equivalent
to the product of the common path, multiplied into the
algebraic sum of all the forces, or into the resultant. But
this latter product is the quantity of work performed by
the resultant. Hence, the quantity of work performed by
the resultant of any number of parallel forces, is equal to
the algebraic sum of the quantities of work performed by
the components.
§ 125. We have seen, § 122, that the product of the
intensity of the resultant of several parallel forces into the
perpendicular distance of its point of application from any
plane, is equal to the sum of the products arising from
multiplying the intensity of each force into the perpen-
dicular distance of its point of application from the same
plane. Denote this latter sum by K, the intensity of the
resultant by R, and the perpendicular distance of its point
of application from a given plane by r, then will

MECHANICS OF SOLIDS.
133
Rr
=
K
whence
K
r =
R
Position of the
resultant of
parallel forces.
and if the given plane be parallel to the direction of the
forces, r will be the distance between it and a second plane
containing the line of direction of the resultant. If we
know the value of K, in reference to another plane, also
parallel to the direction of the forces, the corresponding
value of r, will give the position of a second plane, whose
intersection with the first will give the line of direction of
the resultant. Thus, the principle explained in § 122,
may be employed to determine the line along which the
resultant of several parallel forces acts.
principle of
parallel forces by
§ 126. To illustrate the principle of parallel forces, Illustration of the
let us take the example of the common steelyard, an
instrument employed to ascertain the weight of different the steelyard.
substances. It con-
sists of a bar MN,
which turns freely
about an axis C sus-
pended from a fixed
point; the substance
Q to be weighed, is
placed at one end
A, while a constant
M
Fig. 58.
B
N
P
weight P is placed at a suitable point B, towards the other
end. In order that there may be an equilibrium, it is
necessary that the resultant of the forces P and Q shall
pass through the fixed point C; in other words,
QX AC = PX CB,
from which
BC=
1
X AC;

134
NATURAL PHILOSOPHY.
or, if P be taken equal to one pound, then will
BC= QXA 0.
The scale of the If Q be taken suc-
steelyard
constructed.
cessively equal to 1,
2, 3, 4, &c. pounds,
then will the corre-
sponding values of
B C, become A C
2 AC, 3 AC, 4 A C,
M
Fig. 58.
A
B
N
P
&c. Thus, if a scale of equal parts be constructed on the
longer arm, having its zero at the point C, and the con-
stant distance between the consecutive divisions equal to
AC; the number of the division estimated from C, on
which the weight P is placed to hold Q in equilibrio, will
indicate the weight of the latter.
The construction of the steelyard depends, as we see,
upon very simple principles; it gives rise, however, to
considerations, which will be referred to when we come to
treat of the lever.
VI.
Point of
application of
resultant of
CENTRE OF GRAVITY OF BODIES.
Fig. 59.
C
A
B
YQ
§ 127.-The intensity R, and point of application C, of
the resultant of two par-
parallel forces; allel forces P and Q, do
not depend upon the in-
clination of these forces
to the line AB, which
connects their points
of application, but will
continue the same,
however the direction
of the forces may re-
volve about these points
YP
R
0
D

MECHANICS OF SOLIDS.
135
of application, provided the forces continue parallel to each
other, and preserve unchanged the ratio of their intensities;
for the intensity of the resultant is given by
and the point C, by
R = PQ,
BC=
P.AB
R
through which
the resultant will
which are wholly independent of the angle which the
common direction of the forces makes with the line A B
So, likewise, if there be three forces P, Q, and S, we may
join the point of application D, of the third force S, with
that of the resultant R, and show, in like manner, that the there is one point
position of O, the point of application of the resultant T
of R and S, (that is, of P, Q, and S) is entirely independ- always pass;
ent of the inclination of the forces to the line CD. And
as the same reasoning may be extended to any number of
parallel forces, we conclude, that in every system of paral-
lel forces, there is one point through which the resultant
will always pass.
This point is called the centre of parallel forces.
the centre of
parallel forces.
§ 128.-Every body is composed of an indefinite num-
ber of elementary heavy particles, which are the points of
application of as many vertical or parallel forces, whose
resultant is a force equal to their sum, and is called the
weight of the body. The point of application of the weight Weight of a body;
is obtained by combining the parallel forces in the manner
before explained; this point will be the centre of the sys-
tem, and, because the forces are those which result from
the action of gravity, it is called the CENTRE OF GRAVITY. centre of gravity.
The centre of gravity of any body may be defined, the
point through which the line of direction of the weight always
passes.
§ 129.-The centre of gravity of a body being the
centre of all the vertical forces which solicit its heavy

136
NATURAL PHILOSOPHY.
finding centre of
gravity;
particles, this point must remain invariable, while the
forces, without ceasing to be parallel, revolve about the
Two methods of points of application. Instead of causing the forces to
rotate, let the body revolve. In this motion, the forces
will preserve their vertical direction, and the line of
direction of the weight always passing through the centre
of gravity, there will result two very simple methods of
finding the position of this point as long as the figure
of the body remains unchanged.
first method-
by suspension;
A body being suspended by means of a thread A C,
from the point A, will take such a position, that the effort
exerted along the thread to sup-.
port it, will be in equilibrio
with the weight, and thus, when
the body comes to rest, the di-
rection of the thread will pass
through the centre of gravity G.
If we change the point C, to
which the thread is attached, to
C', the body will assume a new
position, and when it comes to
rest again, we shall have a sec-
ond line C' G, also passing
through the centre of gravity,
and whose intersection with the
Fig. 60.
first, will determine the position of that point.
A
By the same reasoning it follows, that a body will be
supported upon a point, whenever the vertical through
the centre of gravity passes through
this point; and all positions of the
body which satisfy this condition,
give as many lines intersecting at
second method- the centre of gravity. The upper
and lower points, in which any two
of these lines pierce the surface, be-
ing known, and connected by recti-
lineal openings, these openings will
by poising;
Fig. 61.
G

MECHANICS OF SOLIDS.
137
planes through
the centre of
give, by their intersection, the centre of gravity of the
body. To find these upper and lower points, suspend traces of the
the body, by a thread or rope, and when it comes to
rest, suspend a plummet on each side, and in such posi- gravity found.
tions that the plane of their threads shall contain the
suspension line of the body; then, with a pencil, trace
upon the body the intersection of this plane with its
surface. Next, suspend the body from some other point,
and repeat the same operation; the intersections of the
two traces will give two of the points required; and the
same for others.
found by
computation;
§ 130.--This method becomes impracticable in the case Centre of gravity
of very heavy bodies, of those which are fixed, or of such
as do not yet exist, and of which the construction is only
in project. In general, when the form of a body is
defined geometrically, or by a drawing, the centre of
gravity is determined in this wise. Conceive the body to
be divided into small portions by a series of planes; take
the product of the weight of each portion into its distance
from some assumed plane of reference, and take the sum
of these products; this sum is, according to what we
have seen of the principles of parallel forces, equal to
the product of the entire weight of the body into the distance of centre
distance of its centre of gravity from the same plane. of gravity from a
Hence, the distance of the centre of gravity from any plane, is
equal to the sum of the products obtained by multiplying the
weight of each element of the body into its distance from this
plane, divided by the whole weight of the body.
plane;
assumed
Find the distance, given by this rule, from any three from three
arbitrary planes, and the position of the centre of gravity planes;
becomes known. This method, which becomes long and
tedious in many instances, may be abridged according to
circumstances, particularly when the object is to find the process may be
centre of gravity of homogeneous bodies. A body is said
to be homogeneous, when any two of its parts have the homogeneous
same weight under equal volumes.
abridged in the
case of
bodies.

138
NATURAL PHILOSOPHY.
8131.-Experience shows us that a bar A B, of wood,
Centre of gravity metal, or any other material, which is perfectly homo-
of regular and
homogeneous
bodies;
of a bar;
of a bar with
equal spheres at
the ends;
of regular and
homogeneous
bodies, at the
centre of figure;
right prism;
geneous, will remain in
equilibrio in a horizontal
position, if suspended by
its middle point C; and
hence the centre of gravity
of this bar is situated at
the middle of its length.
The bar is also found to
remain in equilibrio when
placed in a vertical posi-
tion, if suspended by the
central point of its end;
and hence the centre of
gravity is situated at the
central point of its thick-
ness. If the bar support
at its ends equal spheres,
M
Fig. 62.
E
C
C
M
it will still remain in equilibrio when suspended by its
middle point, if placed in a horizontal position.
The centre of gravity of a sphere is at its centre of
figure, for when suspended by any one of its points, the
direction of the suspending thread always passes through
centre of gravity that point. And it is a general principle, that the centres
of gravity of all regular and homogeneous bodies are at
their centres of figure. And, hence, a right prism or
cylinder has its centre of gravity at the middle of its
length, breadth, and thickness; a circle at its centre; and
a right line at its middle point.
circle, &c.;
centre of gravity
of a surface; of a
line.
Body
symmetrical in
reference to a
plane;
By the centre of gravity of a surface, is understood
that of a body of extreme thinness, such as paper, tin-foil,
gold-leaf, &c.; and by the centre of gravity of a line, is
meant that of a body whose breadth and thickness are
very small as compared with its length.
§ 132. A body is said to be symmetrical in reference

MECHANICS OF SOLIDS.
139
to a plane, when the latter cuts into two equal parts every
perpendicular which is drawn to it, and which is termina-
ted by the opposite extremes of the body. This plane plane of
is called the plane of symmetry.
symmetry;
A body is symmetrical in reference to a line, when it symmetrical in
has two planes of symmetry passing through the line. reference to a
This line is called a line of symmetry.
line; line of
symmetry;
symmetrical in
reference to a
A surface is symmetrical in reference to a line, when surface
the latter cuts into two equal parts, all the perpendiculars
to it which are terminated on opposite sides by the con- line;
tour of the surface.
In all cases, the centre of gravity of homogeneous symmet- centre of gravity
rical bodies, is situated in their planes, or lines of symmetry. lines of
Consider, for example, a
curve having AB for its
line of symmetry, and of
which we have found the
centres of gravity G and
G, of the two halves
AMB and AM B.
These two halves being
turned about the line of
symmetry till one is ap-
plied to the other, their
A
Fig. 63.
G
G
M
M
B
in planes and
symmetry;
symmetrical
centres of gravity will coincide; that is to say, the centres illustration in
of gravity G and G, were, before the motion, situated case of a
upon a right line GG, perpendicular to the line AB. curve;
Hence, if the curves be supposed concentrated at their
respective centres of gravity, GG becomes a right line,
terminated by two material points whose common centre
of gravity is at the middle point O, on the line of symme-
try. A similar reasoning may be applied to all bodies of
symmetrical dimensions.
of a surface with
two axes of
The centre of gravity of a surface which has two axes centre of gravity
of symmetry, is at the intersection of these axes. The
transverse and conjugate axes of the ellipse, for
ample, being axes of symmetry, cut each other at
ex- symmetry;
the

140
NATURAL PHILOSOPHY.
case of the
ellipse;
rectangle;
centre of gravity of the elliptical
surface. For the same reason, the
centre of gravity of a rectangle is
at the intersection of the right
lines joining the middle points of
its opposite sides.
When a volume has a right
Fig. 64.
line of symmetry, its centre of gravity is on this line. A
volume with one right cylinder, with an elliptical base, has two planes of
axis of
symmetry;
sphere many
axes of
symmetry.
Centre of gravity
of two
homogeneous
bodies, one
within the other.
symmetry, determined by the
longer and shorter axes of
the ellipse, its centre of grav-
ity is, therefore, on the line
G G, joining the centres of
gravity of the bases, and at
its middle point 0.
Fig. 65.
0
Other bodies are divided symmetrically, in an infinity
of ways. Such, for example, is the sphere of which all
the planes of symmetry pass through the centre of figure;
it is for this reason that this point is also its centre of
gravity.
§ 133.-If the regular homogeneous body contain
within its boundary another homogeneous body of dif
ferent density, the centre of gravity of the whole mass is
found, by first regarding it as of uniform density, and the
same as that of the larger body; the centre of gravity 0,
obtained on this hypothesis, gives
rise to a first approximation. We
then conceive the weight w, of the
body supposed homogeneous, to be
concentrated at the centre of
grav-
ity O, and subtracting this weight
w from the total weight W, we ob-
tain a difference W-w, neglected
in finding the point 0. Let O' be
Fig. 66.
0
K
the centre of gravity of the volume corresponding to this

MECHANICS OF SOLIDS.
141
difference; join O with O' by a right line, and divide this
line at the point K, so that
X
w × OK = (W-w) KO';
the point K will be the common centre of gravity.
their centres of
line.
§ 134.-Whenever a body may be divided into parallel when the layers
layers, and the centres of gravity of these are situated on of a body have
a right line, the centre of gravity of the whole body is also gravity on a right
upon
this line. For compounding the weights of any two
of these layers, supposed concentrated at their respective
centres of gravity, and the resultant of these with the
weight of a third, &c., it is easy to see, from the principle of
parallel forces, that the point into which the whole weight
must be concentrated will be on the line in question.
§ 135.-If, for example, the parallelogram A B CD, Centre of gravity
supposed to possess a
small thickness, be di-
vided by planes par-
allel to CD, into an
indefinite number of
strata or layers, the
centre of gravity of
each one will be at
its middle point, and
therefore on the line
C
I
Fig. 67.
D
T
N
B
E
A
FE, joining the middle points of the opposite sides CD
and AB; the centre of gravity of the parallelogram will,
§ 134, also be on this line. In like manner, it may be
shown to be on the line IN, joining the middle points of
the opposite sides C B and D A; it must, therefore, be at
their intersection 0.
of a
parallelogram;
A similar reasoning will show that the centre of gravity
of a parallelopipedon and cube, will be at the common of a
intersection of three right lines joining the centres of and cube.
gravity of their opposite faces.
parallelopipedon

142
NATURAL PHILOSOPHY.
§ 136. The triangle ABC, being divided into very
thin layers, parallel to the side A C, it follows, from what
has just been said, that
Centre of gravity the centre of gravity of
of a triangle;
each layer, and, there-
Fig. 68.
fore, of the whole tri-
angle, will be situated
B
upon the right line BD,
drawn from the vertex
B to the middle of the
side A C. For the
same reason, the centre
of gravity of the trian-
gle will also be on the
line AF, drawn from
E
F
D
the angle A to the middle of the opposite side CB; and
hence it must be at the intersection G.
Join F D. Since the sides A C and B C, are divided
proportionally at the points D and F, the line D F is
parallel to A B; hence the triangles A G B and D G F
are similar, and give the proportion
AG: GF: AB: FD;
but, because the points F and D are at the middle of the
lines B C and A C, it follows that F D is half of A B,
and, therefore, from the above proportion, F G is half
A G; or F G is one third of the whole line A F Hence,
where situated; the centre of gravity of a triangle, is on a line drawn from
one of the angles to the middle point of the opposite side, and
at a distance from this side equal to one third of the line.
common centre
of gravity of
three equal balls.
This point is also the common centre of gravity of
three equal balls, whose centres of gravity are situated at
the angles of the triangle, for the centre of gravity of the
balls A and C is at the middle point D, and this point
being joined with B, the centre of gravity of the three
balls will divide the line BD at the point G, so that B G
shall be double G D.

MECHANICS OF SOLIDS.
143
8137. To find the centre of gravity of any polygon, Centre of gravity
as ABCDEF, draw from any one of the angles, as A,
of a polygon.
the diagonals AC, AD, AE,
&c., and thus divide the polygon
into triangles. Find the centres
Fig. 69.
B
of gravity g, g', g', g'", &c. of
each of these triangles by the
rule above; join the points g and
g' by the right line gg', and de-
note the areas of the triangles
ABC and A CD by a and a',
respectively; then will the centre
0
G
F
E
of gravity of the area A B CDA, be found by the pro-
portion
a + a'
: a: gg': g' G.
In like manner, joining G and g' by a right line, and
denoting the area of the triangle ADE by a", will the
centre of gravity of the area ABCDEA be found from
the proportion,
a + a + a": a" :: Gg":
G G';
and so on to the last triangle; the quantities g' G, G G',
&c., being the only unknown quantities become known
from the proportions.
§ 138.-A series of planes
parallel to the base DBC,
of the triangular pyramid
ABCD, will give rise to a
series of strata or layers per-
fectly similar to the base, and
all their centres of gravity
will be situated upon a right
line joining the centre of
gravity of the base and the
vertex, because they are all
similarly situated to the base.
B
G
Fig. 70.
D
A
A pyramid
divided into
layers parallel
to the base;

144
NATURAL PHILOSOPHY.
its centre of
gravity found;
Fig. 70.
A.
As either of the solid angles may be taken as a vertex
and the opposite face as a base, and as the dividing planes
may be passed parallel to each
of the bases, it follows that
the centre of gravity of the
pyramid must be upon the
four lines drawn from the solid
angles to the centre of gravity
of the opposite faces, and must,
therefore, be at their common
point of intersection.
B
Gr
D
Let G' and G" be the cen-
tres of gravity of the triangu-
lar faces A B D and B CD;
join these points with the
opposite vertices by the right
lines A G" and CG', their
point of intersection G, will be the centre of gravity
of the pyramid. Join G' and G"; then, because the lines
A E and E C are divided proportionally at the points
G' and G", the line G' G" is parallel to A C, the triangles
G GG" and G A C are similar, and give the proportion,
G' G": GG" :: AC: AG;
but G' G" is one third of A C, and hence G G" is one
third of A G, or one fourth of A G". The centre of
where situated. gravity of a triangular pyramid is, therefore, on a line join-
ing one of the angles with the centre of gravity of the opposite
face, and at a distance from this face, equal to one fourth of
The common
centre of gravity
of four equal
balls.
the line.
The same result may be obtained for the common
centre of gravity of four equal balls, whose centres of
gravity are situated at the four vertices of the pyramid.
§ 139. The foregoing reasoning is equally applicable
to a pyramid, of which the base is any polygon. For the

MECHANICS OF SOLIDS.
145
centre of gravity is on a line drawn from the vertex Sto Centre of gravity
the centre of gravity of the base, because it contains the of any pyramid;
centres of gravity of all sections parallel to the base; and
if we conceive the pyramid divided
into triangular pyramids by planes
through this line, and through the
angles A, B, C, D, &c. of the base, the
centres of gravity of these elementary
pyramids, and therefore of the whole
pyramid, will be situated in a plane
parallel to the base, and at one fourth
the distance from the base to the
vertex; it must, therefore, be at the
intersection of this line and plane.
Hence, to find the centre of gravity of
Fig. 71.
B
D
any pyramid, join the vertex with the centre of gravity of the where situated.
base, and lay off a distance from the base on this line equal to
one fourth of its length.
This rule is also applicable to a cone, which may be Centre of gravity
regarded as a pyramid of an indefinite number of sides.
of a cone.
polyhedron.
§ 140.-Since every polyhedron may be divided into of any
triangular pyramids whose weights may be supposed to
act at their respective centres of gravity, and since, from
the principles of parallel forces, the sum of the products
which result from multiplying the weight of each partial
pyramid into the distance of its centre of gravity
from any plane, is equal to the product of the entire
weight of the polyhedron into the distance of its centre
of gravity from the same plane, the distance of the
centre of gravity from three planes may be found, and
thus its position determined.
§ 141.-When a body is terminated by curved surfaces, of a body of any
by planes, or by curve lines, it may be divided into small form;
elementary parts, similar to the figures which have been
already considered-as right lines, triangles, parallelo-
10

146
NATURAL PHILOSOPHY.
the partial
grams, pyramids, parallelo-
pipedons, polyhedrons, &c.;
the sum of the products
products found; which result from multi-
plying the weight of each
into the distance of its cen-
tre of gravity from some as-
sumed plane, or right line,
must be found, and this
E
A
C
sum divided by the entire
Fig. 72.
万
​F
0
B
divided by the
entire weight;
the sum of these weight of the body; the result will be the distance of the
centre of gravity from the plane or line. Let it be required,
for example, to determine the centre of gravity of any plane
area Cab Fdc; draw in its plane any right line A B, and
divide the given area into a series of very thin layers,
perpendicular to this right line. The layer a c d b, may
regarded as a small rectangle, and, supposing its density
uniform, its centre of gravity is at its middle point 0;
denoting the density by D, and the force of gravity by 9,
one of the partial products will be
be
illustration;
D.g.
ac + db.ef.io = D. g.
D.g.ac ef.
2
ac+db
2
ea+ec
2
when the force
of gravity is
The other partial products being found in the same way,
and their sum divided by the product of Dg into the
entire area Ccd Fba C, determined by the method of § 46,
will give the distance of the centre of gravity of this area
from the line A B. Performing the same operation in
reference to another line A E, the centre of gravity is
completely determined, being the intersection of two right
lines, parallel respectively to AB and A E, and distant
from them, equal to the results obtained by the above
process.
It is to be remarked, that when the force of gravity g
is constant, and the density D is uniform throughout the
density uniform; body, these quantities strike out, and leave the distance
constant and

MECHANICS OF SOLIDS.
147
terms of the
of the centre of gravity from the line, or plane, equal to the partial
the sum of the products arising from multiplying the ele- products in
mentary volumes into the distances of their respective volumes.
centres from the line or plane, divided by the entire
volume.
§ 142.-The consideration of the centre of gravity is
very useful in computing certain volumes and surfaces,
which are found with considerable difficulty by the ordi-
nary process. The screw, the curbs of stair-ways, surfaces
of revolution generated by
the rotation of a plane curve
C.DE about an axis A B
situated in its plane, are ex-
amples. Suppose, in the case
of a volume, the generating
area CDE to be divided into
small rectangles, of which the
sides are parallel and perpen-
dicular to the axis A B. Each
rectangle will generate around
the axis an elementary ring,
Fig. 73.
A
B
E
D
and the sum of all these rings will give the volume of the
solid of revolution. Let r denote the distance of the centre
of gravity of one of these small rectangles from the axis;
we know that the volume of the ring, of which the profile
is the rectangle, is measured by the product of the area a
of the rectangle, multiplied by the mean circumference of
the ring, 2 r; for the annular base of such a ring being
developed, will form a trapezoid, the half sum of whose
parallel sides is equal to 2 r r, and hence we shall have for
the value of the ring the expression 2 r a. The volumes
generated by the other rectangles, whose areas are a', a", a"",
&c., will be 2 r' a,' 2 x r' a", 2 r'" a"", &c. And de-
noting by V the total volume generated, we shall have
T
V = 2 (ar + a' r' + a" " + a""" + &c.);
T
Use of the centre
of gravity in
computing
volumes and
surfaces;


148
NATURAL PHILOSOPHY.
relation of
volume to
generatrix and
path of centre of
gravity;
but the quantity within the brackets, is the sum of the
products which result from multiplying the elementary
volumes of the generating
area CED, by the distances
of their respective centres of
gravity from the line AB,
which we know to be equal
to the product of the whole
area CED, into the distance
of its centre of gravity from
the same axis. Denoting the
area CED by A, the distance.
of its centre of gravity from
A B by R, we, therefore, have
A
B
V = 2 RA.
Fig. 73.
£
V
D
E
(38).
If, instead of an area, we had considered a plane curve
CE, the quantities a, a', a", &c.,
would represent the lengths of ele-
rule;
mentary portions of this curve, A
would represent its entire length,
R would be the distance of its
centre of gravity G, from the line
A B, and V would be the value of
the surface generated by the entire
curve about A B. Whence we
derive this rule, viz.: The volume
generated by the motion of any plane,
or surface generated by the motion of
Fig. 74.
4.
B
E
G
any line, is equal to the generatrix, multiplied by the path
described by its centre of gravity; the direction of the motion
being perpendicular to the generatrix.
This rule supposes the body to possess a constant pro-
file, of which the plane is perpendicular to the path of the
centre of gravity.

MECHANICS OF SOLIDS.
149
Example 1st. Let it be required to
find the volume generated by the ro-
tation of the right-angled triangle
ABC, about the side A B. The cen-
tre of gravity G, being found by the
rule already explained, draw GD
perpendicular to AB. Then, in the
triangles E G D and E BC, we have
A
Fig. 75.
ED
B
example-the
volume of a
cone;

CB : GD ::
CE:
GE: 3 1;
whence
and
GD = }CB;
2 GD = 3 CB,
which is the length of the path described by the centre of
gravity. The area of the triangle is
ABX CB;
whence the volume V becomes

V = T CB² X AB,
which is the usual measure of the volume of a cone.
Example 2d. Let it be required to
find the surface generated by the rota-
tion of the line CD, about A B. The
centre of gravity of CD is at its middle
point G; and GD', CA, and D B being
perpendicular to A B, we have
GD' = (AC + BD);
and for the path described by G,
Fig. 76.
example-the
D
surface of a conic
G
frustum;
D'
B
(AC+ BD)
2 GD = 2*
2

150
NATURAL PHILOSOPHY.
and hence,
2 TAC+2 BD
V =
.CD;
2
example-the
volume of a
stairway curb;
excavation from
ditches;
rule holds for
any portion of
an entire
revolution.
which is the usual measure for the convex surface of a
conic frustum.
Example 3d. Let it be required
to find the volume of the curb of
a stairway, of a helical form.
First, compute the area of a
section abcd, perpendicular to a
mean helix gg, or that described
by the centre of gravity; then
multiply this section by the length
by the le
of the mean helix.
The excavation taken from a
ditch, of which the profile is con-
stant, may be estimated in the
same way.
In examples 1st and 2d, the
centre of gravity is supposed to
have described an entire circumfe-
Fig. 77.
rence; but had it moved through only an eighth, tenth, or
any other fractional portion of a circumference, the volume
generated would still, as in example 3d, have been given
by the area of the generatrix into the extent of the path
described.
Motion of
translation;
VII.
MOTION OF TRANSLATION OF A BODY OR
SYSTEM OF BODIES.
§ 143.-A body, or system of bodies, is said to have a
simple motion of translation, when all its elements describe,
simultaneously, equal and parallel paths.
MECHANICS OF SOLIDS.
151

Denote by v the velocity which any motive force com-
municates to all parts of the system during any small
interval of time t. The force of inertia f, of an element
whose weight is p, will be given by the equation
ƒ =
p
g
.
t
and the force of inertia f', of an element whose weight
the measure of
the inertia of an
element;
is p', by
f' =
p'
บ
g
t
10 0010: 31
and so of all the others, provided the degree of velocity
impressed upon all the elements is the same during the
time t. Moreover, as each force of inertia is exerted in
the direction of the path along which the elements respec-
tively move, and as these are supposed parallel, the forces
of inertia are parallel, and give a resultant equal in inten-
sity to their algebraic sum. Denoting the intensity of this
resultant by F, we have
ย
+
·
F = ƒ +ƒ' + ƒ" + &c. = = (p + p' + p" + p'" + &c.
F=ƒ+f'+f"
t
g
&c.);

and replacing the sum of the partial weights by the entire
by the entire mass M of the system, we
P
weight P, and
g
shall finally have
v
FM.
t
of that of the
(39).
entire mass;
It remains to find the invariable point of application of
F. This point is called the centre of inertia. The inten- centre of inertia;
sities of the forces f, f', f", &c., are proportional to the
weights p, p', p", &c., to which they are respectively
applied, and thus the point of application of F, will coin-
cide with that of the resultant of the forces p, p', p", &c.;
152
NATURAL PHILOSOPHY.
measure of
that is to say, with that of the entire weight P, which is
inertia in words. the centre of gravity of the system. Hence, the total
force of inertia of a body, or system of bodies, having a simple
motion of translation, is measured by the mass of the system,
multiplied into the ratio which the small degree of velocity
The communicated bears to the time during which the velocity is
pole impressed. And the total force of inertia has its point of appli-
cation at the centre of gravity.

The force of
gravity being
constant, the
This coincidence of the centre of inertia with the
centre of gravity, results from the assumption that the
force of gravity is the same in its action upon the different
parts of the system. Had it been otherwise, that is to say,
centre of gravity had the force of gravity varied in intensity from one ele-
ment to another, the centre of inertia, being always at the
centre of mass, would be different from the centre of
gravity.
and of inertia
coincide;
these centres
sensibly the same
in bodies on the
earth.
Quantity of
motion of a
body;
The intensity of the force of gravity being regarded as
the same within the limits of a body on the earth's surface,
the centre of inertia and of gravity may be regarded as
coinciding, and hence these terms will be used indis-
criminately.
§ 144.-Let V represent the velocity of a body having
a motion of translation, supposed uniform at any instant;
the quantity of motion of any one of its elements whose
weight is p, is measured by
PV,
g
and of an element whose weight is p',
PV,
g
and so for the other elements; and as these motions are
parallel, their sum will give the quantity of motion of the
entire body. Designating this quantity by Q, we shall
have
MECHANICS OF SOLIDS.
153

Q =
p+p'+p" + &c.
=
g
-V MV. . (40). its measure.
=
Thus the total quantity of motion, in any body having a
motion of translation, is measured by the mass of the
body into its velocity.
body when the
§ 145.-When a certain degree of velocity v, is im- Motion of a
pressed upon all the elements of a body during a very direction of the
short interval of time t, we have seen that the total force motive force
of inertia is given by, Eq. (39),
v
passes through
the centre of
gravity;
F = MX
We have seen, also, that this force of inertia is exerted in
the direction of the body's motion, and through the centre
of gravity. If, therefore, we suppose that at the instant in
which the body has acquired the velocity v, a force equal
to F is applied in a direction contrary to the motion, and
at the centre of gravity, it will
destroy the motion. This being
supposed, if we apply at the
centre of gravity of the body, a
motive force X, it will commu-
nicate to it a simple motion of
translation. For this force X
will be equal and directly op-
posed to the force of inertia F,

X
Fig. 78.
which it develops. This latter force F will be resolved
into as many partial forces of inertia f, f', f", &c., as
there are elementary portions of the body, and the inten-
sities of these partial forces will be proportional to the
respective weights of these elements. Denoting the masses
of the elements by m, m', m", &c., we shall have,
m
J = F
ƒ' = F
m'
M
m"
f" = F, &c.
M

154
NATURAL PHILOSOPHY.
The degree of velocity which each of these forces im-
presses upon the part on which it acts, will, § 82, be
measured by
f.t
f'. t
f". t
m
m'
m"
&c.;
or, replacing f, f', f", &c., by their values as given above,
simply by the expression
F. t
M
will be
that of simple
translation.
force does not
and as this measure is the same as that before deduced,
Eq. (39), for the degree of velocity impressed on the centre
of gravity by the force F, or its equal X, we see that,
to impress a simple motion of translation upon any body, it is
necessary that the line of direction of the motive force, or the
resultant of the motive forces, when there are several, must pass
through the centre of gravity; and, reciprocally, if the line of
direction of the force, or that of the resultant, in the case of
several forces, pass through the centre of gravity, the body will
have a simple motion of translation.
§ 146.-If the force X, were
applied along the right line AB,
Motion when the not passing through the centre
of gravity G, it is easy to see
pass through the
centre of gravity; that the motion cannot be one
of simple translation. For, if
this latter motion obtained, the
partial forces of inertia would
have a resultant of which the
Fig. 79.
F
I
B
line of direction would, from what we have seen, pass
through the centre of gravity G; and if this resultant
were replaced by an equal force F, applied along the same
line and directly opposed to the motion, the latter would
be destroyed, and an equilibrium would result. But it is
impossible that two forces X and F, applied to the ex-
tremities of a physical line or bar A G, can produce an

MECHANICS OF SOLIDS.
155
of rotation at the
equilibrium, unless they act in the direction of the bar. will be that of
Hence, when a body receives the action of a force, of which translation and
the direction does not pass through the centre of gravity, same time;
its motion will not be that of simple translation, but will
be compounded of a motion of translation and of rotation;
that is to say, some one of its elements will move, for the
instant, in a right line, while the others will rotate about
it as a centre.
To find this element C, conceive a plane to be drawn
through it, parallel to the direction of its motion, and per-
pendicular to the planes in which the other elements, for

the instant, rotate,
and let A B be its
trace upon that one
of these planes which
contains the point C,
and its rectilineal
path. Let my be the
projection of some
one element m' upon
this latter plane, and
m2 1
Fig. 80.
m
position of the
element having
a motion of
m2
translation.
B
A
Ki
Cra

take CC to represent the velocity v of translation, and
M2 M3 the velocity of rotation acquired by the element m', in
the small time t. Make my my equal and parallel to CC₁;
then would m, m2 represent the velocity acquired by m',
had the body moved with a simple motion of translation;
but by virtue of the motion of rotation, the actual velocity
acquired by m', in the direction of C's motion, is my m
diminished or increased by the projection of m, mg upon
the line CC, according to the direction of the rotation.
Project the points m, m2, and mg, upon A B, by the
perpendiculars my k₁, ma k₂, mg kg; then will the actual
velocity v', acquired by m', parallel to C's motion, be
my my my 0, or
but
-
v' = v — M₂ 0;
M₂ o = mz M₂ X cos mg m₂ o = Mg mg x cos C₁ ma kaj
156
NATURAL PHILOSOPHY.
=
to denoting Cm Cm₂ = C₁ m₂ by r, m₁ k₁ = mg kg by y
and the velocity of rotation acquired by a point at the unit's
distance from C, called the angular velocity, by V, then will
and
m2 m3
Vir,
cos C₁ mg kg = 11,
r
Relative velocity which substituted above, give
of two elements
of a solid body
v'
= v
in motion;
V₁y
Moreover, m, o is the
velocity of the ele-
ment m' perpendicu-
lar to the direction
of C's motion; and
calling this velocity
v", and the distance
C2, x, we shall
m1
A-
C
k1
Fig. 80.
(41).

an
m2
C1 Kg Ka
-B
the same in a
direction
perpendicular to
the former;
have
v"
=
V₁x,.
(42).
Denoting, as before, the weight of the element m' by p',
and its force of inertia in the direction CC1 by f', we
have
f'
=
p' v'
g
.
t
=
p'
It
(v - V₁y),
and similar expressions for the inertia of the other ele-
ments. Taking the sum of these, and representing the
inertia of the entire mass by F, we have, from the princi-
ple of parallel forces,
1
F=
p"
+
+
g
g
+ &c.)
9
1
p"
g
-}· · (4·3 + 2x + 3+);
(2 y,
&c.);

MECHANICS OF SOLIDS.
157
or, denoting the entire mass of the body by M, and the
masses of the several elements by m', m", m'", &c., this
reduces to
F= M.
v
t
-
1. (my,+ my.. + m'" ... + &c.).
t
Now the first term of the second member, which alone
involves the motion of the point C, is wholly independent
of the figure of the body and of the distribution of its
elements.
It will, therefore, remain the same whatever changes
take place in its figure and size, provided its quantity of
matter remain the same. The place of C, as determined
from any supposition consistent with this last condition,
will, therefore, be its position generally.
This being understood, conceive the whole body to
contract gradually in all directions till it is concentrated
in a single point; this point must, from necessity, be the
centre of gravity which alone remains undisturbed during
contraction, as it will during an expansion, being the
centre of mass. The point C, and the centre of gravity,
not being disturbed by this change of volume, must
coincide, and hence must always remain one and the
same point.
But when the plane in reference to which the products
m' y₁, m'" y,,, &c., are taken, passes through the centre of
gravity, we have
m'y, +m"y", + m"" y + &c.
value for the force
of inertia of a
body;

Y+ &c. = 0;

and the above equation reduces to
F= M.;
which is identical with Eq. (39).
always equal to
the mass into ratio
of the increment
of velocity to that
of the time.

158
NATURAL PHILOSOPHY.
The body will
have a motion of
translation; it
will also rotate
about the centre
of gravity;
We conclude, therefore, 1st, that when a body is acted
upon by one or more forces, its centre of gravity will move
as though the forces were applied directly to it, provided their
directions remain unchanged; 2d, that when the line of di-
rection of the force, or that of the resultant of several forces,
does not pass through the centre of gravity, the body will, in
addition, rotate about this centre.
The law which regulates the motion of the centre of
gravity results from the above equation, for if X represent
the resultant of all the forces, and F the total force of
inertia, we have from the equality of action and reaction,
X= F, which value of F, substituted above gives, after
reduction,
value of the
velocity of
translation.
v=
X.t
M
(42)';
in which is the velocity impressed in the very short
interval t, from which we may pass to the velocity ac-
quired at the expiration of any time, and thence to the
space described.
§ 147.-What has been before explained, applies also
to the total living force possessed by a body having a sim-
ple motion of translation. For being the common
velocity of all the elements, X v2, will be the living
P
g
p
force of that whose weight is p;
X² the living force
g
Living force in a
simple motion of
translation.
of that whose weight is p', &c.; so that the sum of all
these living forces, or the total living force, denoted by L,
p+p+p" + &c.
will be v² X
g
; and representing the
entire mass of the system by M, as before,
L = Mv².
If the body have a motion of rotation as well as of

MECHANICS OF SOLIDS.
159
translation, then will the living force of m', in the direc- If the body have
tion of the motion of translation be, Eq. (41),
m' v¹² = m' (v — V₁y,) = m' v² - 2 v. V₁m'y, + V₁ m'y²;
and in the direction perpendicular to the motion of trans-
lation, Eq. (42),
also a motion of
rotation;

m' v'12
=
m' Vi ;
and similar expressions for the elements whose masses are
m", m'", &c. Taking the sum of these, denoting the
living force, as before, by L, and reducing by the equa-
tions
we find
m'y, + m' y,, + &c. = 0,
y² + x,² = r²,
2
y₁₁² + x, 2 = r²,
&c. &c. =
&c.,
m' + m" + m"" + &c. =M;

L = Mv² + V₁ (m' r² + m" r, 2 + &c.);
or, making
2
m' r² + m" r² + &c. = Σm r²,
L = M² + V². zmr²
(43).
§ 148. The considerations which have now been
developed, show that in the motion of translation of a
body or system of bodies, the computations may be great-
ly simplified, since we are permitted to disregard the
shape of bodies, to suppose them concentrated about
the living force is
equal to that due
to translation,
increased by that
due to rotation.

160
NATURAL PHILOSOPHY.
General theorem
work of weights;
their centres of gravity, and to reason upon these points
as upon the total masses.
§ 149.-We have seen that in all questions affecting
-the quantity of the circumstances of simple motion of translation, we may
regard the mass as concentrated about its centre of grav-
ity. But when the different parts of a body receive
motions which differ from each other, this concentration
is generally inadmissible, since the partial forces of inertia
not being parallel, their resultant will no longer be equal
to their sum. If, however, we desire, in any case of the
coexistence of various motions, to estimate the work per-
formed by the weights of the parts of a body, during a
given time, the action exerted by these latter forces being
parallel, and their resultant or the total weight always pass-
ing through the centre of gravity, we may still reason upon
the motion of this point as though the mass were concen-
trated at it, and disregard the motion of rotation of the
other parts of the body about it. In this case, the quantity
of work expended in every
instance, will be obtained
by taking the product of
the weight into the path
described by the centre of
gravity, estimated in a ver-
tical direction. If, for exam-
ple, the centre of gravity of
any body, as a bomb-shell,
pass from the position G to

equal to the
weight, into the
projection of path
of the centre of
gravity.
Fig. 81.
R
G', describing the curve G G', we obtain the work done
by the weight during the interval of time occupied in
passing from one of these positions to the other, by mul-
tiplying the weight of the shell into G'R, the projection
of the path G G' on the vertical through G'.
This theorem, in regard to the work performed by the
weight, is by no means restricted to the motion of a single
body, but extends to a collection of pieces, such as wheels,
MECHANICS OF SOLIDS.
161
kinds of
machinery;
bars, levers, &c., connected with each other after the man- Applies to all
ner of ordinary machinery. If the quantity of work
performed by each piece be computed, and the algebraic
sum be taken, it will be found to be equal to the quantity
of work performed by the weight of the whole system,
acting at its centre of gravity, computed by the same rule.
to In general, let p, p', p", &c., be the weights of the
several pieces connected together; h, h', h", &c., the verti-
cal distances passed over by their respective centres of
gravity, in passing from one position to another, by virtue
of their connection; P, the sum of all the weights or the
weight of the entire system; and 2, the vertical space
described by the common centre of gravity: then will

P = ph + p'h' + p" h" + &c. . . . (44).
To demonstrate this, let m, m', m", &c., be several
bodies so connected as to be acted upon by each other's
weights. Let P de-
note the weight of
the entire system;
P, p', p", &c., the
weights of the sever-
al bodies m, m', m",
&c.; Z, the distance
of the common cen-
A-
m
Fig. 82.
Ө
772
A
m
tre of gravity from a vead to malaye adli 11-001 ?
B
dadt roitibugo done ni ban
horizontal plane AB;
and H, H', H", &c., the distances of the centres of gravity
of the bodies m, m', m", &c., from the same plane. Then,
from the principle of the centre of gravity, will)
PZ = pH+p'H' + p" H" + &c.;
bus
and for a second position of the system, laitisq od poned
=
mathematical
expression of the
rule;

30 mindthept
albod grand

demonstration of
the rule:
do dos vousob to
PZ, pH, p'H' + p" H" + &c.;
pay
how to
11

162
NATURAL PHILOSOPHY.
conclusion and
rule.
and subtracting the first from the second,
P(Z-Z)=p(H-H)+p' (H-H')+p" (H"-H)+&c.
And supposing the horizontal plane of reference to be
below both positions of the entire system, Z,-Z is the
vertical distance z, through which the common centre of
gravity has ascended or descended, according as Z, is
greater or less than Z; H-H, H'H', H" - H" &c.,
are the corresponding distances h, h', h", through which
the centres of gravity of the bodies m, m', m", &c., have
ascended or descended. Moreover, the products P(Z,-Z),
p (H-H), p' (H-H'), &c., are the quantities of work
due to the entire weight and to the partial weights.
Whence this rule, viz.: The total quantity of work due to the
action of the entire weight of any system, is equal to the sum
of the quantities of work of the weights which ascend, dimin-
ished by the sum of the quantities of work of the weights which
descend.

Equilibrium of
heavy bodies;
VIII.

EQUILIBRIUM OF A SYSTEM OF HEAVY BODIES.
§ 150.-If the system of heavy bodies be so connected,
and in such condition that the common centre of gravity
continue on the same horizontal line, while the bodies are
made to take different positions, then will Z,- Z=2=0,
and Eq. (44) becomes,
ph+p'h'+p" h" + &c. = 0.. (45);
partial quantities hence, the partial quantities of work of the several bodies
of work destroy destroy each other, and, therefore, there must be an equi-
librium in the system, and the least extraneous effort
each other;

MECHANICS OF SOLIDS.
163
give motion.
will impart motion. Such is the condition of equilib-
rium of a system of bodies acted upon only by their
own weights. The equilibrium presents itself under dif-
ferent states according to the positions of the system. If the slightest
the position be such that in a slight derangement the effort sufficient to
common centre of gravity descend, it will tend to descend
more and more, and a certain quantity of work will be
requisite to restore it to its primitive position. Such an
equilibrium is said to be unstable, because the system Unstable
tends of itself, on slight derangement, to depart from it. equilibrium;
On the contrary, if on slightly displacing the system, the
common centre of gravity ascend, this displacement will
require the expenditure of a certain quantity of work
which the weight of the system tends to restore; the
equilibrium is then said to be stable, because the system is stable
urged by its own weight to return to its primitive state
when abandoned or left to itself. Finally, if during a
slight derangement, the centre of gravity neither ascend
nor descend, the quantity of work expended by the sys-
tem is always nothing, the system will have no tendency
of itself to return to, or depart from its first position, and equilibrium of
the equilibrium is said to be indifferent.

§ 151.-Take a rod sus-
pended at one end so as to
turn freely about a hori-
zontal axis A, and support-
ing at the other a body
which is symmetrical in
reference to a line drawn
from the axis A to the com-
mon centre of gravity G.
It is obvious that there will
be an equilibrium when the
rod is vertical. It is more-
over stable; for in deflect-
Fig. 83.
ing the system, the centre of gravity will ascend while
equilibrium ;
indifference.

Illustration of
stable
equilibrium;

164
NATURAL PHILOSOPHY.
describing the arc G G', about A as a centre, and a
certain amount of work will be expended which the
weight will restore as soon as the deflecting cause is re-
shtyllat moved. Indeed, the system will, when abandoned, per-
of issue to form a series of oscillations, whose amplitude about the
vertical A G, will diminish continually till it comes to rest.
collum evig
Now suppose the system inverted; if the rod be per-
fectly vertical, the line of direc-
intention of the weight will pass of fin Fig. 84. Indilimpe
do through the point of support A,
illustration of
unstable
equilibrium;
and there is no reason why the
system should move one way
rather than another. It will there-
to
abinet
200
ITO MEGFOO
G
fore be in equilibrio, but the equi-
librium will be unstable; for,
however slight the derangement, glow my
the centre of gravity G will de-
d begu
scend along the circular path
adgila
GG', described about A as a cen-meup 4 boob 201
tre, and a certain amount of work gaidon evevis ei tod
odlings will be requisite to bring it back to its primitive position.
both kinds
illustrated by
B
doda
A
A
When a cone ABC, resting upon its base BC, is
inclined to the position A'B' C, its centre of gravity G
will ascend and describe an are bora laT IGI &
G G', and if, in this inclined po- o bobsbred
Fig. 85.
sition, it be abandoned by the
disturbing force, it will return.
When the cone is placed up-
means of the on its vertex, with its centre of
de gravity directly above that point,
it will also be in equilibrio as it
was when resting on its base, but
the slightest motion will cause the
centre of gravity to descend. The
first position is one of stable, the
second of unstable equilibrium.
cone;
B
It of
C
Fig. 86.
sont
livery lo
G
el An elliptical cylinder placed upon a horizontal plane
MECHANICS OF SOLIDS.
165

is in s
stable or unstable equilibrium, according as the
smaller or longer axis of its elliptical base, ban oldate
is perpendicular to the plane.
ostao od ned
Fig. 87.
802210
D
Ling
A spherical ball upon a horizontal
plane, is an example of equilibrium of
indifference. The centre of gravity re-
Get
maining at the same level however the common ofT
ball may be displaced, provided it pre-ho Fig. 88.
serve its contact with the plane, the quan-
tity of work necessary to displace it will did
always be inappreciable, and the ball will,
in consequence, have no tendency either
to recede from or return to its primitive
position. A perfectly circular cylinder
on a horizontal plane is an example of
the same kind.
also by an
ellipsoid of
revolution;
A
repeated
dow loupo
Fig. 89.tw
equilibrium of
opods indifference
of
C
has exemplified by
the sphere; /
by some varieties
Some varieties of draw-bridges are but collections of
pieces in a state of equilibrium of indifference. And to of draw-bridges;
insure this state, it is only necessary that the common
centre of gravity of the bridge and appendages, shall ell to doititog odt
serve the same level during the motion, in which case,
the system will be in equilibrio in all possible positions.
pre-

Wagons and carriages should, in strictness, require no
work to move them on a horizontal plane, except to over-
come their inertia, and should, therefore, be so constructed
as to preserve their centres of gravity always on the same
level.
At gair
only fitrosited as of
test (Fig. 90.borg
yd oooiq
α ουσία
ster
1500 eo ad
If, during the motion of a
wheel, it is seen sometimes toto
quicken and sometimes to slack-10.nib
en its motion, it is because the
centre of gravity G is out of
the axis of motion A, and, there-
fore, alternately rises and falls
during the rotation. A wheel
whose centre of gravity is out
of the axis of motion, passes
G
To ent
effect of throwing
the centre of
gravity out of the
onion axis of a wheel.
B
0
oldatsin od lliw comm
sibai

166
NATURAL PHILOSOPHY.
The common
balance:
in the course of a single revolution through the conditions
of stable and unstable equilibrium, the former occurring
when the centre of gravity G crosses the vertical line BC,
through the axis A, at the lowest point 0, and the latter
when it crosses the same line at the highest point O', of its
path.
The common balance consists of a horizontal arm
A B, mounted upon a knife-edge D, resting upon the sur-
face of a circular opening made in the end of a vertical
frame C, which is supported by a hook attached to a
fixed point E. The ends of the balance carry basins of
equal weights, one of
which receives a sub-
stance to be weighed,
and the other the stand-
ard weights previously
determined.

The bal-
ance may
may be stable,
unstable, or indifferent,
the position of its according as it tends
centre of gravity; to return to a horizon-
when stable;
unstable:
indifferent:
tal position when de-
flected from it, to over-
turn, or to retain any
position in which it
may be placed. Refer-
ring the entire system
Fig. 91.
E
D
C
A
F
B'

to any horizontal plane A'B', and taking the sum of the
products which result from multiplying the weight of each
piece by the distance of its centre of gravity from this
plane, and dividing this sum by the weight of the entire
balance; the quotient will give the distance of the com-
mon centre of gravity of the moveable part of the appa-
ratus from the plane A'B'. If this distance be less than
FD, the distance of the knife-edge above the plane of
reference, the balance will be stable; if greater, the bal-
ance will be unstable; and if equal to this distance, the

MECHANICS OF SOLIDS.
167
balance will be indifferent. All of which supposes the centre of gravity
common centre of gravity to fall somewhere on the vertical
line FD, passing through the knife-edge.
on the vertical
through the point
of support.
IX.
EQUILIBRIUM OF SEVERAL FORCES, VIRTUAL
VELOCITIES, AND MOTION OF A SOLID BODY.

forces acting
§ 152. To find the conditions of equilibrium of several
forces, P, Q, R, S, &c., applied to different points of a solid Equilibrium of
body, take in the interior of the body three points a, b, c, upon a free body;
and regard these points as the vertices of an invariable
triangle abc; resolve each force into three components
whose directions shall pass through the
application and the vertices a, b, and c.
In this way we
shall be able to replace the given forces by three groups
of components, the directions of each group having a com-
Each of these groups, having a
mon point at a, b, or c.
common point, may be
replaced by a single
resultant, and thus, the
equilibrium of the giv-
en forces be reduced
to that of three forces.
Call the resultant of
the group having the
common point a, X;
that of the group hav-
ing the common point
b, Y; and that of the
given point of

Fig. 92.
S
Z
P
R
Y
X
Q
the given forces
may be replaced
by three groups
of components;
group having the com-
and these by
mon point c, Z. These three forces being in equilibrio, three single
the equilibrium will not be affected by supposing the three
forces;


168
NATURAL PHILOSOPHY.
y to do lines ab, bc, and ca, to become fixed in succession. The
Leotard o line ab being fixed, the forces X and Y, whose directions
talog sdt doonde
oqqus to intersect it, will be destroyed by its resistance, and if the
an equilibrium
requires these
three to act in the
same plane;
to mud
the resultant
zero;
third force Z, does not
act in the plane abc, it
will cause the system
to turn about ab; the
same may be shown of
the forces X and Y.
The forces, X, Y, Z,
must, therefore, act in
the same plane; and
in order that they may
Fig. 92.

S
Z
KOITOR
Y
aroitiboad b
C
X
P
If the resultant be
be in equilibrio, the re- of boil R101
sultant of either two d dot to gostal sat nå solat bod
of them must be equalov udt en atalog sendt hunger han
and directly opposed to the third; that is to say, the
resultant of the three must be zero.
zero, the quantity of work is zero. The quantity of work
of X, Y, or Z, is equal to the algebraic sum of the quan-
tities of work of the group of which it is the resultant, and
thus the sum of the quantities of work of X, Y, and Z,
be replaced by that of the quantities of work of the
forces grouped about a, b, and c. But these last, taken
three by three, give the quantities of work of the proposed
forces P, Q, R, S, &c.; so that the sum of the quantities
of work of the forces X, Y, and Z, is the same as the
algebraic sum of the quantities of work of the forces P, Q,
the forces will be P, S, &c. Whence we conclude, that several forces, acting
upon the different points of a free body, will be in equilibrio,
algebraic sum of when the algebraic sum of the quantities of work of the forces
is equal to zero.

in equilibrio
when the
the quantities of
work is zero.
may

Now suppose the body to
be slightly deranged from its
state of rest, and let A A' be
egated the path described by the
vd wood ban
end quot
To amb
driog gom
Fig. 93.
ed 4
n
P
point of application A, of ed ton fir uninditin po
AO
A
MECHANICS OF SOLIDS.
169
the force P, in an indefinitely short time t. Draw
A'n perpendicular to A P; An will be the space de-
scribed by the point of application in the direction of the
force, and the quantity of work performed by P during
the derangement will be PX An, or Pp, denoting An
by p.
golige sdr doldw

The path A A' is called the virtual velocity of the force P; Virtual velocity,
An=p, the projection of the virtual velocity; and the projection of
product Pp, the virtual moment of the force P. doidy ho and virtual
Denoting by q, r, s, &c., the projections of the virtual
velocities of the forces Q, R, S, &c., the quantities of
work, or the virtual moments of these forces, will be,
respectively, Q, Rr, Ss, &c; and if the system be in
equilibrio, we have, from the rule just demonstrated,ollo
Pp Qq Rr + Ss + &c. = 0. . (46).
dans mon to ON
virtual velocity
moment;
virtual velocities.
This equation is but the mathematical expression of the principle of
principle, known under the name-virtual velocities, which
consists in this, viz.: when several forces are in equilibrio,
the algebraic sum of their virtual moments is equal to zero.
que to alloy
yod boseliget
sosiy guibooong add to moitos out of botoje vhod
d§ 153. Any mechanical device that receives the
action of a force or power at one point, and transmits A machine;
it to another, is called a machine. oidilippe si od oooiq
od Conceive a machine, composed of wheels whose axes
are sustained by supports, and which communicate motion
to each other, either by bolag

teeth, chains, or straps,
on their circumferen-
ces.
R
de now
Suppose a force ojo
or power to be applied
so as to turn the first
wheel; this wheel will
experience a resistance
from the second; this
b
4
201
resistance, in its turn, dow
A
og off
Fig. 94.
BBA Ad
D
becomes, for the second bro estre
C3
W
to composed of
wheels;
Testy ofs to ma

170
NATURAL PHILOSOPHY.
the process by
which the action
of a moter is
transmitted;
wheel, a power which causes it to rotate also; the second
will experience a resistance from the third wheel, which
resistance becomes a
power to give it mo-
tion, and so on to the
end. But each wheel
experiences a reaction
at the points of sup-
eport which keep it in
position, and it is this
reaction that becomes
the means of transmit-
ting the power to the
following wheel; for
points of support,
replaced by
active forces;
if these points were
sum of the virtual
moments for first
piece;
Fig. 94.

R
B RA
D
ና
201
W
RR
W₂
unsupported, the wheels would cease to act upon each
other and the power first applied could not be trans-
mitted.
Now, replace the supports, by the efforts of reaction
which they exert: each piece or wheel will become a free
body subjected to the action of the preceding piece, the
resistance of the following, and the force of reaction by
which we have replaced its point of support; and if the
piece be in equilibrio, the algebraic sum of the virtual
moments of this action, resistance, and reaction, must be
equal to zero.
Represent the power applied to give motion to the
first wheel A by W₁, the resistance of the second wheel
B by R2, and the reaction at the point of support of the
first wheel, by C; the projection of the virtual velocity
of W₁ by w₁, that of Ra by r2, and that of C by c; then
will


W₁w₁ + C₁₁ + R₂₂ = 0;
denoting the resistance of the third wheel D by Rg, the
reaction at the centre of the second wheel by C; and the

MECHANICS OF SOLIDS.
171
projections of their virtual velocities by rg and c₂, respec-
tively,
R22+ C2C₂+ R3 73 = 0;
and thus we may continue throughout the entire com-
bination till we finally arrive at the last wheel, to which
is opposed, as a final resistance, the work to be done. De-
noting this resistance by We, the resistance of the last
wheel to the action of the preceding by Re, the reaction
of the support of the last wheel by Ce; and the projections
of the corresponding virtual velocities by we, re, and ce
respectively, we shall finally have,
Re re+ Ce ce + We we = 0.
sup-
But from the nature of the connection, the points of
port must not move; their virtual velocities, and there-
fore the projections, must be zero. Hence, C₁₁ = 0,
C₂ C₂ = 0, . . . Ce ce = 0, and the preceding equations
become
C2
W₁ w₁ + R₂ r₂ = 0,
same for second;

also for the last;
R2 2+ Rg 13
R₂ T2
=
0,
virtual moments
(47). of points of
=
0,
support, zero;
Re re+We we= 0,
Subtracting the second from the first, and adding the
third, subtracting from this result the fourth and adding
the fifth, and so on to the last, we finally obtain d
W₁ w₁ + We w₂ = 0
(48);
relation of motive
dito esorb force to the final
resistance;
which shows us that the quantity of work of the final
resistance is equal to the quantity of work of the power,
or that no work is lost. In other words, the quantity no work lost.
of work of the forces which tend to turn the system in
one direction is exactly equal to the quantity of work of
those which tend to turn it in the opposite direction.


172
NATURAL PHILOSOPHY.
bavose
Tool; work of
An examination of Eqs. (47), will show that the same
remark is applicable to each piece of the combina-
otion taken separately, and thus starting from the piece
which first receives the action of the force, and pro-
ceeding to that which does the work, and which, on this
power equal to account, is called the
account, is called the tool, we see that the quantity of
work of the power is equal to that of the tool. In a
word, where forces work upon bodies through the medium
of machinery, we must distinguish the powers from re-
sistances, and we shall always find the work of the first
to be equal to that of the second.
that of tool;
the work of
If the bodies press against each other in a way to
beat out sot out produce a change of figure and friction, new resistances
arise which must be taken into account, and the work
of these must be subtracted from that of the forces to
obtain the work of the tool, and hence such resistances
are, in general, a hinderance to the final work to be
accomplished.org sift bas
friction and that
employed to
change figure;
0

If the equilibrium is to be maintained while the
work estimated machine is at rest, then must the quantity of work be
estimated by the aid of a supposed displacement, as in that
avoid inertia; case, the influence of inertia will be avoided.
by a supposed
displacement, to
To alloy to
Fonas Hoggoe
the same in
If the equilibrium is to exist during a uniform motion
of the machine, the quantity of work must be computed
uniform motion; from the actual motion of the points of application, for
then the inertia will again be excluded.
is variable, the
work of inertia
comes into the
account.
eds
But if the equilibrium is to take place during an
when the motion acceleration or retardation of the motion, the inertia of
the pieces will no longer be zero, and must be compre-
hended among the powers and resistances. The con-
ditions of the motion must, however, always be the same;
that is to say, the work of the powers must be equal to
that of the resistances, augmented by the work of inertia
when the motion is accelerated, and diminished by the
same work when the motion is retarded.tort to our
to shows to Witness of faupo gliozzo
8154.-Whenever the forces applied to a body accel-

MECHANICS OF SOLIDS.
173
work of inertia to
the work of all
erate or retard its motion, the inertia of the body is
developed; and by virtue of the principle that action is
equal and contrary to reaction, this inertia must be in Relation of the
equilibrio with the forces; that is, the quantity of work
of inertia will be equal to the sum of the simultaneous other forces;
quantities of work of the forces which urge the body in
one direction, diminished by the quantity of work of
those which urge it in the opposite direction. But we
have seen, 85, that when the body takes, at different
instants of time, two velocities which differ from each
other, the work of inertia is measured by half the differ-
ence of the living forces possessed by the body at these
instants, or by half the living force gained or lost in the
interval, according as the motion has been accelerated or
retarded. Hence, the total work of several forces acting upon the total work of
a body, during any time, is always equal to half of the
living force gained or lost by the body during the same living force lost

time.
several forces,
equal to half the
or gained;
sonol gaivil
Suppose, for example, a projectile whose weight is P, to
leave the point A with an initial velocity V. If its weight
did not act, the body would pursue its primitive recti
lineal path A T. (gow abod bilt sano eidt ni) odo illustration--the
But by virtue of
the weight, which
would act alone
the
in vacuo,
projectile is con-
tinually deflected
from this path,
and will, in conse-
quence, describe add
Fig. 95.
T
B
soned bus
case of a
projectile;

10
loob of Jaspo
od to how it
A
hissjon
curve line ABD; quirom
and we know,
C
D
of fooler
anolitory ow.
to level to somewhil
112, that when a body describes any curve under the
action of its weight alone, the work is equal to the
weight of the body into the difference of level of its two
positions. Thus, in the case before us, while the projectile

174
NATURAL PHILOSOPHY.
of attorn
is passing from A to B, the work expended by its weight
will be PX BC, or Px H, by making BC= H.
Denoting by V',
the velocity of the
the living force at projectile at B, its
two points:
living force at this
point will be
Fig. 95.
y
T
drops
loss or gain of
living force;
P
g
B
B
Y
and at A, it was
P
g
A
and its loss of living force, in passing from A to B,
P
g
D
equal to double
the work of the
force;
the half of which is the quantity of work of the extraneous
forces, (in this case the body's weight,) in the same time,
and hence
P
1/4
(V² — V²) = PH,
g

or
relation of
velocity to
difference of
level;
the pieces
V2 V3 = 2g H... (49).
-
Thus the difference of the squares of the velocities in any
two positions of the projectile, moving in vacuo, is equal
to the difference of level of the two positions, multiplied
by twice the force of gravity. When the projectile
adyjow
arrives at D, then will
olmi vhod
aliotomy H = 0; and V2 V² = 0;

MECHANICS OF SOLIDS.
175
that is to say, the velocity will be equal to what it velocity same on
was at A.
the same level;
descending)
From Eq. (49), it is obvious that while the projectile velocity on
is on the ascending branch of the curve, its velocity di- ascending and
minishes, and while on the descending branch its velocity, branch of curve;
on the contrary, increases.

hee
The description of the trajectory or curve ABD in
vacuo, is obtained by very simple considerations, founded
upon the independence of the motions of the same body, the curve in
and of the action of forces which solicit it in the directions vacuo found;
of these motions, (§ 106 and 107.) The body may be re-
garded as animated by two motions, one horizontal in the
direction A x, the other vertical in the direction A y. The
initial velocities in the directions of these motions are the
components of the
initial velocity V,
computed by the
principle of the
parallelogram of
velocities. After
the body leaves the
point A, it will be
subjected to the
T
Fig. 96.
B
B
21
D
action of no motive force in the horizontal direction; the
horizontal component of its velocity will be constant, and
the spaces described in this direction in equal times will
be equal. Denote the angle x AT by a; the space de-
scribed in the horizontal direction A x by x, and the time
required for its description by t, then will
X= V cos at.
.. (50).
But in the vertical direction, the weight will, during
equal times, diminish the component of the initial ve-
locity, in that direction, by equal degrees; the motion
will be uniformly varied, and the spaces described in the
direction of the vertical A y, in the time t, will be given
by Eq. (12), after substituting V sin a for a, t for T and
space described
horizontally in
the time t;


176
NATURAL PHILOSOPHY.
space in same g for v₁. Hence, denoting the vertical space by y, we get
time in vertical
direction;
of the projectile
at any instant;
-
BBW
y = V sin at gt... (51).
x
the true position The true positions of the projectile, which are but points
of the curve A B D, are given by the intersections of a
vertical and horizontal line drawn at distances from A,
equal to the spaces y = Ay₁, and a = Aa₁, simultaneously
described in these two directions. To find these distances,
it will be sufficient to substitute a given value of t, in
equations (50) and (51).
at es mit
the equation of
the curve
described--a
parabola;
Eliminating t from these same equations, and re-
ducing, we find
Has an bobing
g
out one anody = tan a.x-
(52);
2 V². cos² a
20
range;
angle of
projection;
which is an equation of a parabola. Hence, the curve
described by a body when thrown in a direction oblique
to the horizon, and acted upon alone by its own weight, is
a parabola.
The horizontal distance intercepted between the point
of projection A, and the point D where the projectile
attains the same level, is called the range. The angle
x A T = a, is called the angle of projection.
dua
To find the range, make y = 0, in Eq. (52), and find
the corresponding value of x. Making y = 0, we have
Iliw somit laupo al soit
-ob oosqa odotan a x
emit odt bus vd
Jaupo od
2 V² cos² a
whence
(08)
yd al
3x= 10,
=AD = range;
grity lliw
bi 2 V2 sin a. cos a
X=
g
and representing by h, the height due to the velocity V,
we have
hns T sol
trov odt to no
V2gher. (1) (53);

MECHANICS OF SOLIDS.
177
and denoting the range by R, and recollecting that bba
2 sin a. cos a =
sin 2 a,
we have, finally,
gronds
(54).
the value of the
range;
Fig. 97.
y
bug
This value for
the range will be
a maximum when
a = 45°, in other
words, the great-
est range corres-
ponds to an angle
of projection equal
R2h. sin 2 a
T
T
to 45°.
Since
sin 2 α =
sin 2 (90° - a),
it follows that the same range may be attained by two
angles DAT and DA T', which are complements of
each other.
If in Eq. (54), we make a =
complementary
angles give the
same range;
whence
- 45°, then will
R = 2h,
h = 1R;
V = √ Rg
√Rg..
and this in Eq. (53), will give
(55).
That is to say, if the range corresponding to an angle of
45° be measured, the initial velocity may be readily
found, being equal to the square root of the product of
this range into the force of gravity. Squaring the above
equation, we obtain
greatest range
given by an angle
of projection
equal to 45°;
value of initial
velocity in terms
of greatest range;
V2 = Rg;
12

178
NATURAL PHILOSOPHY.
effective quantity
of action
impressed upon
the projectile;
eprouvette;
and denoting by W, the weight of the projectile, its living
force on leaving the mouth of the piece from which it is
thrown becomes
(40)
W.V² = RW;
g
Hani oved ov
and the effective quantity of action impressed, denoted
by Q,
Q = } RW. 0
(56).
It is from this relation that are obtained the results
of the eprouvette, a small mortar constructed to test the
relative strength of different samples of gunpowder. For
this purpose, a heavy solid ball is projected from it under
an angle of 45°, with small but equal charges of different
kinds of powder, and the relative strength is inferred
goes from the effective quantity of action impressed.
adi evig colyte
For example, suppose equal charges of two different
examples of its samples of powder, give R=1050 feet, and R = 1086
feet; these values substituted successively in Eq. (56)
give
use;
egret sestav
moltbelong to
Q = W.525
these results but
approximations
in air;
Rule.
Q = W. 543;
so that, the weights of the projectiles being the same,
the strengths of the two samples of powder will be to
each other as 525 to 543.
This supposes the motion to take place in vacuo. If
the trajectory be described in the air, the resistance of this
fluid will diminish the velocity of the projectile, the curve
will cease to be a parabola, and the results above will be
but approximations to the truth. But as the resistance to
the motion of the same body in air varies as the square
of the velocity, these approximations may be made as

MECHANICS OF SOLIDS.
OF SOL
179
close as we please by using small charges and very dense these
projectiles.
approximations
made close by
velocities;
Taking the general case, without limitation as regards giving small
the velocity of a body in air, the curve may still be
described, provided we have a table giving, in pounds or general case in
any other unit of weight, the resistances corresponding to
different velocities of different calibres.
which the
projectile is
thrown into the
mut doldwas
size bax
atmospheric
Thus, knowing the initial velocity and its two com- air; to how od?
ponents, find from this table, in pounds, the value of the mods bed
initial resistance, and its horizontal and vertical com-
ponents at the commencement of the motion. Of these
components, one is the motive force in the horizontal, and
the other, added to the weight of the projectile, the tables of
motive force in the vertical direction. With these forces, resistance;
supposed constant during a very short time, compute by
the laws of uniformly varied motion, the loss of velocity
in these two directions during this short interval; subtract
from the primitive components of the initial velocity, the
loss in their respective directions; the remainders will be
new component velocities, of which, find the resultant,
and take from the tables the corresponding resistance.
This new resistance treated in the same manner as that successive steps
due to the initial velocity, will give a third resistance, and by which to
this a fourth, and so on indefinitely. We thus obtain a of the projectile
series of components, forces acting for a short time with at any instant and
constant intensity in the horizontal and vertical direc-
tions; with these compute, by the laws of uniformly
varied motion, the corresponding spaces described in their
respective directions by the projectile. The total spaces
simultaneously described, obtained by adding together
the spaces corresponding to the same number of con-
secutive intervals from the beginning of the motion, will
give the distances, Ax, and Ay₁, which determine the boob si
points of the curve. The actual space described by the
trajectory will be the development of this curve.
obtain the place
the curve
of
novi
imp into ad ed

180
NATURAL PHILOSOPHY.
ed sools shoes
The work of
forces which turn
a body about a
fixed axis
X.
lemosy odt geble
yood a to foolent
MOTION AND EQUILIBRIUM OF A BODY ABOUT
AN AXIS.
§ 155.-The principle demonstrated in § 113, of the
work of forces acting upon a body, may be extended to
any case whatever. Let us now apply it to that of a
body which is free to turn about a fixed axis with which
it is invariably connected.
Fig. 98.
L
R
Conceive a force R, acting upon the point A of a body
free to turn about a fixed axis LM; resolve this force
into two others, the one Q,
parallel to LM, the other P
in a plane perpendicular to
this line, and passing through
the point of application A.
Doing the same with regard
to all the other forces acting
upon the body, the system
will be reduced to two groups
of forces, of which one will
be parallel to the axis, and
the other in planes at right
angles to it. The algebraic
M
A
P
sum of the quantities of work of the components is equal
to that of the resultants. But the work of the first group,
is equal to the product of their resultant, multiplied by
the path described by the body in the direction of this
resultant, that is to say, in the direction of the axis; but as
is reduced to that the body is invariably connected with the axis, it cannot
move in that direction, and the path described by the
point of application of the resultant of the parallel group
is nothing, and therefore the quantity of work is nothing.
Thus, the total quantity of work of the given forces is
of their
components in
planes
perpendicular to
the axis.

MECHANICS
181
MECHANICS OF SOLIDS.
SOLID
reduced to that of their components, in planes perpen-
dicular to the axis, and passing through the points of
application.
Ieups xixs
pred ow dovit
components
perpendicular to
§ 156.-The quantity of work of forces applied to a The work of the
body which can only have a motion of rotation is always,
as we have just seen, reduced to that of their components the axis;
in planes perpendicular to the axis, or, which is the same
thing, to that of the projections of the forces on these
planes. It remains to determine this work.
Fig. 99.
L
Let P be one of these components, A its point of
application upon the body, C the point of the axis in
which it is cut by the perpendicu-
lar plane containing the component
P. Let fall upon PA, the per-
pendicular CD, and recall what
has been demonstrated in § 116,
viz.: that the quantity of work of
a force is always the same wherever
its point of application be taken
upon its line of direction. The
quantity of work of P, estimated
by the path described by the point
D, is the same as that estimated by
M
D
P
the path of A. But the point D describes, in the short
interval of time t, an arc S, of which CD is the radius,
and, hence, the quantity of work of P will be P. S
As all the points of the body are invariably connected
with the axis and with each other, they will describe
simultaneously equal angles, and consequently arcs pro-
portional to their distances from the axis; hence if S
denote the length of arc described at the unit's distance,
and the distance of the point D from the axis, then will
r
S = r S₁₂
and the quantity of work of P becomes 11 Adi
Pr S
fod
od in To
ads to engis
the quantity of
work of a single
component;

182
NATURAL PHILOSOPHY.
and for forces of which P', P", &c.,
are the projections, at distances from
the axis equal to r', r', &c., respec-
tively, we have the quantities of work
how to quitpop
od to show off measured by, ootd
alasnogovo
the same for
other
components;
P'r' S, Pr" S, &c. &c.
Fig. 99.
L
add for minoib
P
Knowing that the total quantity
the effective work
of all the
components;
conclusion;
signs of the
moments.
To limang od
M
of effective work of the given forces,
which we will denote by Q, is equal to
the sum of the work of those which tend to turn the body
in one direction, diminished by the sum of the work of
those which tend to turn it in an opposite direction, we
shall have
QS(Pr+ P'r' + P" " + &c.) (57).
#sod and
But we recognize Pr, as the moment of the component
P in reference to the axis, and the same of P'r',
P"r", &c.; whence, the effective work of the component, and
consequently of the force itself, is equal to the product arising
from multiplying the arc described at the unit's distance from
the axis, into the moment, in reference to the same line, of the
projection of the force on the perpendicular plane; and
Eq. (57) shows that the effective quantity of work of
several forces, applied to turn a body about an axis, is equal
to the arc described at the unit's distance multiplied by the
algebraic sum of the moments of the projections of the forces
on planes perpendicular to the axis.
bue
The sign of the moments of those forces which tend to
turn the body in one direction, must be different from
the sign of those which tend to turn it in an opposite
direction; in other words, if the sign of the first be
positive, that of the latter must be negative.
§ 157.-If the given forces be in equilibrio about the
olgate a to how axis, their total work will be zero, whether the body be

MECHANICS OF SOLIDS
183
at rest or in motion; a condition that can only be fulfilled
by making, in Eq. (57),
Pr+Pr+ P" " + &c. = 0.
LEB
(58);
edi to stemte
a lo sovo) gaivit
ybod
equilibrio inode
that is to say, several forces will be in equilibrio about a Forcus in
fixed axis, when the algebraic sum of the moments, in reference to a
reference to this axis, of the projections of the forces on per- fixed axis.
pendicular planes, is zero.
forces to a motion
§ 158.-When forces are applied to a body to turn Extension of the
it about an axis, the motion of its particles can only take principle of living
place in planes perpendicular to the axis; if the forces be of rotation;
not in equilibrio, the motion will be either accelerated or
retarded, and will give rise to forces of inertia which act
in the direction of the motion, and of which the quantity
of work will be equal to that developed in the same time
by the motive forces. When all the points of the body
have simultaneously the same velocity, the total quantity
of work of inertia is equal to the product arising from
multiplying half the mass into the difference of the
squares of the common velocity at the beginning and
end of the interval for which the estimate is made. But
when the different points have different velocities during
the same time, which is always the case in a motion.
of rotation, it is necessary to estimate at the beginning
and the end of the interval, the living force of each
element of the body, to take the sum of those at the
beginning, and the sum of those at the end; the difference
of these sums will be the total increment or decrement
of living force during the interval. The half of this
living force being the work of inertia, and this latter
being equal to that developed by the motive forces, or in rotation, the
rather by their projections on planes perpendicular to the work of the
axis, it is easy to perceive that in the motion of rotation components, is
of a body, the work of the perpendicular components of
the forces is half of the increment of the living force living force.g
of the body. The process of estimating the living force dialer alsforda
alugan adt
boley
perpendicular
half the
increment of le
tuga

184
"NATURAL PHILOSOPHY.
Estimate of the
of a body having a motion of rotation will now be
given.
§ 159.-Consider an element m of a body, situated at a
living force of a distance r from an axis of rotation L M. Denote by V the
body turning
about a fixed
axis;
velocity which it has at any in-
stant, and by p its weight, m its
mass=
P. Then will its living
P.
the model force be 2. V² or m V².
g
If S denote the space described
by m during a very short interval
of time t, and S the space de-
scribed in the same time by a point
at the unit's distance from the axis,
we shall have
S = r. S₁₂
and dividing both members by t,
ՉՆ
Fig. 100.
I
7
M
S
t
S
= r.
t
(59);
the angular
velocity;
relation of
angular to
but we have seen that, in any motion whatever, the
velocity is equal to the space described, during a very
short interval of time, divided by this interval, hence
S
t
S₁
t
=
=V,
=
V₁;
in which V₁ is the velocity of the point at the unit's dis-
tance from the axis-in other words, the angular velocity;
and Eq. (59) becomes
absolute velocity; o gaivil
gothy = r. V₁,

MECHANICS OF SOLIDS.
185
and the living force of m becomes m² V. The simul-
taneous living force of m', is m' V₁, and so on of
others; and the total living force of the entire body,
denoted by L, becomes
value of the
Sego
L = V₁² (mr² + m' r'² + m'' r'² + &c.) . . (60). living force of a
In which it is to be remarked, that if the living force
changes, the factor V₁ will alone vary, while the factor
(m r² + m² r²² + m" + &c.) will remain constant, and
of course, appear in the estimate of the new living force.
This quantity, which has been called the moment of inertia,
let us designate by I, and we have
rotating body;
I = mr² + m²r" + m² 2
+ &c. .
.
(60)'
(60)";
L = VI. .
whence we see, that the living force of a body which turns equal to the
about an axis, is equal to the product of the square of its square of the
angular velocity, multiplied by its moment of inertia.
Let us suppose that at the end of a certain interval,
the angular velocity becomes V, the living force L',
will be
angular velocity
into the moment
of inertia;

L' = VI;
and subtracting the preceding equation from this one,
we get
L' — L = I. (V₁² – V₁³)
(61), increment of
har o living force
during any
for the increment of the living force during this interval, interval;
which is double the quantity of work produced by the
motive forces, or their perpendicular components, during
the same interval. Denote by F, the resultant of these
components, and by E, the path described by its point

186
NATURAL PHILOSOPHY.
equal to twice the
of application, estimated in its own direction during the
interval in question; then will
quantity of work thod situs ada
of the motive
forces in same
time.
I. (VV) = 2 F. E.
(62).
From this expression it is easy to deduce the nature
of the quantity I. For if we suppose the change in the
angular velocity to give
entral
vojost oft slidy wy
then will
1
2
V₁ - V₁ = 1,
aiguado
been aid
the moment of
inertia;
its measure.
I = 2 F.E;
What is meant by whence we conclude, that the moment of inertia of any body,
is twice the quantity of work exerted by its inertia, during a
change in the square of its angular velocity equal to unity.
It is measured by the sum of the products which arise
from multiplying each elementary mass into the square
od of lauge of its distance from the axis, Eq. (60)'.
fabroad to
Example
illustrative of
the preceding
principle;
Fig. 101.
§ 160. By the aid of what has just been explained, we
may find the intensity of a motive force which causes a
body to rotate about an axis, when
we know the angular velocity at
any two given instants of time, and
the path described by the point of
application in the interval between
them. And reciprocally, if the
force and the path described by
the point of application be given,
poll we may deduce the angular accel-
eration. Suppose a wheel, for ex-
ample, mounted upon a horizontal
arbor and turned around its axis by od to
a weight P, suspended from a cord
wound around the arbor; required
P
P

MECHANICS OF SOLIDS.
187
the action of a
the angular velocity Vi of the wheel when, moving from wheel turned by
a state of repose, the weight shall have descended through weight;
a vertical height H. Let I denote the moment of inertia
of the wheel, then will the living force acquired be IV,
and we shall have,
I. V₁ = 2P. H;
solibooler
whence
poler talogne odt lo so
odla to often toon
2
=
sila Unouposao bris
2P. Halgod od ta
I
Jink
od to onder a
ed to extorps
elibolov
and consequently
2P.H
V₁ = V
BBROT
afely you molbom odi
value of theiqqa
po angular velocity;
§ 161.-The fly-wheel is a large ring, usually of metal, application to the
of which the circumference is thrown to a considerable fly-wheel;
distance from the arbor upon which it is mounted by
means of radial arms, and is used to collect the work of
a motor when the effort of the latter is greater than
that required to overcome a given resistance, to be given
out again when the resistance becomes greater than the
effort of the motor. It is a kind of store-house in which to
husband work for future use.
Conceive one or more forces to act upon such a wheel
during an interval separating two given instants at which
the angular velocities are V and V. The increment
of the living force of the fly-wheel will be equal to
double the effective quantity of work of the motor, and
we shall have, retaining the notation of § 159,00 novi
doidy cu
or
hoda od ni
I(VV) = 2 FE;
2 F. E
=
;
hoaoggo
Bas boow
2018
od Insomni
ploved leda o
nompros ons
increment of
living force in
any interval;

188
NATURAL PHILOSOPHY.
bond which gives the difference of the squares of the angular
velocities. If the quantity of work developed by the
motor remain the same during the interval, and, by
changing the wheel, the moment of inertia increase, the
fraction
the value of the
difference of the
squares of the
velocities;
2 F. E
I
and consequently the difference of the angular velocities
at the beginning and end of the interval, will be less.
And, as the moment of inertia is in the direct ratio of the
mass into the square of its distance from the axis, it is
the motion may plain that it is always possible so to construct a wheel as
to make its motion approximate to uniformity, even
though the motive force be very great.
be made to
approach
uniformity;
While the motion is accelerated, it is obvious that
the work of the motor will exceed that of the resistance;
the fly-wheel will acquire an increase of living force
which it will retain till, on the contrary, the motion is
use of fly-wheel; retarded, when it will be again given out in aid of the
motor, which now becomes less than the resistance.
exemplified in
the common
saw-mill.
There are certain machines whose tool cannot perform
its work without the fly-wheel. This is strikingly ex-
emplified in the instance of the common saw-mill, in which
it is obvious that the work during the ascent and descent
of the saw is very different;
the work of the motor
exceeds that of the tool or saw during one semi-oscillation,
while the reverse takes place during the other; in the
first case, the saw is merely elevated and the fly-wheel
absorbs living force; in the second, this living force is
given out to aid the motor in overcoming the resistance
opposed to the saw, which, in its descent, sinks into the
wood and is thus made to perform its work.
§ 162.-If the elementary mass m, receive in the short
interval t, the velocity V, and we denote by f its inertia,
we shall have, Eq. (28),

MECHANICS OF SOLIDS.
189
f=
m V
t
Distance from
the axis at which
the resultant
inertia of a
and for any other masses m', m", &c., whose acquired rotating body is
velocities in the same time are V', V",
exerted;
=
m' V'
t
slide vhod
=
m" "
t
&c.
f"
If, moreover, the masses m, m', m', &c., form parts of
a body which has a motion of rotation, their velocities
will be proportional to their respective distances from the
axis. Denoting these distances by r, r', r", &c., and by
V₁ the small degree of velocity impressed upon the point
at the unit's distance from the axis, we shall have
V = r V₁; V' = r' V₁; V" = r" V₁;
which in the above equations give,
f = mr. V;
; f¹ "= m' r'.
V₁;
; f" m" p'
1
value of the
14; &c. partial forces of
But, if this increment of angular velocity V₁, has been
impressed upon the body by a force F, whose direction
perpendicular to the axis, and applied at a distance
from it equal to R, this force is the measure of the inertia
of the body, and will be in equilibrio with all the partial
forces of inertia f, f', f", &c. But these latter act in
directions tangent to the circles described by the masses
m, m', m", &c., about the axis, and hence, § 157,
or
FR-(fr+f' r' + ƒ"r" + &c.) = 0;
FR − 1 (m r² + m' r² + m'p'ª + &c.) = 0;
-
inertia ;
equilibrium of
these with the
motive force
equal to their
resultant;

190
NATURAL PHILOSOPHY.
foldw in size st
ato altroal
but the expression within the brackets is the moment of
inertia I, and therefore
the moment of
Boitopos sody
the inertia
actually exerted;
VITA
F.R= . I
of bite
(63);erv
distinction
whence we see, that the moment of the inertia exerted by a
body while receiving a motion of rotation about an axis, is
equal to its moment of inertia in reference to the same axis,
multiplied into the quotient arising from dividing the small
degree of angular velocity communicated, by the element
of the time during which it is impressed. Notwithstand-
ing the close analogy which exists between the moment of
usually called the the inertia of a body, and what has been called the moment
of inertia, they must not be confounded with each other.
The former is converted into the latter by making
equal to unity.
between this
and what is
moment of
inertia ;
From Eq. (63) we find
Vi
t
value for the
angular velocity;
F. Rxt rode od
V₁ =
(64),
I
how used.
Measures of the
moments of
inertia ;
from which, having given the motive force that im-
presses a motion of rotation upon a body about an axis
perpendicular to its direction, we may find, at each instant
of time, the angular velocity communicated, provided we
can calculate the moment of inertia of the body in
reference to the same axis. And from this, it is possible,
by means of a curve which has for its abscisses the series
of times t, and for its ordinates the velocities V₁ acquired,
to determine all the circumstances of the motion of rota-
tion.
§ 163.-The moment of inertia of a body with refer-
ence to any axis, we have seen, is measured by the sum of
the products which arise from multiplying each elemen-
tary mass into the square of its distance from the axis.

MECHANICS OF SOLIDS.
191
v. those are those in reference
Of all the moments of inertia of the same body,
easiest obtained which refer to axes through the centre of to axes through
gravity. It is, therefore, important to be able to find the easiest obtained;
moment of inertia with
reference to any axis,
by means of that taken
with reference to a par-
allel axis through the
centre of gravity.
Let GH be this
latter axis, LM any
parallel axis, m an ele-
mentary mass of the
body GKH, through
which element conceive
a plane to be passed per-
pendicular to the axes,
and cutting them at the
points a and b. Join
0+1=1
Fig. 102.
G
HO
K
L
b
M
centre of gravity
Jethen ad
that in reference
to any axis, in
terms of the
reference to a
m with a and b, and let fall from m, the perpendicular moment in
me upon ab. Designate mb by r, ma by r, ab by parallel axis
D, and ae by d; we shall have
through centre of
gravity; abiert

p² = r² + D² + 2 Dd,
and multiplying by the mass m,
mr² = mr² + m D2 + 2m Dd;
and for the masses m', m", m'", &c.,
Cotentinos
m' 2/2
livery to
2
=
m² r² + m² D² + 2 m' Dd',
m" 2
bing sdt loaded
m" r,"² + m" D² + 2m" Dd", enta od
&c.,
&c.,
&c.
D, which is the distance between the two axes, remaining
obviously the same in all.

192
NATURAL PHILOSOPHY.
Adding these equations together, and denoting the
A of moment of inertia in reference to the axis G H by I, and
that in reference to L M by I, we find
the sum of all
the partial
moments;
I = I + D² (m + m' + m" + &c.) + 2D (md + m' d'
+m"d" + &c.);
but m+m+m" + &c.
is the entire mass of
Fig. 102.
G
the body, and md+
m'd' + m'd" + &c. is
the sum of the prod-
ucts which result from
multiplying each mass
into its distance from a
plane through the centre
of gravity, which sum
is equal to zero. Hence,
designating the mass by
M, we have
H
K
L
M
resulting value;
conclusion.
of the bodies are
I = I + MD2 . .
. (65);
whence we conclude that, the moment of inertia of a body,
taken with reference to any axis, is equal to the moment of
inertia taken with reference to a parallel axis passing through
the centre of gravity, increased by the product of the entire
mass of the body into the square of the distance from the centre
of gravity to the first axis.
It follows from this theorem, that if the distances
Value, when the of the particles of the body from its centre of gravity
linear dimensions be small in comparison with the distance of this point
from the axis of rotation, we may take, for the moment
of inertia, simply the product of the mass into the square
of the distance of the axis from the centre of gravity.
small, in
comparison with
their distances
from the axis;
Finally, if Eq. (65) be multiplied by the square of

MECHANICS OF SOLIDS.
193
the angular velocity, V₁, with which the body turns about
the axis LM, we shall have
V. I= VI+ M. D' V.
(66);
value of the
living force;

but VI is the living force of the body; VI, is the
living force it would have, if it rotated about a parallel
axis through the centre of gravity with the same angular
velocity V₁; M. D2 V2 is the living force of the same
body supposed concentrated at its centre of gravity.
Whence, the living force of a body which rotates about any
axis, is equal to the living force of the same body concentrated expressed in
at its centre of gravity, augmented by that which it would
possess if it turned, with the same angular velocity, about a
parallel axis through the same centre.
Finally, when the body is so small that I Vi2 may be
neglected in comparison with M. D2 V2, we have simply
V₁I = M. D² V₁²
.
(66)';
words;
value when the
linear dimensions
of the body are
very small as
that is to say, the living force of the body is equal to the
product of its mass into the square, De Vis, of the velocity compared with
of its centre of gravity.
its distance from
the axis.

of bodies in terms
§ 164. Thus far the moment of inertia of a body has
been expressed in terms of its elementary masses. If the Moment of inertia
body be homogeneous and the specific gravity or weight of their linear
of a unit of its volume be denoted by d, its elementary dimensions and
volumes by a, a, a", &c., and masses by m, m', m", &c.,
we shall have
density;
ба
da'
δα"
m=
m'
=
im".
&c.;
g
9
g'
and these in the general expression I, of the moment of
inertia, give
I:
=
g
(a 2² + a' 2¹² + a'' ''² + &c.);
its general value;
13

194
NATURAL PHILOSOPHY.
rule;
ed to siter
moment of inertia
12
112
that is to say, to find the moment of inertia of any homo-
geneous body, find the value of a 2+ ar²² + a" '² + &c.,
and multiply it by the quotient arising from dividing the
specific gravity, or weight of a unit of its volume, by the
force of gravity.
§ 165.-1st. The moment
of a straight bar, of inertia of a straight bar
in reference to a
perpendicular
axis through its
middle;
whose length is a and cross
section b, in reference to an
axis passing through its mid-
dle point A, and perpendicu-
lar to its length, is given by
Fig. 103.
A
δ
I₁ = b. (12 a³),
very nearly.
g
2d. The moment of in-
ertia of a right cylinder
sashodhaving a circular base, with
that of a right
cylinder, in
reference to its
own axis;
respect to an axis through
its centre of gravity, and
coinciding with its axis of
figure, is given by the equation
ச
alhunt to touch
emel al
δ
I
=
Cr4
2
44
Fig. 104.
in which is the radius of the base, c the length of
the cylinder, and the ratio of the circumference to the
diameter of a circle.
3d. The moment of inertia of a circular ring, whose
that of a circular section by a plane through its centre of figure is rectangu-
lar, taken with reference to an axis through its centre of
gravity and perpendicular to its plane, gives
ring, in reference
to a
perpendicular
axis through its
centre;
b2
I₁ = 2 = r ab (r² + ) x 7
X

MECHANICS OF SOLIDS.
195
in which r is the mean radius,
or that of a circle whose cir-
cumference is midway be-
tween the inner and outer
surface of the ring, a the
thickness parallel to the axis,
and b the thickness in the
direction of the radius.
Fig. 105.
Bo doidy ni
eto jedi
afghs butsoust
4th. That of a spherical
segment taken in reference to
a diameter passing through
its centre of gravity, or mid-
dle, gives
odro hundovitogenesend
Fig. 106.
gill
alto that of a spherical
segment, in
reference to its
versed sine;
To ad
binegil
-
I₁ = 𠃳 (} r² — fr + ioƒ³) ×
in which ƒ denotes the versed sine of the segment, and r
the radius of the sphere; and for the entire sphere,
I₁ = 15 π 75 ×
5th. That of a right cone
having a circular base, taken
with reference to the axis of
figure gives,
1₁ = 10
a p4
X
δ
Fig. 107.
of a sphere, in
reference to a
diameter;
that of a cone, in
reference to its
axis;

196
NATURAL PHILOSOPHY.
that of a
and that of a truncated right
truncated right
cone;
cone, having circular bases,
that of an
ellipsoid;
T 2.5
=
a
10
215
X
g
Fig. 108.
7°
in which and r' are the radii of the greater and smaller
bases respectively, and a the altitude.
Fig. 109.
6th. The moment of in-
ertia of an ellipsoid is given
α
by
that of a
rectangular
parallelopipedon;
the same for a
different axis;
δ
I₁ = is abc (b² + c²) x
= =-=;
in which a, b, and c denote the three axes, and the moment
being taken with reference to the axis a.
7th. That of a rectangular
parallelopipedon, of which the
three contiguous edges are a,
b, and c, taken with reference
to an axis passing through its
centre of gravity G, and par-
allel to the edge a,
I₁ = 1½ abc (a² + b²) ×
Fig. 110.
G
6010
The same taken in reference to an axis through the middle
of the face ab and parallel to a,
I = 1₂ abc (b² + 40²). -_-_-
g

MECHANICS OF SOLIDS.
197
8th. That of a right prism
having a trapezoidal base of
which the greater and less
parallel sides are respectively
b and b' and distance between
them c, the altitude of the
200
Fig. 111.
b'
that of a right
a
prism being a, and the moment taken with reference to
an axis through the middle of the side b, and parallel
to the altitude a,
b'
I = ac (b + b²). (b² + b²
2
9th. If the trapezoidal
base of the above prism
be replaced by a segment
of a parabola, of which
c is the length of the
transverse axis, and b
that of the chord per-
24
+
C b+3b'
6b+b'
+
X
Fig. 112.
α
b
g
pendicular to it, and which terminates the parabola, the
moment of inertia, with reference to an axis parallel to
the altitude and passing through the middle of b, is
given by
I = 3abc (3.5.19 +160)
70
δ
X
prism with
trapezoidal base;

the same when
the base becomes
a segment of a
parabola.

§ 166. We shall close this subject with an example Application to
for the sake of illustration, and we shall first take that examples;
of a trip-hammer, whose
weight is P, mounted up-
on a handle in the shape
of a rectangular paral-
lelopipedon which turns
freely about an axis 0, at
right angles to its length.
Denote by R, the distance
B
Fig. 113.
G
of the centre of gravity of the head B from the axis 0.
that of a common
trip-hammer;

198
NATURAL PHILOSOPHY.
If the linear dimensions
maing 3d Fig. 113.
dahalo tedi of the head be small com-
dilw msing
dtable pared with this distance,
its moment of inertia will
the moment of
inertia of the
head;
not differ much from
P
of sox R2,
follamy 9
B
04
gaived
ban & hay d
le od pod
aried meing
and that of its handle is given in reference to an axis
through its centre of gravity by the 7th case, or
its centro of g
that of the handle
with reference to
its centre of
gravity;
δ
abc (a² + b²).
9
to ound
nawet and denoting by K, the distance of the centre of gravity
of the handle from the axis, its moment of inertia, with
todeng reference to the axis 0, becomes, Eq. (65),
10 tomges a
abc
od aloded
X
g
12
(b² + a²) + — — a b c K³,
-
g
of Jefferey
30
or
with reference to
the axis;
P² (x² + ²² + α²);
P'
g
K2
12
d novig
since abc = P', the weight of the handle. The total
not moment of inertia is, therefore, given by
the moment of
the entire
hammer;
P
I =
R2+
Pr' (K² + ¹² + ²).
b2+
12
g
g
qinda
The process for finding the moment of inertia of the
fly-wheel is much simplified by the fact that all its parts
moment of inertia are nearly at the same distance from the axis. Thus, by
of the fly-wheel; calling R the mean radius of the wheel, we may take
P R2 for mr²+m' r²+m" r2+ &c.; and hence,
9
Bizn odr
& bod

MECHANICS OF SOLIDS.
199
Intog
bolh
L₁ = 1/
P
R2;
g
and denoting the angular velocity of the wheel by V₁,
its living force will be, § 159,
its value;
P
X R². Vi².
g
does
living force of
the fly-wheel;
To evibes bas
To find the angular velocity of the wheel, count the num-
ber of its revolutions in a given time, multiply this
number by 2, and divide the product by the number
of seconds in the given time; the quotient will be the angular velocity
angular velocity. Let V equal 9 feet; the weight P
of the wheel 2000 pounds, and the mean radius R, 6 feet;
omitting the fraction in the value of g, the expression for
the moment of inertia becomes
found to eat a
experimentally;
=
2000
X 36
32
11
2250;
and for the living force,
2250 X 81 182,250;
example.
ley you oved
the half of which, or 91,125 pounds, raised through one
foot, is the quantity of work absorbed by the inertia of
the wheel, to be given out when the moter ceases to act.
§ 167.-Resuming Eq. (60)', we may make
mp² + m² p¹² + m" pl² + &c.
=
MK2
gated Centre and radius
pidada di goal of gyration;
of log box
in which M is the entire mass of the body, and motorat al ser
dasash of ti
K = ±V
√m
mr² + m' r'² + m" p'²
2
2
+ &c.
M
og sid
90 9]
godom bonianta
boulente
But this is equivalent to concentrating the entire mass into
a single point whose distance from the axis is K, without

200
NATURAL PHILOSOPHY.
definition;
changing the value of the moment of inertia. This point is
called the centre of gyration, and the distance K, is called
the radius of gyration. As the moment of inertia varies
with the position of the axis, there will be an infinite
number of centres and radii of gyration, or as many
of each as there are possible positions for the axis. When
the axis passes through the centre of gravity, they are
principal centre called the principal centre and radius of gyration.
and radius of
gyration;
moment of inertia
in terms of radius
of gyration.
Denoting the principal radius of gyration by K', we
may write MK2 for I, in Eq. (65), and we have
I = MK² + MD2
(66)".
Central forces;
XI.
CENTRAL FORCES.
§ 168.-Conceive a body, whose weight is P, attached
to a fixed point C by a rigid bar A C, and suppose it to
have any velocity whatever in the direction AT, perpen-
dicular to the bar. If
the body were free, it
would, in virtue of its
Fig. 114.
inertia, move in the di-
M T
rection A T with a con-
stant velocity. But not
a body made to being free, the bar will
revolve about a
fixed point by
means of a bar;
keep it at the same dis-
tance from C and cause
it to describe the circum-
ference of a circle about
C
this point as a centre. There are, then, during this con-
strained motion of the body, two central efforts exerted in
the direction of the bar, the one by the bar to draw the
body from the tangential path A T, the other by the body

MECHANICS OF SOLIDS.
201
to stretch the bar out to that path. These forces are equal
and directly opposed, because action and reaction are
always equal and contrary. The first, or that which
tends to draw the body within the tangent, is called
the centripetal force, and the second, or that which tends centripetal force;
to stretch the bar, the centrifugal force. The centrifu- centrifugal force;
gal force is, then, the resistance which the inertia of a body in centrifugal force
motion opposes to whatever deflects it from its rectilinear arises from
path.
inertia;
We will first suppose that the dimensions of the body
are so small as compared with its distance from the fixed
centre, that it may be regarded as a material point,
animated with a velocity V. For the circle which it
describes, we may substitute a regular polygon ABCDE, substitution of a
of a great number of very small sides and having its polygon for the
angles in the circumference. This being supposed, it is
first to be shown that the material point will describe each
of the sides of this polygon with the same velocity, or
that there will be no loss of velocity in passing from one
side to another.
For this purpose, we remark, that if the body possess
W
Fig. 115.
circle;

E
D
B
-C
C
A
to prove there is
no loss of velocity
from the reaction
of the curve;
the velocity V at the moment of its arriving at B, the
beginning of the side BC, it will be animated, while

202
NATURAL PHILOSOPHY.
describing this side, with two simultaneous velocities.
One of these is the primitive velocity V=B W, in
the prolongation of AB; the other BU, in the direc-
tion BO, is a velocity due to the action of the centripetal
the body has two force while the body is passing from the side A B to the
side BC of the polygon. But we have seen, § 106, that
when a body receives two simultaneous velocities in differ-
ent directions, its resultant velocity will be the same as if it
simultaneous
velocities;
Bod
bazit aut mont
Jalog Inftstour
it doidy obalo sift
will gaived the exte
el si bocogique D
does odhof w
E
Fig. 115.
W
B
C
-C
A
vode nd of al
on of life ough th
the resultant of
which has the
direction of the
side next to be
described;
no loss of
velocity;
possessed them successively, and as though they were
communicated one after the other in their respective direc-
tions. Thus the resultant velocity BC', with which the
side BC is described, coincides in direction with this
side, otherwise the body would take some other path,
which is contrary to the hypothesis. BU and WC' are
equal and parallel, from the parallelogram of velocities.
The radius OB divides the angle ABC into the two
equal angles A BO and OBC; the angle ABO is equal
to the angle B WC', and the angle OBC is equal to
the angle BC'W; hence the angles B C'W and B WC'
are equal, and the side BC' is equal to the side BW; in
other words, the resultant velocity BC', with which the
side BC is described, is equal to the velocity BW which
the body had at the end of the side A B. Whence it

MECHANICS OF SOLIDS.
203
altered by the
10
results, that the velocity communicated to a material point is velocity not
not altered in its circular motion; a result easy to foresee, deflecting cause;
since the centripetal force, acting in a direction perpen-
dicular to the direction of the motion, cannot work effi-
ciently; it can neither accelerate nor retard the motion,
and, therefore, can neither increase nor diminish the living
force of the material point. gaivik odr
症
​Now observe, that BU= WC', is the velocity genera-
ted by the centripetal force in its own direction during
the time the material point is passing to the side B C.
Denote this time by t, and the centrifugal, which is equal
though opposed to the centripetal force, by F, and the
mass of the point by M, then will the value of F, be given,
Eq. (39), by the equation
of of longs
authes td babirib
FM.
W C'
t
=
centrifugal force;
Draw the radius CO; the triangles BOC and BW C' to find value of
are similar, because the angle OCB OBC is equal
to the angle BC' W, and the angle OBC is equal to the
angle B WC'. Hence we have the proportion,
etbod odr
BO BO:: BW: WC'; 001 to
or, denoting the radius BO by R, and replacing B W by
its equal V
R: BC: V: WC';
Ꮃ Ꭴ
whence
al bosgro
Atooler safegue
ads BCX V
0018
WC
=
R
da guitator
and this, in the value for F, gives
lusionsorg
a ol nohastel
BC
V
to chod
oit do
F= M.
t
R

204
NATURAL PHILOSPHY.
Bohol but BC, the element of the space, divided by t, the element
of the time, is equal to the velocity V, whence
the measure of
the centrifugal
force;
F=
M. V2
R
(67).
equal to the
living force
impressed,
divided by radius
of curvature;
The
Such is the expression for the centrifugal force.
numerator is the living force of the body, and the de-
nominator is the radius of the circular arc which the body
is describing for the instant; whence we conclude, that the
centrifugal force of a body of small dimensions, as compared
with its distance from the centre about which it revolves, is
equal to the living force impressed upon the body, divided by
the radius of the circle described by its centre of gravity.
Suppose, for example, that the weight of the body is
100 pounds, that its centre describes a circle whose radius
is 3 feet, with a velocity of 12 feet.
V = 12; V² = 144; R = 3;
M
=
Be32
100
;
illustration;
100 X 144
F=
= 150
150 pounds;
32 X 3
expressed in
terms of the
angular velocity.
Extension to a
body of any
dimensions;
the body, therefore, tends to stretch the bar with an effort
of 150 pounds.
Denote by V₁ the angular velocity; then will
V² = V₁ R²,
and this, in Eq. (67), gives
F = MVR..
§ 169.-Let us next take the
case of a thin layer of matter
DAB, rotating about an axis 0,
perpendicular to its plane, with
an angular velocity V₁. Taking
any one of the elements of the
layer whose mass is m, and de-
B
(68).
Fig. 116.
m
PP
D

MECHANICS OF SOLIDS.
205
noting its distance from the axis O, by r, its living force
will be
and its centrifugal force,
m
2
mr V₁2
= mr
mr Vi;
2
which will act in the direction Om of the radius of the
circle described by m about the centre O. Through the
point O, draw in the plane of the layer, any two rectan-
gular axes, as Ox and Oy. Resolve the centrifugal force
into two components acting in the direction of these axes;
these components and their resultant will be proportional
to the sides and diagonal of the rectangle p 0 qm, and we
shall have, denoting Op by x, Oq by y, the component
parallel to the axis Ox by X, and that parallel to Oy
by Y,
T: x :: mr V: X,
ry:: rv: Y;
mr
whence,
X = mx V₁₂²
Y = my V₁²;
and for any number of small masses m', m", &c., by using
the same notation with accents,
X' = m' x' V₁₂²,
X"
=
m" x" V₁³,
&c.
=
&c.;
centrifugal force
of an element;

resolved into
rectangular
components;

components
parallel to the
axis x, for other
elements;

206
NATURAL PHILOSOPHY.
sendt gaivil
the odd mo
weils at quitou
Y' =
w
components
parallel to the
axis y ;
Y" = m" y" V₁,
&c. =
&c.;
by this process all the centrifugal forces have been re-
reduced to two groups of forces acting upon the point 0,
in the direction of the axes Ox and Oy, and from the
principle of parallel forces, each group will have for its
resultant, denoted by X and Y, respectively,
₁ = V₁² (mx + m' x' + m'' x' + &c.),
m" x" in
Y₁ = V₁² (my + m'y' + m" y" + &c.);
resultants
parallel to the
axes x and y;
measure of the
layer's
centrifugal
force;
that is,
X₁ = VM'x
followy
Y₁ = V₁₁ M'y₁;
in which M' denotes the entire mass of the layer, and
x, and y, the co-ordinates OP and OQ of its centre of
gravity G.
The resultant of the forces X and Y is the entire
centrifugal force of the layer; and this denoted by Fu
is, from the principle of the parallelogram of forces,
2
F₁ = √ V₁ M² y² + V₁ M²x² = V M' √x² + y²;
and making
√x² + y² = OG = r,
F₁ = M'r, Vi;
whence, the centrifugal force of a thin layer of matter
revolving about an axis perpendicular to its plane, is equal
to the square of its angular velocity, multiplied by the product

MECHANICS OF SOLIDS.
207
of its mass into the distance of its centre of gravity from the
axis of rotation. This force is applied to the centre of
gravity, since it acts in the direction O G.
Ingehdass od
Now suppose any body, as ABC, to turn around the centrifugal force
axis LM. Divide the body into thin layers whose planes
are perpendicular to the
axis. These layers will g
give rise to as many cen-
trifugal forces acting at
their centres of gravity,
G, G', G", &c. All
these forces are perpen-
dicular to the axis LM,
without being parallel to
each other. Sometimes
they have a single re-
sultant, sometimes they
will reduce to two forces,
and sometimes they will
reduce to nothing, de-
pending upon the form
and density of the body,
Fig. 117.
C
I
G
0
G
Ꭴ
G
ON
of a body of any
size;
G
0'
G
ON
may reduce to a
GY-
Or
single force, two
forces, or to zero;
OV
B
M
G
pressure on axis;
and the position of the axis. In the last case, viz.: that in the last case no
in which the forces reduce to nothing, there will be no
pressure upon the axis.
If the centres of gravity G, G', G", &c., be all on the
same straight line parallel to LM, the centrifugal forces
will be parallel, will act in the same plane, at the same
distance R from the axis of rotation, and their resultant,
which becomes equal to their sum, will pass through the
centre of gravity of the entire mass, and we shall have
F=
and making
VR (M' + M" + M"" + &c.);
M'+M" + M"" + &c. = M, Janonail
F = V2 R. M.
the centres of
gravity of the
layers on same
line parallel to
the axis;

208
NATURAL PHILOSOPHY.
the centrifugal
force the same
as though the
body were
reduced to centre
of gravity;
examples.
Illustration of the
action of the
centrifugal force;
the horse
travelling in a
circle;
that is to say, the centrifugal force of a body, whose sec-
tions perpendicular to the axis, have their centres of
gravity in a straight line parallel to
the axis, is the same as though the
entire mass were concentrated at the
common centre of gravity. This sim-
plification is peculiar to the sphere,
the cylinder, and surfaces of revolu-
tion generally whose axes of figure
are parallel to the axis of rotation.
Fig. 118.
Fig. 119.
I
M
§ 170.-The centrifugal force accounts for a multitude
of interesting facts. When a horse is made to travel in
the circumference of a circle,
his centrifugal force will
vary as his mass and the
square of his velocity; when
the latter is doubled, his cen-
trifugal force is quadrupled;
when trebled, it is made
nine times as great, &c., so
that it would soon become
sufficient to overturn him
or to cause him to recede
P
from the centre C. It is to resist this effort that horses,
under these circumstances, are seen to incline their bodies
inward, and this inclination is determined by that of the
resultant of his centrifugal force and weight, as the line
of direction of this resultant must pierce the plane
of his path somewhere within the polygon formed by
joining his feet.
If, then, we lay off upon the vertical and horizontal
drawn through his centre of gravity G, the distances
his inclination; GP and GF, to represent his weight and centrifugal
force respectively, and construct the rectangle PGFR,
the diagonal GR will give the inclination sought. Deno-
ting the weight of the horse by P, his distance from the

MECHANICS OF SOLIDS.
209
centre by R, and his actual velocity by V, we have
F
D V²
P
11
g
Ri
and consequently the pressure or
his oblique
pressure on the
ground;
GR = P√√1
V4
1 +
g2 R2
Finally, in order that the horse may not slip, the surface surface of his
BA of his path, must be perpendicular to G R.
When a horseman rapidly turns a corner, he leans
path;
his body towards the centre of the curve which he is a horseman
describing, to bring the resultant of his weight and turning a corner:
centrifugal force to pass between his points of support
in the stirrups.
Fig. 120.
G
When a wagon makes a quick turn, its centrifugal force
tends to overthrow it towards the convex side of the curve
it describes; and the risk of
upsetting is directly propor-
tional to its weight and the
square of its velocity, and
inversely proportional to the
radius of the curve.
is why the exterior of the
roadway is usually elevated
in short turnings, and car-
riages diminish their speed
when approaching them.
This
The sling, the axe, the
sabre, &c., exert upon the
hand, when we give them
a circular motion, a traction
equal to the centrifugal force.
The common wheel is usually
composed of fellies A, A, &c.,
connected with the nave N,
by means of radial arms, 1, 1,
A
A
P
Fig. 121.
A
A
R
A
A
a wagon making
a short and rapid
turn;
inclination of
roadway;
other examples-
the sling, axe,
sabre, &c.;
common
carriage-wheel
14

210
NATURAL PHILOSOPHY.
action upon the
fellies of the
common wheel.
&c., and the centrifugal force is constantly acting when
the wheel is in motion to draw these arms from their
places, to enlarge the circumference, and thus to detach
the fellies from each other; hence the tire not only pro-
tects the wheel from the wear and tear arising from the
roughness of the road, but also counteracts the effect of
the centrifugal force.
§ 171.-We know that the earth revolves about its
axis A A' once in twenty-four hours, and that the cir-
Centrifugal force cumferences of the parallels
at earth's
surface;
of latitude have velocities
which diminish from the
equator to the poles; the
Fig. 122.
centrifugal force will hence
P
R
S
cons
biges be rude e
diminish. To find the laws
of this diminution, let P be
guide the weight of a body on the
surface of the earth in any
that of a body
whose weight is
P;
elgato soto
Fides
nomics
parallel of which R' is the
radius, its centrifugal force
will, Eq. (68), be
P
g
.
E
-Q
Q
R
V2 R';
A
in which V₁ is the angular velocity of the earth. Sub-
stituting M for
P
g'
we have
F=MV2 R'.
Denoting the equatorial radius CE = CP, by R, and
the angle CPC' PCE, which is the latitude of the
place, by o, we have in the triangle PCC',
R'
=
R cos;

MECHANICS OF SOLIDS.
211
which substituted for R' above gives
FMVR cos.
. (69).
of centrifugal
force;
Now, the only variable quantity in this expression, law of variation
when the same mass is taken from one latitude to another,
is q; whence we conclude that the centrifugal force varies as
the cosine of the latitude.
The centrifugal force is exerted in the direction of the
radius R' of the parallel of latitude, and therefore in a
direction oblique to
the horizon TT'.
Lay off on the pro-
longation of this ra-
dius, the distance
PH, to represent
this force, and re-
solve it into two
components PN
and PT, the one
normal, the other
tangent to the sur-
H
Fig. 123.
T'
N
E
R'
C
the centrifugal
force resolved
into a vertical
and horizontal
component;
A
face of the earth;
the first will dimin-
ish the weight P by its entire value, being directly opposed
to the force of gravity, the second will tend to urge the
body towards the equator.
The angle HPN is equal to the angle PCE, which
is the latitude, denoted by ; whence the normal com-
ponent
and
PN= PHX cos = F. cos q = MVR cos 20,
PT PH sin = F. sin o = MV2 R. sin o cos ;
=
value of vertical
component;
horizontal
component;

212
NATURAL PHILOSOPHY.
but
sin q. cos = sin 29;
its value;
therefore
PT = MVR sin 2 o
whence we conclude, that the diminution of the weights of
bodies arising from the centrifugal force at the earth's surface,
varies as the square of the cosine of the latitude; and that all
bodies are, in conse-
quence of the cen-
effect upon the trifugal force, urged
weights of bodies
and figure of the
earth:
towards the equator
by a force which
varies as the sine of
twice the latitude.
At the equator
and poles this latter
force is zero, and at
the latitude of 45°
it is a maximum,
and equal to half
Fig. 123.
T'
R'
H
A
of the entire centrifugal force at the equator.
C
At the equator the diminution of the force of gravity
is a maximum, and equal to the entire centrifugal force;
at the poles it is zero. The earth is not perfectly
spherical, and all observations agree in demonstrating
that it is protuberant at the equator and flattened at the
poles, the difference between the equatorial and polar
diameters being about twenty-six English miles. If we
suppose the earth to have been at one time in a state of
present figure of fluidity, or even approaching to it, its present figure is
readily accounted for by the foregoing considerations.
cause of the
the earth;
The weight of a body which varies, according to the
Newtonian hypothesis, directly as the mass and inversely
as the square of the distance from the centre of the earth,

MECHANICS OF SOLIDS.
213
is, therefore, on account of a difference of distance and of weight of the
the centrifugal force of the earth combined, less at the same body
equator than at the poles.
To find the value of the centrifugal force at the
equator, make, in Eq. (69), M = 1 and cos o = 1, which
is equivalent to supposing a unit of mass on the equator,
and we have
F = V2 R.
The angular velocity is equal to the absolute velocity,
divided by the equatorial radius of the earth. The abso-
lute velocity is equal to the circumference of the equator
in feet, divided by the number of solar seconds in one
siderial day:
Diameter of earth in miles 7925.....
greatest at the
poles and least at
the equator;
centrifugal force
at the equator;
T
Feet in one mile..
.3.1416...
.5280...
Circumference of earth in feet.....
..Log. 3.8989993
.Log. 0.4971507
.Log. 3.7226340
Log. 8.1187840
4.9353259
Length of a sid. day in Sol: seconds, 86400X0.997269, Log.
Absolute velocity in feet.
Radius of earth in feet
Angular velocity V₁.....
2
Square of angular velocity V
Radius of earth in feet....
.......
1
f
Centrifugal force at equator..0.1112.
.....
.Log. 3.1834581 computed;
.Log. 7.3206032
Log. 5.8628549
.......Log.
.Log. 1.7257098
.Log. 7.3206032
9.0463130
Thus the value of the centrifugal force at the equator is its value;
0.1112 of one foot.
By the aid of this value, it is very easy to find the
angular velocity with which the earth should rotate, to to find angular
make the centrifugal force of a body at the equator equal velocity sufficient
to its weight; for by the present rate of motion we weights at the
find
to destroy
equator;
f
0.1112 =
V2 R,

214
NATURAL PHILOSOPHY.
and by the new rate of motion.
32.1937 =
VR;
f
in which 32.1937 is the force of gravity at the equator.
Dividing the second by the first, and we find
32.1937
0.1112
=
=
V2
289, nearly;
whence
result;
V₁ = 17 V₁;
that is to say, if the earth were to revolve seventeen times
as fast as it does, bodies would possess no weight at the
the weight of all equator; and the loss of weights at the various latitudes
from the equator to the poles diminishing in the ratio of
the squares of the cosines of latitude, the weights of all
bodies, except at the poles, would be affected.
bodies affected.
Motion in a
circular groove,
when plane of
groove is
horizontal;
§ 172.-If we now sup-
pose the body, instead of
being connected with the
point C by means of a rigid
bar, to move about the same
point in a circular groove,
the effects, as regards the
centrifugal force, will ob-
viously be the same, since
the body will be constrained,
by the resistance of the
Fig. 124.
H
m
m
V
groove, to remain at the same distance from the centre.
If the plane of the groove be horizontal, the pressure of
the body against the side will be constant and equal to the
centrifugal force, that is to say, to

MECHANICS OF SOLIDS.
215
V2
M.
R
eved
the effect of the
body's weight;
But if the plane of the groove be vertical, the weight
of the body will also exert its influence; for the weight
being resolved into two components, one tangent and the
other normal to the curve at the place of the body, the
latter will sometimes act with, and sometimes in opposi- when vertical,
tion to the centrifugal force, while the former will some-
times increase and sometimes diminish the velocity; so
that the pressure becomes greater or less than the cen
trifugal force depending upon these two circumstances.
Knowing one of the velocities which the body may have, from one velocity
it is easy, by the principle of living forces, to find the
others. Take the body at its lowest point m', and denote
its velocity, supposed known, by V', and let it be re-
quired to find its velocity at any other point m, whose
vertical height above m' is H.
latter point by V, then will the loss of living force in
passing from m' to m be

Denote the velocity at this
to find the others;
M V₁2

—
M V²;
and this being equal to double the quantity of action of
the weight denoted by W, in the same interval, which
quantity of work is 2 WH, we have,
M (V² — V²) = 2 WH;
-
ponge in
di dharath
W
replacing M by its equal
and reducing
V² = 2g H,
V = √ √²² - 2 g H.
value of velocity
at any point;
લો

216
NATURAL PHILOSOPHY.
same in terms of
difference of level
of the points;
Denoting by H', the height due to the velocity V', we
have
V² = 2 g H';
which in the above equation gives
-
V = √2g(H' — H).
Thus, the velocity of the
body will be diminished
by the action of its weight
during its ascent, while, on
the contrary, it will be in-
creased during the descent,
being always the same at
points situated on the same
velocity greatest horizontal line.
The ve-
at lowest point; locity will be greatest at the
lowest and least at the high-
least at the
highest.
living force.
est point. During the de-
Fig. 124.
H
m
V
scent, the body will acquire living force by absorbing the
gain and loss of work of its weight, which living force will again be
destroyed during the ascent because it is opposed to the
weight.
of a body which
describes any
curve;
§ 173.-When a body, in vir-
tue of the motive forces which
act upon it, describes a curve in
Centrifugal force space, the effect is the same as
though it passed over the arcs of
the successive osculatory circles
of which the curve is composed.
If the positions of the centres C,
C', C", &c., of these successive
circular arcs be known, as well
as their radii A C, A' C', A" C",
&c., the curve will be given
C"
Fig. 125.
T"
P
T

MECHANICS OF SOLIDS.
217
by the series of arcs AA', A' A", A" A", &c., de-
scribed about these centres, and terminated by these radii.
And it will be easy, from the consideration of the centrifu-
gal and motive forces, to obtain for every point of the
curve, the position of the centres and the magnitudes of the
radii of the osculatory circles, and, consequently, to trace
the path described by the body.
and centrifugal
Let P denote the resultant of the motive forces which to trace the curve
act upon the body at any particular point as A; M the from the motive
mass of the body; V its velocity, of which the direction forces;
is AT; and the radius A C; then will the centrifugal
force be measured by

MV2
r
But the body, in describing the curve, does not abandon
the small are A A', and must therefore be retained on it
by a force equal and directly opposed to the centrifugal
force; in other words, the motive force AP, being re-
solved into two components, one tangent and the other
normal to the curve, this latter must be equal to the
centrifugal force. Denote the normal component by p,
then will

M V2
P =
T
whence
r=
M V2
p
(70);
value of the
normal
component of the
motive force;
radius of
(71).
Such would be the radius of the initial arc A A', provided
the velocity V were constant during its description. This
condition cannot, however, be fulfilled, since the tan-
gential component of the motive force will either increase
or diminish the velocity. It will be sufficient to make
V
1=
nt n'
25
curvature;

218
NATURAL PHILOSOPHY.
terminal velocity
on the initial arc
to be found;
its value;
Fig. 125.
T
A T
in which n and n' denote the velocities of the body at the
beginning and ending of the arc. The former of these
must be given, being the ini-
tial velocity; the latter must
be found, and for this pur-
pose we remark, that as the
arc is described in a very short
time, say the tenth of a second,
the motive force, and therefore
its tangential component, may
be regarded as constant during
this interval. Denoting the
tangential component by q, and
the time by t, we have, from
the laws of uniformly varied
motion, Eq. (11), and (30)'
n' = n + // t₂
M
CI
P
value of mean
velocity;
value of radius;
To welk
value of arc
described;
and
V =
n + n'
2
= n +
9
2 M
t.
(72);
which, in Eq. (71), gives
2
M (n +
9
2 M
r =
p
(73).
This distance being laid off from the point A, upon the
perpendicular to the tangent A T, will give the centre C.
The length of the arc, denoted by s, is found from Eq.
(10), or
s = nt + }
(74).
The law of the motive force being known, the intensity
of its action on the body at A' becomes known, and its

MECHANICS OF SOLIDS.
219
component perpendicular to the tangent A' T", denoted by
p', will give
or
30
p' =
=
Ꮇ ᏙᎸ
p'
M V,2
r' =
;
p'
vigge
value of normal
component of
motive force;
in which is the radius of the arc A' A", and V', the
mean velocity with which it is described.
Denoting the new tangential component by q', we find,
in the same way as before,
n" = n' +
q'
t,
M
V!
=
n" + n'
2
q'
= n' +
t;
2 M
terminal velocity
on second arc;
which in the equation above gives
p' =
M (n² + 2 24 1)²
q'
2
M
p'
and this being laid off, as before, upon the perpendicular
to the tangent A' T", will give the centre C'.
The length of the arc A'A", denoted by s', will be
found from
q'
t2.
M
8 = n² 1 + 2/2/1/1 P.
n't
radius of second
arc;
length of second
arc;
Finding the value of the motive force at A", its nor-
mal and tangential components p" and q", as well as the
mean velocity V", we obtain the value of the radius
C"A", and the position of the centre C"; the tangential the same process
component and time will give us the length of the new
for other arcs;

220
NATURAL PHILOSOPHY.
application to the
shell thrown into
osculatory arc, and thus the description of the curve may
be continued to the end.
To apply this general case to a particular example,
case of a bomb- take the instance of a bomb thrown into the air. The
forces here are, that arising from the explosive action
of the powder and which gives the initial velocity, the
resistance of the air, and the weight of the bomb.
the air;
Let A be the mouth
of the piece, of which the
axis coincides with the
line A T This line will
be tangent to the path de-
scribed by the bomb at
the point A. Denote the
weight of the bomb by W,
the initial velocity by n,
resistance of air; and the resistance of the
air due to this velocity by
H-
T
Fig. 126.
CK
components of
the weight of the
bomb;
f. The value of ƒ may
W
be taken from a table giv-
ing the resistances corre-
sponding to different velocities and calibres. Through A
draw A H parallel to the horizon, and denote the angle
TAH by a; lay off upon the vertical through A, the
distance A W to represent the weight of the bomb, and
resolve this weight into two components: one, Ac=
normal to the tangent A T; and the other, Am=k, in the
direction of this line. The angle WAc is equal to the
angle TAH = a; and hence,
= p,
p = W cos a,
k = W sin a;
and since the resistance of the air is directly opposed to
the motion, the force in the direction of the tangent, after
the initial impulse, is retarding, and becomes

MECHANICS OF SOLIDS.
221
therefore
9
=
-
k+f= (W sin a +f);
tangential
component;
n' = n-
W sin a + f
M
terminal velocity;
and
V = n -
W sin a + f
2 M
+f. t;
mean velocity;
this value and that of p, in Eq. (71), give
W sin a + f
2 M
M (n
2=
W cos a
radius of initial
arc;
;
and writing, in Eq. (74), for q its value, we find
s = nt
-
12
148
W sin a +f. t.
M
Fig. 127.
Through the
point A, draw an
indefinite perpen-
dicular to the line
AT, and lay off
from A the dis
tance A C, equal
to r; with C as a
centre, r as radius,
describe the arc
A A' equal to s.
This will give the
initial arc.
The linear di-
mension of an arc
at the unit's dis-
H
length of initial
arc;

construction;

222
NATURAL PHILOSOPHY.
length of arc at
tance from C, is
unit's distance
S
from the centre;
and denoting the ratio of the circumference of the circle to
its diameter by, we have
its value in arc;
S
2 T:
360°
: 2,
r
2=
360° X s
2πη
it a
in which z denotes
to the number of de-
grees in this arc,
Fig. 128.
T
or the value of the
T
angle A CA'.
But
A
this angle is equal
H
Α
K
angle of the
tangents at the
initial points of
to that made by the
two consecutive tangents A T and
arcs;
angle of
projection at the
beginning of the
second arc;
A'T" at the extrem-
ities of arc A A';
and the angle which
the tangent at the
beginning of the
second arc, A' A"
makes with the ho-
rizon, or the angle
T'A' H', will be
H---
C
a
- 2 = a'.
Pursuing the same operation as before, we find
p' = W cos a',
I'
=
W sin a';
and taking from the tables the resistance f', corresponding

MECHANICS OF SOLIDS.
223
to the new velocity n', we construct in the same way the
second are A' A", &c., &c.
It is to be remarked, that as the angle denoted succes-
sively by a, a', &c., diminishes in passing from are to arc,
it
projection;
will presently become equal to zero, at the summit, and after-
ward take the negative sign; in the first case, the tangential variation in the
component of the weight of the bomb will be zero, its sign angles of
will then change, and instead of being a retarding, it will
become an accelerating force. Hence, in this curve, three three parts of the
portions are to be distinguished, viz.: the ascending branch, curve;
the descending branch, and that immediately about the
summit.
atmosphere and
in vacuo.
The resistance of the atmosphere to the motion of
bodies in it, is found to vary as the square of the velocity
of the moving body, and some idea of the intensity of this
resistance may be formed from the fact, that a twenty-four range in
pound shot, projected under an angle of 45°, in vacuo,
with a velocity of 2000 feet a second, would have a range
of 125000 feet, while the same ball, projected under the
same circumstances in the atmosphere, would only attain
to the range of 7300 feet; about one-seventeenth of the
former.
§ 174. The laws of the
centrifugal force may be il-
lustrated experimentally by
means of the whirling-table.
This consists of a frame-
work upon which are mount-
ed two vertical axes. Upon
the top of each axis is fast-
ened a circular block B, B,
having a groove cut in the
circumference for the recep-
tion of an endless cord C, C,
C, which also passes round
a wheel W. This wheel is
d
Fig. 124.
Qu
Ow
Whirling-table to
illustrate
centrifugal force;

224
NATURAL PHILOSOPHY.
the parts of the
table;
provided with a crank and handle H, for the purpose
of communicating motion to the whole. The circular
blocks are so made, that their circumferences, around
which the cord passes, may be varied to change the velo-
arrangement of city of rotation. A piece of wood dd, is mounted upon each
of the circular blocks, by means of screws, to support two
polished horizontal metallic bars b, b, along which a small
stage & may slide with as little friction as possible. This
stage is connected with an-
other S', which slides freely
on a pair of vertical bars b', b',
by means of a piece of flex-
ible catgut passing over the
pulleys p, p', in such manner
as to lift the stage S' in a
vertical, when motion is com-
municated to S in a hori-
zontal, direction.
scale and
moveable
weights;
example first;
The stage S is placed
with its centre immediately
over the axis of motion.
On the piece d d is a grad-
uated linear scale, having its
Q
W
Fig. 124.
Qu
a
zero in the axis, for the purpose of measuring the distance
of the stage S from the centre of motion. A series of
weights W', W', in the shape of small circular plates,
complete this part of the apparatus. The weights, being
perforated in the centre, are kept in place by a vertical
pintle rising from the middle of each stage.
Example 1st. Load one of the stages S, with the weight
5, and place it over the division 8 of the scale; load the
other stage S with the weight 2, and place it over the di-
vision 5; make the circumference of the first circular block
double that of the second. The angular velocity of the
first being V₁, that of the second will be 2 V₁. When
motion is communicated, the centrifugal forces will,
Eq. (68), be, respectively,

MECHANICS OF SOLIDS.
225
or
30
5 x 8 V and 2 × 5 × 4 V,
40 V and 40 V;
that is to say, the centrifugal forces will always be equal result;
to each other. Hence, if the stages S" be loaded equally,
they will be drawn up simultaneously.
Example 2d. Retaining the same ratio as before between example second;
the angular velocities, viz., V, and 2 V₁, load one of the
stages S with weight 6, and place it over the division 8 of
the scale; load the other stage S with weight 3, and place it
over the division 7. When rotation takes place, the cen-
trifugal forces will be, respectively,
6 x 8 V₁ = 48 V₁,
3 x 7 x 4 V₁ = 84 V₁,
the ratio of which is
48
84
12
=
21
and hence, if the first stage S' be loaded with 12 weights, result;
and the second with 21, they will rise together, and with a
little care may be kept suspended by properly regulating
the motion.
If the particles of which a body is composed may
move among each other, that is, if the body be soft, a
change may be effected by the action of this force in its
figure.
and of spherical
Such a body of a spherical form, revolving about one when a rotating
of its diameters, acquires a flattened shape in the direction body is soft,
of this diameter or axis, because the parts that lie in the figure, it acquires
plane of the greatest circumference which can be drawn
perpendicular to the axis, that is, in the plane of the body's
equator, have the greatest centrifugal force, while those
a flattened shape;
15

226
NATURAL PHILOSOPHY
in the neighborhood of the poles have the least; the
former will, therefore, recede from and the latter approach
the centre. Hence the inference in regard to the causes
of the flattened figure of the earth.
Example 3d. On the vertical axis ab, is an armillary
sphere, composed of elastic wires, fitting
round the axis by means of a ring,
which holds them all together. By
experimental this contrivance it is possible for the
elastic wires to assume an elliptical
illustration.
Principle of
the areas;
of the central
figure, having a shorter vertical diame-
ter. Screw this apparatus into the
middle of the circular block of the h
whirling table, and give to the whole
a rotatory motion; the wires, instead
of their original form represented by
the dotted lines, will assume, in conse-
Fig. 130.
algen
quence of the centrifugal force, the figure shown in the
dark lines.
8175.-When a body
moves with uniform mo-
tion, it passes over equal
spaces in equal times.
Thus, suppose the body
to start from A, and to
move uniformly in the
direction from A to B; the
line A B being divided
into equal spaces A m',
m' m", m" m"", &c., these
spaces will be described
Fig. 131.
B
0
C
a
n
A
m
m
a body in motion in equal times. If the several points of division be joined
under the action with any point as C, off the line, a series of triangles A Cm',
m'Om", m" Cm"", &c., will be formed, all having a com-
mon vertex and equal bases lying in the same straight line.
The areas of these triangles will, therefore, be equal, and
force;

MECHANICS OF SOLIDS.
227
will have been described in equal times during the motion
of the body by the line joining it with the point C.
led
If when the body arrives at m', it receive an impulse
in the direction from m' to C, which would cause it, if
moved from rest, to describe the path m'n, in the same
time that it would have described m' m" if unmolested,
then will it describe, in the same time, the diagonal m' m,, the forces first
of the parallelogram constructed upon m'n, and m'm" impulsive;
as sides. The line m" m,, being parallel to m' C, the
triangles C'm' m' and C'm' m,, will have the same base
Cm', and equal altitudes; their areas will therefore be
equal; hence the triangles CA m' and Cm' m,, will be
equal. In like manner, if when the body arrives at
t m
it receive another impulse directed towards C, which
would cause it to describe m, n,,, in the time it would
have described m,, O= m'm,, if undisturbed at m,,, it will
describe the diagonal m,, m,,, of the parallelogram con-
structed upon m,, O and m, n,, as sides; the triangle
Cm,,m,,, will be equal to the triangle Cm,, O C'm'm,,
=
0 =
CAm'. These equal triangles are described in equal
intervals of time by the line joining the moving body
with the centre C. If now the impulses towards C be
applied at intervals of time indefinitely small, the force
may be considered incessant, the sides of the polygon then incessant;
Am', m'm,,, m,, m,,,, &c., will become indefinitely small,
and the polygon itself will not differ from a curve. The
line which joins the body and the centre C, is called the
radius vector; and the incessant force acting in the direction radius vector;
of this line towards the centre, is called the centripetal force.
proportional to
Whence we conclude, that when any body having received areas described
a motion, is acted upon by a centripetal force, of which the by radius vector
direction is oblique to that of the motion, its radius vector will
describe equal areas in equal times. qque bas 10
And conversely, if the radius vector of a body moving in
a curve, be found to describe equal areas in equal times about
a fixed point, the body must be urged towards this fixed point
by a centripetal force, for the equality of the triangles
the time of
description;

228
NATURAL PHILOSOPHY.
conversely, the
Cm'm" and Cm'm
Cm,, m
areas being equal Cm,, O and
in equal times,
the force must
&c., depends upon the
tend to the fixed lines m" m,,, Om,,,, &c.,
point;
being respectively paral-
lel to m' C, m,, C, &c.,
drawn from the positions
in which the body re-
ceives the deflecting im-
pulses to the centre C.
Denote the area by
A, and the time in
C
Fig. 131.
B
ՊՆ,
m
A
m
ma
which it is described by t; the ratio of A to t, must,
from what has just been shown, be constant. Denote this
constant by a, and we shall have
ratio of areas to
the times.
or
<= α₂
t
A = at
(74)';
and making t equal to unity, we find
A = a;
from which we conclude, that a denotes the area described
in the unit of time.
§ 176.-Let a body de-
scribe the curve A B under
Measure of the the action of a centripetal
centripetal force; force directed to the cen-
tre C; and suppose m and
m' to be two positions of
the body very near to each
other. Draw the tangent
m Q to the curve at the place
B
Fig. 132.
m
0
糖
​A

MECHANICS OF SOLIDS.
229
m,
the actual
velocity;
and draw m' Q parallel to the radius vector C'm, and
m'n parallel to the tangent. If the centripetal force had
ceased to act at m, the body would have described m Q in
the time that it has actually described m m'. Again, if the components of
body had been moved from rest at m by the centripetal
force alone, it would have described the path mn = m' Q,
in the same time; the path m n is, therefore, the path due
to the action of the centripetal force. The places m and m'
being very near each other, the centripetal force may be
considered as constant during the passage of the body from
the one to the other. Denote the velocity which the cen-
tripetal force can generate in the body at m, in a unit of
time, by v,, then, Eq. (7), will
18
whence
mn = 1 v, ť²,
2₁ =
2 mn;
t2
but, Eq. (74)',
t =
승​:
α
and substituting this for t, we find
=
2a2 X mn
A2
Multiplying both members by the mass of the moving
body, denoted by M, we have
value of the
acceleration due
to the centripetal
force;
Mv,
=
2 Ma² X mn
42
Draw from m', the line m' h perpendicular to Cm, then,
because A is the area of the triangle Cm m', will
A = Cm x m'h,

230
NATURAL PHILOSOPHY.
the intensity of
the centripetal
force;
which in the above equation gives
11 Ansgnat edt os lollerng
bluow bod
iMv, = 8 Ma² X
2
mn
with ban
of Bozen
(74)".
ybod
Fig. 132.
versed sine;
altitude of the
sector;
value of the
intensity of the
force in words.
The distance m n is called
the versed sine of the arc
m m', and m'h the altitude
of the sector; the first mem-
ber, or Mv,, is the quantity
of motion which the centrip-
etal force can generate in
a unit of time, and there-
fore measures its intensity;
ஜம்பதி:
C
B-
m'
Pabiendo
m
A
whence we conclude that, the intensity of the centripetal
force by which a body is made to describe a curve, is always
equal to eight times the mass of the body into the square of the
area described by its radius vector in a unit of time, multiplied
by the versed sine of the elementary arc and divided by the
square of the radius vector into the square of the altitude of
the sector.
ed to unlov
sub nolimpiospe
Jaloghts al of
Phenomena of
the heavenly
bodies;
tol aids gabusi
XII.
MOTIONS OF THE HEAVENLY BODIES.
Jaisa diod
§ 177.-The phenomena of the heavenly bodies may
be divided into three classes: the first, comprehending the
motion of revolution round the sun; the second, the mo-
tion of rotation about their respective centres of inertia;
and third, their figure and the oscillations of the fluids on
their surfaces. It is only proposed to consider the force
which produces the motion of revolution, and the orbits
which the bodies would, if undisturbed, describe.
Observation has established three laws respecting the

MECHANICS OF SOLIDS.
231
motion of the planets, which, from their discoverer, are laws of Kepler:
called KEPLER's laws, viz.:
tuogant out of lelleng
1st. The planets move in plane curves, and the radius 1st law;
vector of each describes round the centre of the sun, areas
proportional to the times of their description.ge
2d. The orbits of the planets are ellipses with the centre of 2d law;
the sun in one of the foci.
3d. The squares of the times of revolution of the different 3d law;
planets are to one another as the cubes of their mean distances
from the sun or semi-major axes of their orbits.
These laws relate only to a motion of translation, and only relate to
must, therefore, be limited to the motion of the centres of
gravity of the planets.
Bliing el
§ 178. From the first of these laws, and the prin-
ciple of areas proportional to the times, explained in § 175,
motion of
translation.
first law;
it follows that, the centripetal force which keeps the planets in Consequences of
their orbits is directed to the centre of the sun, and that this
body is, therefore, the centre of the system.
The consequence of the second law relates to the varia-
tion which takes place in the intensity of the centripetal
force arising from a change in the body's place, and may be
determined thus. Let
m and m', be two con-
secutive places of the
planet moving in an
ellipse of which CA
and CB are the semi-
transverse and semi-
gulliw pogodw
ow m
Fig. 133.
B
m
D
Z
Nh
n
m
ell to oular
modeduced;
Q'
that of the second
A
S
S
F
conjugate axes, and
having the sun,
towards
G
which the centripetal
force is directed, in the
pro-
focus S. Draw m'n parallel to the tangent m Q, and
duce it till it meets m C, drawn to the centre of the ellipse,
in the point ; let fall the perpendicular m' h upon the
radius vector Sm; join the body at m with the other focus

232
NATURAL PHILOSOPHY.
il to
S'; draw S' N and CD
parallel to the tangent
to the curve at G.
m Q, and produce m C
The tangent QQ'
Construction of makes equal angles,
the figure;
Qms and Q'm S',
value of the
versed sine;
value of the
altitude of sector;
with the line drawn
from the place m to
G
the foci, and because
B
D
Fig. 133.
Q
m
m
Nh
L
S
S
A
F
S' N is parallel to this
tangent, the triangle
m S' N is isosceles, making S'm = Nm; and because CD
is parallel to S' N, and CS is equal to CS', the distance
NL is equal to LS; hence mLmS+mS'
2
is equal
to the semi-transverse axis CA = A. Denote the semi-
conjugate axis by B.
In the similar triangles m nv and m L C, we have,
mn : m v :: m L
: m C;
whence, writing A for m L, we have
mn =
A.mv
m Ca
Again, drawing m F perpendicular to DC, we have,
from the similar right-angled triangles m L F and m'hn,
m' h² : m'n² m F²: mĽ²;
::
whence, writing A for m L, we have
2
m' h²
m' n² xm F²
=
A2

MECHANICS OF SOLIDS.
233
and, dividing the last equation by this one, we have
bare
mn
m v
1
ratio of the
=
m'h²
2
A3 X
X
m C
m'n Xm F
versed sine to the
square of altitude
of sector;
The equation of the ellipse, referred to the conjugate
diameters C'm and CD, gives, because the points n and v
will sensibly coincide for consecutive places of the body,
m'n
2
CD2
=
Om
X mv x v G;
which, substituted for m' n above, we find
Pipe teel
C'm
m n
the same, in other
=
A3 X
m'h
xem
CDX m FX v G
terms;
and, because the rectangle of the semi-axes is equivalent
to the parallelogram constructed upon the semi-conjugate
diameters CD and C'm, we have
CD² Xm F2 = A² × B²;
moreover, the points m and m' being contiguous, Gv will
not differ sensibly from 2 Cm. Making these substitutions,
the above equation reduces to
m'ha
m n
2
=
A
2 B2;
and, multiplying both members by
8 Ma² X
m n
m'h² x Sm
8 Ma²
Sm²'
2
4 Ma² A
1
=
X
B2
Sm
its final value;

234
NATURAL PHILOSOPHY.
edt ol
The first member we have seen, Eq. (74)", is the intensity
of the centripetal force at m. Calling this force F and
adult writing r for the radius vector Sm, we finally have
value of the
force;
consequence of
the second law;
to find the
consequence of
the third law;
r,
F=
4 Ma² A 1
X 72
B2
Every thing being constant in the second member but
it follows that, the force which urges a planet towards the
sun, varies inversely as the square of the planet's distance from
that body.
The consequence of the third law is not less important,
and may be evolved thus. Multiply both members of the
last equation by 2 A2 B2, and we have
F2 A2 B2 42 Ma² A3 X
D
=
1
22;
divide both members of this equation by Fa², and there
will result
periodic time;
the value of its
square;
T2 A2 B2
=
a²
42 M
F
1
X A³ X
has
Now, AB is the area of the entire ellipse; a is the area
described by its radius vector in a unit of time; hence
TAB
α
is the number of units of time in one entire revolu-
tion of the planet, called the periodic time. Denote this
TAB
by T, and substitute it for
and we get
α
42 M 1
T2 =
A³.
F
2.2
oda
In like manner for any other planet, whose mass is
M', mean distance A', radius vector r', periodic time T",
and centripetal force F", we have
T²
=
42 M' 1
F

MECHANICS OF SOLIDS.
235
and dividing this equation by the one above all conti
T2
T2
3o dion, doso
ratio of the
squares of
ni botrovni periodic times;
M' F.2
Fre A
=
MF'. 12
fli od vd boenerge
T2
A'
=
But, by the third law, grid
whence
30
or
T2 Д; si bas buid at omse
lb 3o oguedo
otogryd boterdelen
M' F2
MF" '2 = 1,
boterdelen sut tuobe
The featured
F
MX 2
F
=
M'
X
216
getheta
log
centripetal
F
Now is the velocity which the centripetal force can
M
generate in one unit of time, or, which is the same thing,
it is the measure of the acceleration due to the force
F"
which acts upon the planet M; so, likewise, is the
M'
ybod
acceleration due to the centripetal force which acts upon
the planet M'; and resolving the above equation into the
proportion
acceleration;
F
F
:
::
M
M'
14
1
2,21
consequence of
the third law;
boob bow
we see that the forces which urge two different planets
towards the sun, are to each other in the inverse ratio of
the
squares of the distances; so that the same law which
regulates the intensity of the force in a single orbit, also
extends to different planets revolving in different orbits.
If r be made equal to r', then will the accelerations due
to the centripetal force be equal; that is to say, if all the

236
NATURAL PHILOSOPHY.
the centripetal
accelerations are
equal;
at same distance, planets were brought to the same distance from the sun,
each unit of mass would be urged towards that body with
the same intensity; and as the different planets might be
inverted in respect to the order of their distances from
the sun, without the relation of the periodic times as
expressed by the third law being affected, it follows that
the force which acts upon all the planets is absolutely the
same in kind, and is only qualified, in intensity, by a
change of distance. These considerations led Newton to
adopt the celebrated hypothesis which laid the foundation
of physical astronomy, viz.: that all bodies attract each
other with an energy which is directly proportional to their
masses and inversely proportional to the squares of their dis-
tances from each other.
Newtonian
hypothesis of
universal
gravitation;
Starting from this hypothesis, it is easy to solve by a
process not suited to an elementary work like this, the
consequences of converse problem of that which led to the consequence of
this hypothesis; the second law, and to show, that a heavenly body may
have been
ellipses,
describe any one of the conic sections having the sun at
one of the foci, depending upon the relation which subsists
between its velocity and the energy with which the body
and the sun attract each other. The orbit will be a para-
the orbits might bola, an ellipse, or hyperbola, according as the square of the
body's velocity is equal to, less, or greater than, twice the
attractive force, multiplied by the distance from the sun.
ads ofal
parabolas, or
hyperbolas.
The angular
velocity;
§ 179.-Let Cmm' be
the sector described in the
unit of time: take the dis-
tance Cb equal to unity,
and describe, with C as a
centre and Cb as radius,
the arc bd=s,, which will
measure the angular velo-
city. With C as a centre,
d
Fig. 134.
m
m
hK
and C'm' = r as radius, describe the arc m' h'; then will
m'h' rs,.

MECHANICS OF SOLIDS.
237
Supposing the unit of time small, in which case m' will
be very near to m, m'h will be sensibly equal to m'h',
m C to m' C, and we have for the area of the sector C'mm',
1 Cm x m'h' = r²s, = a;
whence
S =
2 a
2.2
its value;
from which we find that, the angular velocity of a planet law of its
about the sun, varies inversely as the square of its distance or
radius vector.
Supposing the planet
to describe the ellipse
ABPD, having the sun
at the focus S, the ex-
tremities A and P of the
transverse axis are call-
ed, the former the Aphe-
lion, and the latter the
Perihelion. The angular
velocity of the planet is
the least at aphelion and
greatest at perihelion.
Again, denote the angle
Cm Q by a, and suppose
the motion of the body on
the small arc m m' uniform,
which we may do without
sensible error, the length of
m m' will measure the ve-
locity of the planet at m,
since it is described in a
unit of time. Hence
P
S
Fig. 135.
B
D
A
variation;
aphelion;
perihelion;
angular velocity
greatest at
perihelion and
least at aphelion;
Fig. 136.
absolute velocity;
m m' sin a = m' h = V. sin a,
m

238
NATURAL PHILOSOPHY.
and the area of the triangle or sector Cm m' will be
V. sina xr; whence
or
V. r sin a
<= αig
2
X
value of the
absolute velocity;
V
=
Draw the tangent m Q to
to the curve at the point m,
and from Clet fall the per-
pendicular CQ, then in the
right-angled triangle CQm,
will
2 a
r. sin a
Sadt Ban
Fig. 136.
m'
onsig
C
CQ = r. sin a = p,
which substituted above gives
the same in
different terms;
2 a
V=
Panque par
m
its law of
variation;
greatest at loads
perihelion and
least at aphelion.
that is to say, the velocity of a planet in its orbit, varies in-
versely as the length of the perpendicular let fall from the
centre of the sun upon the tangent drawn to the orbit at the
body's place.
From this it follows that the velocity of the planet will
be greatest at perihelion and least at aphelion.
§ 180.--It will be found convenient when we come
to discuss the nature of light, to know that when a body
Centripetal force describes an ellipse under the action of a force directed
towards the centre of that curve, the force will vary
elliptical orbit; directly as the length of the radius vector, and that the
periodic time will be the same for all ellipses, great and
directed to the
centre of an
small.

MECHANICS OF SOLIDS.
239
Let the body, under
the action of a force di-
rected to the centre C,
describe the ellipse of
which CA and CB are
the semi-axes, denoted
respectively by A and
B; and suppose m and
m' to be two of its con-
secutive places. Draw
the tangent m Q at the
G
D
Fig. 137.
B
ed gaibivib
m
Q
margola
D'
point m, and parallel to this tangent draw the diameter
D D', perpendicular to which, draw from m the line m K.
From m' draw m'n parallel to the tangent till it meets the
radius vector C'm in n, and let fall upon the same radius
vector the perpendicular m' h.
The equation of the ellipse, referred to its conjugate
diameters C'm and CD, gives
CD2
to find the law;
m' n²
=
2
x mn x n G;
Cm
whence
mn =
2
m'n² x 7m²
CDXnG
Because m'n and m'h are respectively perpendicular to
the lines m K and m C, the angles h m'n and Cm K are
equal, and the angles at K and h being right angles, the
triangles m'nh and Cm K are similar, and give the pro-
portion
value of the
versed sine;
Tay
2
m'n²
:
m'h ::
2
Cm²: m K²;
Touch scholing
whence
m' ha
2
m'n² × m Kod quib
Cm²
value of the
ivid
square of sector's
altitude:

240
NATURAL PHILOSOPHY.
dividing the last equation by this one, we have
ratio of the
versed sine to
square of sector's
altitude;
m n
Cm
4
=
m'h
CD2 x m K2 x n G
But the rectangle of the semi-axes is equivalent to the
parallelogram described upon the semi-conjugate diame-
ters, hence
same in different
terms;
CD2 x m K2
=
A2 x B2;
moreover, n G is sensibly equal to 2 Cm; making these
substitutions above, there will result
m n
Cm³
=
2 A2 B2
m' h
multiplying both members by 8 Ma², and dividing by
Cm, we have, Eq. (74)",
m n
4 Ma²
8 Ma² X
=
F:
=
× Cm,
Cm² x m'h
X
A2 B2
in which M is the mass of the body. Finally, writing r
for Cm, we find
value of the
centripetal force;
the law of its
variation;
to find the
periodic time;
F=
=
4 Ma²
A2 B2X;
that is to say, the centripetal force which will cause a body to
describe an ellipse when directed to the centre of that curve,
varies directly as the radius vector.
Multiply both members of the last equation by
2A2B2, and we have
F.2 A2 B2 = 42 Ma² X r.
Dividing both members of this equation by Fa, and we
have

MECHANICS OF SOLIDS.
241
2A2 B2
a²
M
=
42
F
r = 42 X
r
Fi
doldw digo
Molang
i bandi od vlieso
TAB
taking the square root, and recollecting that, is the
α
oqati levit
mulubang all and W
periodic time = T, we find
To ostro/4r Tudt
T = T
F
olieo doua ni
Jod si Delivery
M
goleang onil leoit
d fliw ti L
Jest on end
anastalo
value of the
periodic time;
F
The quotient is the measure of the acceleration due
M
to the centripetal force, which we have just found to vary
directly as the radius vector. This makes the radical ex-
pression constant; hence T must also be constant.
200
1800
Juod
Whence we conclude, generally, that when any number
of bodies are solicited towards a fixed point by forces which
vary directly as the distances of the bodies from that point, they conclusion.
will describe ellipses, or circles, one of the varieties of the
ellipse; and that they will all perform their revolutions in the
same time.
to urgh
han molabaton
sad toangs aids
To abom
Toolsdogene
XIII.
opia to obom
stig of as
THE PENDULUM.
fiat oldiezog
Fig. 138.
M
he Compound
pendulum;
8181.-A body MQN, suspended
from a horizontal axis A, about which
it
may swing with freedom under the
action of its own weight, is called, in
general, a compound pendulum. When
the body is reduced to a material
heavy point, and the medium of con-
nection with the axis is without
weight, it is called a simple pendu-
lum.
behind
Q
N
simple
pendulum;
dod
16

242
NATURAL PHILOSOPHY.
has no real
existence;
The simple pendulum is but a
mere conception, and yet the ex-
pression for its length, which may
easily be found in a manner soon
to be explained, is of great prac-toos
tical importance.
When the pendulum is at rest,
in such position that its centre of
gravity G is below and on the ver-
tical line passing through the axis
A, it will be in a state of stable equi-
librium, § 151; but as soon as it is
deflected to one side, as indicated
effect of friction in the figure, and abandoned to
and resistance
of air;
figure of
pendulum and
mode of
suspension;
knife-edge and
bob;
itself, it will swing back and forth
about the position of equilibrium,
into which it will finally settle in
consequence of the resistance of
the air and friction on the axis.
If these causes of resistance were
removed, the pendulum would con-
tinue its motion indefinitely; but
this cannot be accomplished in
practice, and hence such figure and
mode of suspension are resorted to
as to give these impediments the.
least possible influence.
The pendulum is usually mount-
ed upon a knife-edge A as an axis,
resting upon a well-polished plate
of metal, or other hard substance,
B; and the figure of the pendulum
is that of a flat bar C, supporting
at its lower end a heavy lenticular-
shaped mass D, called a bob.
One entire swing of the pen-
dulum, by which its centre of
Q
Fig. 139.
Mi
ON
A
Fig. 140.
P
Fig. 141.
A
B
B
D
L
M
C
σ
D

MECHANICS OF SOLIDS.
243
gravity is carried from the extreme limit G of its path,
on one side of the vertical AL, to G" on the other, is oscillation;
called an oscillation.
To find the time of
a single oscillation, call
the weight of the entire
pendulum, W; its mass,
M; its angular velocity
at any instant, V₁; its
moment of inertia with
reference to the axis of
suspension, I; the dis-
tance of its centre of
gravity from the axis,
D; the vertical distance
P.G', through which the
centre of gravity must
Fig. 142.
to find time of a
single
oscillation;
Djalog
IG
bidw
notation;
G
G'
L
doldw
descend from its highest point G to arrive at any point
G', y.
The living force of the pendulum when the centre of
gravity reaches the point G' will, § 159, be
I. V,
and the quantity of work of the weight will be
and hence
living force;
Wy = Mgy,
Inspo M. L. guide
Jeff
work of the
weight;
IV = 2 Mgy.
900
The point C on the line A G at the unit's distance from
A, will, during the motion, describe an are similar to
G G', and the vertical distance G, P, denoted by y
through which this point will fall while G is passing to
G', will be given by
y =
Dyi
fall of the centre
of gravity;

244
NATURAL PHILOSOPHY.
square of the
angular velocity;
square of the
time required to
describe a very
small arc;
and this, in the above equation, gives
afto
Wolving
I. V 2 Mg Dy,;
ballag
whence
V2
=
To bylow siff
M.D.2gy, lubang
I
Denoting by s, the small distance described by the
point during the very short interval t, succeeding the
instant at which the angular velocity is V₁, we shall have
C
=
which, in the preceding equation, gives
Janit my silt: C
Hold w dynerds.
in yhary to ontmo
Jatown in vita
M.D
el mot booedly
=
.2gy,i
porot malvil bu
whence
I
8,2
12:
=
M.D
2gy,
Toyahnoop sit bas
to find the arc
described in the
small time;
How halow
Taking A Mequal to
unity, let CBC" be
the arc described in
one oscillation by the
point M, and MN the
small arc s, described
in the time t, immedi- q
ately succeeding the
instant at which the
angular velocity is V1.
Draw ME perpendi-
cular to the vertical
Fig. 143.
P
D
N
Ο
E
B

MECHANICS OF SOLIDS.
245
A B, and NQ perpendicular to ME: then, in the similar
triangles A ME and MN Q, we have
QN EM :: MN: AM;
and because A M is unity, and MN is s,
QN
纸
​one value for
balahledus did the arc;
EM
S₁ 1=
But from the property of the circle
EM=√2AB.EB
EM=2AB. EB - EB
-
養生
​E B² = √√ 2 EB – EBª,
and if we take the arc CBC" very small, the versed sine
EB will be a very small fraction, and its second power
may be neglected in comparison with the first. Whence
EM= √2EB;
which, in the value of s, above, gives selgust of
QN 49 = **
S₁
=
V2EB
and this, in the value for t², gives
another value for
the arc;
I
1
QN2
12
=
another value for
the square of the
M.D 4g y, BE
un bus time:
Upon BD as a diameter, describe a semi-circumference
Dmn B, and through the points M and N, the extremities
of the arc s,,
draw the horizontal lines Mm and Nn, cut-
ting this semi-circumference in the points m and n. Draw
the radius Om, and the vertical nq. From the property
of the circle we have
m Ea
=
=
BEX ED = BEX PM = BEX yi
1

246
NATURAL PHILOSOPHY.
M. fum & A
olgant
A
09
whence
vertical distance
from last point;
BE
=
m E2
y,
Fig. 143.
oral ow 9 Ma
Pad which, substituted for
BE in the equation
above, gives
P
D
O
M
E
N
B
value of the
element of the
time;
sof seday toda
I
1
QN 2
t2 =
M.D
obie borney edi
4g
m E21
and, taking the square root,
lama yoy a od by W
you
I
QN
t
=
V
9. M.D
X
m E
The two triangles m OE and m qn are similar, and give
gn= QN :
mE ::
n m
: 0m;
whence
QN
=
nm
m E Omi
day adi ai gidi bo
and this substituted above in the value of t, gives
the projection of
I
n m
t = V
X
9.M.D
Om
proportional to Such is the value of the time required to describe the
arc on the circle elementary are MN, which we see is proportional to
whose diameter the arc mn, or to the projection of MN on the semi-
circumference described upon DB as a diameter, every
other quantity in the second member of the equation being
is versed sine of
arc of oscillation;

MECHANICS OF SOLIDS.
247
constant; and hence, the time required to describe the
whole arc CMB, which is obviously the sum of all the the time of
elementary times of describing the elementary ares MN,
&c., must be equal to
making a
semi-oscillation
found;
글 ​V
I
1
Sad ethver
X
9. M. D Om'
into the sum of all the projections of MN, &c, on the
semi-circumference Dm B; but this sum is the semi-
circumference itself; and denoting the time from C to B,
or that of a semi-oscillation, by T, we have
T= V
I
g. M. D
X
Dm B
;
its value;
Om
but
Dm B
== 3.1416,
ด้านได้
Om
the ratio of the circumference to the diameter; whence,
T = T
I
9.M.D
wondled time of a single
(75).
oscillation;
From this formula we see that the duration is inde-
pendent of the amplitude of the oscillation, when this
amplitude is small; and a pendulum slightly deflected
from its vertical position and abandoned to itself, will
oscillate in equal times whatever be the magnitude of the
arc, provided it be inconsiderable. Such oscillations are isochronal
said to be Isochronal.
If the number of oscillations performed in a given in-
terval, say ten or twenty minutes, be counted, the duration
oscillations;
of a single oscillation will be found by dividing the whole time of a single
interval by this number.
oscillation found
from
Thus, let e denote the time of observation, and N the observation;
number of oscillations, then will

248
NATURAL PHILOSOPHY.
edt odieob offerin
T=
10 se odr.odt lie to me
I
N
V
9. M. D'
lodw
guiding
band if the same pendulum be made to oscillate at some
other location during the same interval, the force of
gravity being different, the number N' of oscillations will
be different; but we shall have, as before, g' being the
new force of gravity,
oda otai.
the same for a
second place;
0
I
=T
N'
g'. M. D
சய
Squaring and dividing the first by the second, we find
N12 g
=
N2
g
(76);
are as squares of
number of
forces of gravity that is to say, the intensities of the force of gravity, at
different places, are to each other as the squares of the
number of oscillations performed in the same time, by the
same pendulum. Hence, if the intensity of gravity at one
station be known, it will be easy to find it at others.
oscillations in
same time.
Simple
pendulum;
§ 182.-Resuming the general value for I, Eq. (65), we
have
Borellab digile I = I + D² M;
which value of I, in Eq. (75), gives
Inmordant 91S
T=&
+ D2 M
9. M.D
(77).
mass
If, now, we suppose the entire mass of the pendulum to
concentrated in a be concentrated into a single point, and this point con-
nected with the axis by a medium without weight, we
single point;
have

MECHANICS OF SOLIDS.
249
I₁ = Σm r² = 0;
moment of inertia
in reference to
the centre of
since the centre of gravity must also go to that point, and gravity;
r = r' = r'' = &c. = 0; whence, writing 7 for the new value
assumed by D, which now becomes the distance from the
axis to the single heavy point, we have
T = T
g
(78);
time of oscillation
of the simple
151 pendulum;
which is the expression for the time of oscillation of a
simple pendulum of which 7 is the length.
If the time of oscillation of the simple, be the same as
that of the compound pendulum, we shall have, from Eqs.
(75) and (78),
30
or
I
adow and
T
g.M.D
=T
I
I₁ + MD2
1=
(79);
M.D
MD
in which case 7 is called the equivalent simple pendulum; equivalent simple
that is to say, the length of a simple pendulum which will
oscillate in the same time as a compound pendulum whose
moment of inertia in reference to the axis of suspension is
I, whose mass is M, and of which the axis of suspension is
at a distance from the centre of gravity equal to D.
pendulum;
oscillation;
The point situated on a line drawn through the centre centre of
of gravity of the pendulum, perpendicular to the axis of
suspension, and at a distance from that axis equal to l, is
called the centre of oscillation; and is that point of which
the circumstances of oscillation would in nowise be altered
were the entire pendulum concentrated into it, or were it
disconnected from the other points of the pendulous mass,
its connection with the axis being retained.

250
NATURAL PHILOSOPHY.
Axes of
suspension and
of oscillation are
reciprocal;
§ 183.-A line drawn through the centre of oscillation, and
parallel to the axis of suspension, is called the axis of oscilla-
tion. The axes of suspension and of oscillation are reciprocal.
en Let D' denote the distance of the axis of oscillation
from the centre of gravity; then will
duiog vood olentia ad
1 =D + D'.
(87)
Invert the pendulum and make the axis of oscillation the
axis of suspension, take l' for the new equivalent simple
pendulum, then will
new equivalent
simple
ap mot ovad lady =
I+MD2
pendulum;
M. D'
(81) ban (1)
the simple
pendulum the
same;
but we have, from the foregoing equation,
D' = 1- D;
and this, in the preceding value for l', gives
V'
=
I₁ + M(1 - D²
M. (D)
Again, from Eq. (79), we have
I
supo 7-D =
;
MD
substituting this in the above value for l', we finally get
I + MD2
V'
=
=
MD
1;
that is to say, when the axis of oscillation is taken as the

MECHANICS OF SOLIDS.
251
simple pendulum
axis of suspension, the old axis of suspension becomes the
new axis of oscillation. In other words, these axes are conclusion;
reciprocal. This furnishes an experimental method for
finding the length of any equivalent simple pendulum, equivalent
which is the more valuable in view of the great difficulty found from
of computing the moment of inertia of a compound pendu- experiment;
lum by the ordinary calculus, owing to the peculiar forms
of that instrument rendered necessary by the circumstan-
ces under which it is employed. But before proceeding
to the explanation of this method, it will be proper to
premise, that the time of oscillation of a compound pendu-
lum will be a minimum, when, in Eqs. (78) and (79),
joning out for M
I
I + D2 M
MD
M
+ D2
=
= 1,
D
ban
value of
equivalent simple
pendulum;
is the least possible; or replacing
by its value K,
deduced from Eq. (66)" by making D = 0, the expression
K² + D2
D
must be the least possible.
wit o ofer
But it may easily be shown, either by trial, or by a
simple process of the calculus, that this expression is a
minimum when
K' = D,
od) to noteb
and consequently
1 = 2 K';
bang
length of the
shortest
equivalent simple
that is to say, the time of oscillation of a pendulum will pendulum;
be the least possible when the axis of suspension passes
through the principal centre of gyration, and the length

252
NATURAL PHILOSOPHY.
usual form of the
compound
pendulum;
of the equivalent simple pendulum is twice the principal
radius of gyration.
Fig. 144.
Let A and A' be two acute
parallel prismatic axes firmly con-
nected with the pendulum, the
acute edges being turned towards
each other. The oscillation may
be made to take place about either
axis by simply inverting the pen-
dulum. Also, let M be a sliding
mass capable of being retained in
any position by the clamp-screwight f
device to change H. For any assumed position of
the position of
the centre of
gravity;
M, let the principal radius of gyra-
tion be G 0; with G as a centre,
G C as radius, describe the circum-
M
ference CSS'. From what has been explained, the time
of oscillation about either axis will be shortened as it
approaches, and lengthened as it recedes from this circum-
ference, being a minimum, or least possible, when on it.
By moving the mass M, the centre of gravity, and there-
fore the gyratory circle of which it is the centre, may be
thrown towards either axis. The pendulum bob being
made heavy, the centre of gravity may be brought so near
one of the axes, say A', as to place the latter within the
gyratory circumference, keeping the centre of this circum-
centre of gravity; ference between the axes, as indicated in the figure. In
this position, it is obvious that any motion in the mass M
would at the same time either shorten or lengthen the
duration of the oscillation about both axes, but unequally,
in consequence of their unequal distances from the gyra-
tory circumference.
position of
pendulum made
same time;
The pendulum thus arranged, is made to vibrate about
to oscillate during each axis in succession during equal intervals, say an hour
or a day, and the number of oscillations carefully noted;
if these numbers be the same, the distance between the
axes is the length 7 of the equivalent simple pendulum;

MECHANICS OF SOLIDS.
253
if not, then the weight M must be moved towards that
axis whose number is the least, and the trial repeated, till
the numbers are made equal. The distance between the distance between
axes may be measured by a scale of equal parts.
the axes
measured;
From this value of 1, we may easily find that of the
simple second's pendulum; that is to say, the simple pendu- simple second's
lum which will perform its vibration in one second. Let pendulum;
N be the number of vibrations performed in one hour
by the compound pendulum whose equivalent simple
pendulum is 7; the number performed in the same time
by the second's pendulum, whose length we will denote by
l', is of course 3600, being the number of seconds in 1 hour,
and hence, from Eq. (78),
quie
1h
=
T = T
N
1h
= T' = 7
3600
and because the force of gravity at the same station is
constant, we find, after squaring and dividing the second
equation by the first,
で
​=
1. N2
(3600$)2
(80).
its length;
Such is, in outline, the beautiful process by which KATER
determined the length of the simple second's pendulum
abnises signi
at the Tower of London to be 39.13908 inches, or 3.26159 value at London;
feet.
English system
As the force of gravity at the same place is not sup-
posed to change its intensity, this length of the simple
second's pendulum must remain for ever invariable; and, basis of the
on this account, the English have adopted it as the basis of weights and
of their system of weights and measures. For this purpose, measures;
it was simply necessary to say that the 15th part of
the simple second's pendulum at the Tower of London shall
6159

254
NATURAL PHILOSOPHY.
English linear
foot;
the gallon;
1728
be one English foot, and all linear dimensions at once re-
sult from the relation they bear to the foot; that the gallon
shall contain 231th of a cubic foot, and all measures of
volume are fixed by the relations which other volumes
bear to the gallon; and finally, that a cubic foot of distilled
avoirdupois Water at the temperature of sixty degrees Fahr. shall weigh
one thousand ounces, and all weights are fixed by the rela-
tion they bear to the ounce.
ounce;
apparent force
of gravity at
London;
diy
It is now easy to find the apparent force of gravity at
London; that is to say, the force of gravity as affected by
the centrifugal force and the oblateness of the earth. The
time of oscillation being one second, and the length of the
simple pendulum 3.26159 feet, Eq. (78) gives
bas
whence
ft.
3.26159
1 = T
;
g
g= r² (3.26159) = (3.1416). (3.26159) = 32.1908 feet.
From Eq. (78), we also find, by making T one second,
and assuming
length of the
simple second's
pendulum, a
function of the
latitude;
we have
9 = 21,
T2
1 = x + y cos 2+,
7/0
T2
= x + y cos 2+... (81).
Now starting with the value for g at London, and
causing the same pendulum to vibrate at places whose
latitudes are known, we obtain, from the relation given in
Eq. (76), the corresponding values of g, or the force of

MECHANICS OF SOLIDS.
255
found at different
places;
gravity at these places; and these values and the cor- force of gravity
responding latitudes being substituted successively in
Eq. (81), give a series of equations involving but two un-
known quantities, which may easily be found by the method
of least squares.
In this way it has been ascertained that
x =
32.1808 and 2.y =
0.0821;
whence, generally,
alquie
f
-
g = 32.1803 0.0821 cos 2+.
(81)'; Force of gravity
and substituting this value in Eq. (78), and making
T= 1, we find
f
in any latitude;
7 = 3.26058 -0.008318 cos 2 + . (82). length of simple
-
second's
pendulum in any
Such is the length of the simple second's pendulum at latitude;
any place of which the latitude is 4.
If we make = 40° 42' 40", the latitude of the City-
Hall of New York, we shall find
ft.
in.
7 = 3.25938 = 39.11256.
length at City
Hall of New
York;
The principles which have just been explained, enable
us to find the moment of inertia of any body turning
about a fixed axis, with great accuracy, no matter what its moment of inertis
figure, density, or the distribution of its matter. If the found by means
of simple
axis do not pass through its centre of gravity, the body pendulum;
will, when deflected from its position of equilibrium,
oscillate, and become, in fact, a compound pendulum; and
denoting the length of its equivalent simple pendulum by
7, we have, Eq. (79),
M.D.1 = I;

256
NATURAL PHILOSOPHY.
or, since has
W
M
g
bodiom
yd bo
dold
its value;
=
(83);
g
simple second's
pendulum known
from latitude;
Wang Jo
the body's
equivalent
simple
pendulum;
W.D.1 I
in which W denotes the weight of the body.
Knowing the latitude of the place, the length ' of the
simple second's pendulum is known from Eq. (82); and
counting the number N of oscillations performed by the
body in one hour, Eq. (80), gives
1=
guideor bae (8T)
7'.(3600)2
N2
bah
Ball ow
elqmle to digest
at malang
distance from
centre of gravity
to axis found;
To find the value of D, which is the distance of the
centre of gravity from the axis, attach a spring or other
balance to any point of the body, say its lower end, and
bring the centre of gravity to a horizontal plane through
the axis, which position
will be indicated by the
maximum reading of the
balance. Denoting by a the
distance from the axis C
to the point of support R,
and by b the maximum in-
dication of the balance, we
have, from the principles
of moments,
LA
R
Fig. 145.
=
ba WD.
Haugh
value of the
moment of
inertia ;
The distance a may be measured by a scale of equal parts.
Substituting the values of WD and 7 in the expression
for the moment of inertia, Eq. (83), we get
b. a. l'. (3600) 2
9. N2
=
I. ... (84).

MECHANICS OF SOLIDS.
257
If the axis pass through the centre of gravity, as, for the moment of
example, in the fly-wheel, take Eq. (79),
inertia found
when the axis
passes through
the centre of
I+MD2
7=
MD
refied
algan gravity;
whence
IM.D.1-MD2.
Mount the body upon a
(85).
parallel axis A, not pass-
ing through the centre of
gravity, and cause it to vi-
brate for an hour as before;
from the number of these
vibrations and the length
of the simple second's pen-
dulum, the value of 7 may
be found as before; M is
known, being the weight
Fig. 146.
4
A
W divided by g; and D may be found by direct measure-
ment, or by the aid of the spring balance, as already
indicated; whence I becomes known.
8 184.-When a body,
BQ NO receives a motion
of rotation about an axis
A, which is here supposed
perpendicular to the plane
of the paper, each elemen-
tary mass m, will develop
a force of inertia whose di-
rection is perpendicular to
the shortest line connecting
it with the axis, and whose
intensity will be measured by
m.r.
Fig. 147.
B
9
V₁
t
N
G
0
example of the
fly-wheel;
X
Find the point of
application of the
resultant inertia
of a rotating
body;
inertia exerted by
an elementary
mass during an
elementary time;
17

258
NATURAL PHILOSOPHY.
notation;
co-ordinate
planes;
B
in which is the distanced Fig. 147.
of m from the axis, and Vi
the elementary amount of
angular velocity generated
in the very small portion
of time denoted by t.
Through the axis A,
pass two planes at right
angles to each other, and
let their traces on the paper
be A x and A y. Deno-
9
m
ข
N
0
ting the co-ordinates Ap and Aq of m, referred to these
planes, by x and y, respectively, we shall have
cos m Ap
=
8/2
cos m A q
y
=
r
Resolve the force of inertia, above given, into two compo-
nents in the directions of these planes. The component
parallel to the plane of which the trace is A y, will be
component of the
inertia parallel to
the plane Ay;
V₁ x
mr
-
t r
=m.x
and that parallel to the plane whose trace is A x, will be
that parallel to
the plane Ax;
V₁ y
mr.
=
t
my. I;
t
and for other elementary masses m', m", &c., of which the
co-ordinates are x'y', x'y', &c., we shall have the com-
ponents
m'x'.
"
V₁
&c.,
the same for
other elementary
masses;
m'y. my", &c.;
T

MECHANICS OF SOLIDS.
259
the resultant of the components parallel to the plane Ay,
will be
V₁
Μα,
71 (mx + m'x' + m"a" + &c.) = Ma
t
and of the components parallel to the plane A x,
T¹ (my + m'y' + m" y" + &c.)
=
Vi Myi
in which M denotes the entire mass of the rotating body,
and x and Y the co-ordinates of its centre of gravity.
And the intensity of the general resultant will, from the
parallelogram of forces, be
2
V₁ M√x;² + y² = ₂ M.D;
t
resultant of the
components
parallel to Ay,
resultant of those
parallel to Ax;
resultant of the
whole;
in which D represents the distance of the centre of gravity
G, of the whole mass, from the axis. The direction of its direction:
this resultant will be perpendicular to AG, drawn
through the centre of gravity perpendicular to the axis,
as will readily appear by reference to its components
parallel to the planes A y and Ax found above.
The moment of this force, with reference to the axis,
will therefore be its intensity multiplied into some dis-
tance as A0 = L, on this line, or
V₁ M. D. L.
t
But, Eq. (63), the sum of the moments of all the forces of
inertia actually exerted, in reference to the axis A, is equal
to the product of the entire moment of inertia I, multiplied
by the ratio, therefore
its moment;
V₁. M.D.L = I.
t
V₁

260
NATURAL PHILOSOPHY.
or
I
L =
M.D
.
(86);
point at which
the resultant
inertia of a
whence we conclude that, the point at which the resultant
inertia of a rotating mass is exerted, is on a line drawn
rotating mass is through its centre of gravity perpendicular to the axis, and at
a distance from the axis equal to the moment of inertia divided
by the product of the mass into the distance of the centre of
gravity from the axis.
exerted;
This being understood, suppose a
force Fapplied at the point C' in a di-
rection perpendicular to the line A 0,
and immediately opposed to the direc-
Ootion of the motion; this force would
obviously tend to bend the line A 0,
the point A being retained by the
axis, and the point O being urged
onward by the inertia concentrated
CONDO
shock
experienced
by the axis when at it. If the force be suddenly ap-
the body is
struck;
plied, the axis must receive a shock,
and to estimate its intensity S, de-
Fig. 148.
C
G
0
F
note by X the distance A C; then, from the principles of
parallel forces already explained, we have
L: L-X:: F: S;
its intensity;
whence
S=F.
F.L-X = F(1-4)
(87),
or, substituting the value of L, Eq. (86),
S = F(1 - MD. X). . . (88).
I
If we suppose the body at rest, and desire to apply the

MECHANICS OF SOLIDS.
261
force F so as to communicate no shock, we make
S = 0,
a condition that can only be satisfied by making
1-
-
MD
I
x X = 0; liv
the blow applied
so as to
communicate no
shock to the axis;
whence
I
X =
=
= LA 0.
MD
distance from the
axis at which it
must be applied;
There being no shock to the axis, it can oppose no resist-
ance to the motion of rotation, and hence we infer that,
this latter will be the same as though the body were per-
fectly free. The point O is, on this account, called the
centre of percussion, which may be defined, that point of centre of on-af
a body retained by a fixed axis, at which it may be struck in a
direction perpendicular to the plane of the centre of gravity
and axis without communicating any shock to the axis.
The centre of per-
cussion may be found
experimentally thus:-
lay the axis Cupon a
support A A, and per-
mit the body to fall up-
on a moveable edge B,
resting on a horizontal
plane; when this edge
P
B
Fig. 149.
percussion
defined;
A 30
centre of
For percussion found
experimentally;
is placed in such position that the axis C will not move
when the body falls upon it, the centre of percussion will
be immediately above the point struck. Since the dis-
tance of the centre of percussion from the axis is equal to
I
MD'

262
NATURAL PHILOSOPHY.
to put a
pendulum in
motion, the
it must be at the centre of oscillation. To move a pen-
dulum without communicating action to its axis, the force
force should be must be applied at the centre of oscillation.
applied to centre
of oscillation,
The shock may
be positive,
nothing, or
negative;
centre of
spontaneous
rotation;
distance of centre
of spontaneous
rotation from
§ 185.-Resuming Eq. (87),
we see that the shock upon the
axis A will be positive, that is to
say, will act in the direction of
the impressed force F, as long as
X is less than L: when X is equal
to L, there will be no shock;
when X is greater than L, there
will again be a shock, but with a
negative sign, which indicates
that it will be exerted in a direc-
tion opposite to that of the im-
pressed force. Now these shocks
Fig. 150.
A
E
in opposite directions, with a neutral point A, can only
arise from an effort of the particles, which are situated on
opposite sides of the axis, to move in contrary directions
when the body is struck at the centre of oscillation; and
as the effect upon the neutral point A is the same in this
latter case, whether the body be retained by an axis or a
force, it follows that every free body, when struck, in gen-
eral, begins to move for the instant, but only an instant,
about a single point. This point is called the centre of
spontaneous rotation.
If the blow be impressed at any
point, as O, the centre of spontaneous rotation will be upon
the axis corresponding to the point 0 as a centre of oscil-
lation, and hence its distance from the latter will be given
by
L = √D
I
MD
.
(89);
direction of blow;
and since the centre of oscillation and axis of suspension
are reciprocal, I will denote the moment of inertia taken

MECHANICS OF SOLIDS.
263
with reference to an axis through the point 0, and D the relation of
distance of the latter from the centre of gravity.
centre of
spontaneous
Referring to Eq. (88), if the axis be supposed to pass rotation to centre
through the centre of gravity, D will be equal to zero, and
of oscillation;
S = F;
always
that is to say, no matter where the force F be applied, its the entire shock
entire effect will be communicated to the centre of gravity, communicated to
which is a confirmation of the result given in § 146.
centre of gravity;
impact pass
If the line of direction of the force pass through the
centre of gravity, D, in Eq. (89), will be zero, and the dis- if direction of
tance of the centre of spontaneous rotation will be at an
infinite distance from the point of impact; in other words
the body will not rotate, which is another result of § 146.
B
Fig. 151.
§ 186.-Let Q be a body god (08).
suspended from an axis A
perpendicular to the planet
of the figure. This body be-
ing at rest, suppose it to be
struck at the point 7 by an-
other body P, moving in the
direction TL at right angles
to the surface of contact,
and in a plane perpendicu-
lar to the axis A. Denote
by m and w the mass and
weight of the impinging
body, and by V its velocity
before the impact. At the
instant of meeting there will
be developed a force of com-
pression F, which will act
I
D
Q
T
P
高​師
​equally upon each body along the line TL, but in oppo-
site directions. The pressure upon both bodies, which is
nothing when they begin to touch each other, will aug-
through centre
of gravity, the
rotate.
body will not
Collision of a
body having e
motion of
translation
against another
retained by a
fixed axis;

264
NATURAL PHILOSOPHY.
the action and
ment by degrees as they ap-
reaction variable; proach to the state of great-
est compression; so that F,
although always represent-diveng
ing a number of pounds
weight, is, nevertheless, not
a fixed, but a variable quan-
tity. We may disregard for
a moment the body Q, and
suppose the force F applied
to the body P, considered as
free; the force will deprive
this body of a series of small
degrees of velocity denoted
by v, each in the small time
t, so that its measure at any
instant will, Eq. (39), be given by
I
B
Fig. 151.
D
G
gaht
P
С
measure of the
force of reaction;
moment of action
equal to moment
of reaction:
m v
F =
t
But the force F also acts upon the body Q, and turns it
about the axis A, generating in it, during the same in-
terval of time t, an angular velocity v,; and the forces of
inertia thence arising, must be in equilibrio with the force
F; in other words, the sum of the moments of the first in
reference to the axis A, must be equal to the product of
the force F into the perpendicular A C, drawn from the
axis to the line of direction TL. Hence, Eq. (63),
F. AC= I.
and substituting the value of F above, and dividing by
A C, which we will represent by the single letter P,
m.v I.v,
tp.t
=

MECHANICS OF SOLIDS.
265
or, finally,
1=
p.m.v = I. v,.
Denote by v', v", "", &c., the small degrees of velocity
lost by the body P, during the first, second, third, &c.,
intervals of time t, supposed to be always of the same
length; and by v,, v", v", &c., the angular velocities ac-
quired by the body Q during the same intervals; we
shall have
p.m.v' = Ιυ,
result for a single
instant of time;
p.m v":
=
Iv",
&c.
=
&c.;
the same for
other instants of
time;
by taking the sum of the whole,
p (v' + v''+ v'''+ viv + &c.) m = I (v,' + v," + v,'""'+ &c.);
and denoting by U the whole velocity lost by the body
P, and by V₁ the whole angular velocity gained by the
body Q during the entire action, we shall have
U = v' + v" + v"" + viv + &c.,
V₁ = v,+ v," + v,"" + v," + &c.;
whence, by substituting above,
1
p.m. U IV
the sum of the
whole;
velocity lost;
angular velocity
gained;
result for the
(90). entire duration
20/1900 of the impact;
If the bodies be not elastic, it will only be necessary to
consider the impact from the instant in which they first
come in contact, to that in which the body P has lost its
excess of velocity over that part of Q into which it be-
comes imbedded; for, as soon as the body P has taken the

266
NATURAL PHILOSOPHY.
not elastic, they
if the bodies be angular velocity of the other about the axis, there will be
no effort to regain lost figure, and the two bodies will
turn about A as though they constituted but a single
will ultimately
constitute a
single one;
one.
But the angular velocity of Q about A being V₁, the
velocity of P will be p V₁, and we shall have di
U = V − p V₁;
—-
substituting this value of U in Eq. (90), we find
pm (V - pv) = IV;
- =
whence
angular velocity
generated by the
impact.
V₁ =
p.m.V
m.p² + I
(91);
Application to
the balistic
pendulum;
which gives the angular velocity of the body struck, after
the impact, in terms of its moment of inertia, the mass
and velocity of the impinging body, and the distance from
the axis to the path described by its centre of gravity.
8 187.-In artillery,
the initial velocity of
projectiles is ascertained
by means of the balistic
pendulum, which consists
of a mass of matter sus-
pended from a horizon-
tal axis in the shape of
a knife-edge, after the
manner of the compound
its construction; pendulum. The bob is
either made of some
unelastic substance, as
wood, or of metal pro-
vided with a large cavity
T
0
G
Fig. 152.
A
26
R
G

MECHANICS OF SOLIDS.
267
filled with some soft matter, as dirt, which receives the
projectile and retains the shape impressed upon it by the
blow.
Denote by V and m, the initial velocity and mass of the
ball; V, the angular velocity of the balistic pendulum after notation;
the blow, I and M its moment of inertia and mass. Also
let r represent the distance of the centre of oscillation of
the pendulum from the axis A. That no motion may be
lost by the resistance of the axis arising from a shock, the the pendulum
ball must be received in the direction of a line passing centre of
through this centre and perpendicular to the line A O. oscillation;
This condition being satisfied, we have
bre
must be struck at
and Eq. (91) becomes
from which we find
p = r, 1
eovig, ponot
sonodw
=
rm V
m p² + I
(m² + I)V
value for the
=
(92);
mr
the velocity V becomes known, therefore, when V₁ is
known, since all the other quantities may be easily found
by the methods already explained. To find V₁, denote
by H the greatest height to which the centre of gravity
of the pendulum is elevated by virtue of this angular
velocity; then, since the moment of inertia of the ball is
mr, we have, from the principle of the living force,
(I + m r²) V₁² = 2 (M + m) g H;
velocity of
projectile;
equation of living
force;
whence
(I + m r²) V₁₂³
(M + m) g
= 2H.

268
NATURAL PHILOSOPHY.
time of a single
oscillation of
balistic
pendulum;
angular velocity
of the pendulum;
moment of inertia
of the whole;
time of oscillation
of the equivalent
Denoting by T the time of a single oscillation of the
pendulum after it receives the ball, we have, Eq. (75),
Way I + mp²
T
T = √ (M + m) D. g'
τα
D being the distance from the axis to the centre of grav-
ity; whence,
I + m r²
(M + m) g
DT2
=
T2
and this value, substituted in the equation of the living
force, gives
whence
also
DT² v² = 2 H;
T2
T
2 H
V₁ = T
I + m² (M + m) g. D. T²;
and because, Eq. (78),
=
T2
simple
T = T
g'
pendulum;
we find
length of this
pendulum;
Tag
r=
T2
Substituting these values of V₁, I+ m² and r in Eq. (92),
we find
V
√2HD.
M+ m
m
;

MECHANICS OF SOLIDS.
269
or, replacing the masses by the weight divided by the
force of gravity,
T
W + w
V =
√2H.DX
T
พ
in which W and w denote the weights of the pendulum
and ball respectively.
Observe that H is the height to
which the centre of gravity rises in
describing the arc of a circle of
which D is the radius. Let G G'K
be half of the circumference of
which this arc is a part, G and G'
the initial and terminal positions of
the centre of gravity during the
ascent; draw G'R perpendicular to
KG. Then, because A G= D, and
GR=H, we have, from the proper-
ty of the circle,
RG
=
√ H(2D - H);
K
A
R
Fig. 153.
H
and if the pendulum be made large, so that the arc GG'
shall be very small, which is usually the case, I may be
neglected in comparison with 2 D, and therefore
simpler value for
velocity of
projectile;
to find the radical
part of this
value;
RG
=
V2H.D;
√2 HD is half the chord of the are described by the
centre of gravity in one entire oscillation. Denoting this
chord by C, and substituting above, we have
V = }
.
W + w
C
T
w
From this equation, we may find the initial velocity V;
and for this purpose, it will only be necessary to have the
value of radical
part found;
velocity of
projectile in
terms of the
chord of the arc
of vibration;

270
NATURAL PHILOSOPHY.
vibration;
duration of a single oscillation, and the amplitude of the
are described by the centre of gravity of the pendulum.
The process for finding the time has been explained. To
to find the arc of find the arc, it will be sufficient to attach to the lower ex-
tremity of the pendulum a pointer, and to fix on a perma-
nent stand below, a circular graduated groove, whose
centre of curvature is at A; the groove being filled with
some soft substance, as tallow, the pointer will mark on
it the extent of the oscillation. Knowing thus the arc,
denoted by 4, and the value of D, found as already
described, § 183, we have
RGC = D. sin 0;
its value found;
whence
and finally
C = 2 D. sin;
final value of
velocity.
W + w
V
=
D.
sin.. (93).
T
พ
A machine
defined.
SIMPLE MACHINES.
§ 188.-A machine is any device by which the action
of a force is received at one set of points and transmitted
to another set, where it may either balance or overcome
the action of one or more opposing forces and perform its
effective work. The force impressed is usually called the
power, and that overcome, the resistance. We proceed to
discuss the simple machines, so named because some one
or more of them enter as elements into the composition of
all machinery.

MECHANICS OF SOLIDS.
271
XIV.
FUNICULAR MACHINE.
machine;
inertia small as
§ 189. This consists of an assemblage of cords or bars; Funicular
the former united by knots, and the latter by joints or
hinges. The cords are supposed, for simplification, per-
fectly flexible, the bars perfectly rigid, and both inexten-
sible, without weight, and devoid of inertia. The weight
and inertia of the several parts of every machine, are
usually small when compared with the intensity of the weight and
power and resistance; and when this is not the case, they
may be estimated and taken into the account by the the power and
methods already explained. The hypothesis of inextensi- the resistance;
bility is also admissible, because when a cord or bar is ex- inextensibility
tended or the latter compressed under the action of one or admissible;
of several forces, the maximum change of dimensions is
soon attained, after which the figure remains unaltered
during the subsequent action.
Let the extremities
of the straight cord
AB be solicited by
several forces. Each
force may be resolved
into two components,
one in the direction
of the cord, the other
X
A
Fig. 154
compared with
case of a single
B
cord;
at right angles to it. Since the cord is perfectly flexible,
if it be in equilibrio, the perpendicular components at each conditions of
end must destroy each other, otherwise they would pro- equilibrium ;
duce flexure. The components in the direction of the
cord must reduce to two forces, which are equal in in- forces must act
tensity and immediately opposed. They must also act to to stretch the
stretch the cord, for compression would only bend it, and
cord;

272
NATURAL PHILOSOPHY.
in the case of a
bar, the forces
may also act to
compress it;
action of the
molecular
springs;
the tension the
same throughout,
except when
vertical;
cords never
equally strong
throughout;
the action of one force could not be transmitted to the
point of application of the other.
If instead of a cord we suppose a bar, the conditions
of equilibrium will be the same, only that the bar being
inflexible, the forces in the direction of its length may act
either to stretch or to compress it. By recalling what was
said of the physical constitution of bodies, we may regard
the molecular forces as so many springs which, as soon as
an effort is made to disturb the particles from their posi-
tions of rest, are extended or compressed everywhere
equally by the equal and contrary forces which act at the
ends of the cord or bar. Hence the tension, that is, the
effort by which any two consecutive elements are urged to
approach each other or to separate, in the direction of the
cord or bar, must be equal throughout, and equal to one
of the equal forces in question, except when the cord or
bar is vertical; in which case, the tension at any point
is increased by the weight of all the particles below it.
When a cord or bar is subjected to a force of traction,
it stretches, and may even break. If it be equally strong
throughout, the rupture ought to take place simulta-
neously at all its points, and yet this is never found to be
the case in practice, and it is because bars and cords
are not homogeneous, and break at the weakest point.
When two pieces of cord of the same kind, are of the
same length, no reason can be assigned why one should
break rather than the other under the same resistance;
but when of unequal length, the chance of rupture is
in practice, cords greater for the longer; and this is the reason why cords
and ropes, which to all external appearances are the
same in kind, are generally found to be weaker as they
are longer.
and bars are
weaker as they
are longer.
§ 190. We have seen that when forces which act
upon the extremities of a cord are in equilibrio, the re-
sultant of those acting at one end, must be equal and
directly opposed to that of those acting at the other; and

MECHANICS OF SOLIDS.
273
forces which act
that their common line of direction must coincide with
that of the cord. The work of these resultants must The work of the
be equal, and hence we conclude that the work of the at the ends of a
forces which act at one end of a cord is equal to the cord must be
work of those which act at the other. The work of each
resultant must also be equal to that of the tension of the
cord at any one of its
points, as C; and to
find the value of this
work, it is only neces
sary to multiply this
tension by the path
described by the point
C in the direction of
Fig. 155.
σ
equal;
the tension. Thus the
In
the of the forces
applied to one
quantity of work of several forces applied to one end of a quantity of work
cord, is equal to the quantity of work of its tension.
example of the common device for ringing large
it is usual to attach to one end A of a rope, which
nects with the ma-
chinery of the bell,
several cords C, upon
each of which a man
may pull. It would
be difficult to estimate
the work performed
by each man, because
his effort, as well in
intensity as direction,
varies at each instant;
but there is a general
tension exerted upon
A
a
B
Fig. 156.
bells, end of a cord is
equal to that of
con- the tension;
example of the
bell-ropes;
the main rope, and the
quantity of work of this tension is equal to the sum of the
effective quantities of work of the several men. The effort
of each man is resolved into two components, one in the
direction of the main rope A B, the other perpendicular to
18

274
NATURAL PHILOSOPHY.
it. The perpendicular components must be in equilibrio,
while the parallel components are alone effective in pro-
ducing useful work. The perpendicular components only
produce fatigue, and
exhaust uselessly the
effect of
components
perpendicular to
the main rope of
the bell;
strength of the men.
And, although the to-
tal quantity of work is
transmitted to the main
rope, yet the disposi-
tion of inclined cords
is a source of real loss,
which is the greater in
proportion as the incli-
nation is greater. It
is for this reason that
effect of a hoop. a rigid hoop mn is so
Equilibrium of
several cords
meeting in a
point;
introduced as to sepa-
rate the cords, and give
m
C
B
A
Fig. 157.
the portions to which the efforts are immediately applied
parallel directions.
§191.-When several forces act upon cords which meet
in a point and are united by a knot, the tension of any
one is equal to the resultant of the efforts exerted upon
the others, and the equilibrium requires that this same
tension shall be equal and directly opposed to the force
which solicits the cord in question. Hence, when forces
are applied to cords which meet in a knot, the condition.
of their equilibrium requires that the effort of any one
shall be equal and directly opposed to the resultant of all
the others.
When a force P is applied to a point D, which may
slide along a cord whose ends A and B are fixed, the
equilibrium of a equilibrium of the point D requires that the direction of
the force P shall bisect the angle ADB formed by the
portions of the cord separated by the bend at D; for
sliding knot;

MECHANICS OF SOLIDS.
275
Fig. 158.
the force P must be equal and directly opposed to the
resultant of the tensions on DA and DB; but the
whole cord ADB be-
ing continuous, these
tensions must be equal,
since the tension is the
same throughout; if,
therefore, the distance
DC be laid off on
PD produced, pro-
portional to the inten-
sity P, and from C, the
lines Cm and Cn be
drawn parallel to DB
and DA respectively,
direction of the
force applied to
the knot, must
bisect the angle
of the two parts
D
of the cord;
P
the figure Cm Dn will be a rhombus, because Dm and
Dn, which represent the tensions, must be equal.
Fig. 159.
B
An example of this mode of action is furnished by the
manner of suspending a common lantern L from a small
pulley D, of which the
groove receives the
cord A DB, whose
ends are fastened to
hooks at A and B.
The weight of the lan-
tern will cause the pul-
ley to move till the
direction of the weight
bisects the angle made
by the branches of the
cord; the pulley will
722
P
example in the
mode of
suspending the
common lantern;
be in stable
equilibrium when
then come to rest and remain in a state of stable equilib- the pulley will
rium. The equilibrium will be stable because, being a
heavy system, the centre of gravity is the lowest possible; at the lowest
and to show this, it will be sufficient to remark that the
length of the entire cord being constant, the point D will,
when in motion, describe an ellipse of which A and B are the
point;

276
NATURAL PHILOSOPHY.
position of the
horizontal
tangent;
foci, and as the direc-
tion PC, of the weight
of the lantern, bisects
the angle AD B, it will
be perpendicular to the
tangent to the curve at
D, which must therefore
be horizontal, and no
point of the curve can
lie below it.
A
222
Fig. 159.
D
T
P
AI
when the pulley
is replaced by a
knot;
If the pulley be re-
moved and the lantern
be attached by a knot
arbitrarily to some point
Fig. 160.
(B
as D, the freedom of
motion will be destroyed,
tension will not the tension will no lon-
be the same
throughout;
the equilibrium
ger be the same through-
out, and the conditions
of equilibrium will be
those of forces applied
to three cords meeting
P
D
B
at a single point. Produce the vertical PD, and lay off
the conditions of DC to represent the weight of the lantern. Denote its
will be the same weight by W; the tension on DA by a, and that on DB
as those of three by b; the angle ADB by o, and ADC by 0; then,
drawing On and Cm, parallel respectively to DA and
DB, we have, from the parallelogram of forces,
oblique forces;
W: a::
:: sin
sin : sin (9-0),
tension on one
branch;
W: b ::
sin : sin 0;
whence
W. sin (9-0)
(94),
a=
sin

MECHANICS OF SOLIDS.
277
b =
W. sin 0
sin
(95).
tension on the
other;
If be less than -8, a will be greater than b; that is
to say, the tension will be the greater upon that branch with branch most
which the direction of the weight makes the least angle.
If the cord ADB be drawn into a straight horizontal
line, will become equal to 180°, the sine of which is zero,
and the tensions a and b will become infinite; in other
Φ
inclined has the
greatest tension;
words, there is no force sufficiently great to bring the no force sufficient
whole cord to a horizontal position.
to make the cord
horizontal.
$192.-Let us
now consider a
Fig. 161.
polygon ABCD,
N
A
composed of an
assemblage
of
cords or bars, and
acted upon at the D'
angular points by
the forces P, Q,
R, S. Moreover,
S
let N and N' be
R
B
To find
conditions of
A
equilibrium of
P
the funicular
polygon;
two forces draw-
ing on the points
A and D, in the
directions A A' and DD', respectively; these latter forces
will represent the efforts exerted at the two extremities
where the polygon is attached to fixed supports. The con-
ditions of equilibrium about each of the several angles are
the same as in the preceding case, and the figure formed by
the sides, in turning about the angular points to satisfy
them, is called a funicular polygon. This figure must be
such that the equilibrium will subsist at each angle.
therefore, any one of the forces, as R, be resolved into two
components in the directions of the sides DC and BC,
adjacent to its point of application, these components will
If, equilibrium must
subsist at each
angle;

278
NATURAL PHILOSOPHY.
length of sides;
and will subsist
when the sides
are zero;
be equal and directly opposed to the tensions of the
is independent of sides. The equilibrium is entirely independent of the
length of the sides, and will subsist when these are re-
duced to zero, in which case, all the forces and tensions
will be transferred parallel to their primitive directions to
the same point; and as each side is drawn by two equal
and contrary tensions, these latter will disappear or
destroy each other, so that the conditions of equilibrium
of several forces applied to a funicular polygon is, that these
forces shall remain in equilibrio when transferred parallel to
their primitive directions and applied to a single point.
conditions of
equilibrium in
words.
§ 193.-If all the forces P, Q, R, &c., be weights, and
the polygon in equilibrio, since the force R will be in
the plane of the
sides BC and
CD, adjacent to
the angle C; the
force equally
in the plane of
the sides BC and
When the forces AB; the sides
are parallel, the
polygon and
AB, BC, and
direction of forces CD, will be in
are in same
plane;
the polygon a
collection of
heavy bars;
the plane of the
parallel forces Q
and R. In the
same way it
may
be shown that the
entire polygon
and the forces
applied to it are
Fig. 162.
D'IN'
N
A
B
P
U
M
R
Ꮴ
D
S
in the same plane. If the polygon be a collection of
heavy bars, each side will be solicited by its own weight
in addition to the weights applied to the angles. Denote
by w the weight of the bar AB; this weight must pass
through the centre of gravity of A B. Resolve it into two

MECHANICS OF SOLIDS.
279
components acting at the extremities of the bar. If the
bar have the same cross section throughout and be of
homogeneous density, the components at A and B will be
resolved into
w. In like manner, if w' be the weight of the side BC, weight of sides
the components at B and C will be w', and so on for the parallel
other sides. Thus the angles B and C will be acted upon components;
by the weights (w+w') and (w'w') respectively,
that is, by the half sum of the weights of the adjacent
sides. The extreme ends will each be acted upon by half
the weight of the adjacent side; and thus we have but to
consider the polygon as without weight and solicited by
forces applied to its angular points. Since all the weights
P, Q, R, S, and the weights w, w', w', &c., are maintained
in equilibrio by the reaction N and N' of the fixed points,
which are equal to the tensions of the sides A'A and DD'
respectively, the resultant of these tensions must be equal resultant of
and directly opposed to that of all the weights. If, there-
fore, the lines A A' and DD' be produced, their inter- opposed to that
section O will give one point through which the resultant
of the weights P, Q, R, S, and that of the polygon, will
pass; and this resultant being vertical, if the distance OM
be laid off, by any scale of equal parts, so as to contain as
many linear units as there are pounds in P+Q+R+S+
extreme tensions,
equal and
of all the
weights;
w + w' + w", &c., and two lines MU and MV be drawn value of extreme
through M parallel respectively to A A' and DD', the dis- tensions found.
tances OV and OU will give, by the same scale, the
tensions at A' and D', or the values of N and N'.
If the polygon be only subjected to the action of its
own weight, the line OM may be drawn vertically
through its centre of gravity.
the tensions of
the sides;
§ 194.-It is often of great practical importance to Method of finding
know the tensions on the sides of a funicular polygon
subjected to the action of weights, in order to proportion
the dimensions of its several parts.
Let ABCDE be a polygon in equilibrio, under the
action of the weights P, Q, R, S, T, including the

280
NATURAL PHILOSOPHY.
weights of the sides, and
the extreme forces N
and N', of which the di-
funicular polygon rections are A A' and
in equilibrio
under the action EE', respectively. De-
of weights;
note the tension of the
side AB by t, that of
Fig. 163.
E
N
AN
A.
E
D
2
B C
t
13
BO by to, that of CD by
ta, &c. Since the equi-
librium subsists about
P Q R S
T
each angle, as A for example, the force N which acts from
A to A', is equal and directly opposed to the resultant of
the two forces P and t; and if An be taken on the pro-
longation of A'A to represent N, the parallelogram Apno,
constructed on An as a diagonal, will give Ap for the
determination of weight P, and pn for the value of the tension t. This
a single tension; being understood, draw the horizontal line a' e, upon which
lay off the distances a' a, ab, bc, cd, de, proportional to the
general
construction for
finding the
tensions;
weights P, Q, R, S, and
T. From the point a'
draw a' S perpendicular
to A A', and proportional
in length to the tension N,
and join S with the sev-
eral points a, b, c, d, and
e; then will a S, bs, cs,
d S, and eS, represent,
respectively, the ten-
sions t₁, to, to, t, and N'.
For the two triangles
Apn and a' Sa are sim-
N
Fig. 164.
S
P
N
N
ilar, because a' S and a' a are respectively perpendicular to
An and Ap; hence the angles Sa'a and p An are equal;
moreover, the sides about these equal angles are propor-
tional by construction and we, therefore, have
An
=
N pn t₁ ::
=
ta'S: Sa;

MECHANICS OF SOLIDS.
281
Ap
and if a' S represent the tension N, Sa must represent the
tension t. For the same reason, ab being proportional to demonstration;
Q, the third side b S, of the triangle a Sb, will be propor-
tional to to, since the three forces t, Q, and to, are in
equilibrio about the point B. Finally, since a a' and a' S
are perpendicular to the directions A p and An of the
forces P and N, a S will be perpendicular to the side A B
of which it measures the tension t. It will be the same
of B C and b S, and so on. Therefore, when a funicular lines which
polygon is in equilibrio under the action of weights, if a represent the
series of distances be taken on a horizontal line propor- perpendicular to
tional to these weights, the lines drawn through the points
of division perpendicular to the corresponding sides of polygon;
the polygon will meet in a point, and the lengths of these
perpendiculars, included between the common point of
intersection and the horizontal line, will measure the
tensions of the sides of the polygon. The point S is point of tensions.
called the point of tensions.
8195.-The sides of the polygon may be very short
tensions,
the sides of the
funicular
and only subjected to the action of their own weight, The catenary;
which would be the case with a heavy chain A CB sus-
pended from its
extremities. The
polygon of equi-
librium then be-
comes a curve,
called the catenary.
This curve is em-
ployed to give
form to arches and
domes. The use
of the catenary for
such
purposes may
be illustrated by
conceiving a series
ood
Fig. 165.
C'
its use in the
arts;
B
of equal spherical balls held together by mutual attrac-

282
NATURAL PHILOSOPHY.
illustration by a tions, but with perfect freedom to slide the one over the
other. Such a collection of balls would resemble a string
string of balls;
sides of the
polygon, the
chords of the
balls;
of beads, and if
supported at the
ends would, under
the action of their
own weights, as-
sume the form of
the catenary, or
rather funicular
polygon, of which
the sides would be
the chords of the
spheres joining the
points of contact.
If the whole ar-
rangement be re-
A
Fig. 165.
B
versed, and the balls, instead of being suspended, be sup-
ported upon the ends as fixed points, after the manner in-
the string of balls dicated in A' C'B', the figure will remain unchanged and
reversed;
points of contact
extended to
tangent planes;
arch-stones or
voussoirs ;
Fig. 166.
the balls will still be in equilibrio; for, the action of the
weights will be the same as before, and the reciprocal action
of the balls upon each other will simply be changed from a
force of extension to one of compression. If we now suppose
the points of contact to be extended into tangent planes,
and the spaces
between filled up
with solid matter,
as wood, stone, or
metal, we shall
have a perfect sys-
tem of youssoirs
or arch-solids in
equilibrio under
the action of their
own weight, re-
quiring no aid from friction or any other principle of sup-

MECHANICS OF SOLIDS.
283
joints;
port. The tangent planes or joints of the voussoirs will be position of the
normal to the curve. The catenary is also employed in
suspension-bridges supported upon two or more parallel also used in
chains stretched across a river. In the construction of suspension-
bridges.
such catenaries it is important to determine the tension at
the ends, in order to secure an adequate resistance at
those points.
catenary;
§ 196.-The catenary A CB, suspended from two General
points A and B, is nothing more, as we have seen, than properties of the
a heavy polygon in equilibrio, and whose sides are indefi-
nitely small; so that, if upon a horizontal line, a length
A'B' be taken proportion-
al to its weight, and this
length be divided into a
number of equal parts,
there will exist a certain
point & such, that all the
right lines drawn from it
to the points of division,
will be perpendicular to
the small successive sides
or elements of the cate-
nary, and that the lengths
SA', SF', SC', &c., of
these lines, are propor-
tional to the tensions of
the same elements. Of all
the tensions, the least is
given by the line SC',
drawn perpendicular to
the horizontal line A' B'.
But the element of the cat-
enary to which this tension
corresponds being itself
horizontal, it will occupy
Fig. 167.
B
I
D-
N
bd ģ
A
B' D' G
6.
a
d
the lowest point of the curve. This length becoming greater
construction to
find the tension
of the different
points of the
catenary;
least tension at
lowest point;

284
NATURAL PHILOSOPHY.
and greater in proportion as the oblique lines SF', &c.,
recede from the perpendicular SC', the tensions of the
elements of the catenary will increase in proportion as
tension greater they are at a greater distance from the lowest point.
Whence it follows, that
as the element is
at a greater
distance from the the tension is the greatest
lowest point;
possible at the extremi-
ties A and B. Two equal
tensions SF' and SG',
appertain to two elements
equally distant from the
lowest point C: moreover,
elements of equal these elements form equal
tensions form
equal angles with
the vertical
through lowest
point;
catenary
symmetrical in
reference to this
line;
are equal;
angles with the vertical
LC passing through this
point; hence, these ele-
ments, M and N, are situ-
ated on the same horizon-
tal line MN, and the chord
MN, as well as all similar
chords, will be divided
equally by this vertical
line. The catenary is,
therefore, a symmetrical
curve in reference to
a vertical line passing
through its lowest point.
It follows, also, that when
Fig. 167.
B
I
D
N
M
bd g
B'D' G
6.
ogoty
to enclaust
H10
a
A
the extremities or attached points A and B are on the
on same level the same horizontal line, the extreme tensions are equal, and
extreme tensions that the point of meeting which determines the tensions
position of the is upon the perpendicular drawn through the middle
point of tensions. of the horizontal line A'B', which is proportional to
the weight of the catenary. A and D being, for exam-
ple, the two points of suspension, and A'D' being the
length proportional to the weight of the catenary ACD,
SC', perpendicular to A'D' and passing through the

MECHANICS OF SOLIDS.
285
point S, will divide A' D' into two equal parts A' C'
and C' D'.
independent of
size;
§ 197.-Two catenaries, (last figure), A CB and a cb similar
are similar when the points of suspension A and B of the catenaries;
one, and a and b of the other, are situated upon parallel
right lines, and when their lengths A CB and acb are
proportional to the distances A B and ab, between their
points of suspension. If the equilibrium subsists in the
catenary A CB, this equilibrium will not be disturbed if the
length of its elements and its other dimensions be propor-
tionally diminished indefinitely, § 192. Therefore, when equilibrium
ACB is reduced to the size a cb, the equilibrium will not
only exist, but there will be no one of its parts which will
not be parallel and proportional to the corresponding part
of the original. But since the elements of the smaller
catenary acb are parallel to those of the larger A CB,
all the tensions of the former are comprised within the
angle A'SB', which contains the different tensions of the
latter. We have, then, but to find in this angle, the posi- tensions of one
tion of a line a'b' parallel to A' B', which represents the catenary found
weight of the smaller catenary, as A'B' represents the similar one;
weight of the larger, and the slightest consideration will
show that the two tensions Sf' and SF" situated upon the
same line converging to S will appertain to parallel ele-
ments of the two curves. These are called homogeneous homogeneous
tensions. But because A'B' and a' b' are parallel, we
have the proportion
Sf': : SF :: a'b': A'B';
from those of a
tensions;
whence we conclude that, in two similar catenaries, the ten- tensions of
sions of elements similarly situated are to each other as the
weights of the catenaries.
elements
similarly situated
are as the weights
of the entire
curves.

286
NATURAL PHILOSOPHY.
$198.-Let A'B'
To construct the be a horizontal line pro-
catenary from its
weight, length,
portional to the weight
and the point of of the catenary, S the
tensions;
to draw a tangent
to any point of
the cater ry.
point of tensions. Di-
vide the line A' B', and
the length of the cate-
nary into the same and
a great number of equal
parts; those of the cate-
nary may be regarded
as its elements, and
those of A'B' their
A
Fig. 168.
AS
A
13
B
corresponding weights. Draw the lines SA, S1, S2',
S3'... SB'; these will be perpendicular to the different
elements of the catenary. From any point A, on SA,
draw A 1 perpendicular to SA and equal to an element
of the catenary; from the point 1 draw 1-2 perpendicular
to S1' and equal to an element; again 2-3 perpendicular
to S2', and equal to an element, and so on to the end.
The polygon A-1-2-3... B, will approximate to the re-
quired catenary the nearer in proportion as the number of
divisions is greater.
The point of ten-
sions S gives the means
of drawing a tangent
to the catenary at any
point. Let E be the
given point, and let A'e
represent the weight of
the portion A E of the
catenary; through e
and S draw the indefi-
nite line e G, and from
E draw EG perpen-
dicular to e S, E G will
be the tangent line.
2
Fig. 169.
B
A
E
G
S
B

MECHANICS OF SOLIDS.
287
199.-The point
of tensions in the cate-
nary depends upon the
intensity and directions
of the extreme ten-
sions. For A'B' being
A
Fig. 170.
B'
the horizontal line proportional to the weight of the entire
catenary, if from the extremities A' and B' arcs be de-
scribed with radii proportional to the extreme tensions,
their intersection S will give the point of meeting.
The process for find-
ing the extreme ten-
sions must of course
depend upon the data
given. Let us first sup-
pose the catenary A CB
to be given and traced
out. It is evident from
the conditions of equi-
librium, that the verti-
Fig. 171.
A
I
B
G
B'
A
0
cal O L drawn through the intersection O of the extreme
tangents AO and BO, will pass through the centre of
gravity of the catenary. If, therefore, a distance O G be
taken on this line to represent the entire weight of the
catenary, and the parallelogram OB' GA' be constructed
upon the tangents, the sides O A' and O B' will represent
the tensions at A and B respectively.
But if only the two
points A and B of
suspension, the weight,
and entire length of the
catenary be given, the
process for finding the
extreme tensions is as
follows, viz.: Take a
small chain and sus-
pend it against a ver-
BA
D
d
Fig. 172.
C
A
Determination of
the point of
tensions;
to find the
extreme tensions
from the curve
traced;
to find the
extreme tensions
from the points
of support, the
weight, and
length of the
curve;

288
NATURAL PHILOSOPHY.
figure found by
means of a small
chain;
the tensions
found by
construction;
tical plane from two
Fig. 172.
points a and b, situated
B
RA
upon a right line paral-
lel to A B, and whose
distance apart shall be
to the distance from A
to B, as the length of
the smaller chain is
to the length of the
longer. The smaller
chain being thus sus-
pended, measure by
D
2
d
0
a
means of a spring balance the tension exerted at the points
a and b. The tensions on the points A and B produced by
the larger chain, will be equal to the tensions at a and b,
multiplied by the number of times which the weight of the
larger chain contains that of the smaller. § 197.
Fig. 173.
K
0
Instead of measuring with a spring balance the tensions
at the ends of the catenary, we may proceed as follows:
Draw through the low-
est point of suspen-
sion a, a horizontal
line cutting the oppo-
site branch of the small
chain in the point d.
Upon a horizontal line
take the distance a' b'
to represent the weight
of the entire chain, and
lay off the distance a' d'
proportional to the
length a cd. The por-
tion a cd of the cat-
enary would be in
b'
d'
equilibrio if the point d were fixed and the remainder db
removed; the point of tensions for a c d, and therefore for
a cb, will, from what has already been explained, be found

MECHANICS OF SOLIDS.
289
somewhere on the perpendicular C'K' drawn to the mid-
dle of a' d'; assume it at O, and by means of this point-
and the line a' b', construct a catenary after the manner construction of
of § 198, and let a e be the resulting distance between its an approximate
points of support. Through O draw a perpendicular to
C'K', and lay off upon it from the point 0, the distance
-
Og =αe a b, to the right when a e is greater than ab,
and to the left when the reverse is the case. Assume
another point as O' below 0, and do the same as before; we
shall find a new point g', say to the left of C' K'; repeat
the process with points between 0 and O' several times,
and pass through the points g, g', g', &c., thus determined,
a curve; its intersection S with C'K' will be the true
point of tensions. The distances Sa' and Sb' will repre-
sent the extreme tensions.
$ 200.-We have seen that in the
catenary the tensions at the different
points are different, and that the small-
est'tension is at the lowest point. This
is still true when the catenary becomes
a vertical chain loaded with a weight.
For the lowest link supports only the
attached weight Q; the link C' only
supports the weight and link C,
and so on to the topmost link, which
supports all below it; so that if the
chain were proportioned to the tension
of its different parts, it would be made
stronger above than below.
$201.-The point S being
the point of meeting of the
tensions, and A' B' a hori-
zontal line representing the
weight of the catenary, we
have seen that the tension at
Fig. 174.
Fig. 175.
S
C"
C
C
D'
C B
curve.
The smallest and
greatest tension
of a vertical
chain.
Direct measure of
the tension on
any point of the
catenary;
19

290
NATURAL PHILOSOPHY.
construction;
D is represented by the length D'S, and that at C
the lowest point, by SC', perpendicular to A'B', the
lengths A' D' and D'C' representing, respectively, the
weights of the portions
AD and DC of the
the tension at any curve; that is to say,
the tension at any
point is the
hypothenuse of a
right-angled
triangle;
point D, is represent-
Fig. 175.
S
tension in
horizontal
direction;
in vertical
direction;
ed by the hypothe-
nuse of a right-angled of
triangle, of which one
side represents the ten-
sion at the lowest point
of the curve, and the
other the weight of
that portion of the cate-
nary included between
the lowest point and the
point whose tension is
to be found. Hence, the
tension at any point of
the curve, estimated in a
horizontal direction, is
A
D'
C'
B'
Fig. 176.
A
DM
N
B
constant and equal to the entire tension at the lowest point;
and estimated in the vertical direction, is equal to the weight of
that portion of the catenary included between this point and the
lowest point.
The horizontal tensions at A and B are therefore the
same, although they may be situated on very different
levels. If the catenary be suspended from the tops of
piers, the vertical components will promote their stability
tensions on piers. by pressing them down, while the horizontal components
will tend to overturn them.
effect of these
§ 202.-It is comparatively easy to compute the ex-
treme tensions of the catenary when the versed sine of its
arc is small. Let A CB be a catenary, of which CD, the

MECHANICS OF SOLIDS.
291
tensions when
distance of the lowest point below the horizontal line BA, To find extreme
is very small. The curve being in equilibrio, the equi- the versed sine of
the curve is
small;
Fig. 177.
B
D
A
G
H
G'
C
librium of the part BC will not be disturbed by taking the
point C as fixed, and regarding it and the point B as the
points of suspension. But because of the smallness of
DC, the curvature must be very small, and the centre of
gravity of BC may, without sensible error, be regarded as
at the middle point G. The tangents CH and BG', at the notation;
points of suspension, will intersect at G' on a vertical line
drawn through the point G. Denote by T, the tension at
B; by To, the tension at C; and by p, the weight of the
portion BC.
Because the three forces p, T, and To, are in equilibrio
about the point G', we have
P: To :: BH: HG',
whence
PT::
BH: BG';
HG'
T = PBH'
T
=
P
.
BG'
BH
Observe that BH is the versed sine, which denote by f;
and, because B G C may be regarded a right line, HG' is
half the semi-space BD, which semi-space denote by 7.
Then, since the triangle B G'H is right angled,
72
BG' =VBH + G'H³
=
√√F + I
tension at lowest
point;
tension at highest
point;

292
NATURAL PHILOSOPHY.
horizontal
tension or thrust;
Substituting these quantities in the above equations, we
find
tension at highest
point.
T =
Application to
suspension-
bridge;
To
=
72
4
pl
2f'
=
72
p
+
The first expresses the tension at the lowest point, which
we have seen is equal to the horizontal thrust at the points
of suspension. The second gives the entire tension at the
same points, which must be known in order to adjust the
dimensions of the chain.
§ 203. To conclude the subject of the catenary, and
show the application of the preceding principles, take the
case of a bridge suspended from two parallel chains
extended from one bank of a river to the other.
To the different
points, A, B, C, &c.,
of the catenaries, or
Fig. 178.
rather to the angles of
*
the funicular polygons
thus formed, are attach-
A
suspending rods; ed vertical suspending
rods, which are united
at the bottom in pairs
by transverse pieces
called sleepers; these
receive a set of longi-
tudinal joists, which, in their turn, support the floor plank.
The distances between the suspending pieces in longitudi-
nal direction are supposed equal. These equal portions
of the roadway included between two consecutive sleep-
ers, are called sections. Each sleeper is loaded with half
the section which precedes and half that which follows it;
that is to say, with the weight of an entire section. This
joists;
sections;

MECHANICS OF SOLIDS.
293
weight is known, and
determines the cross sec-
tion of the suspending
rods. The weight of the
suspenders being small
compared with that of
the roadway, may be
neglected, and thus the
weight of the bridge
will be equally distribu-
ted.
Draw a horizontal
right line, and take
uv proportional to the
weight of the bridge;
let S be a point such
U
Q
00
Fig. 179.
TT
P
A B
K
N
M
A B' D'
S
v
το
that Su shall be perpendicular to the side UA, and
proportional to its tension. Take upon uv, the portions
ua, ab, &c., proportional to the weights supported at the
angles A, B, &c.; the converging lines a S, b S, &c., will be
proportional to the tensions on the sides A B, &c., and the
perpendicular Sd, to the tension on the horizontal side of
the polygon. First, find the difference of level between any
two consecutive angles, as A and B. Draw the horizontal
line BA", and the two triangles A A"B and Sad, will be
similar and give
geinained bas
AA"
whence
mateib
A" B: ad: Sd;
ad: Sd;S OR T
each pair of
suspending rods
supports the
weight of one
section;
tensions on the
sides of the
funicular
polygon;
AA"
=
A" B
a d.
Sd
A han
difference of level
between two
Because of the equality of distances between the sus-
pending rods, A" B will be constant. Moreover, a d and
Sd being proportional respectively to the weight of the
portion A' D', and the tension to upon the horizontal side,
consecutive
angles;

294
NATURAL PHILOSOPHY.
ratio of weight of
half the bridge to
the horizontal
tension;
if we denote by the weight of a unit of length of the
bridge,
a d
S'd
=
wA'D'
to
;
which in the preceding gives
WA" B.A' D'
A A"
=
;
to
but w A" B is the weight of a section of the bridge
Denoting this by p, we have
A A"
1=1
P. A' D';
to
value of the
difference of level
of two
consecutive
angles;
and denoting the constant ratio of the weight p to the
tension to at the lowest point by k,
AA" k. A' D';
=
from which we conclude, that the difference of level
of two consecutive angles, is equal to the constant ratio
k, multiplied by the horizontal distance of the higher of
the two angles from the lowest angle of the funicular poly-
gon. Denoting by the constant length of a section,
and beginning at the lowest angle K, the horizontal
distances will be successively 7, 21, 31.
nl, for the
1st, 2d, 3d,... nth, angle to the right and left. Thus the
difference of level between the lowest angle K and the
next in order C, is kl; between
C and B, 2 kl; between B
and A, 3 kl, &c. The heights
difference of level of the angles C, B, A, &c.,
above the lowest point K, will
of the angles of
the polygon
above the lowest be respectively kl, kl+2 kl,
angle;
kl+2kl+3 kl, kl + 2kl+
3 kl+4 kl, and, in general,
M
A
•
Fig. 180.
B
K
N

MECHANICS OF SOLIDS.
295
height of the nth
angle above the
lowest one;
if there be n sections between the lowest angle and that un-
der consideration, the height of the latter above the former
will be given by the expression
n=
kl (1+ 2+ 3+ 4.... + n) = kl.n
n+1
od 2
In this expression, if we make successively n = 1, n = 2,
= 3, n = 4, &c., we have kl, 3 kl, 6 kl, 10 kl, &c.,
for the heights of 1st, 2d, 3d, 4th, &c., angles above the
horizontal side of the funicular polygon.
The locus of all these
angles is a parabola, for
if y = KP MU de-
=
note the height of one
of these angles above
the lowest point K, n
U
0
Fig. 181.
P
being the number of its
the locus of the
angles is a
parabola;
place from the latter, we
have
M
H
KK'
N
2n+1
y = kl.n
(96);
2
and making
or
30
nl = x = KM,
x
= k. (" + ¹) ∞ = 1/2 (x + 1) 17/7;
y =
2
X=
k x²
y =
k
+
27
,;
this is the equation of a parabola, of which the vertex is
equation of the
locus of the
angles;

296
NATURAL PHILOSOPHY.
place of the
vertex of the
locus curve;
to the right of the point K, and at a distance from it
equal to
X=
-
7
2
it is below the horizontal side by the distance
HK' = y=
-
7cl
8
a quantity so small that
it may be neglected in
practice.
Moreover, from the
property of the parabola,
the squares of the ordi-
nates are to each other
to find the point as the abscisses; that is
Fig. 182.
Q
B
0
TT
A
P...
in which the
M
K
vertical through
to say,
the vertex cuts
the line of
supports;
Ꮲ AP² :
: BQ
MA
MA: NB;
and from the similar triangles obtained by joining A
and B,
A P²
2
:
BQ PO: Q02;
: P02
whence
P03
Q02: MA: NB;
or
but
P02 × NB = Q02 × MA;
X
PO OK - KP OK - MA,
=
–
QO= QK - OK = NB - KO,

MECHANICS OF SOLIDS.
297
which, substituted above, give
(OK-MA) x NB = (NB - KO). MA;
developing the squares and reducing, we get
K02 = MAX NB.
vertex;
That is to say, the distance KO, at which the vertical line distance of this
drawn through the vertex of the curve cuts the chord point above the
joining any two of its points, is a mean proportional be-
tween the heights of these points above the vertex. This
property furnishes an easy method of finding the lowest
point K on the level MN. For this purpose, join the points
of suspension U and V,
by the cord UV; draw
the horizontal line UP
through the lower point
U, and produce it till it
cuts the vertical VN in
P'. Upon the distance
P' V describe the semi-
circle VTP', and from
the point N draw the
tangent NT; with N as
Fig. 183.
Q
V
0
T'
T
P
P
M
K
N
construction for
finding the
position of the
lowest point;
a centre and NT as a radius, describe the arc TT" till it
cuts VN in T", and through the point T' draw a hori-
zontal line; this line will cut the cord UV in the point 0,
through which draw a vertical line OP, and its intersection
with the horizontal side will give the lowest point K.
Taking this point as the extremity of the horizontal side,
and laying off on the line MN the equal lengths of the and the abscisses
sections; the points of division will correspond to the of the angular
vertical ordinates kl, 3 kl, 6 kl, . . . n.
n+1
2
kl. This
last appertaining to the point U, whose height h is given,
points;

298
NATURAL PHILOSOPHY.
ratio of the
weight of a
section to
horizontal tension
found;
from which the
lengths of the
suspenders are
known;
and horizontal
tension found;
we have
whence we have
n. (n + 1) k l = h;
2
2h
k =
n(n + 1); •
(97);
and hence the lengths of the several suspenders k 1, 3 kl.
&c., are known.
We have seen that
and therefore
p
to
2 h
-
k
=
n (n + 1) l'
=
n(n + 1). l.p
2 h
(98);
the tension on the horizontal side is, therefore, also known.
The tension on the side next in order to the horizontal
side is
√₁₁² + p²,
tension on side
next in order;
that of the second in order
that on the
√t + (2p)²,
second in order;
that of the third
that on third;
√t² + (3 p),
and so on to
that on the nth in
order;
√t² + (np)²,

MECHANICS OF SOLIDS.
299
which is the tension on the nth side from the horizontal
one.
log
If the points U and V be on the same level, it is
obvious that the curve or polygon becomes symmetrical
in reference to the vertical O K, in which case it is only
necessary to find the lengths of the suspenders for one data necessary to
half the bridge. Having given the points of suspension, make known the
their horizontal distance apart, and the level of the lowest bridge.
side of the funicular polygon, it is easy to determine the
dimensions of every part of the bridge.
dimensions of the
XV.
OF BODIES RESTING UPON EACH OTHER, AND
UPON INCLINED PLANES.
in the
Let us
reaction of bodies
§ 204.-When two bodies touch and compress each Action and
other, there is immediately a depression or yielding in a forced into
direction perpendicular to the surfaces at the point of apparent contact;
contact, which indicates that the reaction of the two bodies
takes place in the same direction; that is to say,
direction of the normal common to both surfaces.
suppose one of the two bodies as A to
be solicited by forces of which the re-
sultant shall coincide with this nor-
mal, and that the other body A' is
fixed; it is plain that the reaction of
the latter body will destroy this resul-
tant, and that the body A will remain
at rest. But the equilibrium will also
subsist if the body A' be replaced by a
Fig. 184.
A
A
action and
force equal to the reaction which it exerts on the body A,
while this latter body is perfectly free to move and acted
upon by this new force in conjunction with the given forces.
This property of all bodies, by which they resist the re-
reaction of two
bodies;

300
NATURAL PHILOSOPHY.
the reaction of
two bodies
extends to
several.
ciprocal action of each other in directions normal to both
the principle of surfaces at the common point of contact, extends to the
general case of a single body pressing upon two or more
bodies at the same time. The reaction of these last are so
many real forces which may be substituted for the resisting
bodies at the several points of contact, and in virtue of this
substitution, the conditions of equilibrium of the first body
will be the same as though it were free to move in any
direction whatever. Let us examine the circumstances of
the simple case of a body resting upon a plane, and having
first but one point of contact, then two, three, &c.
Illustration;
$ 205.-Let us consider a
sphere subjected to the action
of its own weight, and rest-
the bodies having ing upon a level plane A B
but one point of with a single point of contact
contact;
Fig. 185.
F
G
m. Since the reaction takes
place in the direction of the
perpendicular to the plane
through the point of contact,
A
B
m
and must be in equilibrio with the weight W of the
sphere, the centre of gravity G must be upon a vertical
line, in order that the weight and reaction may destroy
each other. In like manner, when a body rests upon any
plane whatever, and is solicited by forces, no matter how
directed, their resultant must be perpendicular to the plane,
and pass through the point of contact; for if the resultant
were oblique, it might be resolved into two components,
one normal, and the other parallel to the plane; the first
would be destroyed by the reaction of the plane, while the
latter would put the body in motion. In order, therefore,
conditions which that a body, supported against a plane, and having a
will keep a body single point of contact with it, shall be in equilibrio, it is
necessary, 1st, that the resultant of the forces which act upon
it be perpendicular to the plane; and 2d, that this resultant
pass through the point of contact.
at rest against a
plane.

MECHANICS OF SOLIDS.
301
§ 206. But when the body
has two points of contact, A
and B, with the plane, it is not
necessary that the resultant of
the forces shall pass through
either. It will be sufficient if
it meet the line AB in any
A
Fig. 186.
If the body have
two points of
contact the
resultant need
not pass through
either;
B
point between A and B, and be perpendicular to the
plane. For the reaction of these points of support being
both perpendicular to the plane, their resultant, which is
parallel to them, will also be perpendicular to it: this
resultant and that of the forces acting upon the body
must be in equilibrio; they must, therefore, be equal and
directly opposed; in other words, the resultant of the
forces acting upon the body must admit of being resolved it must be normal
into two components, respectively equal and directly op- intersect the line
posed to the resistances at the points of support. But joining the points
these latter act in the same direction, so also must the for-
mer, and hence their resultant will have its point of appli-
cation between A and B; and this resultant being parallel
to its components, will be perpendicular to the plane.
to the plane, and
of contact ;
horizontal.
If the body be laid on a horizontal plane, the equi- when the plane is
librium will subsist whenever the vertical drawn through
the centre of gravity intersects the line joining the points
of support somewhere between them.
more points;
§ 207.-Now let us suppose three or more points of
contact. The resistances of these points are perpendicular case of three or
to the plane, and cannot maintain the forces which act upon
the body in equilibrio unless the resultant of the latter
may be decomposed into components which are respectively
equal and directly opposed to these resistances; this
resultant must, therefore, be perpendicular to the plane,
and as its components must act in the same direction, resultant still
its point of application will, from the principles of parallel
forces, be within the polygon formed by joining the points polygon of
of contact. If the line of direction of the resultant, pierce
normal, and
within the
contact;

302
NATURAL PHILOSOPHY.
if the resultant
the plane in a point m, ex-
pierce the plane terior to the polygon which
without the
polygon of
connects the points of support,
contact, the body the body will tend to overturn
will overturn;
around the edge ab of this
Fig. 187.
m
polygon nearest to m; if the
line of contact be a curve, the
body will overturn about the
tangent nearest to m. The ef-
fort by which the body will
be urged to overturn is meas-
ured by the intensity of the
resultant of the forces, into
the shortest distance from its
the body is urged line of direction to that about
effort by which
to overturn.
Examples;
which the motion of rotation takes place.
Fig. 188.
Fig. 189.
§ 208. The conditions of equilibrium of a heavy
sphere, resting upon a horizontal plane, have already
been considered. Let us ap-
ply the same principles to
other examples, and take first
the case of a heavy body
resting upon a table having
but three feet. If the feet be
upon a horizontal plane and
when the feet are in the same right line, and the
vertical line through the
centre of gravity be not in
table having but
three feet;
in same right
line;
will overturn
pass through this
line;
the vertical plane passing
through this line, the table
unless the weight will overturn towards the
side on which the centre of
gravity is situated, and with
an effort equal to the product
of the weight into the distance
Ag of the projection of the
Fig. 190.
G
AJ

MECHANICS OF SOLIDS.
303
centre of gravity from the line a a' of rotation.
This
product is called the moment of stability. If the distance moment of
Ag is zero, the weight will pass through the line of stability;
support, and there will be an equilibrium; but it will
be unstable, since the centre of gravity will be at the
highest point.
If the three feet be not in the same right line, and the if the feet be not
weight pass within the tri-
angle formed by joining the
feet, the table will be in
equilibrio. But if the line
of direction of the weight
pass without the triangle of
the feet, the table will over-
turn about the nearest edge
a b.
In the first case, the
equilibrium is stable, because
no derangement can take
place about the line of either
two of the feet without caus-
ing the centre of gravity to
ascend. And, generally, if
the table have any number
of feet, there will be stable
equilibrium whenever the
line of direction of the weight
passes within the polygon
formed by joining them.
The effort with which the
table or any other body will
resist a cause which tends to
upset it, is measured by the
product of its weight into the
shortest distance Ag from the
line of direction of the weight
to the line ab about which
the motion is to take place;
a
A
B
Fig. 191.
in same right
line;
Fig. 192.
stable
equilibrium;
G
Fig. 193.
a
A
Fig. 194.
2
in case of any
number of feet
the resultant
must pass within
the polygon;
effort by which
a body resists a
cause to overturn
it;

304
NATURAL PHILOSOPHY.
moment of
stability of a
heavy body;
the same
principles apply
to solids resting
and this effort will be smaller in proportion as the dis-
tance Ag is less. For this reason, the moment of stability
of a heavy body is the smallest moment of its weight
taken with reference to the different lines of its polygonal
base.
The conditions are the
same if the body rest upon
a plane face bounded by a
on plane faces; polygon or curve. The equi-
librium will exist when the
line of direction of the weight
passes within the base. Such,
for example, is the case with
the cube resting upon a level
plane; also with a right prism,
whatever its height, only that
its stability diminishes as the
height increases; for, in pro-
portion as the centre of grav-
ity G is more and more ele-
vated, the angle GAB be-
comes less and less, and the
centre of gravity will not have
to be raised so much above its
position of rest when the body
is overturned about the edge
example of the
cube and right
prism;
stability
diminishes as the
centre of gravity
is higher;
Fig. 195.
Fig. 196.
B
a a', as it would if the angle G AB were greater, or the
centre of gravity lower. In proportion as the centre of
gravity is placed higher and higher above the same base,
the body will approach more and more to the condition of
unstable equilibrium.
An inclined prism will pre-
serve its equilibrium as long
Inclined prism; as the direction of its weight
falls within its base. The dif-
ficulty of overturning it will
be less in proportion as the
Fig. 197.

MECHANICS OF SOLIDS.
305
distance Ag becomes smaller. blbow
When g falls without the base,
the prism will overturn of
itself. The Tower of Pisa,
though considerably inclined,
preserves its equilibrium be-
cause the line of direction of
its weight passes within its
base. A pile of dominos or
bricks, in which each one pro-
jects beyond that immediately
below it, will preserve its
equilibrium till the line of
direction of the weight of the
entire pile falls without the
A
Fig. 198
will overturn
when weight falls
without the base;
Tower of Pisa;
Fig. 199.
inclined pile of
brick;
increases and as
domino or brick at the bottom, when it will overturn. stability increases
We see, therefore, that the natural stability of bodies in- as the base
creases as their bases increase, and the heights of their the centre of
centres of gravity decrease; and that it is the greatest possi- gravity is lower;
ble when the centre of gravity is at the centre of figure of
the base. This is the reason why walls are usually made
of elements like brick, cut-stone, &c., placed with their
faces vertical, and laid upon large bases, called foundations.
solicited by other
forces than their
If the heavy bodies are solicited by other forces than
their weights, the resultant of the whole, weight included, heavy bodies
must act in the direction of a line passing within the base.
The resultant of the extraneous forces may unite with the weights;
weight and increase the stability of the body. Thus an
inclined prism, the direction
Gg of whose weight falls
without the base AB, would,
if abandoned to itself, over-
turn; whereas, if it were act-
ed
upon by a force in the di-
rection GE, of such intensity
as to give, with the weight,
a resultant which intersects
Fig. 200.
E
G
Ᏼ Ꮽ
these may act to
increase the
stability;
20

306
NATURAL PHILOSOPHY.
equilibrium
stable;
w the base at 0, it would be
supported, and the equilib-
rium would be stable. Re-
ciprocally, the weight W of
Fig. 200.
the prism is opposed to their EG
force GE=F, when the latter
acts to turn the solid about
the edge A. The measure of
this opposing effort is
A 0 B
moment of
stability;
W. Ag;
foregoing in the
construction of
principle of
counterforts;
and in this view, we see that the moment of the natural
stability will increase as Ag increases. od to
Fig. 201.
B
In walls destined to support an embankment of earth or
idad a head of water, in order to resist the thrust with greater
illustration of the effect, the lower exterior edge A is thrown as far as con-
venience will permit from the
sustaining walls; vertical line Gg of the weight.
This is done either by an
exterior slope B A, or by
masses of masonry C, called
counterforts, attached to the
back of the wall. It will be
sufficient, in general, for the
stability of the wall, if the
resultant of its weight W and
the pressure against it, inter-
sects the base A D. The
moment of natural stability
natural stability; of such structures is always
equal to the product of the
weight into the distance Ag;
and therefore the figure of
the cross-section of the wall
may be varied at pleasure
without injury to the sta
moment of
A
g
D
Fig. 202.
B
G
A
ID

MECHANICS OF SOLIDS.
307
bility, provided this product remain the same. Hence the
and weight;
external slope may be suppressed, if the thickness of the external slope
wall be so increased that its augmented weight shall com-
pensate for the diminution in A g.
If the ground upon which
the wall rests be compressible,
it will not be sufficient that
the resultant of the weight
and pressure pass within the
base; it must also pass through
its centre of figure; otherwise
there would be more pressure
on one side of this point than
on the other, and the wall oviy
would incline in that direc-
tion.quiq
If the load of a two-
wheel cart be such that
the direction of its weight
does not intersect the axle-
tree, it will tend to overturn
on the side of the weight,
and will either exert a pres-
sure upon the horse or an
effort to lift him from the
ground, according as the
weight passes in front or in
rear of the axle-tree. If the
centre of gravity of the load
be immediately above the
axle-tree on a level road,
then, when the cart is as-
cending a slope, the weight
will pass behind, and the ten-
dency of the load will be to
lift the horse; while, on the
contrary, when the cart is bu
Fig. 203. 0088
A
F G
A
ed on
0
D
Fig. 204.
digrol
G
Fig. 205.
of ybod ant
Sote iw dilglow
Fig. 206.
when the ground
is compressible,
the resultant
should intersect
middle of the
base;
case of a loaded
cart on a level;
on an inclined
road ascending
the tendency is
to lift the horse;
descending;

308
NATURAL PHILOSOPHY.
the tendency is descending a slope, the tendency of the load will be to
to press upon the throw a pressure upon him. If the centre of gravity be
on the axle-tree, the horse will experience no effort of the
kind referred to.
horse.
$209.-Let A B represent
the section of an inclined
Inclined plane; plane in the direction of its
greatest declivity. Although
the plane be indefinitely pro-
longed, it will be sufficiently
defined by the relation of the
of height to base; base AC to the height CB,
corresponding to a given
length A B.
defined by ratio
a body on an
inclined plane;
the body may
slide or roll;
conditions of
equilibrium;
G
Fig. 207.
B
G
m
Conceive a heavy body resting upon this plane, and of
which G is the centre of gravity. The equilibrium of this
body requires, 1st, that its weight shall intersect the plane
within the polygon formed by joining the points of con-
tact; 2d, that the weight shall be perpendicular to the
plane. This last condition cannot be satisfied for any but
a horizontal plane, since the weight is always vertical. If
the weight be replaced by its two components, one perpen-
dicular and the other parallel to the plane, the former will
be destroyed by the resistance of the plane, while the latter
will cause the body to move in the direction of its length
BA. If the direction of the weight meet the plane within
the polygon of contact, the parallel component will cause
the body to slide, otherwise it will cause it to roll. This
last will happen in the case of a spherical ball, since the
weight will not meet the plane in the single point of con-
tact m.
Let a force P be applied in the direction GS, next figure,
to prevent the body from moving down the plane. Since
the body must be in equilibrio under the action of its weight
W and the force P, these must have a resultant, and this
resultant must be perpendicular to the plane and intersect

MECHANICS OF SOLIDS.
309
be applied;
it within the polygon of contact, or in the case of the plane in which
sphere, at the point m. The force P must, therefore, be the force must
applied in a vertical plane which passes through the
centre of gravity, and which is, at the same time, perpen-
dicular to the inclined plane.
Fig. 208.
S
Lay off on the vertical through the centre of gravity
G, the distance G G' to represent the weight W, through
the same point draw G M
perpendicular to the inclined
plane, and through G', the
line G' M parallel to the di-
rection of the force P; from
the point M draw MQ paral-
lel to G G'; the distance G Q
will represent the intensity
of the force P, and GM that
of the resultant, R, of W and
P. From the principle of the
parallelogram of forces, we have
A
TIL
M
B intensity of the
C
force found by
construction;
intensity of the
W: R: P:: sin QG M: sin G' GQ: sin G'GM; force found
but G G and GM being respectively perpendicular to
A C and A B, the angle A is equal to the angle G' G M,
and we have
analytically;
sin G' G M
11
sin BAC=
=
BC
A Bi
and this substituted in the foregoing proportion gives, after
reduction,
W: R: P:: A B. sin QG M: AB. sin G' GQ: BC;
from which we find
P = W.
Ᏼ Ꮯ
AB. sin QG M
(99);
value of the
force;
=
R W
sin G' GQ
sin QG M
value of the
(100).
pressure against
the plane;

310
NATURAL PHILOSOPHY.
ability at any If the power P be ap-
os to dogy
Fig. 209.
power applied
T
S
Briga
Q
parallel to the
plane;
plied parallel to the
plane, the angle QG M
=
90°; and the angle
G' GQ becomes the sup-
plement of the angle ords
ABC; whence we have
M
A
IC
sin Q G M = sin 90°
sin G' GQ = sin ABC
=
=
iniog
1;
A C
ABi
which, in the above equations, give
value of force;
value of the
pressure against
the plane;
Ᏼ Ꮯ
PW.
AB'
A C
R = W
AB
That is to say, when the power is applied parallel to the
relation of power, plane, 1st, the power will be to the weight as the height of the
weight, and
resistance of
plane;
power applied
parallel to the
base;
plane is to its length; 2d, the resistance of the plane will be to
the weight as the base of the plane is to its length.
If the power be ap-
plied parallel to the base
Fig. 210.
B
of the plane, the angle D
QGM becomes equal to
the angle ABC, because
GQ and G M are respec-
tively perpendicular to
BC and AB; and the
angle G'GQ becomes
90°, whence
G
e
M
A
relation of the
angles;
AC
sin Q G M = sin ABC =
AB'
sin G' GQ = 1;

MECHANICS OF SOLIDS.
311
which, in Eqs. (99) and (100), give
P = W
.
BC
A C'
AB
R = W
.
A C
value of power;
to noi
pressure on
plane;
That is to say, when the power is applied parallel to the base
of the plane, 1st, the power will be to the weight as the height relation of power,
of the plane is to its base; 2d, the resistance of the plane will weight, and
be to the weight as the length of the plane is to its base.
resistance;
which the power
will be less than
In the application of the power parallel to the plane,
the power will always be less than the weight. When
applied parallel to the base, the power will be less than limits within
the weight, while the inclination of the plane is less than
45°. When the inclination is 45°, the power and weight weight.
will be equal. When the inclination exceeds 45°, the
power will be greater than the weight.
§ 210.-Let us now
consider the motion of a
heavy body on the in-
clined plane. The body
being acted upon by its
weight G G alone, this
may be resolved into two
components, the one GM,
perpendicular, the other
G N, parallel to the plane.
The first will be totally
NK
Fig. 211.
B
G
M
G
A
C
Motion of a
heavy body on
an inclined
plane;
destroyed by the resistance of the plane, while the second
will be effective in giving motion. Denote the weight of
the body by W, the height B C of the plane by h, and its to find the
length A B by 7; then, from the similarity of the triangles component of
ABC and G G' N, will
W: GN: 1: h;
the weight
parallel to the
plane;

312
NATURAL PHILOSOPHY.
whence
its value;
GN=
and because the incli-
nation of the plane is
the same throughout, the
h
ratio will be constant,
ī
from the top to the bot-
tom; whence we see that
the motion is that the motion of the same
arising from the
action of a
body down the plane, is
constant force; that arising from the ac-
=
tion of a constant force.
h
W;
A
Fig. 211.
B
G
N
M
C
it will be
uniformly
varied;
It will, therefore, be uni-
formly varied, and the circumstances of motion will be
given by the laws of constant forces.
Substituting Mg for W, we have
h
GN =
Mg;
component of the
force of gravity in
direction of the
plane;
the motion may
be regulated by
varying the
inclination of the
plane;
and making M equal to unity, and denoting by g' the
corresponding value of the component G N, we find
h
g'
=
T. 9.
Such is the intensity of the force of gravity in the direc-
tion of the inclined plane. This may be varied at pleasure
by changing the ratio
h
in other words, by altering the
inclination of the plane.
pressed during the first unit of time on the same body,
moved from rest, are proportional to the forces producing
them, the motion may be made as slow as we please by
Now, since the velocities im-
diminishing
h
T
It was in this that Galileo discovered
way

MECHANICS OF SOLIDS.
313
Galileo
the laws which regulate the fall of heavy bodies. These in this way
being the same as for bodies moving on an inclined plane, discovered the
it was easy so to regulate the inclination of the plane as to laws of falling
enable him to note and compare the spaces described, times
elapsed, and velocities acquired, with each other.
If the body be
mounted upon wheels,
as in the case of the
loaded cart referred to
in § 208, it will be
urged to roll along the
inclined plane by an
effort of which the
measure is
W.D;
in which W denotes
the weight of the cart
and its load, and D
G
כא
b
Fig. 212.
TIL
bodies;
when the body
is mounted on
wheels it will
roll;
example of the
loaded cart;
the perpendicular distance m b from the point of contact moment of the
m, to the line of direction Gb of the weight W.
effort by which
rotation is
produced.
XVI.
FRICTION AND ADHESION.
§ 211.--When two bodies are pressed together, expe- Friction;
rience shows that a certain effort is always required to
cause one to roll or slide along the other.
This arises manifested when
surfaces of two bodies are
pressed together
almost entirely from the inequalities in the
contact interlocking with each other, thus rendering it and one is moved
necessary, when motion takes place, either to break them over the other;
off, compress them, or force the bodies to separate far
enough to allow them to pass each other. This cause of
resistance to motion is called friction, of which we distin-

314
NATURAL PHILOSOPHY.
sliding and
rolling friction;
the measure of
its intensity.
Intensity
measured by
spring balance;
guish two kinds, according as it accompanies a sliding or
rolling motion. The first is denominated sliding, and the
second rolling friction. They are governed by the same
laws; the former is much greater in amount than the
latter under given circumstances, and being of more import-
ance in machines, will principally occupy our attention.
The intensity of friction, in any given case, is measured
by the force exerted in the direction of the surface of
contact, which will place the bodies in a condition to resist,
during a change of state, in respect to motion or rest, only
by their inertia.
8212. The friction between two bodies may be meas-
ured directly by means of the spring balance. For this
purpose, let the
surface CD of
one of the bod-
ies M, be made
perfectly level,
so that the oth-
er body M',
when laid upon
C
M'
Fig. 213.
B
E
A
F
M
D
it, may press
with its entire weight. To some point, as E, of the body
M', attach a cord with a spring balance in the manner
the indication of indicated in the figure, and apply to the latter a force F
the balance, when
the motion is
uniform is the
measure;
of such intensity as to produce in the body M' a uniform
motion. The motion being uniform, the accelerating and
retarding forces must be equal and contrary; that is to say,
the friction must be equal and contrary to the force F, of
which the intensity is indicated by the balance.
The experiments on friction which seem most entitled
to confidence, are those performed at Metz by M. Morin,
the most valuable under the orders of the French government, in the years
1831, 1832, and 1833. They were made by the aid of
contrivance, first suggested by M. Poncelet, which is on
of the most beautiful and valuable contributions th
experiments are
those of M. Morin;

MECHANICS OF SOLIDS.
315
theory has ever made to practical mechanics. Its details where these
are given in a work by M. Morin, entitled "Nouvelles experiments may
Expériences sur le Frottement." Paris, 1833.
The following conclusions have been drawn from these
experiments, viz.:
be found;
these
experiments;
The friction of two surfaces which have been for a conclusions from
considerable time in contact and at rest, is not only differ-
ent in amount, but also in nature from the friction of sur-
faces in continuous motion; especially in this, that the
friction of quiescence is subjected to causes of variation
and uncertainty from which the friction during motion is
exempt. This variation does not appear to depend upon
the extent of the surface of contact; for, with different
pressures, the ratio of the friction to the pressure varied
greatly, although the surfaces of contact were the same.
accompanies
considered;
The slightest jar or shock, producing the most imper- in machinery,
ceptible movement of the surfaces of contact, causes the the friction which
friction of quiescence to pass to that which accompanies motion to be
motion. As every machine may be regarded as being
subject to slight shocks, producing imperceptible motions
in the surfaces of contact, the kind of friction to be em-
ployed in all questions of equilibrium, as well as of motions
of machines, should obviously be this last mentioned, or
that which accompanies continuous motion.
The LAWS of friction which accompanies continuous the laws of this
motion are remarkably uniform and definite. These laws friction are
are:
uniform and
definite;
1st. Friction accompanying continuous motion of two first law;
surfaces, between which no unguent is interposed, bears a
constant proportion to the force by which those surfaces
are pressed together, whatever be the intensity of the force.
2d. Friction is wholly independent of the extent of the second law;
surfaces in contact.
3d. Where unguents are interposed, a distinction is to be
made between the case in which the surfaces are simply third law;
unctuous and in intimate contact with each other, and that
in which the surfaces are wholly separated from one another

316
NATURAL PHILOSOPHY.
influence of
unguents;
an apparent
exception to
second law;
three conditions
of the surfaces in
respect to
friction;
fourth law;
remarkable
instance of the
uniformity of
these laws;
by an interposed stratum of the unguent. The friction in
these two cases is not the same in amount under the same
pressure, although the law of the independence of extent
of surface obtains in each. When the pressure is in-
creased sufficiently to press out the unguent so as to bring
the unctuous surfaces in contact, the latter of these cases
passes into the first; and this fact may give rise to an
apparent exception to the law of the independence of the
extent of surface, since a diminution of the surface of con-
tact may so concentrate a given pressure as to remove the
unguent from between the surfaces. The exception is
however but apparent, and occurs at the passage from one
of the cases above-named to the other. To this extent,
the law of independence of the extent of surface is, there-
fore, to be received with restriction.
There are then three conditions in respect to friction,
under which the surfaces of bodies in contact may be
considered to exist, viz.: 1st, that in which no unguent
is present; 2d, that in which the surfaces are simply
unctuous; 3d, that in which there is an interposed stratum
of the unguent. Throughout each of these states the
friction which accompanies motion is always proportional
to the pressure, but for the same pressure in each, very
different in amount.
4th. The friction, which accompanies motion, is always
independent of the velocity with which the bodies move;
and this, whether the surfaces be without unguents or
lubricated with water, oils, grease, glutinous liquids,
syrups, pitch, &c., &c.
The variety of the circumstances under which these
laws obtain, and the accuracy with which the phenomena
of motion accord with them, may be inferred from a single
example taken from the first set of Morin's experiments
upon the friction of surfaces of oak, whose fibres were
parallel to the direction of the motion. The surfaces of
contact were made to vary in extent from 1 to 84; the
forces which pressed them together from 88 to 2205

MECHANICS OF SOLIDS.
317
pounds; and the velocities from the slowest perceptible
motion to 9.8 feet a second, causing them to be at one time
accelerated, at another, uniform, and at another, retarded;
yet, throughout all this wide range of variation, in no result;
instance did the ratio of the pressure to the friction differ
from its mean value of 0.478 by more than 4 of this same
fraction.
Denote the constant ratio of the normal pressure P, to
the entire friction F, by f; then will the first law of fric- first law
tion be expressed by the following equation,
expressed by an
equation;
F
p = f
whence
(101);
F= f. P.
friction;
This constant ratio f is called the coefficient of friction, coefficient of
because, when multiplied by the total normal pressure,
the product gives the entire friction.
Assuming the first law of friction, the coefficient of
friction may easily be obtained by means of the inclined
plane. Let W denote the
weight of any body placed
upon the inclined plane
AB. Resolve this weight
GG' into two components,
one GM perpendicular to
the plane, and the other
parallel to it. Because the
Fig. 214.
G
N
angles G' GM and BAC
are equal, the first of these
components will be
Λ
W. cos A,
M
B
its value found
by means of the
inclined plane;
component of the
weight
perpendicular to
the plane;

318
NATURAL PHILOSOPHY.
that parallel to
the plane;
and the second,
W. sin A,
Fig. 214.
the friction on
the plane;
friction and
parallel
in which A denotes the
angle BA C.
The first of these com-
ponents determines the total
pressure upon the plane,
and the friction due to this
pressure, will be
(101)
words
B
G
N
M
A
of eit
G
f. W cos A.
The second component urges the body to move down
the plane. If the inclination of the plane be gradually
increased till the body move with uniform motion, the
total friction and this component must be equal and
opposed; hence
f. W. cos A = W. sin A;
lof.
component
equal;
whence
value of the
coefficient of
friction;
sin A
f=
=tan A.
cos A
We, therefore, conclude, that the unit or coefficient of
friction between any two surfaces, is equal to the tangent
of the angle which one of the surfaces must make with the
horizon in order that the other may slide over it with a
constant velocity, the body to which the moving surface
belongs being acted upon by its own weight alone. This
angle of friction; angle is called the angle of friction or limiting angle of
limiting angle of
resistance;
resistance.
The values of the unit of friction and of the limiting
angles for many of the various substances employed in the
art of construction, are given in the following tables:

MECHANICS OF SOLIDS.
319
TABLE I.
EXPERIMENTS ON FRICTION, WITHOUT UNGUENTS. BY M. MORIN.
The surfaces of friction were varied from .03336 to 2.7987 square feet,
the pressures from 88 lbs. to 2205 lbs., and the velocities from a scarcely
perceptible motion to 9.84 feet per second. The surfaces of wood were
planed, and those of metal filed and polished with the greatest care, and
carefully wiped after every experiment. The presence of unguents was
especially guarded against.
SURFACES OF CONTACT.
FRICTION OF
MOTION.*
FRICTION OF
QUIESCENCE.t
Coefficient
of Friction.
Limiting
Angle of
Resistance.
Coefficient
of Friction.
Limiting
Angle of
Resistance.
Oak upon oak, the direction of the
fibres being parallel to the motion
Oak upon oak, the directions of the
fibres of the moving surface being
perpendicular to those of the quies-
cent surface and to the direction of
the motion+
Oak upon oak, the fibres of both sur-
faces being perpendicular to the
direction of the motion
Oak upon oak, the fibres of the
moving surface being perpendicular
to the surface of contact, and those
of the surface at rest parallel to
the direction of the motion -
Oak upon oak, the fibres of both sur-
faces being perpendicular to the
surface of contact, or the pieces f
end to end
Elm upon oak, the direction of the
fibres being parallel to the motion
Oak upon elm, dittos
Elm upon oak, the fibres of the mov-
ing surface (the elm) being perpen-
dicular to those of the quiescent
surface (the oak) and to the direc-
tion of the motion -
0.478 250 33' 0.625 320 I'
0.324 17 58 0.540 28 23
0.336 18 35
0.192 10 52 0.271 15 ΙΟ
0.43
23 17
0.432 23 22
0.246 13 50
0.694 34 46
0.376 20 37
0.450 24 16 0.570 29 41
*The friction in this case varies but very slightly from the mean.
The friction in this case varies considerably from the mean. In all the experi-
ments the surfaces had been 15 minutes in contact.
The dimensions of the surfaces of contact were in this experiment .947 square feet,
and the results were nearly uniform. When the dimensions were diminished to .043, a
tearing of the fibre became apparent in the case of motion, and there were symptoms of
the combustion of the wood; from these circumstances there resulted an irregularity in
the friction, indicative of excessive pressure.
It is worthy of remark that the friction of oak upon elm is but five-ninths of that
of elm upon oak.

320
NATURAL PHILOSOPHY.
TABLE I.-continued.
SURFACES OF CONTACT.
FRICTION OF
MOTION.
FRICTION OF
QUIESCENCE.
Coefficient
of Friction.
Limiting
Angle of
Resistance.
Coefficient
of Friction.
Limiting
Angle of
Resistance.
Ash upon oak, the fibres of both sur-
faces being parallel to the direction
of the motion
Fir upon oak, the fibres of both sur-
faces being parallel to the direction
of the motion
Beech upon oak, ditto
0.400 21° 49' 0.570 29° 41'
0.355 19 33 0.520 27 29
27 56
Wild pear-tree upon oak, ditto
0.360 19 48
0.370 20 19
Service-tree upon oak, ditto
0.400 21 49
0.53
0.440 23 45
0.570 29 41
Wrought iron upon oak, ditto*
0.619 31 47
0.619 31 47
Ditto, the surfaces being greased and
well wetted -
0.256 14 22
0.649 33 O
0.252 14 9
Wrought iron upon elm -
Wrought iron upon cast iron, the
fibres of the iron being parallel to
the motion --
Wrought iron upon wrought iron, the
fibres of both surfaces being par-
allel to the motion -
Cast iron upon oak, ditto
Ditto, the surfaces being greased and
Cast iron upon elm
Cast iron upon cast iron
Ditto, water being interposed be-
tween the surfaces -
Cast iron upon brass -
Oak upon cast iron, the fibres of the
wood being perpendicular to the
direction of the motion
Hornbeam upon cast iron-fibres par-
allel to motion -
Wild pear-tree upon cast iron-fibres
parallel to the motion
Steel upon cast iron -
Steel upon brass
Yellow copper upon cast iron
0.194 10 59 0.194 10 59
0.138
7 52 0.137 7 49
0.490 26
7
0.646 32 52
0.195 11 3
0.152 8 39 0.162
0.314 17 26
9 13
0.147 8 22
2 23
0.372 20 25
0.189 10 49
0.394 21
31
0.436 23 34
0.202 II 26
0.152
8 39
oak
0.617 31 41 0.617 31 41
0.217 12 15
Brass upon wrought iron, the fibres of
0.161 9 9
0.172 9 46
0.201 II 22
Ditto
Brass upon cast iron -
the iron being parallel to the mo-
tion
Wrought iron upon brass
Brass upon brass. -
* In the experiments in which one of the surfaces was of metal, small particles of
the metal began, after a time, to be apparent upon the wood, giving it a polished
metallic appearance; these were at every experiment wiped off; they indicated a
wearing of the metal. The friction of motion and that of quiescence, in these experi-
ments, coincided. The results were remarkably uniform.

MECHANICS OF SOLIDS.
321
SURFACES OF CONTACT.
TABLE I.-continued.
Coefficient
of Friction.
FRICTION OF
MOTION.
Limiting
Angle of
Resistance.
Coefficient
of Friction.
FRICTION OF
QUIESCENCE.
Limiting
Angle of
Resistance.
Black leather (curried) upon oak*
Ox hide (such as that used for soles
and for the stuffing of pistons) upon
oak, rough
0.265 14° 51' 0.74 36° 31'
0.52
27 29 0.605 31
II
23 17
Ditto ditto ditto, smooth 0.335 18 31 0.43
Leather as above, polished and har-0.296 16 30
dened by hammering -
Hempen girth, or pulley-band, (sangle
de chanvre,) upon oak, the fibres of
the wood and the direction of the
cord being parallel to the motion
Hempen matting, woven with small
cords, ditto -
Old cordage 14 inch in diameter, ditto
Calcareous oolitic stone, used in build-7
ing, of a moderately hard quality,
called stone of Jaumont-upon the
same stone -
Hard calcareous stone of Brouck, of)
a light gray color, susceptible of
taking a fine polish, (the muschel-
kalk,) moving upon the same stone
The soft stone mentioned above, upon
the hard - -
0.52
27 29 0.64
32 38
0.32
17 45 0.50
26 34
0.52
27 29 0.79
38 19
0.64
32 38 0.74
36 31
0.38
20 49 0.70
35
0
0.65
33 2 0.75
36 53
The hard stone mentioned above, up-
on the soft
0.67
33 50 0.75
36 53
Common brick upon the stone of Jau-
mont -
-0.65
33 2 0.65
33 2
Oak upon ditto, the fibres of the wood
being perpendicular to the surface
0.38
20 49 0.63
63 32
32 13
of the stone
Wrought iron upon ditto, ditto
0.69
Iron,
Common brick upon the stone of Brouck 0.60
Oak as before (endwise) upon ditto
34 37 0.49
26 7
30 58 0.67
33 50
0.38
20 49 0.64
32 38
ditto
ditto -
0.24
13 30 0.42
22 47
The friction of motion was very nearly the same whether the surface of contact
was the inside or the outside of the skin.-The constancy of the coefficient of the friction
of motion was equally apparent in the rough and the smooth skins.
+ All the above experiments, except that with curried black leather, presented the
phenomenon of a change in the polish of the surfaces of friction-a state of their sur-
faces necessary to, and dependent upon, their motion upon one another.

322
NATURAL PHILOSOPHY.
TABLE II.
EXPERIMENTS ON THE FRICTION OF UNCTUOUS SURFACES.
By M. MORIN.
In these experiments the surfaces, after having been smeared with an
unguent, were wiped, so that no interposing layer of the unguent prevented
their intimate contact.
SURFACES OF CONTACT.
MOTION.
FRICTION OF
FRICTION OF
Coefficient
of Friction.
Limiting
Angle of
Resistance.
Coefficient
of Friction.
QUIESCENCE.
Limiting
Angle of
Resistance.
Oak upon oak, the fibres being paral-0.108
lel to the motion
Ditto, the fibres of the moving body
being perpendicular to the motion
Oak upon elm, fibres parallel -
Elm upon oak, ditto -
Beech upon oak, ditto
Elm upon elm, ditto -
Wrought iron upon elm, ditto
Ditto upon wrought iron, ditto
Ditto upon cast iron, ditto -
6° 10' 0.390 21° 19'
0.143 8 9 0.314 17 26
0.136
7 45
0.119 6 48 0.420 22 47
0.330 18
18 16
0.140 7 59
0.138
7 52
0.177 IO 3
0.118 6 44
0.143 8 9
0.160 9 6
Cast iron upon wrought iron, ditto
Wrought iron upon brass, ditto
Brass upon wrought iron
Cast iron upon oak, ditto
-
Ditto upon elm, ditto, the unguent
being tallow
Ditto, ditto, the unguent being hog's
lard and black lead -
Elm upon cast iron, fibres parallel
0.166
9 26
0.107 6 7 0.100 5 43
0.125 7 8
0.137
7 49
Cast iron upon cast iron
Ditto upon brass -
Brass upon cast iron
Ditto upon brass -
Yellow copper upon cast iron
Copper upon oak
Leather (ox hide) well tanned upon)
cast iron, wetted
Ditto upon brass, wetted
0.135 7 42 0.098 5 36
0.144 8 12
0.132
7 32
0.107 6 7
0.134 7 38 0.164 9 19
0.100 5 43
0.115 6 34
0.229 12 54 0.267 14 57
0.244 13 43
The distinction between the friction of surfaces to
which no unguent is present, those which are merely
unctuous, and those between which a uniform stratum of
the unguent is interposed, appears first to have been
remarked by M. Morin; it has suggested to him what

MECHANICS OF SOLIDS.
323
between the
and of Coulomb;
appears to be the true explanation of the difference be-
tween his results and those of Coulomb. He conceives, cause of the
that in the experiments of this celebrated engineer, the discrepancy
requisite precautions had not been taken to exclude un- results of Morin
guents from the surfaces of contact. The slightest unc-
tuosity, such as might present itself accidentally, unless
expressly guarded against-such, for instance, as might
have been left by the hands of the workman who had
given the last polish to the surfaces of contact-is sufficient
materially to affect the coefficient of friction.
illustrative of
this;
Thus, for instance, surfaces of oak having been rubbed example
with hard dry soap, and then thoroughly wiped, so as to
show no traces whatever of the unguent, were found by
its presence to have lost ds of their friction, the co-
efficient having passed from 0.478 to 0.164.
upon surfaces
This effect of the unguent upon the friction of the effect of friction
surfaces may be traced to the fact, that their motion upon without
one another without unguents was always found to be at- unguents;
tended by a wearing of both the surfaces; small particles of
a dark color continually separated from them, which it was
found from time to time necessary to remove, and which
manifestly influenced the friction: now with the presence
of an unguent the formation of these particles, and the
consequent wear of the surfaces, completely ceased. In-
stead of a new surface of contact being continually pre-
sented by the wear, the same surface remained, receiving
by the motion continually a more perfect polish.

324
NATURAL PHILOSOPHY.
TABLE III.
EXPERIMENTS ON FRICTION WITH UNGUENTS INTERPOSED. BY M. MORIN.
The extent of the surfaces in these experiments bore such a relation to
the pressure, as to cause them to be separated from one another throughout
by an interposed stratum of the unguent.
SURFACES OF CONTACT.
FRICTION FRICTION
OF
OF
MOTION. QUIESCENCE.
UNGUENTS.
Coefficient
of
Friction.
HH
Coefficient
of
Friction.
Oak upon oak, fibres parallel
0.164
Ditto
ditto
0.075
0.440
0.164
Dry soap.
Tallow.
Ditto
ditto
0.067
Ditto, fibres perpendicular
0.083
0.254
Tallow.
Ditto
ditto
Ditto
ditto
9000
0.072
0.250
Hogs' lard.
Hogs' lard.
Water.
Ditto upon elm, fibres parallel
0.136
Ditto ditto
0.073
0.178
Dry soap.
Tallow.
Ditto
ditto
0.066
Hogs' lard.
Ditto upon cast iron, ditto -
0.080
Tallow.
Ditto upon wrought iron, ditto
0.098
Tallow.
Beech upon oak, ditto
0.055
Tallow.
Elm upon oak, ditto
0.137
0.411
Ditto ditto
0.070
0.142
Ditto
ditto
0.060
Dry soap.
Tallow.
Hogs' lard.
Ditto upon elm, ditto
0.139
0.217
Ditto upon cast iron, ditto -
0.066
Dry soap.
Tallow.
Wrought iron upon oak, ditto
0.256
0.649
Ditto ditto ditto
0.214
Ditto ditto
ditto
0.085
0.108
Ditto upon elm,
ditto
0.078
Ditto ditto
ditto
0.076
Ditto ditto ditto
0.055
Ditto upon cast iron, ditto -
0.103
Ditto ditto ditto
0.076
Ditto
ditto ditto
0.066
0.100
Ditto upon wrought iron, ditto
0.082
Ditto
ditto
ditto
0.081
Ditto ditto ditto
0.070
0.115
Wrought iron upon brass,
fibres parallel-
0.103
Ditto
ditto
ditto
0.075
Ditto
ditto
ditto
0.078
Cast iron upon oak,
ditto
0.189
Ditto ditto
ditto
0.218.
0.646
Ditto ditto ditto
0.078
0.100
Ditto
ditto ditto
0.075
Ditto ditto
ditto
0.075
0.100
Ditto upon elm,
ditto
0.077
Greased, and
saturated with
water.
Dry soap.
Tallow.
Tallow.
Hogs' lard.
Olive oil.
Tallow.
Hogs' lard.
Olive oil.
Tallow.
Hogs' lard.
Olive oil.
Tallow.
Hogs' lard.
Olive oil.
Dry soap.
(Greased, and
saturated with
water.
Tallow.
Hogs' lard.
Olive oil.
Tallow.

MECHANICS OF SOLIDS.
325
TABLE III.-Continued.
SURFACES OF CONTACT.
FRICTION
FRICTION
OF
OF
MOTION. QUIESCENCE.
Coefficient
of
Friction.
Coefficient
of
Friction.
UNGUENTS.
Cast iron upon elm-fibres
0.061
parallel-
Ditto ditto ditto
0.091
Olive oil,
Hogs' lard and
plumbago.
su
Ditto, ditto upon wrought iron
0.100
Tallow.
Cast iron upon cast iron
0.314
Water.
Ditto
ditto
0.197
Soap.
Ditto
ditto
0.100
0.100
Tallow. l
Ditto
ditto
0.070
0.100
Hogs' lard.
Ditto ditto
0.064
Olive oil.
Ditto ditto
0.055
Ditto upon brass
0.103
Ditto ditto
0.075
Ditto ditto
0.078
Copper upon oak, fibres parallel
0.069
0.100
Yellow copper upon cast iron
0.072
0.103
(Lard and
plumbago.
Tallow.
Hogs' lard.
Olive oil.
Tallow.
Tallow.
Ditto ditto
0.068
Hogs' lard.
Ditto ditto
0.066
Olive oil.
Brass upon cast iron
0.086
0.106
Ditto ditto
0.077
Ditto upon wrought iron
0.081
Ditto
ditto
0.089
Ditto
ditto
0.072
Ditto upon brass -
0.058
Steel upon cast iron
0.105
0.108
Ditto ditto
0.081
Ditto
ditto
0.079
Ditto upon wrought iron
0.093
Ditto ditto
0.076
Ditto upon brass -
0.056
Ditto
ditto
0.053
Ditto
ditto
0.067
Tanned ox hide upon cast iron
0.365
Ditto
ditto
0.159
Ditto ditto
0.133
0.122
Ditto upon brass
Ditto ditto
Ditto upon oak
Hempen fibres not, twisted,
moving upon oak, the fibres
of the hemp being placed in
a direction perpendicular to
the direction of the motion,
and those of the oak parallel
0.241
0.191
0.29
0.79
0.332
0.869
water.
Tallow.
Olive oil.
Tallow.
(Lard and
plumbago.
Olive oil.
Olive oil.
Tallow.
Hogs' lard.
Olive oil.
Tallow.
Hogs' lard.
Tallow.
Olive oil.
(Lard and
plumbago.
Greased, and
saturated with
water.
Tallow.
Olive oil.
Tallow.
Olive oil.
Water.
Greased, and
saturated with
to it

326
NATURAL PHILOSOPHY.
TABLE III.-continued.
FRICTION FRICTION
OF
OF
SURFACES OF CONTACT.
MOTION. QUIESCENCE.
UNGUENTS.
The same as above, moving
upon cast iron
Ditto ditto
Soft calcareous stone of Jau-
mont upon the same, with a
layer of mortar, of sand, and
lime interposed, after from
10 to 15 minutes' contact
Coefficient
of
Friction.
0.194
0.153
Coefficient
of
Friction.
0.74
Tallow.
Olive oil.
conclusions in
regard to olive
oil and lard;
tallow not so well
A comparison of the results enumerated in the above
table leads to the following remarkable conclusion, easily
fixing itself in the memory, that with the unguents hogs'
lard and olive oil interposed in a continuous stratum between
them, surfaces of wood on metal, wood on wood, metal on
wood, and metal on metal, when in motion, have all of them
very nearly the same coefficient of friction, the value of that
coefficient being in all cases included between 0.07 and 0.08,
and the limiting angle of resistance therefore between 4°
and 4° 35'.
For the unguent tallow the coefficient is the same as the
suited to metal. above in every case, except in that of metals upon metals; this
unguent seems less suited to metallic surfaces than the others,
and gives for the mean value of its coefficient 0.10, and for its
limiting angle of resistance 5° 43'.
Adhesion;
§ 213.-Besides friction, there is another cause of re-
sistance to the motion of bodies when moving over one
another. The same forces which hold the elements of
bodies together, also tend to keep the bodies themselves
together, when brought into sensible contact. The effort
by which two bodies are thus united, is called the force of
Adhesion.

MECHANICS OF SOLIDS.
327
the force of
adhesion;
Familiar illustrations of the existence of this force are illustrations of
furnished by the pertinacity with which sealing-wax, wa-
fers, ink, chalk, and black-lead cleave to paper, dust to
articles of dress, paint to the surface of wood, whitewash
to the walls of buildings, and the like.
olden fur
depends upon the
The intensity of this force, arising as it does from the its intensity
affinity of the elements of matter for each other, must vary extent of the
with the number of attracting elements, and therefore with surface of
the extent of the surface of contact.
contact;
This law is best verified, and the actual amount of ad-
hesion between different substances determined, by means measured by the
of a delicate spring-balance. For this
purpose, the surfaces of solids are reduced
to polished planes, and pressed together to
exclude the air, and the efforts necessary
to separate them noted by means of this
instrument. The experiment being often
repeated with the same substances, having
different extent of surfaces in contact, it
is found that the effort necessary to pro-
duce the separation, divided by the area
of the surface, gives a constant ratio.
Thus, let S denote the area of the surfaces
of contact expressed in square feet, square
inches, or any other superficial unit; A,
the effort required to separate them, and
a the constant ratio in question, then will
Fig. 215.
O
spring balance;
mode of
operation;
or,
A
S
= αy
A
= a. S.
inamele
ilon
The constant a is called the unit or coefficient of adhesion, coefficient of crozib
and obviously expresses the value of adhesion on each adhesion;
unit of surface, for making
aidil at
S = 1,

328
NATURAL PHILOSOPHY.
we have
adhesion between
solids and
liquids:
mode of
ascertaining its
amount in any
case;
A =α.
Fig. 216.
To find the adhesion between solids and liquids, sus-
pend the solid from the balance, with its polished surface
downward and in a horizontal posi-
tion; note the weight of the solid,
then bring it in contact with the hor-
izontal surface of the fluid and note
the indication of the balance when
the separation takes place, on draw-
ing the balance up; the difference
between this indication and that of
the weight will give the adhesion;
and this divided by the extent of sur-
face, will give, as before, the coeffi-
cient a. But in this experiment two
opposite conditions must be carefully
noted, else the cohesion of the ele-
ments of the liquid for each other
may be mistaken for the adhesion of
the solid for the fluid. If the solid
precaution to be on being removed take with it a
observed;
layer of the fluid; in other words, if the solid has been
wet by the fluid, then the attraction of the elements of the
solid for those of the liquid is stronger than that of the
elements of the liquid for each other, and a will be the
attraction of fluid unit of adhesion of two surfaces of the fluid. If, on the
contrary, the solid on leaving the fluid be perfectly dry,
those of solids; the elements of the fluid will attract each other more pow-
erfully than they will those of the solid, and a will denote
the unit of adhesion of the solid for the liquid.
elements for each
other and for
diversity in the
in this respect;
It is easy to multiply instances of this diversity in the
action of bodies action of solids and fluids upon each other. A drop of
water or spirits of wine, placed upon a wooden table or
piece of glass, loses its globular form and spreads itself

MECHANICS OF SOLIDS.
329
diversity;
over the surface of the solid; a drop of mercury will not
do so. Immerse the finger in water, it becomes wet; in
quicksilver, it remains dry. A tallow-candle or a feather illustration of this
from any species of water-fowl remains dry though dipped
in water. Gold, silver, tin, lead, &c., become moist on
being immersed in quicksilver, but iron and platinum do
not. Quicksilver when poured into a gauze bag will not
run through; water will: place the gauze, containing the
quicksilver in contact with water, and the metal will also
flow through.
surfaces with
lard, oil, &c.;
from a tumbler;
Solids which become wet on being immersed in a fluid, effect of covering
lose this property if covered with any matter not similarly
affected by that particular fluid. A drop of water placed
upon a wooden table or piece of glass, smeared with oil or
tallow, will not spread, but retain its globular shape and
roll off, if the surface be sufficiently inclined. Pour water
from a clean common glass tumbler nearly full, and it will
run along the exterior surface; smear the rim with hogs' illustrated in the
lard or tallow, and the fluid will flow clear of the tumbler. flow of water
The living force with which the elements of the water in
contact with the glass tend to leave the tumbler by the
pressure from behind, is, in a great measure, overcome by
the attraction between the glass and water, and they are
thus made to flow along the surface, while the viscosity of explanation;
the water, or the attraction of the fluid particles for
each other, drags the remote elements after them; and thus
the water, under the combined action of its living force,
adhesion for the glass and viscosity, becomes spread out
into a sheet of which the plane is normal to the surface of
the tumbler. When the tumbler is smeared with grease,
the adhesion is so much reduced as to offer but feeble
opposition to the living force with which the water reaches
the edge of the tumbler, it will, therefore, pass the edge
after the manner of a projectile. Quicksilver poured out case of
of a glass or wooden vessel will, in like manner, flow clear quicksilver
of the outer surface; but the contrary will happen if a tin different kinds of
vessel be used.
poured from
vessels;

330
NATURAL PHILOSOPHY.
effect of
interposing a
surfaces in
contact;
The adhesion of solids is apparently increased by intro-
ducing a liquid between them. The fluid fills up the ex-
fluid between isting inequalities of the surfaces, and thus, by increasing
the number of points of contact, increases the adhesion by
an amount equal either to that of the fluid particles for
each other, or to that of the fluid for the solid for which
it has the least affinity, depending upon whether the solids
are wetted or not by the interposed fluid. This is strikingly
exemplified by means of common window-glass, blocks of
wood, metallic plates, and the like.
it is difficult to
find the adhesion
between the
of machinery;
It is difficult to ascertain the precise value of the force
of adhesion between the rubbing surfaces of machinery,
rubbing surfaces apart from that of friction. But this is attended with little
practical inconvenience, as long as a machine is in motion.
The experiments of which the results are given in the table
of § 212, and which are applicable to machinery, were
made under considerable pressures, such as those with
which the parts of the larger machines are accustomed to
move upon one another. Under such pressures, the adhe-
sion of unguents to the surfaces of contact, and the oppo-
sition to motion presented by their viscosity, are causes
whose influence may be safely disregarded as compared
with that of friction. In the cases of lighter machinery,
except in watches however, such as watches, clocks, and the like, these con-
siderations rise into importance, and cannot be neglected.
this adhesion
may be
disregarded;
and the like.
Friction on a
plane;
normal
component of
the weight;
8 214.-Let any body M, rest with one of its faces in
contact with the inclined plane A B. Denote its weight
by W, and suppose it to be solicited by a force F in
the direction GQ, making with the inclined plane the
angle QGq', which denote by o. Denote the inclination
BAC of the plane to the horizon by a. Resolve the
weight WG G' into two components, Gp and G p', one
perpendicular and the other parallel to the plane. The
angle G' Gp being equal to the angle BA C, the first of
these components will be,
W. cos a;

MECHANICS OF SOLIDS.
331
M
Fig. 217.
g
B
A
a
C
P
and the second,
W. sin a.
In like manner, resolve the force F= GQ, into two com-
ponents Gq and Gq', the first normal and the second
parallel to the plane. The first of these will be,
parallel
component of
the weight;
F. sin;
and the second
F. cos q.
The total pressure upon the plane will be
W. cos a
-
- F. sin o;
and the friction thence arising
normal
component of
the force;
its parallel
component;
pressure upon
the plane;
f(W. cos
COS a
-
F. sin );
in which ƒ denotes the coefficient of friction. The force
which solicits the body in the direction of the plane
corresponding
friction;

332
NATURAL PHILOSOPHY.
whole force in
direction of the
will be,
plane;
F. cos o
W. sin a.
This will tend to
accelerate the body; the friction will
tend to retard it. When they are in equilibrio, the body
will either have a uniform motion or be just on the eve
of motion; which condition will therefore be expressed by
force necessary to
hold the body in
equilibrio, or to
keep it in
uniform motion
up the plane;
to find under
what angle to the
F. cos o
-
W sin a = f(W cos a
f(W cos a F. sin q);
- -
whence
F=
W(f cos a + sin a)
cos +f. sin q
.
(102).
Here the force F will be the smallest possible, or will be
applied under the most advantageous circumstances, when
the denominator is the greatest possible, since all the
quantities in the numerator are constant. To ascertain
the relation between the quantities of the denominator
to satisfy this condition,
draw GQ making with
the plane A B the angle
QGB equal to ; from
9;
G lay off the distance
Gb equal to unity, and
draw be perpendicular
to A B; then will
Fig. 218.
A
h
d
eB
plane this force
may be applied
to greatest
advantage;
G c = cos 9,
bc = sin q.
Take the distance Ge equal to f, and we have
analy
ed = fsin o. dt silos

MECHANICS OF SOLIDS.
333
Make Gh equal to ed, and there will result
h c = cos o + f. sin o,cogr
which is the value of the denominator in Eq. (102). Draw
Gk perpendicular to GQ, and erect at h a perpendicular
to A B, then, because the angle k Gh is the complement
of BGQ=9, will
kh Gh cot ;
or, substituting the value of Gh, as given above,
consda
value of the
denominator;
kh = f.sin o.
cot o = ƒ cos q.
pro-
Join k and b, and it will be obvious that he is the
jection of the line kb on AB, and that this projection
will be the greatest possible when kb is parallel to AB;
that is, when kh and be are equal; which condition is
expressed by the equation,
ƒ cos o = sin
or
f=
sin
COS O
= tan o;
the value of the
tangent of this
angle;
that is to say, the power will be applied to the greatest ad- conclusion;
vantage, when its direction makes with the inclined plane an
angle of which the tangent is equal to the coefficient of the
friction between the plane and the body on it.
If the plane be horizontal, the angle a will be zero, and
Eq. (102) reduces to
F=
Wf
cos of sin p
value of the force
when plane is
horizontal;

334
NATURAL PHILOSOPHY.
when in
equilibrium on
eve of motion
Finally, if the body is to be retained in equilibrio on
the eve of motion up the plane, the condition for this
down the plane; purpose is given by Eq. (102) as it stands, but if the equi-
librium is maintained on the eve of motion down the plane,
the friction will act in aid of the force F, and the equation
becomes
the value of the
force;
W (sin a
α
—
- f cos a)
F' =
(103);
cos of sin o
whence it follows, that there are an indefinite number of
different values for the force between F and F'' which will
maintain the body in equilibrio on the plane. If the body
be in motion up the plane, the force whose intensity is F
infinity of forces will make it uniform; if in motion down the plane, the
that will maintain force whose value is F" will make it uniform. The im-
portance of this will be perceived when we come to treat
of the screw.
the equilibrium.
on the inclined
plane;
§ 215. The inclined plane is one of the most useful
machines employed in the arts, and facilitates the trans-
Quantity of work portation of the heaviest burdens to considerable eleva-
tions. To build a stone wall, for instance, to any height,
the labor of many men would be required to elevate the
necessary materials in a vertical direction, whereas that
of a few accomplishes the same end over a ramp or
inclined plane whose slope is sufficiently gentle to admit
the easy passage of
men, horses, carts, &c.
Burdens are convey-
ed up inclined planes
usual direction of by applying the power
the power;
parallel to its length,
and the force for this
purpose is given by
Eq. (102), after ma-
king the angle equal
Φ
Fig. 219.
B
C
A

MECHANICS OF SOLIDS.
335
to zero, that is by
F=
=
W (sin a+fcos a).
its value;
Multiplying both members by AB, the distance through
which F is exerted, we have,
FX AB W [A B sin a +f. AB cos a];
which reduces to
=
FX AB W.BO+f. W. A C.. (104).
=
its quantity of
work;
The first member is the quantity of work performed by
the power in moving the burden from the bottom to the
top of the plane; and this, we see, is equal to the quantity
of work which the weight of the burden would have per-
formed if raised vertically through the same height, in- this value
creased by the quantity of work which the friction due to words;
a pressure equal to the entire weight, would have exerted
through a distance equal to the horizontal projection of
the plane.
If the burden be rolled, in which case the friction may
be disregarded, or if it be transported in any way to avoid
the friction, ƒ would be zero, and we should have
F.AB- W. BC.
expressed in
value when the
body is rolled;
plane;
That is to say, the work in the direction of the plane is
equal to the work in the vertical direction. What, then,
is gained by the use of the plane? Why nothing more advantage of the
than the ability, which it gives, of putting in motion by a
feeble power, applied in the direction of its length, a burden
which the same power could not move vertically upward.
Resuming Eq. (104), we shall find that what is true
of an inclined plane is equally true of a curved surface,

336
NATURAL PHILOSOPHY.
such as that of a
all equally true of common road or rail-
inclined curved
surfaces;
road over an undula-
Fig. 220.
L
ting piece of ground.
b
B'
B
For, portions of the
62
road, as Ab, bb', b'b",
b
forces on
elementary
portions of the
surface;
&c., may be taken so
short as to differ in-
sensibly from a plane,
in which case we
M
C
shall have, by denoting the intensities of the forces on these
several elementary planes by F", F", F", &c.
FX Ab = W.bc+f. W. Ac,
Fx bb' W. b'c' + f. W. bc',
=
F" x b'b"
=
W.b"c"+f. W. b'c",
&c.,
11
&c., +
&c.
total quantity of
work on entire
surface;
Adding these equations together, and denoting the first
member, which will be the total amount of work in the
direction of the surface, by Q', we have
Q' = W [be + b'c' + b"c" + &c.] + f W[Ac + be'
+ b'c' + &c.];
and supposing the burden to reach the highest point L,
we shall have
bc + b'c' + b" c" + &c. = LM,
Ac+be+b'c" + &c. = AM;
which, in the above equation, give

MECHANICS OF SOLIDS.
337
Q' W.LM+f. W. AM... (105). quantity of work
=
After passing the highest point L, the weight acts in
favor of the force applied in the direction of the plane, and
the first terms of the second members will all change their
signs; and denoting the quantity of work in the direction
of the plane from L to B by Q", we shall have, by the
same process,
in the ascent;
Q" = — W.LB' +f. W. B' B.. (106); quantity in the
--
adding this to Eq. (105), and denoting the total quantity
of work in the direction of the planes from A to B by Q,
we find
Q = Q'+Q" = W [ML-LB']+fW [AM+ BB'],
descent;
or
Q = WX BC+f. W.AC.. (107).
quantity in the
ascent and
descent;
Now it is to be remarked, that every trace of the path
actually described by the burden whose weight is W, has
disappeared from this value for the quantity of work; this
latter is, therefore, wholly independent of this path, and
for the same burden, only depends upon the difference of
level from A to B, and the horizontal distance A C between
these points; so that, the work would be the same as quantity of work
though the load had been transported from A to B along
one continuous plane. Nothing is said here of the resist- had been
ance of the atmosphere, which, like the friction, would be
a cause of opposition to the motion.
the same as
though the path
straight.
§ 216.-We are now prepared to measure the tension Tension of cords;
of a cord arising from the action of its own weight. For
22

338
NATURAL PHILOSOPHY.
this purpose take the.
cord PAMF, resting
the tension of a
upon any surface of
cord arising from
which A is the high-
its own weight;
est point, and con-
sider the part AF
which tends by its
weight to move in
the direction from A
to F. Omit the con-
sideration of friction
for the present, and
F
N
N
G
the question will con-
Fig. 221.
A
α
P
77
TV
weight of a given
portion;
this weight
resolved into
components;
sist in this, viz.: to
find a force which, acting in the direction of its length,
will keep the cord in equilibrio. This force must be
equal and directly opposed to the tension on the part
AF. Designate by W, the weight of a unit of length
of the cord; then considering the element whose length is
MN, its weight will be
W.MN.
Through the centre of gravity 0 of this element, draw the
vertical O G to represent this weight, which resolve into
two components GQ and Q0, the one perpendicular
and the other parallel to the cord. The first will be
destroyed by the reaction of the surface; the second will
act to move the cord in the direction of its length, and
therefore to produce tension. Draw MN' perpendicular
and N N' parallel to the horizon; then will the triangles
GQO and MNN' be similar, both being right-angled
triangles, and the angle QGO of the one, equal to the
angle MNN' of the other, because the side GQ is per-
pendicular to MN, and OG to NN'; hence the pro-
portion,
QO OG :: MN: MN;

MECHANICS OF SOLIDS.
339
whence
bud
Q0=
bas OGX MN'
MN
Denote the tension by t, which will be equal to Q0;
OG represents the weight, equal to WX MN; and pro-
jecting the points A, M, N, F, P, upon the vertical by the
horizontal lines Aa, Mm, Nn, Ff, and Pp, we have
MN' equal to mn, and the last equation becomes,
component of
weight parallel
to the cord;
t =
Wx MN X m n
MN
= W x mn.
cord
value of the
tension for a
single element;
The second member is the weight of a portion of the cord
equal in length to the vertical projection mn of the element
MN. Now the length AF is composed of a number of
elements, each one of which produces, in like manner, a
tension equal to the weight of a portion of the cord of the
same length as its vertical projection. The tension on
each element is transmitted in the direction of the cord to
the elements above. Hence, the entire tension at any
point of the cord, is measured by the weight of a portion tension at any
equal in length to the vertical projection of all the
below it. Thus, if F be the end of the cord, the
sion at A will be measured by the weight of a portion
the cord equal to af, provided no motion take place. In
like manner, the tension at A, arising from the weight of
AP, will be measured by the weight of a portion equal to
ap, so that if the cord have no fixed
point it will move in the direction of
the lower end F, under the action of
a force equal to
W (af- a p).
If the ends of the cord be upon
upon the
same level, or if the cord be endless, it
will be in equilibrio.
Fig. 222.
point is measured
ten- the vertical
of projection of all
by the weight of
the cord below it;
an endless cord is
in equilibrio.

340
NATURAL PHILOSOPHY.
§ 217.-We shall now take into consideration the fric-
Friction of a cord tion of a cord when sliding around any body, say a fixed
sliding around a
fixed cylindrical
beam;
cylindrical beam in
a horizontal position.
Let the cord support
at one end a weight
W, and be subjected
to the action of a
force F applied at
the other end. If
the force communi-
cate motion, it must
not only raise the
weight W, but must
also overcome the
7
F
Fig. 223.
Ъ
tu
tz
1b
W
friction between the
cord and solid. If
construction of
the figure and
notation;
to find the
tension on a
single element of
the cord;
the surface were perfectly polished, the friction would be
zero, and the force F would be equal to the weight W, in
the case of an equilibrium. Divide the enveloping portion
of the cord, a, t, to, tg, &c., into an indefinite number of
very small and equal parts, and draw through the points
of division, t, ta, ta, &c., tangents to the cord; these
tangents will intersect, two and two, at the points b, b', b",
&c., and the extreme ones will coincide with the straight
portions of the cord to which the force and weight are
applied. The points of division being extremely close,
the arcs will be sensibly confounded with their chords
a t₁, t₁ to, to tz, &c. The tension of the cord on the tangent
ab, with which the cord sensibly coincides, is obviously
equal to W, if we neglect the weight of the cord. Let t
be the tension which acts at t, on the second tangent bb';
this tension must overcome the weight W and the friction
on the arc ta, comprised between the points of contact.
Denote by p the pressure exerted by this element upon
the cylinder, and by f the coefficient of friction, then
will

MECHANICS OF SOLIDS.
341
=
W+fp.
its value;
To find the pressure, we will still disregard the weight
of the cord, and remark that the two tangents ab and bt
are equal. Moreover, if we construct the rhombus amtb,
and consider ab as proportional to the weight W, this
same side will represent the tension of the cord from a to
t. The diagonal bm, will be normal to the chord a t₁, and
therefore to the surface of the cylinder, and being the to find the normal
resultant of the tensions at a and t will be the pressure from the tension;
arising from the tension, and consequently equal to p.
The triangles a Ot, and m ab are similar, because they are
both isosceles, and the angle O of the one is equal to mab
of the other; hence
mb : at₁ :: ab: 0a;
mb represents the pressure p; at, may be taken equal to
the arc of which it is chord, which denote by s; ab
represents the weight W; and Oa is the radius of the
cylinder, which denote by R, and the proportion may
be written
whence
ps:: W: R;
s. W
p = R
and this, substituted in the value of t₁, gives
4 = w(1+1).
W
R
pressure arising
value of this
normal pressure:
al value of the
tension on first
Denoting by t, the tension along the third tangent element nearest
b" to, and at the third point of division to, this tension must the resistance;

342
NATURAL PHILOSOPHY.
overcome the tension
t and friction pro-
duced by the elemen-
to find tension on tary arc to t₁, equal in
next element in
order;
length to a t₁, or s.
In a word, to will be
circumstanced in re-
spect to t as t was in
regard to W.
Hence
Fig. 228.
Ъ
N
its value;
values for the
successive
tensions in order
around the beam;
F
f.
4-4(1+420)
and if to, t, to・・・
=
R
W
tn be the tensions on the consecutive
tangents, and at the points to, ts, to,... tn in order
around the beam, we shall have
=(1+),
ts =
4-6(1+4)
=
R
t = t -1 (1+√2).
R
Multiplying these equations together and dividing out the
common factor, we have
value of the
tension on the
last element of
contact;
tn
=
W(1+)".
R
The tension t, being the last in order, brings us to the
straight portion of the cord to which F is applied, and,
therefore, t, must be equal to F; whence

MECHANICS OF SOLIDS.
343
F = W. (1+)".
Developing this by the rules for the binomial theorem,
we have
relation between
the power and the
resistance;
F= W[1+n.
F = W [1 + n
fs
—
+
R
n(n − 1) f² s²
1.2 R2
this value
developed;
+
R3
+ &c.]
n (n - 1) (n-2) f³ så
1.2.3
It must be remembered that s was taken indefinitely
small, and therefore for any definite extent of contact
between the cord and cylinder, n must be indefinitely
great; hence the numbers 1, 2, 3, 4, &c., connected with n
by the sign minus, may be neglected in comparison with
n; this gives
F = W [1 +
nfs
na fasa
+
+
R
2. R2
+ &c.]:
2.3. R3
under a different
form;
but ns is equal to the entire arc enveloped. Denote this
by S, and the above becomes
F = W [1 +
fS
f2 S2
+
+
R 2. R2
f3 S3
2.3. R3
+ &c.]:
the quantity within the brackets is the development of the
fs
function e R; whence
in which e =
logarithms.
F=Wxe
fs
R
final relation
between the
(108),
power and the
resistance;
2.71825, the base of the Nap. system of

344
NATURAL PHILOSOPHY.
example to
illustrate;
Suppose the cord to be wound around the cylinder
three times, and f= 3, then will
S = 3.2 R = 6 x 3.1416. R
18.849 R,
and
or
F = W x et X 18.849
=
WX (2.71825)6.2832;
importance of
friction;
its absolute
necessity.
F= W. 535.3;
that is to say, one man at the end W could resist the com-
bined effort of 535 men, of the same strength as himself,
to put the cord in motion when wound three times around
the cylinder. This explains why it is that a single man,
by a few turns of her hawser around a dock-post, is
enabled to prevent the progress of a steamboat although
her machinery may be in motion. Here friction comes in
aid of the power, and there are numerous instances of
this; indeed, without friction many of the most useful
contrivances and constructions would be useless. It is
by the aid of friction that the capstan is enabled to do its
work; the friction between the rails of a railroad and the
wheels of the locomotive enables the latter to put itself and
its train of cars in motion. But for the friction between
the feet of draft animals and the ground, they could per-
form no work; nor, indeed, could any animal walk or
even stand with safety, if it were deprived of the aid of
this principle.

MECHANICS OF SOLIDS.
345
XVII.
sybow
THE WEDGE.
§ 218. Thus far we have only considered the cases of The wedge;
a body pressing against a single surface. The same body
may also act against two or
more surfaces at the same time.
Fig. 224.
A
B
description and
use;
Such, for example, is the case
with the Wedge, which consists
of an acute right triangular
prism ABC, usually employed
in the operation of separating
and splitting. The acute di-
hedral angle ACb, is called the
edge; the opposite plane face us
Ab, the back; and the planes
Ac and Cb, which terminate in
the edge, the faces. The more
common application of the
wedge consists in driving it, by
definitions;
C
a blow upon its back, into any substance which we wish to
split or divide into parts, in such manner that after each common
wedge;
advance it shall be supported against the faces of the application of the
opening till the work is accomplished.
$219. The blow by which the wedge is driven for-
ward will be supposed perpendicular to its back, for if it
were oblique, it would only tend to impart a rotary motion,
and give rise to complications which it would be unprofit- the blow upon
able to consider. And to make the case conform still fur- the wedge should
ther to practice, we will suppose the wedge to be isosceles, to the back;
be perpendicular

346
NATURAL PHILOSOPHY.
to find the
resultant of the
reactions on the
faces;
The wedge ACB being inserted in the opening ahb,
and in contact with its jaws at a and b, we know that the
resistance of the latter will be
perpendicular to the faces of the
wedge. Through the points a
and b, draw the lines aq and bp
normal to the faces AC and
BC; from their point of inter-
section O, lay off the distances
Oq and Op equal, respectively,
to the resistances at a and b.
Denote the first by Q, and the
second by P. Completing the
construction and parallelogram Oqmp, Om will
notation;
represent the resultant of the
resistances Q and P. Denote
this resultant by R', and the
angle A CB, of the wedge, by
8, which, in the quadrilateral
a Ob C, will be equal to the
supplement of the angle a Ob
Fig. 225.
B
e
=p0q, the angle made by the directions of Q and P.
From the parallelogram of forces we have,
R2 = P²+Q2+2 PQ cos p Oq = P2+Q2-2PQ cos;
or
value of the
resultant;
R' = √ P² + Q2 - 2 PQ cos 0.
The resistance Q will produce a friction on the face A C
equal to fQ, and the resistance P will produce on the face
to find resultant BC, the friction fP; these act in the directions of the faces
of frictions;
of the wedge. Produce them till they meet in C, and lay
off the distances Cq' and Cp' to represent their intensities,
and complete the parallelogram C'q' O'p'; CO' will repre-

MECHANICS OF SOLIDS.
347
sent the resultant of the frictions. Denote this by R",
and we have, from the parallelogram of forces,
30
or
R = ƒ Q²+f2 P2 + 2f2 P Q cos &;
R" = ƒ √ P² + Q² + 2 PQ cos 0.
value of the
resultant of the
frictions;
The wedge being isosceles, the resistances P and Q will be
equal, their directions being normal to the faces will inter-
sect on the line CD, which bisects the angle C=0; and
their resultant will coincide with this line. In like manner the wedge being
isosceles;
the frictions will be equal, and their resultant will coincide
with the same line. Making Q and P equal, we have,
from the above equations,
R'
=
P√2 (1 cose),
—
But
RfP V2(1 + cos 0).
1- cos =
8 2 sin² 0,
1 + cos 0 =
2 cos²;
whence we obtain, by substituting and reducing,
R'
=
2 P. sin 0,
R" 2f. P. cos;
=
and further,
AB
sin 10 = 1
A C'
CD
cos 10 = 0;
these values
result;
or these;
circular functions
in terms of
elements of the
wedge;

348
NATURAL PHILOSOPHY.
final value of
these resultants;
therefore,
AB
R' =
P
A C'
CD
1
R" 2 f. P.
A C
value of the blow
when the wedge
is on the eve of
moving forward;
Denote by F the intensity of the blow on the back of the
wedge. If this blow be just sufficient to produce an equi-
librium bordering on motion forward, call it F'; the fric-
tion will oppose it, and we must have,
FR'+R" = P.
AB
AC
+2f⋅ P.AC
CD
.
(109).
If, on the contrary, the blow be just sufficient to prevent
the wedge from flying back, call it F""; the friction will
aid it, and we must have,
value, when on
the eve of moving
back;
F" P.
A B
A C
-2f. P.
CD
AC
(110).
limits within
which the blow
may vary to
produce no
motion;
The wedge will not move under the action of force
any
whose intensity is between F' and F". Any force less
than F", will allow it to fly back; any force greater than
F'' will drive it forward. The range through which the
force may vary without producing motion, is obviously,
FF" 4fP.
=
CD
AC
. . . (111);
which becomes greater and greater, in proportion as CD
and A C become more nearly equal; that is to say, in pro-
AC
portion as the wedge becomes more and more acute.
be
The ordinary mode of employing the wedge requires
that it shall retain of itself whatever position it may
driven to. This makes it necessary that, Eq. (110),

MECHANICS OF SOLIDS.
349
AB
P.
A C
=2f. P.
CD
A C
AC
or PAB <2f. P.
CD
AC
conditions that
the wedge may
retain the place
or, omitting the common factors and dividing both members
of the equation and inequality by 2 CD,
to which it is
driven;
AB
CD
= f, or
AB
CD <f;
but
AB
CD
is the tangent of the angle A CD; hence we
conclude, that the wedge will retain its place when its semi- conclusion;
angle does not exceed that whose tangent is the coefficient
of friction between the surface of the wedge and the sur-
face of the opening which it is intended to enlarge.
Resuming Eq. (110), and supposing the last term
of the second member greater than the first term, F" be-
comes negative, and will represent the intensity of the
force necessary to withdraw the wedge; which will obvi-
ously be the greatest possible when AB is the least pos-
sible. This explains why it is that nails retain with such why nails retain
pertinacity their places when driven into wood, &c.
$220.-One of the
most important uses
of the wedge, is in its
application to what
is called the Wedge
Press. This, in its
simplest form, con-
sists of a truncated
wedge ABC, which,
by a blow upon its
back, is made to slide
between two blocks,
B' and B"; one of
K
Fig. 226.
A
D B
B'
B
K
C
their places.
Application to
the wedge
press;

350
NATURAL PHILOSOPHY.
description,
uses, &c.;
sup-
these blocks rests
against a fixed
port K, and the
other against some
yielding substance
K' to be pressed.
This machine is fre-
quently employed
to pack goods, wool,
cotton, skins, and
the like; and to
express the vege-
table oils and juices
from seeds, fruit,
&c. The quantity of
work performed by
the power will obvi-
ously be the prod-
quantity of work uct of the intensity
of the power;
of the force Finto
the distance, in a
direction perpen-
dicular to the back,
through which the
wedge has been
driven. Call this
distance x, by
which multiply
both members of
Eq. (109), and wri-
ting F for F' we have
its value;
40
Fig. 226.
D
B
K
B'
B"
Fig. 227.
Ao A4
B
D
BB
1
Fx P. x.
AB
AC
CD
+2f. P.x.
(112).
A C
To obtain from this expression for the quantity of work
of the power, a relation which will enable us to compare

MECHANICS OF SOLIDS.
351
between the
power and
the intensity F with the reaction of the substance K', to find relation
let ABC be the primitive position of the wedge, and
A B C any subsequent position; letting fall from A, the resistance;
perpendicular A, o on the back; A, o will be equal to x.
Moreover, BB is the distance through which the whole
wedge has been moved towards the yielding substance K',
and will, therefore, be equal to the distance through which
the reaction of the latter has been exerted.
tance s, and the intensity of the reaction S.
parallel to CB; then, in the triangles A A, A,
the sides A Ag, and A, A, are respectively equal and paral-
lel to BB, and BB, and, consequently, AA, will be equal
to BBs. The two triangles A,AA4, and ABC, are
similar, and give the proportion
Ago AA :: CD: AB;
Call this dis-
Draw A3 A4 notation and
and B BB,
construction;
or
whence
x: 8: CD: AB;
::
CD
AB'
If there were no friction, there would be no obstruction
to the free transmission of the effect of the force F to the
substance to be compressed. But, making f zero, we have,
Eq. (112),
relation of the
elementary paths
of power and
resistance;
Fx = P. x.
A B
40
and, from the principle of virtual velocities,
work of the
power without
friction;
FxPx.
AB
AC
1
Ss;

352
NATURAL PHILOSOPHY.
whence
ydrastai odi
od & Id
Px = Ss X
AB
for
Now substituting this value for Px, and the above value
X, in Eq. (112), and it reduces to
relation of
quantities of
F.s.
work, of power,& bag
CD
AB
=Ss+2f.S.s.
CD
AB
(113).
resistance, and
friction;
Tappe
quantity of work
absorbed and
lost;
The first term of the second member is, obviously, the
effective quantity of work done, being the reaction of the
yielding body multiplied into the distance through which
this reaction has been exerted, or through which the body
has been compressed. This, we see, is less than the quan-
tity of work of the power F, by the quantity
2f. S.sx
CD
AB'
which has been totally absorbed, and therefore lost, in
consequence of the friction. This loss is often very great,
and to illustrate, suppose the reaction S to be 1000
pounds, and that the back of the wedge AB is
length CD; then will
20
of its
lbs.
lbs.
illustration by an
example;
20. F.s
S=
1000.s+ 40.f. s. 1000;
and, taking f= 10,
lbs.
lbs.
20. F.s=1000. s + 4000 s.
numerical loss; Assigning any particular value to s we please, it appears
that the useful effect is only 1000s, while the loss from
friction is 4000 s, and that the work performed by the

MECHANICS OF SOLIDS.
353
force F is 5000 s. Dividing the above equation by 20s,
we get
tommal od: lo
Stow edT
lbs.
lbs.
lbs.
F= 50+ 200 111
250;
Baivil
How numerical value
of the power;
machine.
which is much less than 1000, the value of the reaction S
Hence we see that the advantage of the wedge press con- defect of the
sists in this, viz.: by its aid the work may be executed
with comparatively a feeble power. The machine is, how-
ever, defective, on account of the large amount of work.
absorbed by its friction.
in the use of the
lo sgando
§ 221. As before remarked, the wedge is driven Effect of the blow
forward by a blow on its back. This mode of employing wedge;
force is an additional source of loss of work. When a
hammer strikes the wedge, two periods are to be dis-
tinguished, viz.: the first corresponds to the duration of
the shock, that is to say, from the instant the hammer
touches the wedge to that in which the greatest com-
pression of the wedge and hammer takes place; the sec-
ond follows immediately and includes the interval during
which the reaction of the body to be pressed gives rise to
the resistance called S, and to the frictions due to the
pressures P and Q. While the wedge is acquiring motion
under the blow, during the first period, its inertia acts as
a resistance; in the second period, the inertia becomes a
power to overcome the resistance S. The blow develops explanation of
at each instant, between the hammer and wedge, real
pressures, which are measurable in pounds; and these
pressures are greater, for the same effect, in proportion as
the duration of the shock or blow is shorter. The wedge
will, in the first period, have a motion from the action of
these pressures in consequence of its lateral compression;
the inertia due to this motion being opposed by the lateral
parts of the machine will give rise to friction, which fric-
tion, together with the inertia exerted by the wedge in
the effect;
23

354
NATURAL PHILOSOPHY.
loss of living
force;
its measure;
loss of living
force from
permanent
change of
figure.
The wedge in
universal use;
acquiring an increased velocity under the continued action.
of the hammer, will be in equilibrio with these pressures.
The work of these frictions, during the first period,
will be absorbed by the machine, and therefore lost to the
inertia or living force of the wedge when this living force
becomes, in the second period, a power to overcome the
resistance S. The quantity of action, or half the living
force, preserved by the wedge at the close of the first
period, and with which it enters upon the work to be
performed during the second, will be given by the rule
furnished in Eq. (113), and from which it appears, that
this quantity of action will be equal to the quantity of
action of the hammer on the back of the wedge during
the first period, diminished by that consumed by the fric-
tion due to the wedge's inertia within the same period.
But this is not all. A part of the work of the hammer
is consumed by the permanent change of figure of the
wedge arising from the violence of the action. Thus we
see, that a considerable portion of the living force with
which the hammer begins its work, is lost by change of
figure, and by friction due to the sudden development of
inertia; neither of which would take place under a force
of gradual and ordinary pressure.
§ 222.-Notwithstanding the disadvantages arising
from the great and wasteful consumption of work which
accompanies the employment of the wedge, this machine
is in universal use. It has not, however, always the
Fig. 228.
its different
figures;

MECHANICS OF SOLIDS.
355
prismatic figure. It sometimes has the form of a pyramid it may be a
with three, four, or more edges; in which case, the rela- pyramid;
tions between the power and resistance when in equilibrio
are altogether analogous to those of the prism; the power
applied to the back is equal and directly opposed to the
resultant of the resistances and the frictions against the
faces. The wedge may also have the form of a truncated a truncated
pyramid or prism. Often it is nothing more than a cone
at the extremity of a cylinder.
pyramid, or even
a cone;
tools;
the degree of
acuteness.
Examples in tools. Almost all the tools employed in examples in
the arts have some relation to the wedge; such as the
different kinds of knives, axes, shears, scissors, files,
chisels, saws, hoes, ploughs, &c., &c. All wedges, of what- rules in regard to
ever kind and however employed, are destined to act by
their pointed ends, and the shape of this should be regula-
ted with special reference to the object in view. If too
acute, it will break off; if too obtuse, it will not penetrate;
and the angle adopted is generally the result of a com-
promise between these difficulties, determined by the
nature of the material of which the wedge is made and
that of the substance to be worked. If
the substance to be worked be hard,
as cold iron, copper, &c., the basil angle
abc should be large; this angle in
the chisel of a carpenter's plane, which
is only intended for wood, is about 30°;
it is made still more acute in knives
employed to cut the softer substances,
meat, bread, and the like.
Fig. 229.
§ 223.--All rotating pieces, such as wheels supported Friction of
upon other pieces, give rise by their motion to friction, rotating pieces;
This is an important element in all computations relating
to the performance of machinery, and cannot safely
be neglected. It seems to be different according as
the rotating pieces are kept in place by trunnions or by trunnions;

356
NATURAL PHILOSOPHY.
#
pivots; pivots. By trunnions
definition and
description of a
trunnion;
boxes;
definition and
description of
pivots and their
sockets;
are meant cylindrical
projections a a from
the ends of the arbor
AB of a wheel; they
are usually made as
small in diameter as
may be found consist-
ent with the requisite
strength, and are so
placed that their axes
coincide with that of
the arbor which is
perpendicular to the
plane of the wheel.
The trunnions rest on
the concave surfaces
of cylindrical boxes
CD, with which they
usually have a small
surface of contact m,
the linear elements of
both being parallel.
Pivots are shaped like
the trunnions, but
support the weight of
the wheel and its ar-
bor upon their circu-
lar ends, which rest
against the bottom
of cylindrical sockets,
A
Fig. 230.
C
Fig. 231.
α
m
Fig. 232.
D
B
B
E
F
I
K
EFGHIK. If the forces which give motion to the
wheel press its pivot against the cylindrical surface of the
socket, the friction will partake of the nature of that due
to the trunnion as well as the pivot; but this is usually
prevented by special arrangements in the mounting of the
wheel. Of the two frictions here referred to, one takes

MECHANICS OF SOLIDS.
357
friction on the
pivot;
place between the end of the pivot and the circular bottom nature of the
of the socket, and is in all respects similar to that of two
surfaces sliding over each other. The friction due to the
motion of the trunnion has been found by Coulomb to be
much less than that of the pivot; and there is also less friction on
adhesion on account of the smallness of the surface of
contact. A table of the coefficients of friction which ac-
companies the motion of trunnions will be given in its
proper place.
trunnions less
than on pivots.
the pivot against
§ 224. It is not sufficient in case of rotary motion, To find the value
to know the ratio of the friction to the pressure; we must of the friction of
also know how the friction arising from the peculiar ar- the bottom of its
rangements of the rubbing parts as just indicated, acts with socket;
respect to the other forces. We shall first take the case
of the pivot turning around
its axis. Let N denote the
force, in the direction of the
axis, by which the pivot is
pressed against the bottom
of the socket. This force
may be regarded as passing
through the centre of the
circular end of the pivot,
and as the resultant of the
partial pressures exerted up-
on all the elementary sur-
faces of which this circle
is composed. Denote by A
the area of the entire circle,
Fig. 233.
N
A
C
403
B
事
​then will the pressure sustained by each unit of surface be
N
pressure on unit
of surface;
and the pressure on any small portion of the surface

358
NATURAL PHILOSOPHY.
pressure on a
single elementary
surface;
the
corresponding
friction;
moment of this
friction;
denoted by a, will obviously be
a. N
A
and the friction on the same will be
f.a. N
A
This friction may be regarded as applied to the centre of
the elementary surface a; it is opposed to the motion, and
the direction of its action is tangent to the circle described
by the centre of the element. Denote the radius of this
circle by r, then will the moment of the friction be
f.a. N.
r.
A
to find the
moment of the
friction on a
single sector;
Fig. 233.
N
If we now consider all the elementary surfaces within the
sector A CB, of which the angle at C is very small, we
may regard the frictions on
these elements as parallel to
each other, and perpendicular
to the radius O C, which
bisects the angle A CB; in
virtue of their parallelism,
their resultant will be equal
to their sum; and, because
of their equality on equal
elementary surfaces, the line
of direction of this resultant
will pass through the centre
of gravity of the sector ACB.
But this sector being very
A
0
B
D
acute, will not differ from an isosceles triangle, of which

MECHANICS OF SOLIDS.
359
the equal sides, and perpendicular drawn to the base from
the vertex C, will be sensibly equal to each other and to
R, the radius CA of the circle: whence the distance of
the resultant friction on the sector from the centre C will
be R. Substituting the small sector for a, and R for
T, in the foregoing expression, and we have, for the moment
of the friction on the sector,
N
f.
sect. A CB x R;
A
the value of this
moment;
and the same is true of any other sector. If the moments
be taken for all the sectors which make up the circle, and
these be added together, we shall have the moment of the
entire friction. The quantity f.
N
A
•
R, is constant, sum of all the
similar moments;
and hence the sum of these moments will be
N
f.
RX (sum of the sectors A CB);
A
but the sum of all the sectors is equal to the area of the
circle, or A; whence the moment of the friction on the
entire base of the pivot is
f.N.R.
(114);
moment of the
friction on the
entire end of the
pivot;
whence we conclude, that, in the friction of a pivot, we may
regard the whole friction due to the pressure as acting in a
single point, and at a distance from the centre of motion equal
to two thirds of the radius of the base of the pivot. This mean lever of
distance is called the mean lever of friction.
friction;
It may happen, that the extremity of the pivot, in-
stead of rubbing upon an entire circle, is only in contact when the friction
with a ring or surface comprised between two concentric is on an annular
circles. This occurs when the arbor of a wheel is urged
surface;

360
NATURAL PHILOSOPHY.
find friction
in the direction of its melbogey Fig. 234. Lepo odt
against a ring; length by a force N a-
alds
po vidi
gainst a shoulder de ba.
Denoting, as before, the
area of the ring which
sustains the pressure
by A, the moment of
the friction on the ele-
mentary sector ABC
is, as before found, A
NW
A
D D
B
B
moment on a
sector of the
larger circle;
N
f.
•
RX sector A CB;
A
kapato al in which R denotes the radius of the larger circle. Again,
the moment of the friction on the sector A' CB' is
that on the
smaller;
that on an
element of the
ring;
ि
f..R' sector A' CB';
X
and the difference, Terp
N
A
3f. [R.X sector A CB-R' sector A'CB'],
at will be the moment of the friction on the surface, A'B'BA.
moment for the
entire ring;
Taking the moments for the remaining elementary surfaces
which make up the ring, and adding them together,
observing that the sums of the sectors make up the areas
of the circles to which they respectively belong, we find,
for the moment of the friction on the whole ring,
22
3f.
N
A
(RX area of circle R-R' X area of circle R']
But the area of the circle whose radius is R, is

MECHANICS OF SOLIDS.
361
T R²;
ibropo
that of the circle whose radius is R', is
area of one circle;
T R²;
soll sadoq sids d
that of the other;
and the area A of the ring, is
Dop off of langa
-
(R² — R²).
area of the ring;
Substituting these values in the above expression, we find
hoop oft has
R3
R3
moment of
f. Nx
friction on the
R2
R/2
ring;
Finally, denote by 7 the breadth of the ring, that is, the
distance A' A; by r, its mean radius or distance from C to
a point half way between A' and A, and we shall have
R = r + ½ ¹,
R' = r — l;
substituting these values above and reducing, we have
72
f. Nx [+]. (115);
r
same in different
form;
and making
72
r +
=
12 r
we obtain, for the moment of the friction on the entire ring,
f.N.r,
(115)'.
The quantity, is called the mean lever of friction for a
ring. Since the whole friction f N, may be considered as
applied at a point whose distance from the centre is R, or
mean lever
arm;

362
NATURAL PHILOSOPHY.
72
r₁ = r +
according as the friction is exerted over an
12'
entire circle or over a ring; and since the path described
by this point lies always in the direction in which the
work consumed friction acts, the quantity of work consumed by it will be
equal to the product of its intensity fN into this path.
Designating the length of the are described at the unit's
distance from C by s,, the path in question will be either
by friction;
its value for an
entire circle;
its value for a
ring;
Rs, or r₁s,;
and the quantity of work either
for an entire circle, or
R.s.f.N
ƒ. N(r + 1227),
f.
for a ring. Let Q denote the quantity of work consumed
by friction in the unit of time, and n the number of revo-
lutions performed by the pivot in the same time; then will
S
= 2π X n;
work consumed
by friction in a
unit of time;
for a ring;
and we shall have
for the circle, and
Q..R.f.N.n... (116)
Q=2x. f. N. (r+2).n. (117)
for a ring; in which = 3.1416.
12
The coefficient of friction f, when employed in either
of the foregoing cases, must be taken from the tables in
$212.

MECHANICS OF SOLIDS.
363
From these expressions, it ap-
pears that the quantity of work con-
sumed by friction, in a given time,
augments with the radius of the piv-
ot, or mean radius of the ring; and
as this work is always opposed to the
motion, there is an advantage in re-
ducing these radii as much as possi-
ble, consistently with the strength of
the pivot. With this view, the pivots
are sometimes made in the form of
a truncated cone, and often with a
convex ellipsoidal or spherical ter-
mination, and the socket having a
corresponding shape, it will only be
necessary to consider the small cir-
cle of contact which arises from the
compression of the material.
Fig. 235.
Referring to the expressions (114) and (115)', we see,
work consumed
by friction
proportional to
mean lever;
pivots should be
as small as
possible;
conical and
spherical
terminations;
the friction;
that to obtain the moment of friction, in the case of the the moment of
pivot, either for an entire circle or ring, we multiply the
coefficient of friction, as given in the table of § 212, by the
pressure, and this product by the mean lever. And referring
of the friction;
to Eqs. (116) and (117) we find, that the quantity of work is quantity of work
obtained by multiplying the moment of friction into the path
described by a point at the unit's distance from the centre of
motion.
Example. Required the moment of the friction on a example of the
pivot of cast iron, working in a socket of brass, and which circular base;
supports a weight of 1784 pounds, the diameter of the
circular end of the pivot being 6 inches. Here
R = 2 = 3
lbs.
N = 1784,
f = 0.147;
in.
ft.
=
0.25,
numerical value
of elements;

364
NATURAL PHILOSOPHY.
value of the
moment;
work consumed
in unit of time;
annular base;
which, substituted in expression (114), gives
lbs.
0.147 × 1784 × × 0.25= 43.708.
x x 2 x
And to obtain the quantity of work in one unit of time,
say a minute, there being 20 revolutions in this unit, we
make n = 20, and T = 3.1416 in Eq. (116), and find,
Q = x 3.1416 x 0.25 x 0.147 x 1784 x 20
5492.80;
that is to say, during each unit of time, there is a quantity
of work lost which would be sufficient to raise a weight
of 5492.80 pounds, through a vertical distance of one foot.
Example. Required the moment of friction, when the
example of the pivot supports a weight of 2046 pounds, and works upon
a shoulder whose exterior and interior diameters are
respectively 6 and 4 inches; the pivot and socket being
of cast iron, with water interposed.
1=
6 4
2
= 1 inch,
r = 2 + 0.5
=
2.5 inches,
ft.
=2.5333 = 0.2111,
numerical value
r =
2.5 +
(1)2
in.
of the elements;
12 X 2.5
value of the
moment;
N = 2046 pounds,
f = 0.314;
which, substituted in expression (115), gives for the mo-
ment of friction,
lbs.
0.314 x 2046 × 0.2111
2111
=
135.62.
The quantity of work consumed in one minute, there

MECHANICS OF SOLIDS.
365
being supposed 10 revolutions in that unit, will be found
by making in Eq. (117), as before, * = 3.1416 and
n = 10,
Q= 2 x 3.1416 x 0.314 x 2046 x 0.211 x 10
8517.24;
work consumed
in unit of time;
that is to say, friction will, in one unit of time, consume a
quantity of work which would raise 8517.24 pounds
through a vertical distance of one foot. The quantity rule for finding
of work consumed in any given time would result from
multiplying the work above found, by the time reduced
to minutes.
timore to
the work in any
time.
trunnions and
8 225. The friction on trunnions and axles, which we Friction on
now proceed to consider, gives a considerably less co- axles;
efficient than that which accompanies the kinds of motion
referred to in the tables of § 212. This will appear from
the following table, which is the result of careful experi-
ment, viz.:-
TABLE IV.
FRICTION OF TRUNNIONS IN THEIR BOXES.
Ratio of friction to
pressure when the
unguent is renewed.
KINDS OF MATERIALS.
STATE OF SURFACES.
By the
Or, con-
ordinary tinuously.
method.
Trunnions of cast iron
and boxes of cast iron.
Unguents of olive oil, hogs' 0.07)
lard, and tallow -
The same unguents moist-
ened with water-
0.054
10.08
0.08
0.054
Unguent of asphaltum
0.054
0.054
批號
​Unctuous
0.14
Unctuous and moistened
with water
0.14
Unguents of olive oil, hogs' (0.07)
lard, and tallow -
to
0.054
Trunnions of cast iron
and boxes of brass.
0.08
Unctuous
0.16
Unctuous and moistened
with water
0.16
Very slightly unctuous* -
0.19
*The surfaces began to move about.

366
NATURAL PHILOSOPHY.
18
TABLE IV.-continued.
KINDS OF MATERIALS.
Ratio of friction to
pressure when the
unguent is renewed,
STATE OF SURFACES.
By the
Or, con-
ordinary tinuously.
method.
Trunnions of cast iron
and boxes of lignum-3
vitæ.
Trunnions of wrought
iron and boxes of cast
iron.
Trunnions of wrought
iron and boxes of
brass.
Without unguents*
Unguents of olive oil and
hog's' lard -
Unctuous with oil and
hogs' lard --
Unctuous with a mixture
of hogs' lard and plum-
bago
0.18
0.090
0.10
Unguents of olive oil, tal-0.07)
low, and hogs' lard -
0.14
to
0.054
0.08
Unguents of olive oil, hogs'
lard, and tallow -
0.07
to
0.054
0.08
0.08)
Old unguents hardened
0.09
Unctuous and moistened
with water
0.19
Very slightly unctuous+ -
0.25
Unguents of oil or hogs'
lard-
O.II
Unctuous
0.19
Unguent of oil
0.10
0.09
(0.045
Trunnions of wrought
iron and boxes of lig-
num-vitæ.
Trunnions of brass and
boxes of brass.
Trunnions of brass and
boxes of cast iron.
Trunnions of lignum-
vitæ and boxes of
cast iron.
Trunnions of lignum-
vitæ and boxes of
lignum-vita.
Unguent of hogs' lard-
Unguents of tallow or of
olive oil
Unguents of hogs' lard
Unctuous
Unguent of hogs' lard
to
0.052
0.12
0.15
0.07
Let us now examine the part performed by friction in
to investigate the connection with the forces which give motion.
friction of
trunnions;
We have
seen that the contact of the trunnion with its box is
along a linear element, common to the surfaces of both.
A section perpendicular to its length would cut from
the trunnion and its box, two circles tangent to each
other internally. The trunnion being acted on only by
*The wood being a little unctuous.
+ The surfaces began to move about.

MECHANICS OF SOLIDS.
367
Fig. 236.
its weight, would, when at rest, give this tangential
point at o, the lowest point of the section poq of the box.
If the trunnion be put in motion by the application of a
force, it would turn
around the point of
contact and roll in-
definitely along the
surface of the box,
if the latter were
level; but this not
being the case, it will
ascend along the in-
clined surface op to
P
N
772
some point as m,
where the inclina-
tion of the tangent
umv is such, that
the friction is just
position of
trunnion when it
begins to slide on
its box;
sufficient to prevent the trunnion from sliding. Here let
the trunnion be in equilibrio. But the equilibrium requires
the resultant
intersects the
that the resultant of all the forces which act, friction in- point in which
cluded, shall pass through the point m and be normal to
the surface of the trunnion at that point. The friction is ap- surface of box;
plied at the point m; hence the resultant N of all the other
forces must pass through m in some direction as md; the
friction acts in the direction of the tangent; and hence, in
order that the resultant of the friction and the force N shall tangential
be normal to the surface, the tangential component of the component of
latter must, when the other component is normal, be equal and opposed to
and directly opposed to the friction.
Take upon the direction of the force N, the distance
md to represent its intensity, and form the rectangle
adbm, of which the side mb shall coincide with the
tangent; then, denoting the angle dma by o, will the com-
ponent of N perpendicular to the tangent be
N. cos ;
resultant, equal
friction;
normal
component;.

368
NATURAL PHILOSOPHY.
friction due to
this component
tangential
component;
and the friction due to this pressure will be now ui
zod art to go no
to
tulog
goltaodique.odt vf. N. cos q. od doinnid sdt 1
bloow ti post
The component of N, in the direction of the tangent,
will be
N. sin ;
Hor
and as this must be equal to the friction, we have
To nobo
od nulous!
f. N. cos. o =
N. sin q
(118);
whence
value of the unit
of friction;
f = tan o;
that is to say, the ratio of the friction to the pressure on
the trunnion, is equal to the tangent of the angle which
the direction of the resultant N of all the forces except
the friction, makes with the normal to the surface of the
djde af trunnion at the point of contact. This gives an easy method
of contact ;
of finding the point of con-
to find the point tact. For this purpose, we
have but to draw through
the centre A, a line AZ, par-
allel to the direction of N,
and through A the line An,
making with AZ an angle
of which the tangent is f;
the point m, in which this
line cuts the circular section
of the trunnion will be the
point of contact.
Because madb, last fig-
ure, is a rectangle, we have
N2
=
72
Fig. 237.
N
N2 cos²+N2 sin²;
A
20

MECHANICS OF SOLIDS.
369
and, substituting for N2 sin² its equal f2 N2 cos q, we to find the value
have
N² = N2 cos² + ƒ² N² cos² o
=
N2 cos² (1 +ƒ³);
of the total
friction;
whence
1
Ncos = N x
√1+fa
and multiplying both members by f
f
f. N. cos o = N.
(119); its value;
√1+f2
but the first member is the total friction; whence we con-
clude, that to find the friction upon a trunnion, we have but to
multiply the resultant of the forces which act upon it, by the rule;
unit of friction, found in Table IV, and divide this product by
the
square root of the square of this same unit increased by
unity.
This friction acting at the extremity of the radius R of
the trunnion and in the direction of the tangent, its moment
will be
N.
f
√1 + ƒ²
X R... (120).
moment of the
total friction;
And the path described by the point of application of the
friction being denoted by Rs, the quantity of work of
the friction will be
N. R.s, X
f
√1 + √²
(121); quantity of work
of friction;
in which s, denotes the path described by a point at the
unit's distance from the centre of the trunnion. Denoting,
24

370
NATURAL PHILOSOPHY.
domas in the case of the pivot, the number of revolutions per-
formed by the trunnion in a unit of time, say a minute, by
n; the quantity of work performed by friction in this time
by Q; and making = 3.1416, we have
quantity of work
in a unit of time;
and
S
s₁ = 2%.n;
f
(122).
√1 + f²
Q, 2. R.n. N.
When the trunnion remains fixed and does not form part
of the rotating body, the latter will turn about the trun-
nion, which then takes
axle;
the name axle, having the
centre of motion at A, the
Fig. 238.
centre of the eye of the
wheel; in this case, the
lever of friction; lever of friction becomes
the radius of the eye of
the wheel. As the quan-
tity of work consumed by
friction is the greater, Eq.
(122), in proportion as this
trunnion better
than the axle in
regard to friction;
A
radius is greater, and as the radius of the eye of the wheel
must be greater than that of the axle, the trunnion has the
advantage, in this respect, over the axle.
The value of the quantity of work consumed by fric-
friction same for tion is wholly independent of the length of the trunnion or
axle, and no advantage is therefore gained by making
shorter or longer.
long and short
trunnions and
axles.
§ 226.-If we examine Eq. (122), we find that, all other
things being equal, the value of the work consumed by
friction will depend upon the radius R of the trunnion, and
Trunnions should that as the latter diminishes, in the same proportion will
this consumption diminish. The trunnion should, therefore,
be made as small as possible, and of the hardest and
be small as
possible;

MECHANICS OF SOLIDS.
371
work by friction
strongest material, as steel. The consumption of work by consumption of
friction may also be diminished by lessening the force N; lessened by
but with these two exceptions there is no way of avoiding diminishing the
the effects of friction for
given materials. When
the trunnion is employed to
support a piece which sim-
ply oscillates through an
arc, as in the case of the
pendulum and weighing-
balance, the knife-edge
may be used to great ad-
Fig. 239.
pressure;
advantages of the
vantage, for, in that case, the radius R is reduced to the knife-edge as an
smallest conceivable length, and the work of friction
almost nothing.
8227.-There is ano-
ther species of friction yet
to be mentioned, viz: that
which arises from the roll-
ing of one body over
another. As the surfaces
of contact are in this case
applied to each other, and
separated in a direction
perpendicular to that of
the motion, there would,
at first view, appear to be
no friction, nor would
there if the surfaces were
perfect that is to say,
to
axle.
Fig. 240.
Friction of
rolling motion;
no friction in
rolling motion if
surfaces were
perfect;
free from all irregularities. But there can, in practice, be
no such surface; when bodies are brought in contact in
the manner here referred to, the slight protuberances on
the surface of one will enter into the corresponding cavi- but in actual
ties on that of the other, after the manner of so many friction;
wedges, and cannot be again withdrawn without giving
practice there is

372
NATURAL PHILOSOPHY.
this kind of
friction, as well
be neglected;
friction of the
second kind;
illustration;
rise to an amount of friction due to their dimension, depth
of insertion, and nature of material. Here adhesion
as adhesion, may assumes a value which is appreciable, as compared with
this friction, but both together are found, in practice, to
be exceedingly small, and generally, adhesion in rolling, as
well as in sliding friction, may, without much error, be neg-
lected. In general this friction, called friction of the second
kind, is less in proportion as the diameter of the rolling
body is less. A wheel of two feet diameter, loaded with
a weight of 100 pounds, and rolling over a piece of level
and smooth ground, only gives rise to a friction of 0.03
of the pressure-that is to say, to only three pounds. The
wheels of carriages meet often with considerable resist-
ance when rolling over compressible or rough ground, but
this is because the carriage must be raised over the in-
clined planes formed in front by the sinking of the wheels,
or over obstacles which project above the common surface.
The little resistance to motion arising from friction of the
second kind, is well illustrated by the comparative facility
with which heavy blocks of stone are often transported
upon rollers over considerable distances. A roadway
is first usually made by
placing straight pieces of
timber along the ground
to prevent the rollers from
sinking into it; the stone
illustration of its is then mounted upon the
practical
substitution for
sliding friction.
rollers, which are placed
upon these pieces at right
angles to their length, and
drawn in the direction of
the road by the applica-
Fig. 241.
tion of any convenient power. As fast as a roller is de-
tached from behind, it is brought forward and interposed,
in time to prevent the stone from tipping forward in con-
sequence of its centre of gravity getting in advance of the
leading roller. The quantity of work necessary to con-

MECHANICS OF SOLIDS.
373
vey a stone over any considerable distance, in this way, is
incomparably less than if it were to rest with its face
against the ground.pixa et et ola boliqqa
soral ad
soned bas
228. The different kinds of friction may be so com-
bined as to diminish both its intensity and the quantity of Employment of
its work. Thus, let
moood
Fig. 242.
rolling to
diminish sliding
noblot of busfriction;
a pair of wheels CD
be mounted upon an
axle, and suppose a
force F, applied to
the latter, parallel to
a level plane A B, to
put it in motion. De-
note the weight of the
axle and its load by
A
F
D
eda otono
Bengst
W, that of the wheels by w. Suppose, for a moment, that
the wheels are firmly connected with the axle and that they
cannot rotate, but, when put in motion, must slide along
AB; the force F, requisite to impart motion and keep it
uniform, will be given by the equation
F= (Ww) f
ding T
case of a common
cart;
Das swer force necessary to
29
3(123);
in which ƒ is the coefficient of sliding friction between the
wheels and plane AB. Next, suppose the wheels capable
of turning about the axle, and the force requisite to keep
the motion uniform to be denoted by F. This force,
acting to communicate motion, will give rise to friction
between the circumference of the wheel and the plane
AB, and also between the axle and the inner surface of
the eye: the latter will yield first, and the whole will
move forward, the wheels having a rotary as well as a
progressive motion. The friction at the axle will, Eq.
(119), be
keep it in uniform
motion when
sliding;
N
f
√1 + ƒ²
friction on axle
when rotating;

374
NATURAL PHILOSOPHY.
normal pressure
on axle ;
friction on the
axle;
in which f, denotes the coefficient of friction at the axle.
The weight W is thrown upon the axle and acts vertically;
the force F,, applied also to the axle, acts horizontally,
and hence
N = VW + F2;
and the friction at the axle becomes
f
√W² + F2 X
√1
1 + £2
Denote the radius of the wheel by R, and that of its eye
by r, and the space described by a point at the unit's dis-
tance from the centre of motion by s,; then will the quan-
tity of work of this friction be
f
quantity of work
of friction;
√ W² + F2 X
work of the
power;
√1 + fi²
Xr.s.
The path described by the point of application of the
power, and in the direction of the power, will be equal to
the development of the arc of the circumference of the
wheel corresponding to the arc s,, that is to say, to Rs
and hence the quantity of work of the power will be
FRs;
whence we have
fi
FRs,
1
VW² + F2 X
X r.si
√1 + f
power to keep
the cart in
from which we find
uniform motion F,, = W f
when the wheels
are turning;
r
R
√(1 +59) (1-
1
fa
شر
X
1 + f²
R2

MECHANICS OF SOLIDS.
375
The value of f, being a small fraction, as will appear from
Table IV., the fraction
1
√(+£9 (1
f2
p2
this factor equal
-
1+f2
R2
to unity;
will differ but slightly from unity, and hence may be
replaced by unity, which will give
2110.0
=
r
F., WR
8010
Dividing this by Eq. (123), we find
(124).
practical relation
of power to load;
3110
F
=
F
W.f
(W+w) f
X
R
. . (125).
relation of the
powers to
produce sliding
tad and rolling
motion;
noiseiamon
Here W is less than W+w; f, is, by the tables, less than
f, and r is usually very much less than R, so that the
second member must be a small fraction, and F, conse-
quently, much greater than F. This is the theory of theory of the
carriage-wheels of every kind, of castors, rollers for smooth- carriage-wheel,
ing ground, and the like.
300 TSVOIT 265
&c.;
Tol Solutiiedua
:sol no alsodw
Example. Suppose a carriage with four wheels, whose
joint weight is 50 pounds, to be loaded with 2040 pounds,
the weight of the axle-trees and body being together
equal to 320 pounds. Let the wheels be of cast iron, the
axles of wrought iron, the radius of the eye half an inch,
that of the wheel one foot and a half, and suppose an un-example-
guent of tallow and the carriage placed upon a rail-track carriage on aurpe
of wrought iron. Here
railway; to how
1owog to bas

376
NATURAL PHILOSOPHY.
numerical value
of the elements;
ratio of the
forces;
gw=2040 + 320
W = 2040 + 320 9900 lbs.
=
2360,
lbs.
w=
50,
lbs.
W+w=
2410,
f = Table IV. .
C.
. +0.08,
f = Table I..
0.194,
r
ed year com
your
and Eq. (125),
F
F
ㄠ​ˋ
R=.
ft.
0.042, nearly,
ft.
1.5;
2360
0.08
0.042
=
X
X
=
2410 0.194
1.5
0.0113;
Bun
(821) pit vd and guil
value of one in
terms of the
other;
or
conclusion;
runners
substituted for
wheels on ice;
F = 88.49. Fi
that is to say, the force requisite to put the carriage in mo-
tion when its wheels are free to rotate, is only about one
eighty-eighth part of that which would be necessary to
drag it, were its wheels locked.
If we examine Eq. (125), we shall find that by taking
f equal to zero, the force F, will be vastly greater than
F, and the wheels will not turn. Now, although this
extreme case can never occur in practice, yet, when a car-
riage is placed upon ice, we approximate to it; and this is
why runners are usually substituted for wheels, under
such circumstances. The same equation explains why it
is that so much more advantage arises from large wheels
than small ones.
Multiplying both members of Eq. (124), by Rs, we find
equality of the
work of friétion
and of power;
FRs,
11
W.f.r.s;

MECHANICS OF SOLIDS.
377
the second member is obviously the work performed by
friction, as the first is that performed by the power F.
Denoting by n the number of revolutions performed by
the wheels, we have
S₁
=
2.n;
which, in the above equation, gives
FRS,
=
alos
W.f, x 2 r. n.
Lupo
og boto
idylla af inge
to author oft
Denote the distance travelled by d, then will
mogy
path described by
point at unit's
distance from
axis of axle ;
d
dw 10 bas D
number of
revolutions of the
n =
Jon Binco afoodyr
2 T R'
budge
and
dog wheel;
gobiont odt
nofani
ed blow nois
r
d. . (126).
work of power
and of friction;
F, Rs, = W⋅f R
If we make d equal to one mile
=
5280 feet, and take
the dimensions and other elements the same as in the last
example, we shall find
FR.s,
=
Mary Hiv
2360 X 0.08 x 0.042 x 5280 000
1.5
=
27912, nearly;
example;
in words, the work expended in moving the carriage one
mile, or the work consumed by its friction, is equivalent
to that which would raise 27912 pounds through a vertical
height of one foot.
still further
When a trunnion is destined to support considerable method by which
weight, its dimensions must be proportionably large; but as friction may be
the radius of the trunnion increases, the effect of friction will reduced;
increase in the same ratio. To avoid the inconvenience

378
NATURAL PHILOSOPHY.
device for
diminishing
friction;
work of friction
when wheels
cannot turn;
friction on the
smaller
trunnions;
work of friction;
Fig. 243.
A
that would arise from this, when great freedom of motion
is desirable, we may have recourse to the following device.
Conceive the trunnion A to
rest upon the circumferences
of two equal wheels, sup-
ported upon smaller trun-
nions C, C', whose distance
apart is slightly greater than
the radius of the wheels.
Resolve, by the parallelo-
gram of forces, the pressure
upon the larger trunnion
N
N'
C'
into two components, normal to the circumferences of the
wheels; these will be transmitted to the smaller trunnions
C and C', where they will be supported. Denote these
components by N and N'. If the wheels could not turn,
the friction between their circumferences and the larger
trunnion would be ƒ Nand f'N'; and the quantity of work
consumed by this friction would be
(081)
(fN+f'N') R, s,;
in which R, denotes the radius of the larger trunnion, and
S₁
the arc described by a point at the unit's distance from
its axis. If, on the contrary, the wheels may turn, the
frictions on the trunnions C and C' will yield before that at
the circumference of their wheels, and from what has just
been shown, Eq. (124), the frictions there become
r
Nf. and N'ffe
R
R
in which r and r' denote the radii of the smaller trunnions,
and R and R' the radii of their corresponding wheels;
and thus the quantity of work of friction will become
γ
.
(Nf. + N'f) R, §,;
R.

MECHANICS OF SOLIDS.
379
a quantity obviously much less than that obtained comparison of
above.
If the wheels and their trunnions be of the same size,
and the trunnions as well as their boxes be of the same
material, the above expression becomes
results;
f(N + N')
r
R
·
Rs,;
work of friction,
when wheels
same size and of
same material;
the value of this expression may be made as small as we
please, indeed inappreciable, in a practical point of view,
by selecting surfaces and unguents for which ƒ is the least
possible, and making r very small. A beautiful applica- used in Atwood's
tion of this principle is exhibited in Atwood's machine,
which will be referred to hereafter.
machine.
XVIII.
STIFFNESS OF CORDAGE.
$229.-Let us now con-
sider a wheel turning freely
about an axle or trunnion,
and having in its circum-
ference a groove to receive
a cord or rope. A weight
W, being suspended from
one end of the rope while
a force F is applied to the
other extremity to draw it
up, the latter will experience
a resistance in consequence
of the rigidity of the rope,
F
Fig. 244.
Resistance from
stiffness of
cordage;
W

380
NATURAL PHILOSOPHY.
measure of the
rigidity of
cordage;
the value of
this measure;
stiffness on a
wheel whose
diameter is
unity;
stiffness on a
wheel of any
diameter;
which opposes every effort to bend it around the wheel.
This resistance must, of necessity, consume a portion of
the work of the force F
The measure of the resist a flaw
ance due to the rigidity of o
cordage has been made the
subject of experiment by
Coulomb; and, according to
him, it results that for the
same cord and same wheel,
this measure is composed of
two parts, of which one re-
mains constant, while the
other varies with the weight
W, and is directly propor-
tional to it; so that, designa-
ting the constant part by
Бле
Fig. 244.
F
W
K, and the ratio of the variable part to the weight W
by I, the measure will be given by the expression
K+I. W;
in which K represents the stiffness arising from the natural
torsion or tension of the threads, and I the stiffness of the
same cord due to a tension resulting from one unit of
weight; for, making W= 1, the above becomes
K+ I.
Coulomb also found that on changing the wheel, the stiff-
ness varied in the inverse ratio of its diameter; so that if
K+I. W
be the measure of the stiffness for a wheel of one foot
diameter, then will
K+I. W
2 R
be the measure when the wheel has a diameter of 2 R. A

MECHANICS OF SOLIDS.
381
to find the
table giving the values of K and I for all ropes and cords data from which
employed in practice, when wound around a wheel of one numerical
foot diameter, and subjected to a tension arising from a measure of
unit of weight, would, therefore, enable us to find the stiff-
ness answering to any other wheel and weight whatever.
stiffness;
But as it would be impossible to anticipate all the dif-
ferent sizes of ropes used under the various circumstances of
practice, Coulomb also ascertained the law which connects these data
the stiffness with the diameter of the cross-section of the abridged;
rope. To express this law in all cases, he found it neces-
sary to distinguish 1st, new white rope, either dry or moist;
2d, white ropes partly worn, either dry or moist; 3d, tarred different
ropes; 4th, packthread. The stiffness of the first class he kinds of rope;
found nearly proportional to the square of the diameter of
the cross-section; that of the second, to the square root of
the cube of this diameter, nearly; that of the third, to the
number of yarns in the rope; and that of the fourth, to the laws which
the diameter of the cross-section. So that, if S denote the govern them;
resistance due to the stiffness of any given rope; d the
ratio of its diameter to that of the table; and n the ratio
of the number of yarns in any tarred rope to that of the
table, we shall have for
New white rope, dry or moist.
S = d2
K+I.W
2 R
(127).
that for new
white rope;
Half worn white rope, dry or moist.
S = d3. K + I. W
2 R
(128).
old white rope;
Tarred rope.
S = n.
K+I. W
2 R
(129).
tarred rope;
Packthread.
S= d.
K+I. W
2 R
(130).
packthread;

382
NATURAL PHILOSOPHY.
TABLES
OF WEIGHTS NECESSARY TO BEND DIFFERENT ROPES AROUND A WHEEL
ONE FOOT IN DIAMETER.
No. 1. WHITE ROPES-NEW AND DRY.
Stiffness proportional to the square of the diameter.
Diameter of rope
in inches.
Natural stiffness,
or value of K.
Stiffness for load of
1 lb., or value of I.
lbs.
Squares of the ratios
of diameter, or val-
ues of d².
for new white
ropes, dry;
0.39
lbs.
0.4024
0.0079877
0.79
1.6097
0.0319501
1.57
6.4389
0.1278019
3.15
25.7553
0.5112019
Squares
Ratios d.
d².
No. 2. WHITE ROPES-NEW AND MOISTENED WITH
WATER.
1.00
1.00
1.10
1.21
I.20
1.44
Stiffness proportional to square of diameter.
1.30
1.69
1.40
1.96
Diameter of rope
in inches.
Natural stiffness,
Stiffness for load of
1.50
2.25
or value of K.
1 lb., or value of I.
1.60
2.56
for new white
ropes, moist;
0.39
0.79
3.2194
1.57
12.8772
3.15
51.5111
lbs.
0.8048
lbs.
1.70
2.89
0.0079877
1.80
3.24
0.0319501
1.90
3.61
0.1278019
2.00
4.00
0.5112019
No. 3. WHITE ROPES-HALF WORN AND DRY.
Stiffness proportional to the square root of the cube
of the diameter.
Natural stiffness,
or value of K..
Stiffness for load of
1 lb., or value of I.
lbs.
Square roots of the
cubes of the ratios
of diameters, or val-
ues of d
Diameter of rope
in inches.
old white ropes,
dry;
0.39
lbs.
0.40243
0.79
1.13801
1.57
3.21844
0.0079877
0.0525889
0.0638794
3.15
9.10150
0.1806573
Ratios or
d.
No. 4. WHITE ROPES-HALF WORN AND MOISTENED
WITH WATER.
Stiffness proportional to the square root of the cube
of the diameter.
1.00
1.10
d3
Power
dž.
or d
1.000
1.154
I.20 1.315
33
21
1.30 1.482
Diameter of rope
in inches.
Natural stiffness, Stiffness for load of
or value of K. 1 lb., or value of I.
1.40
1.657
1.50
1.837
1.60 2.024
old white topes,
moistened;
lbs.
lbs.
0.39
1.70
2.217
0.8048
0.0079877
1.80
2.415
0.79
2.2761
1.57
0.0525889
1.90
2.619
6.4324
3.15
0.0638794
2.00
2.828
18.2037
0.1806573

MECHANICS OF SOLIDS.
383
No. 5. TARRED ROPES.
Stiffness proportional to the number of yarns.
[These ropes are usually made of three strands twisted around each other, each
strand being composed of a certain number of yarns, also twisted about each other
in the same manner.]
No. of yarns.
Weight of 1 foot in
length of rope.
Natural stiffness, or
value of K.
Stiffness for load of
1 lb., or value of I.
668
lbs.
0.0211
lbs.
lbs.
for tarred ropes;
0.1534
0.0085198
15
0.0497
0.7664
0.0198796
30
1.0137
2.5297
0.0411799
For packthread, it will always be sufficient to use the
tabular values given above, corresponding to the least
tabular diameters, and substitute them in Eq. (130). An
example or two will be sufficient to illustrate the use of
these tables.
Example 1st. Required the resistance due to the stiff- examples to
ness of a new dry white rope, whose diameter is 1.18 illustrate the use
inches, when loaded with a weight of 882 pounds, and
wound about a wheel 1.64 feet in diameter.
Seek in Table No. 1 the diameter nearest that of the
given rope; it is 0.79; hence
d
=
1.18
0.79
=
1.5 nearly;
and from the table at the side,
d2
= 2.25.
From Table No. 1, opposite 0.79, we find
of Table No. 1;
elements
obtained from
the tables;
K =
=
1.6097,
I
=
0.03195;
which, together with the weight W
2 R
1=
ft.
1.64, substituted in Eq. (127), give
882 lbs., and

384
NATURAL PHILOSOPHY.
lb.
result;
S = 2.25
lb.
1.6097 +0.03195 × 882
1.64
lbs.
11
40.817,
got me k
example to
illustrate Table
No. 4;
which is the true resistance due to the stiffness of the rope
in question.
Example 2d. What is the resistance due to the stiffness
of a white rope, half worn and moistened with water,
having a diameter equal to 1.97 inches, wound about a
wheel 0.82 of a foot in diameter, and loaded with a weight
of 2205 pounds?
The tabular diameter in Table No. 4, next below 1.97,
is 1.57, and hence
data from the
table;
1.97
d =
=
1.3 nearly;
1.57
the square root of the cube of which is, by the table at
the side,
d2 = 1.482.
Hody
In Table No. 4 we find, opposite 1.57,
69.0
K = 6.4324,
I = 0.06387;
ft.
which values, together with W= 2205 lbs., and 2 R = 0.82,
in Eq. (128), give
lb.
6.4324 + 0.06387 X 2205
lbs.
result;
S = 1.482 X
0.82
lbs.
=
266.109,
which is the required resistance.
Example 3d. What is the resistance due to the stiffness
of a tarred rope of 22 yarns, when subjected to the action

MECHANICS OF SOLIDS.
385
illustrate Table
No. 5;
of a weight equal to 4212 pounds, and wound about a example to
wheel 1.3 feet diameter, the weight of one running foot of
the rope being about 0.6 of a pound?
of
By referring to Table No 5, we find the tabular number
yarns next below 22 to be 15, and hence
22
n=
=
15
1.466 nearly.
In the same table, opposite W, we find
K = 0.7664,
data obtained
from the table;
I =
0.019879;
ft.
which, together with W
(129), give
=
4212, and 2 R = 1.3, in Eq.
S = 1.466
1.3
0.7664 +0.019879 x 4212
lbs.
= 95.188. result;
Example 4th. Required the resistance due to the stiff-
case of
ness of a new white packthread, whose diameter is 0.196 example to
inches, when moistened or wet with water, wound about a illustrate the
wheel 0.5 of a foot in diameter, and loaded with a weight packthread;
of 275 pounds.
The lowest tabular diameter is 0.39 of an inch, and
hence
0.196
d
=
=
0.5 nearly.
0.390
Safor
In Table No. 2 we find, opposite 0.39,
lb.
K = 0.8048,
I = 0.00798;
data from Table
No. 2;
525

386
NATURAL PHILOSOPHY.
result.
=
275, and 2 R = 0.5, we find, after sub-
which, with W
20.onstituting in Eq. (130),
Work due to
stiffness of
cordage;
S = 0.5
foodr
barogs to 0.0 trods quid eqor eft
0.8048 + 0.00798 x 275 lbs.
275inus. Ly
= 2.999.
0.5 0
§ 230. The resistance just found is expressed in
pounds, and is the amount of weight which would be
necessary to bend any
given rope around a ver-
tical wheel, so that the
I adleggo oldet enes
portion A E, between 0.0 A
the first point of con-
tact A, and the point E, 2010.0
where the rope is attach-
ed to the weight, shall ere
be perfectly straight.
The entire process of
bending takes place at
the bending takes this first or tangential
place at the first
point of contact;
point A; for, if motion
be communicated to the
wheel in the direction
indicated by the arrow-
Fig. 245.
1087.0
m
D
m
rear)
E
S
head, the rope, supposed not to slide, will, at this point,
take and retain the constant curvature of the wheel, till
it passes from the latter on the side of the power
F
When, therefore, by the motion of the wheel, the point m
of the rope, now at the tangential point, passes to m', the
working point of the force S will have described in its own
path described by direction the distance A D. Denoting the are described
by a point at the unit's distance from the centre of the
wheel by s,, and the radius of the wheel by R, we shall
the working
point;
T
have
8108.0
ADRs,;
and representing the quantity of work of the force 8, by

MECHANICS OF SOLIDS.
387
L, we get
012
L = S. Rs,;
asgrist bangomily
replacing S by its value in Eqs. (127) to (130),
q
boot bae olxs odt of
K+I.WS
L = Rs,d,
2 R
(131);
Hope
work of the
stiffness;
in which d, represents the quantity d2, da, n, or d, in Eqs.
(127), (128), (129), or (130), according to the nature of the
rope.
Example. Taking the 2d example of § 229, and sup- examples;
posing a portion of the rope, equal to 20 feet in length,
to have been brought in contact with the wheel, after
the motion begins, we shall have
મૂવી
lbs.
L = 20 x 266,109
=
5322.18;
result;
that is, the quantity of work consumed by the resistance
due to the stiffness of the rope, while the latter is moving in words.
over a distance of 20 feet, would be sufficient to raise a
weight of 5322.18 pounds through a vertical height of
one foot.
XIX.
nd Hedy on My
woln
Fig. 246.
WHEEL AND PULLEY.
$231.-A plane wheel,
free to turn about its
trunnions or axle, support-
ed in a fixed box, may
be moved in either di-
rection by two forces F
and Q, which act in its
F
A
on
B
Wheel and
pulley;

388
NATURAL PHILOSOPHY.
equilibrium of
upon the
plane, and tangent to its
circumference at A and B.
These forces, acting in the
same plane, perpendicular
to the axle, and tending
two forces acting to turn the wheel in oppo-
circumference of site directions, will be in
a wheel;
equilibrio when the ele-
F
Fig. 246.
A
B
Vials
Q
mentary quantity of work
developed by each is the same with contrary signs. But
the points of application A and B, belonging to the same
circumference, the paths which they simultaneously de-
scribe will be equal; and since the product of these paths
the forces must by the forces F and Q must be equal, it follows that when-
ever the forces are in equilibrio, they must also be equal.
be equal;
This supposes the wheel free to turn, without obstruc-
tion of any kind. But if we consider the friction at the
trunnion or axle, then, supposing the equilibrium still to
exist, but the wheel on the eve of motion in the direction
of the force F, the elementary quantity of work of the
when friction is latter must be equal to that of the resistance Q, increased
by that of the friction; in which case F and Q will not be
equal; and denoting the radius of the wheel by R, that
of its trunnion or eye by r, and the resultant of F and Q
by N, we shall have
taken into the
account;
relation of work
of power,
resistance, and
friction;
FRS,
QRS, + fN.rs, . . (132);
in which ƒ is the coefficient of friction at the axle or trun
nion, and s, the arc described by a point at the unit's dis
tance from the axis during motion. Dividing by Rs,, we
find
relation of these
forces;
r
F = Q+ fN
R
(133).
From which we might conclude the value of F, but that
N is unknown, being the resultant of F and Q.

MECHANICS OF SOLIDS.
389
Now, two cases may arise, viz.: either the value of r
may be very small in comparison with R, or it may not. two cases may
In the first case, any error committed in the determination
of N would but slightly affect the value of F, since only
r
the small fractional portion of N is taken. We may,
R
therefore, be content with an approximate value for N.
To obtain this, we first
omit the consideration of
friction, which will make
f= 0, in the above equa-
tion, which then reduces
Fig. 247.
M
arise;
to find the anlay
resultant of the
power and hon
resistance;
to
F= Q.
9
Denote by the angle
AMB, which the two
forces Fand Q make with
each other; then, from the
parallelogram of forces,
will
F
fow to lineup
D
A
B
C
N = √F2+ Q² + 2 FQ. cos ;
its value;
and, because Fand Q are, in this case, equal,
N=QV2 + 2 cos o;
or this;
but
whence
2+2 cos q = 4 cos²;
and of on100
90000
N = 2 × 2 cos;
and finally this;

390
NATURAL PHILOSOPHY.
but joining A and B by a right line, as also M and C, we
www have the angle AMC, equal to 9,
and
value of resultant
under a more 1
convenient form;
rule;
pino
cos 19
1=1
sin MCA
AD AD
=
=
A C
Tom W
R
which, substituted above, gives insinco od
A Bobi
N = Q
Ꭱ
since 2 A D = A B. That is to say, the resultant N is
obtained by multiplying the resistance Q, by the chord of
the arc between the tangential points, and dividing the
product by the radius of the wheel. This value of N,
substituted in Eqs. (132) and (133), gives
quantity of work;
value of the
force;
FRS,
=
QRs, +f.rs, Q
AB
R
(134),
AB
X
R
F = 9+√. Qx4 (135);
Q
the first of which will
give the quantity of work
of the power, and the
latter the relation of the
power F to the resistance
Q, necessary to produce
inferences; an equilibrium. The first
shows that the work of
the power is equal to the
work of the resistance, in-
creased by that consumed
by friction.
We now come to the
second case, viz.: that in
which r is not very small
F
Fig. 247
M
Бар
D
B
C

MECHANICS OF SOLIDS.
391
small;
in comparison with R.
ith R. And first we remark, that F is case in which
always greater than Q, and that the resultant obtained trunnion is not so
under the hypothesis of F being equal to Q, is, therefore,
too small. Calling N, this latter resultant, we have
AB
N₁ = Q
R
first
(136);
approximation to
the resultant;
and this value, substituted in Eq. (133) for N, gives
(01)
F = Q+
r
R
first
f. N₁ = F₁.. (137). approximation to
Now if N, be too small, it is obvious that Fi will also be
too small. But this value of F is greater than Q, and if
we find the resultant of two forces each equal to F₁, or
make
the power;
N
=
FAB
R
second
=
N... (138);
approximation to
resultant;
it is obvious that N, will be too great, and so of the value
r
F = Q + ·ƒ⋅ N₂ = F₂-
R
Thus the true value of F is greater than F1, and less than
F2, and as these two values will not differ much, we may
take the true value of F to be an arithmetical mean
between them, that is,
F = F₁₂+ F₁₂
2
second
approximation to
the power;
mean of the
approximations;
or
30
F = Q +/-ƒ. N + N;
R
f.
2
value of the
power;

392
NATURAL PHILOSOPHY.
final value of
power;
quantity of
work;
conclusion.
Pulley;
and eliminating N₁ and N2, by means of Eqs. (136), (137),
and (138), we find
r
AB]..
r AB
RR
F = Q + √ / Q // [1 + √ √ ].. (139);
R
and multiplying each member by Rs,
AB
R
FRs, = QRs, +frs,94 [1+14]. (140).
AB
R R
The first will determine the condition of the equilibrium,
and the second the quantity of work.
8232.-The pulley is a small wheel having a groove
in its circumference for the reception of a rope, at one
end of which is attached the power F, and at the other
Fig. 248.
description;
and mode of
applying the
power;
61) 10
F
Q
the resistance Q. The pulley may turn either upon trun-
nions or about an axle, supported in what is called a

MECHANICS OF SOLIDS.
393
block. This is usually a solid piece of wood, through block;
which is cut an opening large enough to receive the
pulley, and allow it to turn freely between its cheeks.
Sometimes the block is a simple framework of metal.
When the block is stationary, the pulley is said to be fixed pulley;
fixed. The principle of this machine is obviously the
the wheel;
same as that of a simple wheel, and to the discussion principle the
of § 231 we have but to add the consideration of the same as that of
stiffness of the rope, to have all the circumstances of
its action. The quantity of work due to the stiffness of
the rope is given by Eqs. (127) to (130) inclusive.
Now, when the motion is uniform, or when the pulley
is about to turn in the direction of the power F, the quan-
tity of work of the latter must be equal to the work of
the resistance Q, increased by that of the friction and stiff-
ness of the rope; and denoting the radius of the pulley
by R, that of the trunnion or eye of the pulley, as the
case may be, by r, and the arc described at the unit's dis-
tance from the axis by s,, we must have
FRS,
=
QRs, +d,
.
K+IQ
2 R
•
Rs, + fN.r.s,; quantity of work
of power;
in which d, denotes either da, d, d, or n, in Eqs. (127) to
(130), according to the kind and condition of the rope;
and N, the resultant of all forces except friction.
Dividing by Rs,, we obtain
F = Q + d
K+I.Q
2 R
value of the
+
N.
power;
Make
Q + d
K+IQ
2 R
=
and the above becomes

394
NATURAL PHILOSOPHY.
different form of gont boo F = Q, +ƒ N;
same;
and replacing N by its value
R
ai dolilw.
wolls ban wolly
hins al yoling out
resultant of all
the forces except
friction;
√ F¹² + Q² + 2 FQ, cos q,
Φ
in which denotes the angle AMB,
made by the branches of the rope
not in contact with the pulley, and
we get
the most general
value for the
power;
Fig. 249.
M
B
C
F
Q
yaling
F= Q, +ƒV F² + Q² + 2 F Q, cos o.
R
Transposing Q, squaring and solving the equation with
reference to F, and we have
[1+() con
o
Q
(141).
F=
1-(5/2)²
Va+
(1+ cos 4) [2-(ƒ) (1-cos ()]
of (121) ap
Taking the upper of the double sign, because the motion
takes place in the direction of F; replacing Q, by its value,
and calling the angle A CB, enveloped by the rope, e, in
which case,
we finally obtain
K+IQ
2R
Q+d,
the same in
F=
known terms;
1-()
2
R
cos = cos e,
[1-(-) cos 0
+f (1- cos 0) [2-(f) (1+ cos
R
R
(142)
0)]
Firia

MECHANICS OF SOLIDS.
395
gaibh ai 9 to boshui
When the two branches of the
rope are parallel, then will
0 = 180°; cos 4= -1; and the
equation becomes,
Fig. 250. volqmo
C
Q
F
= (2+d, K + 10)
2 R
.
r
1+f-
R
1
r
—
f
R
value when the
(143). branches of the
body
If the rope be perfectly flexible, and the friction be
zero, then will K=0, I=0, f= 0, and
F = Q;
that is, the power will always equal the resistance in the fixed
pulley, when there is neither friction nor stiffness of cordage.
To obtain the quantity of work, multiply both members
of Eq. (142) by Rs,, and there will result
rope are parallel;
[1-(-) cos 0
FRS,=
QRS,+d, (K+IQ) s,
03 (144).
1-(-1/2)*
2
R
+ √(1-cos 0) [2-(f) (1+ cos 0)]
general value for
the quantity of
work of the
power;
In finding the value of N, the weight of the pulley
was not considered, and for the reason that in practice
it is usually small; the friction arising from its action weight of the
may, therefore, in general, be neglected. Should it be pulley generally
desirable, however, in any case, to take it into account,
neglected;
it is easily done. For this purpose, find, by the paral-
lelogram of forces, the resultant of the weight of the but it may be
pulley and the force Q, both of which are known, and taken into the
account;

396
NATURAL PHILOSOPHY.
employ this resultant instead of Q in finding the value
of F.
Example. Requi-
example for
red the quantity of
work necessary to
Fig. 251.
illustration;
raise 500 pounds of
coal, through a ver-
tical elevation of 50
feet, by means of a
any rope passing over a
fixed pulley, in such
a position that the
conditions of the power F shall be ap-
proposition;
tabular elements;
707
numerical values
of the data;
plied in a horizontal
direction; the pul-
ley, which is of lig-
num-vitæ, is 1.25 feet in diameter; the radius of its is
eye
0.05 feet; the axle of wrought iron, lubricated with hogs'
lard; the rope is white, half worn, and has a diameter of
one inch.
Here= 90°, and cos
= 0; in Table IV. § 225,
1
ƒ = 0.11; Table No. 3, § 229, d, = d³ = (079)=
f
(1.2) nearly = 1.315; K = 1.13801; I = 0.0525889;
R = 0.625; r = 0.05; Rs, = 0.625 X s, = 50; whence
50
0.625
S₁ =
80 feet; Q = 500 lbs.; and f
=
R
These data in Eq. (144) give
0.0084.
quantity of work; FRs,= (500 x 50+ 1.315-
1.13801+0.05259 X 500
80) (1+0.0084 √2-(0.0084)2;)
or
FRS,
= 26250.17.
If there were no friction, or stiffness of cordage, then
would

MECHANICS OF SOLIDS.
397
FRS,
=
QRS,
=
25000.0;
whence 26250.17 - 25000 = 1250.17 is the loss due to
value without
friction and
stiffness;
stiffness of cordage and friction, which would be sufficient loss due to
to raise 1250.17 pounds through 1 foot of altitude, or
1250.17
50
= 25 pounds through the given height of 50 feet;
a result well calculated to impress us with the necessity of
including these resistances in all estimates of work.
stiffness and
friction;
F=
=
26250.17
Rs,
=
26250.17
50
lbs.
1=
525 nearly.
numerical value
of the
power.
§ 233. Thus far the axis of the pulley is supposed
to have remained immoveable. We shall now consider Moveable pulley;
the case in which the pulley is supported upon a rope in
its groove, one end of the rope being attached to a fixed
hook A, while the other is acted upon by the force F. The description;
pulley is embraced by a kind of iron or other metallic
fork whose prongs are per-
forated near the ends for the
reception of the axle, and
whose shank terminates in a
hook to which the resist-
ance W is attached. The
pulley is, in this case, said
to be moveable. Denote the
resistance to be overcome
and put in motion, by W;
the tension of the rope be-
tween the fixed hook and
tangential point H by Q;
let the other notation be the
same as in the case of the
fixed pulley.
A
Re
Fig. 252.
F
notation;
H
W
The quantity of work of F must be equal to that of the

398
NATURAL PHILOSOPHY.
bas
tension Q, increased by the work due to the stiffness of
the rope and friction; that is,
Troper
quantity of work; FRS, =
=
FRS, QRS, +d,
K+IQ
2 R
Rs, +rf Ws, .. (145).
Dividing both members by Rs,,
chavoy of
03
value of the
power;
FQ+d,
K+IQ
r
+
2 R
R
W
pallay oldes
The pulley being sup-
posed either on the verge
of rotary motion in the di-
rection of F, or rotating
uniformly, it is obvious that
W will be equal and di- leg
rectly opposed to the result-
ant of F and Q; and that
Q will be equal and directly
opposed to the resultant of
to find the tension F and W. This latter re-
of rope at the
fixed end;
the same
found by
approximation;
sultant being found by the
parallelogram of forces, Eq.
(31), and in its value that of
F, in last equation, substitu-
ted for F, the force Q will be-
come known in terms of W,
the friction, and stiffness of
cordage; and this value of
Q, being substituted in Eq.
(145), will give the work in
terms which are known.
The method here indi-
cated is perfectly rigorous,
but is somewhat long, and
may be avoided by resort-
Fig. 252.
A
F
H
yd
Q
W
ed
A
Fig. 253.
10 W
B

MECHANICS OF SOLIDS.
399
ing to an approximation which in practice is sufficiently
accurate. If F and Q be supposed for an instant equal,
we have seen that
bredo edt tel bus
R.W
Q =
=
AB
rolling out to approximate
od
value for tension
od of somendion ya
which, substituted for Q in Eq. (145), gives
FRS,
=
R
W
Rs,
AB
K+I. W.
AB
+d,
•
2
dividing by Rs,,
+r.fW.s,
10.0
s,.. (146); quantity of work;
06
08.0
pil ni barasitedna doidy
Ꭱ
K+I.W.
R
F=W.
AB+d
AB r
2 R
Ꭱ
R
+fW.. (147). value of the
power;
If we suppose the stiffness of the rope and friction zero,
there will result,
or
Ꭱ
F=W.
AB
power, when
stiffness and
friction are zero;
search
F: W:: R: AB;
that is to say, the power is to the resistance as the radius of relation of power
the pulley is to the chord of the arc enveloped by the rope.
and resistance;
Example. Let the pulley be of cast iron and turn example;
upon a wrought-iron axle, greased with tallow; the di-

400
NATURAL PHILOSOPHY.
ameter of the pulley 1.3 feet, and that of its eye 0.045
feet; the diameter of the rope, which is new, white and
dry, 1.4 inches; the weight W, 3462 pounds; the height
40 feet, and let the chord AB be equal to the diameter
of the pulley.
By reference to the proper tables, we find
d, = d² = (1.4)³ = (1.8)ª nearly = 3.24;
data from the
tables;
f = 0.07; d, = d²
K = 1.6097;
and from the given data,
0.79
I = 0.0319501;
R = 0.65;
2 = 0.0225; AB = 1.3; Rs, = 40;
data of the
example;
S =
40
0.65
= 61.538 nearly; and W
which, substituted in Eq. (146), give
=
3462;
(.65)2
3462 X
1.30
X 61.538
numerical result; FRs, =
1.60970.031950 X 3462 X
0.65
1.3
= 71279.35;
+3.24 X
-X 61.538
2
+0.07 x 0.0225 X 3462 X 61.538
with neither friction nor stiffness of cordage, the quantity
neither stiffness of work would be simply
same with
nor friction;
FRS,
=
3462
·
(0.65)2
1.30
X 61.538
=
69239.5;
work of stiffness the difference 71279.35-69239.5=2039.85 is the loss
and friction.
due to the causes just named.

MECHANICS OF SOLIDS.
401
$234.-The Muffle is a
collection of pulleys in two
separate blocks or frames.
One of these blocks is at-
tached to a fixed point A,
by which all of its pulleys
The muffle;
Fig. 254.
A
d quibivib
Aj
become fixed, while the other S
block is attached to the re-
sistance Q, and its pulleys
thereby made moveable. A
rope is attached at one end
to a hook h at the extremity
of the fixed block, and is
passed around one of the
moveable pulleys, then about
one of the fixed pulleys, and
so on, in order, till the rope
F
is made to act upon each
pulley of the combination. Hide
The power F is applied to
the other end of the rope,
and the pulleys are so pro-
portioned that the parts of
the rope between them, when
by ging A
200
Ch
t t
3
definition and
description;
Q
arrangement of
the rope;
F to
application of the
power and
relative size of
stretched, are parallel. Now suppose the power
communicate uniform motion to the resistance Q. Denote the pulleys;
the tension of the rope between the hook of the fixed
block and the point where it comes in contact with the
first moveable pulley, by t; the radius of this pulley by
R; that of its eye by r₁; the coefficient of friction on
the axle by f; the constant and coefficient of the stiffness notation;
of cordage by K and I, as before; then, denoting the ten-
sion of the rope between the last point of contact with the
first moveable, and first point of contact with the first
fixed pulley, by to, the quantity of work of the tension ta
will, Eq. (145), be
26

402
NATURAL PHILOSOPHY.
work of the
tension on first
ascending
branch;
It
to Rs, = t Rs, + d + R₁s, + f (t+t2) 18,;
dividing by s,
moment of this
tension;
t₂ R₁ = t₁ R₁+d,
2 R₁w
10 Hoold struga
in ei albold oasilt to on0
A soloq bozita of bado
K+ Itavolingi lo lla doidy
K + It . P₁ + ƒ (t₁ + ta)rı · · (148).
.
2 R₁
•
91 ody of bedsatte et hoold
Again, denoting the tension of that part of the rope which
boliate passes from the first fixed to the second moveable pulley
olighoob by ts; the radius of the first fixed pulley by R, and that
of its eye by r2, we shall, in like manner, have
moment of
tension on second
descending
branch;
sloods of
ad!
od to ono bobou
1
tz R₂ = t₂ R₂+d,
K+ Ita
2 R2
bos
R₂ + f (ta + ta) r₂ .. (149).
And denoting the tensions, in order, by t, and to, this last
being equal to F, we shall have oltenidos, si to
bollage at A wowog of
bas
moment of
tension on second
ascending
t4 Rs = t3 Rs + d
K+ Its
2 R3
•
R3 +f(t + ts) 13.. (150),
branch;
tarkt benoitoy
same on third
descending ordly
branch evitalot
K+ It
med
FR₁ = t₁ Rd,
2 RA
RA+ft+F) 4.. (151);
lood
components of
the resistance;
baxit odt
so that we finally arrive at the force F, through the ten-
sions which are as yet unknown. The parts of the rope
being parallel, and the resistance Q being supported by
their tensions, the latter may obviously be regarded as
equal in intensity to the components of Q; hence
Jodi diwaħ + to + ts + t₁ = Q.. (152);
To how to be
d rollng berk
which, with the preceding, gives us five equations for the
determination of the four tensions and power F. This

MECHANICS OF SOLIDS.
403
would involve a tedious process of elimination, which
may be avoided by contenting ourselves with an approxi-
mation which is found, in practice, to be sufficiently
accurate. siv noitroqorq gniwollot od yd A as noienst
If the friction and stiffness be supposed zero, for the method of
moment, Eqs. (148) to (151) become
approximation;
sill mod
steminongga
Bolsos
to R₁
= t₁ R₁₁
anoisuot od: la dose
qullum quid oma odt at doidy To
ound on od lliw tophotz Ra
=
ta Ra
stadoo oft vd barot
-itedna pot and
t4 R3
1
To send evig low (101) pa mi botus friction and
to R₂, do 197 anolenst
viovilegess of Bar
stiffness zero;
FRA = ta Rai
olestanill
00.0 23.0 08.0 39.0
totemsib
Jool 80.0 = 89YO
from which it is apparent, dividing out the radii R1, R2, the tensions
Rg, &c., that to t₁, t3 = t2, t₁ = t3, F= t; and hence, Eq.
1
yelling silf esqqque Bas; 1000120.0
(152) becomes
vollng sir
00.0
oda doidy to sort grow to olzs
t
abrog 004 = 2;
become equal;
whence
gabroo Yo
2000
t₁
=
sonataisox of Bas
bas goitoin arodi W
∞1+
Q
the denominator 4 being the whole number of pulleys,
moveable and fixed. Had there been n pulleys, then
would
resistance equal
to tension on one
branch multiplied
by the number of
pulleys; xosqe
h = 2.
tr
n
de od bas to ouley oft grittitedva
With this approximate value of t, we resort to Eqs. (148)
to (151), and find the values of to, ta, ta, &c. Adding all
these tensions together, we shall find their sum to be
greater than Q, and hence we infer each of them to be too
general value for
the tension;

404
NATURAL PHILOSOPHY.
to find the true
from the
approximate
tension;
large. If we now suppose the true tensions to be propor-
tional to those just found, and whose sum is Q1 > Q, we
may find the true tension corresponding to any erroneous
tension, as t₁, by the following proportion, viz.:
Q
21: 2: t₁:
Q1
or, which is the same thing, multiply each of the tensions
Q
found by the constant ratio
the product will be the true
Qi
example to
iliustrate;
tensions, very nearly. The value of t, thus found, substi-
tuted in Eq. (151), will give that of F.
the
=
r4
Example. Let the radii R1, R2, R3, and R4, be respectively
0.26, 0.39, 0.52, 0.65 feet; the radii T1 =
T2 = 13 = r₁ of
eyes 0.06 feet; the diameter of the rope, which is
white and dry, 0.79 inches, of which the constant and
coefficient of rigidity are, respectively, K = 1.6097 and
I = 0.0319501; and suppose the pulley of brass, and its
axle of wrought iron, of which the coefficient f = 0.09,
and the resistance Q a weight of 2400 pounds.
Without friction and stiffness of cordage,
approximate
value of first
tension;
t₁ =
2400
4
lbs.
=
600.
Dividing Eq. (148) by R₁, it becomes, since d, = 1,
K+ It
t₂ = t +
+
2 R₁
1 f (4 + t₂).
Substituting the value of R, and the above value of ty
and regarding in the last term to as equal to t, which
we may do, because of the small coefficient
find
f
f, we

MECHANICS OF SOLIDS.
405
600
or soda ar
1.6097 +0.0319501 x 600
+
t₂ =
2 X (0.26)
= 628.39.
approximate
value of second
tension;
0.06
+
0.26
× 0.09 × (600 + 600)
Again, dividing Eq. (149) by R2, and substituting this
value of t, and that of R, we find 00
lbs.
ts
1
673.59.
approximate
aquablam ba value of third
tension;
Dividing Eq. (150) by R, and substituting this value of to,
as well as that of Rg, there will result
lbs.
t₁ = 709.82;
whence
oligodily and approximate
000 od blow
lady pountsler
I 09.080
value of fourth
tension;
and
600
Q₁ = t₁+to+ts + t₁ =
+628.39
+ 673.59
+709.82
=
2611.80;
bas
resultant of these
tensions; ab
Q1
10
=
2400
2611.80
1=
0.919;
ratio of the
approximate to
dold the true
resultant;
which will give for the true values of
t = 0.919 x 600 = 551.400
Towog ad
food w
to
=
0.919 x 628.39
=
577.490
=
ts
=
0.919 X 673.59 =
0.919 X 709.82 = 652.324
619.029
true tension;
652.324
2400.243

406
NATURAL PHILOSOPHY.
elewizorgge
binoose to onla
nolsost
The above value for t = 652.324, in Eq. (151), will give,
after dividing by R, and substituting its numerical value,
.08.828
652.324
1.60970+ 0.03195 x 652.324
2 X 0.65
+
F =
aidt gaitutiedua 0.06
+
X 0.09 × (652.324 + F);
0.65 DAN OW
aisy A
10
To suley
Θυραίκοτητε
ids to lay and making in the last factor F= t = 652.324, we find
molanot
numerical value
of the power;
diwal to salay
lbs. bulbs. (001) puspaibivi
F652.324 + 17.270 + 10.831 = 680.425.
糖
​stamizongga Thus, without friction or stiffness of cordage, the intensity
nols of F would be 600 lbs.; with both of these causes of
work absorbed
by friction and
friction
stiffness of
cordage.
edito olier
allosos
description;
resistance, which cannot be avoided in practice, it becomes
680.425 lbs., making a difference of 80.425 lbs., or nearly
one seventh; and as the quantity of work of the power is
proportional to its intensity, we see that to overcome fric-
tion and stiffness of rope, in the example before us, the
motor must expend nearly a seventh more work than if
these sources of resistance did not exist.
Wheel and axle ; 8 235.-Wheel and Axle is a name given to a machine,
eurt on which consists of a wheel mounted upon an arbor, supported
at either end by a trunnion resting in a box. The plane
of the wheel is at right angles to the axis of the arbor;
the power F is applied to a rope wound around the
wheel; the resistance Q is applied to another rope, wound
in the opposite direction about the arbor, and also acts in
a plane perpendicular to the axis of motion. The power
is generally applied in the plane of the wheel, other-
wise, being oblique to the axis, it would be necessary
to resolve it into two components, one perpendicular
and the other parallel to that line; the latter compo-
and application
of power and
resistance;
notamot sa

MECHANICS OF SOLIDS.
407
application ups
nent would press the shoulder of the arbor against the effect of an ento
face of the box, and increase the effect of friction by oblique istina
increasing its "lever arm." It may happen, however, that of the power; om
the particular object to be accomplished will sometimes
make it inconvenient to Aizs out tocaroni de process where
satisfy this condition of or od od nailaor eidt evisono power does not
Fig. 255.
keeping the action of neda lliw rateпoqos wheel
the power in the plane or ift bus outd
of the wheel, in which dazol edt loogeboo
event, it will be easy to
find the pressure arising
from the parallel com-
ponent of the power or
resistance, and to com-
T
act in plane of
no griton edno is loffatago m
ol to equorg own fliw godt
Ingorod D
to fadt
tadsd
A
pute the friction by vol
the rules already given.
Supposing the power
and resistance to act in
planes at right angles to
the axis, we remark, that
the plane of the wheel
in which the power acts,
E
Q1
B
enolitant odt
si no noisin
ali bor spoinden
it ods
aginar
Bus M. V
10
when the power
acts in the plane
F
T
of the wheel;
odt lo tad of
ogabroo to saprile
Toy
ti
bas
bud foodw od) to anibar
too
9.14
and the plane perpen-
dicular to the axis, through the direction of the resistance,
will cut from the arbor equal circles. Through the point
E, at which the rope is tangent to the circle in the latter of
these planes, and the axis, conceive a plane to be passed;
it will cut the circle in the plane of the wheel on the
opposite side of the arbor in E', and the line joining E
and E' will intersect the axis in I, making EI = E' I
At the point E' apply two opposite forces Q1 and Q2, construction;
parallel and each equal to the resistance Q.These forces
will produce no effect upon the system. The resultant of clans
the two equal and parallel forces Q and Q₁ will be equal
to their sum, will pass through I, will be resisted by the
axis, and produce no work, except what may arise from
the friction due to its action on the trunnion. The equi-
21
Jo golisin
bollgas

408
NATURAL PHILOSOPHY.
equilibrium ore
forces which batte librium, if the machine be at rest, or its uniform motion,
maintain the if at work, must, therefore, be maintained by the power F,
motion uniform; the force Q2, the friction, and the stiffness of cordage. To
this end, the resultant of F, Q2, and stiffness of cordage
must intersect the axis. At the point of intersection,
to soob we conceive this resultant to be replaced by its primitive
and eas
To all
pressure upon
the trunnions;
components, and there will then act upon the axis the
forces F, Q2, Q+ Q1, and the resistance due to stiffness of
cordage. Each of these forces being resolved into two
parallel components acting on the trunnions A and B,
there will result two groups of forces, one applied to each
trunnion. Denote the resultant of the group acting on
the trunnion A by M, that of the group acting on the
trunnions and its trunnion B by M', then will the frictions be respectively
fM and f' M'; and, employing the usual notation, the
quantities of work will be f Mrs, and f' M' r' s,, the radii
soog of the trunnions, and their friction being unequal.
friction on the
work;
soul on fald
Moody edito
The quantity of work of the power F, must be equal
to that of the resistance Q2, augmented by the work of the
stiffness of cordage and friction, and hence, denoting the
radius of the wheel by R, and that of the arbor by R',
F.RS, Q₂R's, +d,
=
K+I. QR's,+ƒMrs,+f'M' r's,;
2 R'
but if the trunnions and boxes are supposed of the same
size and material,
10
quantity of work; FRs, = Q2 R's, +d,
friction of
effect wherever
applied;
活
​H
30
K+ IQ R's, +ƒ(M+ M') rs,
2 R'
The quantity M+M', being the sum of the pressures
upon
the trunnions, the last term shows that the friction is
trunnions, same the same as though the resultant of all the forces were
applied to a single trunnion in any arbitrary position, and,
therefore, at the centre of the wheel. But this would re-
duce all the forces to the same plane, in which case Q would
take the place of Q2, and Q1 and Q would disappear from

MECHANICS OF SOLIDS.
409
the system. Hence, denoting the resultant of the entire
system of forces by N, and writing Q for its equal Q2, the
above equation becomes
=
FRS, QR's,+d,
K+IQ
2 R'
R's, +ƒN. r · s, . . (153); quantity of work;
mot oft bonimo
and, dividing by Rs, suit 3d101 flama oot
F= Q
R'
R
+d,
K+IQR'
2 R' R
r
+fN.
(154).
R
value of the
power;
Now, N being the resultant of all the forces of the system
except friction, it is the resultant of F, Q, and d,
K+IQ.
2 R'
act in the same direction, it is
or, since Q and d, K+1Q
2R'
K+IQ
the resultant of F and Q+d,
2 R'
pursue the method explained in § 232.
To find N, we will
find the resultant
of all the forces
but friction;
Make
bas gal
Q + d
K+IQ
=
Q1.. (155);
2 R'
then wil.
R'
F= 21
R
r
R
If we neglect the consideration of friction for a moment, by
and find the resultant N₁ of F and Q1, or of
+f.N
.
•
(156).
R'
Q₁
and Q1,
R
approximation;
we shall have, denoting the inclination of the power to the
resistance by 9,
N=
first
1+1+2 cos).. (157); approximation
R12
R*+2Q,
R
R
cos =Q
R
R'/R'
R
for resultant;

410
NATURAL PHILOSOPHY.
first
approximation
for power;
and this for N, in Eq. (156), gives b
offt 19 Leupo att
Leupo all rot 9 gudist bas
R
F = Q+ƒN=Fi
f R
R
malaya oft
30 mojave
(158).
Now the value of N was too small for N, because we
the firstmation Now th
approximation
generally
sufficient;
ol to outer
when it is not,
a second
omitted the term fNR
in the value for F; and, hence, Fi
is too small for F; but the deficiency is less and less, in
r
proportion as the fraction f is smaller and smaller. In
R
ordinary practice there will be but little difference between
the true value of F and that given by Eq. (158).
In cases wherein r is considerable in comparison with
approximation R, a further approximation will be necessary; and to ac-
must be made;
complish this, we remark, that F, is greater than Q1
R
R'
and
Q1 therefore less than F1 and that if this latter be
therefore
combined with F₁, to
obtain a second re-
bonialipro
sultant N, this last
Fig. 256.
sole M
will be too large, and
N
art of When substituted in
0142
geometrical
indication;
Eq. (156), for N, will
give a value F for
F, which will also be
FA
too large. The mean
QR
of the two values of
F₁ and F, will be the
second
approximation
for resultant;
second
approximation to
value of the
power;
practical value of F be
The value of N is given by the equation,
of 191
νόσ
N₁ = F₁ √1+
√1 +
R R
+ 2 cos o
RR
cos).. (159);
and
R
r
F₁ = Q+JN
F2 Q1 fN₂
R
R

MECHANICS OF SOLIDS.
411
(160), two
amitositados bas 0=200 gaplam (Tel) pa ni gidt bas
whence
0001 X 8888980.0 4 10881.1
98.800
$10.2000-
F=
Fi+Fa R'
Q1
2
=
+f.
R R 2
r N₁+ N₂
mean of the
approximations;
FRs, = QR's, +d, K+IQ
R's,
s, N₁ N₂
To find the quantity of work, multiply both members
by Rs, replace Q1 by its value, and we have
2 R'
s+fr. 8, +N. (161). quantity of work;
2
Example. Required the quantity of work necessary to
raise two tons of coal from the bottom to the top of a pit
which is 80 feet deep,
by means of the
wheel and axle. The
diameter of the wheel
is 4 feet; that of
the axle, 1 foot;
that of the trunnion,
which is of wrought
iron, working in cast-
iron boxes and lubri-
cated with hogs' lard,
1.5 inches; that of
the rope, which is
white, half-worn, and
dry, 1.5 inches; and
the
acts in
power acts in a
horizontal direction.
08.0 X
Fig. 257.
diw bejuttiedna doidy
example to
Op illustrate;
+ Tv 200,00fr
soner
Here R = 2 feet; R' = 0.5 feet; r = 0.125 feet; data of the
question and
2.619; tables; untor
3
f = 0.07; d,
d2 =
(1.5) (1.9)*
K = 1.13801; I
R's,
= 80 feet; s, =
or cos y = 0.
These data, substituted in Eq. (155), give
=
0.79
0.0525889;
Q = 4000 lbs. ;
800
=
160 feet; and =
Φ 90°,
0.5
estoffure ed

412
NATURAL PHILOSOPHY.
lbs.
Q₁ = =4000+2.619.
(001)
and this, in Eq. (157),
for its value
R
R'
lbs.
lbs.
4553.89;
1.13801+0.0525889 X 4000
0.5
=
2
1
making cos = 0, and substituting
0.25
0.25 feet, we find
value of first
rosultant;
(101)
N₁ =
4553.89 V1+ (0.25)
=
4694.04.
R
This and the values of Qf, and
in Eq. (158), give
R'
=4553.89 X 0.25 +0.07 x 4694.04 X 0.06251159.008;
F=
R
which, substituted with the values of
and cos o = 0, in
R
o alqu
Eq. (159), gives
value of second
resultant;
N₂ = 1159.008 V1 + (4)²
v1
=
4778.68;
hence,
N = N₁ + N₂
4694.04 + 4778.68
lbs.
=
=
4736.36;
2
2
which, with the values already found for Q1, in Eq. (160),
gives
lbs.
value of power; F = 4553.89 x 0.25 + 0.07 X
0.125 lbs.
4736.36 1159.19.
=
2
Here it may be proper to direct the attention to the slight
difference between the values of F and F₁, showing that the
first approximation, as given by Eq. (158), will generally
be sufficient.

MECHANICS OF SOLIDS.
413
Finally, from Eq. (161), we obtain
1
lbs. ft.
FRS, 4000 x 80 + 44311.20 + 663.07364974.27. quantity of work;
=
The first term of the second member
320000, is the
value of the work without any resistance from friction and
stiffness of cordage; the sum of the remaining terms
= 44974.27, is the work of friction and stiffness of rope;
hence it appears, that the loss arising from the latter
causes, is nearly one seventh of the work which, without
them, would be required to accomplish the object. This
loss would be sufficient, without the hinderance from fric- loss of work by
tion and stiffness of cordage, to raise more than a quarter stiffness of
of a ton through the given height.
If, in Eq. (154), we make f = 0, and disregard the
friction and
cordage.
stiffness of cordage, we find
R
F = Q
(162);
R
that is to say, in the wheel and axle, the power is to the
resistance as the radius of the axle is to that of the wheel.
wheels;
endless bands,
chains;
§ 236.-Wheels are often so combined in machinery as Combination of
to transmit the motion impressed upon some one of them,
according to certain conditions, determined by the object motion
to be accomplished. This is usually done by one or other transmitted by
of the following means, viz.: 1st. By endless ropes, bands, ropes, and
or chains, passing around cylindrical rollers, called drums,
mounted upon arbors; 2d. By the natural contact of these by natural
drums; 3d. By projections called teeth or leaves, accord- contact;
ing as these projections are upon the surfaces of wheels or
arbors. The communication of motion by these means is
always accompanied by friction, which it is important in
practice to know, since it may not be disregarded.
and by teeth.

414
NATURAL PHILOSOPHY.
$237.-When two isido ow (Fig. 258. o
Resistance due to wheels are connected
stiffness of bands with each other by
and ropes hay means of an endless
the bands and
drum;
band or rope dcbe,
passing around the
0A
F
d
e
ort To
o B
b IT
drums A and B, el que trotiv o odular
mounted upon the
arbors of the wheels,
a sufficient force Fap-
plied to one of them
To mua silt eneboo to Q
nottorit to show ads af S
geof oil Jeds grasas al esned
to dneva ore
will put it in motion; this motion will be communicated to
friction between the other as long as the friction between the band and drums
is sufficient to prevent the former from sliding over the
9 latter, and thus a resistance Q, applied to the second wheel,
may be overcome. The motion of the drum B is ob-
viously due to the difference of the tensions in the two
branches dc and eb; and applying the power as indicated
in the figure, the tension of dc must be greater than that
of eb. Denoting the first of these by T, and the latter by t,
the force which moves the drum B will have an intensity
equal to T-t; and the quantity of its work must be
equal to that of Q, increased by the work of friction on
the trunnions of the common arbor. Denote the radius
of the drum B by Ra; that of the wheel to which Q is
to noted applied by R"; that of its trunnion by ra; the arc de-
scribed by the point at the unit's distance from the axis of
motion due to
difference of
tension;
al motion by s2, &c., then will
batma
work of doll
difference of qephud
tension;
ta
(T-t) RSQ R" Sq + f N2 72 82.. (163).
The action of the force F produces the difference of ten-
ssion T
t, and its work must, therefore, be equal to that
-
of T-t augmented by the work of friction on the trun-
nions of the arbor of the wheel to which F is applied.
Denote the radius of this wheel by R', that of its drum
by R₁, that of its trunnion by r, the are described at the

MECHANICS OF SOLIDS.
415
unit's distance by 81, and we have mban doidw mont
an od bis FR's₁ = (T− t) R₁₁ + ƒ N₁. ₁₁.. (164). work of the
Hoswood noilor stilupor el gobong
goubong of oldisso es llame
emih odt ban bund out
Adding these equations together, we get
FR's,+(T-t) R, S,= (T-t) R, s,+QR" 8, + fN₂ T₂+ fN₁r,,;
To ody toban guibila a qet dgnano terged
but because all parts of the band have the same velocity,
the circumferences of the drums must move at the same
rate; hence
power;
bloods
an
of tuntura
OMAR
R2 S2 = R1 81;
11
which will reduce the above equation to
Lotiation bus band
beodwooth
ture ed of boilgga
bas
bas 19 an toofto
de od tamm
esdoned of
FR' s₁ = Q R" S₂+ fNa 2 S2 + fN₁₁ si.. (165).
(801) pi
ni botutifedua
ot broose od bas
Whence we see that the work of F is equal to the work
circumferences of
the drum have
the same
velocity;
work of the
power;
of Q, increased by that of the friction upon the two sets of inferences;
trunnions; and the same may be shown of any number of
wheels thus connected.
In this equation, N, is the resultant of the forces Q, T,
and t; and N₁ of F, T, and t. To find these resultants it
will be necessary to know T and t.
oved
hat
The difference T-t only exists while the system is in
motion; when at rest, and the force does not act, this dif-
ference is zero, or T is equal to t. In passing from rest
to motion, we may assume that one increases just as much
as the other diminishes, and if the common tension at rest
be
represented by T₁, and the increment of the one and
decrement of the other in passing from rest to motion
be denoted by H, then will
value of the
T=T+H, and t=TH.. (166); tensious;

416
NATURAL PHILOSOPHY.
tension at rest
arbitrary;
should be just
sufficient to
prevent sliding;
rclation of the
two tensions;
from which T and t may be found when T, and I are
known
The tension T, is entirely arbitrary. It should be as
small as possible, to produce the requisite friction between
the band and the drums to avoid sliding during the mo-
tion, for if greater than this, it will only increase the
pressure and, therefore, the friction on the trunnions, un-
necessarily. In general, it will be sufficient if this friction
be great enough to prevent sliding under the effect of Q,
at the surface of the drum of the wheel to which Q is
applied. But this effect, neglecting friction on the trun-
nions and stiffness of cordage, is Q
R
R₂
That is to say,
a force whose intensity is given by this expression, when
applied to the surface of the drum, will produce the same
effect as Q; and the friction between the drum and strap
en t
must be at least equal to this force to prevent sliding.
The branches dc and eb of the band are solicited respect-
ively by the two forces T₁+H, and T₁-H; and these
substituted in Eq. (108), the first for F and the second for
W, we find,
fs
T₁₂+ H = (T₁ - H) eRa;
subtracting T-H from both members of this equation,
and we have
difference of
tensions;
fs
e'
T + H - (T-H) = (TH) ea - (TH);
the first member reduces to 2 H; that is to say, to the dif
ference of tensions on the two branches of the band, which
must be equal to the effect of Q at the surface of the drum;
whence
R"
2 H= Q.
(167),
R2

MECHANICS OF SOLIDS.
417
R"
Q
Ra
=
fs
same in terms of
(TH) (-1). (168); the friction, &c.;
from which two equations we may compute H and T₁, and
therefore, Eq. (166), T and t; and, finally, the resultants
N, and N₁ by the rules for the composition of forces.
tension at rest;
Example. Required the tension of a band necessary to
produce friction enough to move a wheel, when subjected
to a resistance of 1000 pounds, the radius of the wheel example;
being 0.5 foot, and that of the drum 2 feet, and the arc of
the drum enveloped by the band 180°. Let the band be
of black leather, and the surface of the drum of oak.
Here R 2 feet; R"
Q = 1000 lbs. ;
=
=
0.5 feet; Q
f = 0.265, (see Table I, § 212;) S = TR₂
=
3.1416 Ra;
R
Q.
1000 X
Ra
0.5
2
lbs.
=
250;
R
H = } Q ·
1=
lbs.
125;
Ra
half difference of
tensions ;
R"
Q.
R
250
TH=
=
fs
(2.7182818)0.265 × 3.1416 -1
lesser tension;
-
(eR₂ - 1)
The first term of the denominator may be easily found by
the aid of logarithms, as follows:
Log [(2.7182818)0.83251] = Log 2.718281 x 0.83251
=
0.4342942 X 0.83251
value found by
the aid of
logarithms;
= 0.361554 nearly;
the natural number of which is 2.2991, whence
TH=t=
250
lbs.
=
=
192.44.
value;
250
2.2991-1- 1.2991

418
NATURAL PHILOSOPHY.
greater tension;
to find the
resultant;
Adding 2 H=250 lbs., we have
lbs.
T₁₂+ H = T= 442.44.
The arc of the drum enveloped by the band being 180°
the tensions T' and t must be parallel, and their resultant
T=T+t=634.88 lbs., which being combined with Q=
1000, according to the principles of the composition of
forces, will give N2, and with F will give N₁, whence every
thing required to determine the quantity of work in Eq.
(165) is known.
If Eq. (165) be divided by R's,, it becomes
R"
value of power;
F= Q·
R
but
•
S2
52 + f⋅ N₂⋅
S1
T2 S2
R'
+fN
$1
R
velocity of the
circumferences
equal;
11
=
R2 S2 R1 81;
whence
S2
$1
=
R₁.
R₂
and by substituting above,
final value of
power when the
F= Q
R. R
R. R+f. Na
12
R₁
T1
+f. N
R'
R R
motion begins in
its direction;
without friction;
which is the relation subsisting between F and Q, in case
of an equilibrium bordering on motion in the direction of
F, or in the direction of uniform motion.
If we disregard the friction, then will
R". R
FQ·
R. R₂

MECHANICS OF SOLIDS.
419
When the motion from one wheel and axle is communi- combination of
cated to a second machine of the same kind, by passing without friction;
the band about the
axle of the wheel
to which the power
F is applied, and
the wheel of that
to whose axle Qis
applied, then will
R₁ be the radius of
the first axle, and
R that of the sec-
ond wheel, and the
preceding equation
gives us this rule,
viz.:
914
F
Fig. 259.
wheels and axles
19dtoms sa
to the resistance.
When the friction is so small that it may be disregarded, relation of power
the power F will be to the resistance Q, as the product of the
radii of the axles to that of the radii of the wheels, in the case
of an equilibrium or uniform motion.
छ di un plaif
bands may be
neglected;
§ 238.-In the preceding discussion, no mention is made Rigidity of
of the resistance arising from the stiffness of cordage.
When the connection or gearing is made by bands, these
are so thin as to possess considerable flexibility, and their
opposition to bending may, in practice, be safely neglected.
If the connection
be made by an end-
less rope, the op-
position to motion
d
Fig. 260.
T
takes place at the
points where the
rope bends in pass-
ing on to the drums,
and not at all at
the points where
it leaves the latter
F
a
R"
R
b
rigidity of ropes;

420
NATURAL PHILOSOPHY.
value of the
resistance at one
point;
at another;
at another;
and becomes straight. Thus at the point a, the resist-
ance is
at the point b it is
K+I. Q
d.
2 R"
d.
K+ It
2 R2
and at the point d it is
d,..
KIT
2 R
rigidity of chains;
and, finally, at the points f and e it is nothing. These
resistances must be included among those to be overcome
by the power F.
If the connection be made by an endless chain, each
link, as it turns in the next one in order, may be regarded
e
Fig. 261.
En
d
En
F
e
Dn
A
B
E
H
each link a
trunnion in its
box;
as a trunnion revolving in its box; and each, as it comes
to be applied to the drum, revolves about the next one
through an angle E'HE, equal to DCD', the angle
through which the drum revolves to produce the contact;
and taking the sum of all these angles, it is obvious that,

MECHANICS OF SOLIDS.
421
although each link revolves through a very small angle,
yet this sum must be equal to the angle through which the
drum has turned to produce it. Houd of you
Denoting byr the radius of the inner circular arc in notation, &c.;
which the end of each link is shaped, s, the arc described
by the point at the distance of unity from the axis of the
drum B, f the coefficient of friction, and T and t the ten-
sions on the two branches of the chain, then will the work
of friction among the links at the points f and b respec-
tively, (figure before the last,) be
fTr S
and ftrsai
10.0 work of friction
among the links,
at one set of
and denoting by s₁ the arc described by a point at the dis-
tance of unity from the axis of the drum A, the work of
friction at the points d and e will be, respectively,
f Tr su and ftrs;
and the whole amount of this kind of work will be
fr (T+t) (82 +81).
Recollecting that the points on the surfaces of the drums
must have the same velocity, viz.: that of the different
links of the chain, we have
R2 S₂ = R181;
in which R and R, are respectively the radii of the drums
Band A. From this relation we find
points;
the same for
another set;
whole work of
this friction;
velocity of
circumference of
drums equal;
$1 = 821
Ra
R₁
which, substituted above, gives
fr s₂ (T + 1) (1
9) (1+).
(169).
value of the work
of friction among
the links;

422
NATURAL PHILOSOPHY.
example;
data;
quantity of work
Example. Let T and t have the values of the last ex-
ample, (that of the strap,) and supposer= 0.03, the chain
of wrought iron, for which we find in the table of § 225,
(assuming that f is the same for trunnions of wrought iron
in boxes of the same material, as for trunnions of wrought
iron and boxes of cast iron,) f= 0.07; also let the radius
of the drum B be four times that of the drum A; then will
the expression (169) for a single revolution of the drum B,
in which case s₂ = 2 x 3.1416 = 6.2832, become not to
lbs.
1
lbs.
of friction. 0.07 x 0.03 x 6.2832 (442.44+ 192.44) (1+4)= 41.88,
adell si
10 201
that is, the work lost in consequence of the friction among
the links of the chain, during one revolution of the drum
of the wheel to which the resistance is applied, is sufficient
to raise a weight of nearly 42 pounds through one foot of
vertical height.
§ 239.-Let us now suppose the circumferences of the
Resistance from wheels to be furnished with teeth, which interlock with
friction on the
each other, so that
teeth of wheels;
30
conditions of
construction of
the teeth;
a force being im-
pressed upon one
wheel, it cannot
move without com-
municating motion
to the other.
The teeth are
usually curved, and
as to
so shaped as
have a common
normal D₁ D2, at
their point of con-
tact m, where the
action of one and
Fig. 262.
O
my
Di
Q
the reaction of the other take place; and although the
point of contact alters its position, as the wheels rotate,

MECHANICS OF SOLIDS.
423
yet the place of
this normal does
not change, but re-
mains stationary,
and the point of
contact is always
on it. We will not
stop to explain the
constructions
by
which this is ac-
Fig. 263.
D₁
D
side conditions of
preserving a
constant normal
at the point of
contact;
complished; it will be sufficient for our present purpose
to be assured of its practicability, and that we may pro-
ceed on the supposition that it has been executed in the
case under consideration.
From the centres C and C₂ of the wheels, let fall upon
the normal D₁ D2, the perpendiculars C, D, and C2 D2.
wheels;
The points D, and D, must, during the rotation of the relative velocities
wheels, have the same absolute velocity, and therefore the of the two
number of revolutions of the wheel whose centre is C, in
a given time, must be to that of the wheel whose centre is
C2, in the same time, inversely as the perpendiculars C₁ D1,
and C₂ Da; or, because of the similar triangles CBD, and
CBD, inversely as the distances C, B and C, B. The
circles described about C and C as centres, with radii
CB and C, B, respectively, are called the primitive circles. primitive circles;
These circles and their radii may be easily found from the
consideration just named. It will be our object to find a
force which, applied tangentially to these circles at B, will
produce the same effect as friction on the teeth.
Denote by the resistance acting at a
from the axis of the wheel whose centre is C.
distance R
The effect
of this resistance acting at D, in the direction of the
normal D, Da, will, from the principles of the wheel and
axle, be Q1, given by the relation
R
2₁ = Q
(169)';
C₂ Da
effect of the
resistance at the
distance of
common normal;

424
NATURAL PHILOSOPHY.
value of the
friction on the
and this Q will be the pressure at the point m. Its fric
tion will be
f Q1
teeth;
to obtain the
acting in the direction 9192, tangent to both teeth at their
point of contact. The elementary quantity of work of
this friction will be
equal to its intensi-
ty, multiplied into
the elementary dis-
tance by which the
rubbing points now
at m, separate in
the direction of this
tangent; which dis-
quantity of work tance is obviously
equal to that by
of this friction;
the value of this
work;
which the points
Fig. 263.
Di
D
91 and 92 the extremities of the perpendiculars let fall
from C and C₂ upon the common tangent, approach to or
recede from each other. Denoting the elementary path
described by a point at the unit's distance from C₁ by $1,
and that described at the same distance from C₂ by Sa the
paths described by q1 and q will be, respectively, C₁ 91 X 811
and C2 X S2; and because the points q, and q2 must move
in the same direction when the tangent 42 91 passes between
the centres C and C2, the elementary path of friction will
be equal to the difference of these paths, and its elemen-
tary quantity of work will equal
f Q1 [Ca 92 X Sq
C₁₁ X 81].
Designating the radii of the primitive circles whose cen-
tres are C and C₂ by R, and R2, respectively, we have,
because of the equal velocities of the circumferences of
these circles,

MECHANICS OF SOLIDS.
425
whence
R₁ $1
S2 =
=
R2 Sai
d
R₁
$1.
Ra
Moreover, drawing through the point B the line 2, 21,
parallel to the tangent 42 91, and denoting the angle m BC,
which is the complement of the angle CB21, by 9, and
the distance m B by h, we find
Ca la = R₂ cos + h,
C₁ 1
=
R₁ cos o h;
-
relation of the
paths at unit's
distance from the
two centres;
these values of s2, C2 42, and C1 91, substituted in the ex-
pression for the elementary work of friction, give
fQh & (+1)
Denote by the intensity of a force which, applied
tangentially to the primitive circles at B, will produce the
same effect as the friction. Its elementary work will be
w R1 S1, and hence
w - R₁₂ &₁ = ƒ Q₂h &₂ (R² + R₁);
si R2
or
lever arms of the
friction;
work of friction;
tangential force
R₁+ R₂
w = ƒ Q₁h
(170).
at the
R₁ Ra
Represent the angle B Cm by 8. In practice, the
angle m B C does not differ much from 90°, and we may
take
circumference of
primitive circle;
h =
R₁tan 8.

426
NATURAL PHILOSOPHY.
another form for
tangential force
at primitive
circumference;
and because is generally very small, the tangent may be
replaced by the arc, and
=
h = R₁0;
which, substituted above, gives
w = f Q1
R+R₂
R₁ R
X R₁0 . . (171).
The value of varies from a maximum to zero on one
side of the line of the centres CC, and from zero to a
second maximum on the opposite side of this line; the
first maximum corresponds to that position of m in which
any two teeth come first in contact, and the second to that
in which the contact ceases; the intermediate or zero value
occurs when m is on the line of the centres. The quantity
being thus variable, it must be replaced by a constant,
and this constant must be a mean of all the values between
to find the mean the two maxima. Designating the first of these by 41, and
value of the
angular distance
of point of
contact;
the second by 02, lay
off the distance AE
=0; erect at A the
perpendicular AG=
01; draw GE: then
will the ordinates of
this line which are
parallel to A G repre-
sent the different val-
Fig. 264.
0'
0
G
G
A
EX
B
ues of 4, and the area
altitude of a mean
triangle;
of the triangle EAG will be the sum of all the values of
between 0 and zero. Again, make EB = 0; erect at
B the perpendicular BG""; draw G" E: the area
of the triangle EBG"" will be the sum of all values of
between zero and 42. Make
BO=
022 +0,2
01 + 02'

MECHANICS OF SOLIDS.
427
complete the rectangle B O', and draw A 0; then will the construction;
triangle ABO be equivalent to the sum of the triangles
AEG and EBG"", and therefore equivalent to the sum
of all values of between 1 and 2, the mean of which is
obviously the middle ordinate.
0,2 +0,2
2(81+82)
=
T
(0₂+02)2-2010201 +03 0102 mean value of
2(01 +62)
y=}BO=
Neglecting the last term as insignificant,
xy
=
01+02
2
2
0102 angular distance
of point of
contact;
Multiplying by R, we find that R (+82) is the interval
between the place of the first and last point of contact of
the same pair of teeth, estimated on the circumference of
the primitive circle; denoting this interval by a, and sub-
stituting in Eq. (171), we find
13
fQ1
R₁+ Ra
R₁ R2
a
a
X
1
+
2 R2
R
2 P₁).
Denote the number of teeth on the wheel whose centre is
C by n₁, and the number on the wheel whose centre is Ca
by na; then, because the teeth and intervals between them
must be the same on each circumference, in order to work
freely,
tangential force
at primitive
circumference;
2 R₁
α=
=
n1
2 R2
no
which, substituted above, gives
f.
w=5.2 (+1)
T=
= f.. Q₁
nng
Replacing Q₁ by its value given in Eq. (169)', and recollect-
distance from the
place of first to
that of last point
of contact;

428
NATURAL PHILOSOPHY.
final value of
tangential force
which is
equivalent to
friction;
its quantity of
work;
example;
data;
work;
result.
ing that, within the limits supposed, C, D, becomes R2, we
finally have
+2).. (172).
w = fx · Q · R (1₂+ m²).
R2
To find the quantity of work, multiply both members of
this equation by Rs2, which will give
w Rig Sq = ft S₂ Q R . n² + m
R₂
n2
no ni
(173).
Example. Required the work consumed in each revo-
lution by friction on the teeth of a wheel whose arbor is
subjected to a resistance equivalent to 1000 pounds, the
number of teeth on the wheel being 64, and that of the
connecting wheel being 192; let the teeth be of cast iron,
and suppose the radius of the arbor equal to 0.8 foot.
Here, R=0.8; Q=1000 lbs. ; S2 = 2 x 3.1416; *=
3.1416; f=0.152; n = 64; n₁ = 192; and, therefore,
lbs.
w R₂ 8, 0.152 X 3.1416 X 6.2832 X 1000 X 0.8
64+ 192
64 X 192
=50;
that is to say, the quantity of work consumed in one revo-
lution by friction on the teeth, in the case supposed, is suf
ficient to raise 50 pounds through a vertical distance of one
foot.
The screw.
XX.
THE SCREW.
The Screw, regarded as a mechanical power, is a device
by which the principles of the inclined plane are so
applied as to produce considerable pressures with great
steadiness and regularity of motion.

MECHANICS OF SOLIDS.
429
Fig. 265.
§ 240.--To form a clear idea of the figure of the screw Screw with
and its mode of action, conceive a right cylinder ak, with square fillet;
circular base, and a rectangle abcm having one of its sides
ab coincident with a surface element, while its plane passes
through the axis of this cylinder.
Next, suppose the plane of the rect-
angle to rotate uniformly about
the axis, and the rectangle itself
to move also uniformly in the di-
rection of that line; and let this
twofold motion of rotation and of
translation be so regulated, that in
one entire revolution of the plane,
the rectangle shall progress in the
direction of the axis over a distance
greater than the side ab, which
is in the surface of the cylinder.
D
10
mode of
m."
bm'
Sure generating;
un
d
The rectangle will thus generate a projecting and winding
solid called a fillet, leaving between its turns a similarly the fillet, channel,
shaped groove called the channel. Each point as m in the and the helix;
perimeter of the moving rectangle, will generate a curve
called a helix, and it is obvious, from what has been said,
that
every helix will enjoy this property, viz.: any one of
its points as m, being taken as an origin of reference, as
well for the curve itself as for its projection on a plane
through this point and at right angles to the axis, the dis-
tances d' m', d" m", &c., of the several points of the helix
from this plane, are respectively proportional to the circu- properties of a
lar ares md', md", &c., into which the portions mm', mm",
&c. of the helix, between the origin and these points, are
projected.
helix;
The solid cylinder about which the fillet is wound, is
called the newel of the screw; the distance mm", between newel;
the consecutive turns of the same helix, estimated in the
direction of the axis, is called the helical interval. The helical interval;
surfaces of the fillet which are generated by the sides of
the rectangle perpendicular to the axis, are each made up

430
NATURAL PHILOSOPHY.
and inclinations
of the different
helices;
w of a series of helices, all of which have the same interval,
though the helices themselves are at different distances from
relative position the axis. The inclination of the different helices to the
axis of the screw, increases, therefore, from the newel to
the exterior surface of the fillet, the same helix preserving
its inclination un-
changed throughout.
The screw is re-
ceived into a hole in
a solid piece B of
metal or wood, called
a nut or burr. The
surface of the hole
through the nut is
furnished with a wind-
the nut;
fillet of the nut; ing fillet of the same
screw and nut;
shape and size as the
channel of the screw,
which it occupies;
while the fillet of the
latter fills up the
channel of the nut,
formed by the turns
of its fillet, whose
inner surface is thus
brought in contact
with the newel.
screw,
ni Fig. 266.
990in
Q
D
Go
B
F
H
In
From this arrangement it is obvious that when the nut
relative motion of is stationary, and a rotary motion is communicated to the
the latter will move in the direction of its axis; also
when the screw is stationary and the nut is turned, the nut
must move in the direction of the length of the screw.
the first case, one entire revolution of the screw will carry it
longitudinally through a distance equal to the helical inter-
val, and any fractional portion of an entire revolution will
carry it through a proportional distance; the same of the
nut, when the latter is moveable and the screw stationary.

MECHANICS OF SOLIDS.
431
resistance and
The resistance Q is applied either to the head of the screw,
or to the nut, depending upon which is the moveable ele-
ment; in either case it acts in the direction DC of the
axis. The power F is applied at the extremity of a bar application of the
GH connected with the screw or nut, and acts in a plane
at right angles to the axis of the screw. Denote the per-
pendicular distance of the line of direction of F from the
axis of the screw by R, and the helical interval by h; then
will the quantity of work of the power F, in one revolu-
tion, supposing it to retain the same distance from the axis,
be
power;
does moit
work of the
FX 2R;
lap power in one
revolution;
and the quantity of work of the resistance will be
Q x h.
work of the
resistance;
The power F and resistancee Q, both act to press the fillet
of the screw and that of the nut together, the first acting
at right angles to, and the latter in the direction of the
axis. To find the work of friction thence arising, it will
be necessary to find a force F, parallel to F, whose effect
at the fillet is the same as that of F, acting at the distance
R from the axis, and to resolve both F1 and Q into two
components, one normal and the other parallel to the
common surface of the pressing fillets. But the surfaces
being warped, the normals at their different points will be
oblique to each other, and so inclined to the axis that the
normal components of the resistance Q, near the newel,
will be less than those towards the outer surface of the intermediate
fillet, while the reverse will be the case with the power F1.
The resolution must, therefore, be made with reference to a
normal at a helix midway between the newel and outer sur-
face. This helix, like all others, is situated upon the sur-
face of a cylinder of which the axis coincides with that of
the screw. Denote the radius of this cylinder Cmv by r
helix;

432
NATURAL PHILOSOPHY.
construction;
Conceive a tangent plane to this cylinder at any point, as
mv, and two cutting-planes normal to the axis, and at a
Fig. 267.
F
projection of
intermediate
helix;
R
N
C
distance from each other equal to a helical interval, and
equally distant from m. If we now develop the portion
of the cylindrical surface, included between the cutting-
planes, on the tan-
gent plane, the sur-
face of the cylinder
will become a rectan-
gle whose base A E
development of is equal to 2 r, and
the intermediate
Fig. 268.
K
I
helix;
whose altitude EB is
D
B
equal to h; and the
I
Fi
helix will become the
diagonal A B.
De-
A
E
note the length of the
resolution of the helix AB by 7. Then
power and
resistance into
components;
=
draw the normal mv L, and resolve Q and F, as before
stated. Since Q mv K is perpendicular to A E, and
Lm perpendicular to A B, the angles Lm K and EAB
are equal; also, since F₁ = Im¹v is perpendicular to BE,
the angles Im L and A BE are equal, and the triangles
ABE, Im O, and Lm K, being right angled, are similar,
and give the proportions
7: 2πr :: Q: Lm™,
1: h :: F₁: m™ 0;

MECHANICS OF SOLIDS.
433
component of
resistance;
whence
Zoldy
Lm² = 2xrQ
dog or normal
7
miv =
O
h. Fi
7
and the total pressure, which is equal to the sum of mv 0
normal
component of
power;
and m L, becomes
2 = r² +
h Fi
total normal
gairightful pressure;
and the friction
(2 TrQ + h F).
f
7
friction;
and since in one revolution the path described by this
friction is the diagonal A B =
will be
=
7, its quantity of work
f(2rQ+h F₁);
and because the work of the power F must equal the
work of the resistance Q, increased by that of the friction,
we have
2 R. F = Qh + f(2rQ + F₁h).
But the effect of F and F being the same, their quantities
of work must be equal, and hence
2 RF = 2r Fi;
its quantity of
work in one
revolution;
work of power
equal that of
resistance
increased by
work of friction;
whence
28
8888
R
F₁ = F;

434
NATURAL PHILOSOPHY.
work of power;
value of the
power;
work of power;
work absorbed by
friction;
which substituted in the preceding general equation,
we get
2« RF = Qh + ƒ(2«r Q + FR );
and finding the value of F,
r
F = Q
hr + 2ft p²
2 Rr-fRh
(174).
Multiplying both members by 2 R; then adding and
subtracting Qh, in the second member of this equation,
we find
2 R.F= Qh+fQ
h² + 42 p2
2πr-fh
(175),
in which the work absorbed by friction is given by the
last term; that is to say, by
fQ.
h² + 422
2r-fh
relation of power
and resistance
without friction;
If we neglect the consideration of friction, or make f= 0,
we find, from Eq. (174), simply
h
F = 2 × 2 R
Q X
T
that is, the power is to the resistance as the helical interval
is to the circumference described by the extremity of the
perpendicular, drawn from the axis to the direction of
of
the power. From which it is obvious that the power
the screw may be increased, either by diminishing the
stated in words; distance between the thread or fillet, or by increasing the
distance of the power from the axis.
If we examine the expression

MECHANICS OF SOLIDS.
435
h² + 4²²
f Q
2πr-fh'
we shall find that the numerator of the fractional factor
increases more rapidly than the denominator for any
increment in the value of r, the radius of the mean helix.
For this reason, r should be made as small as possible radius of
consistently with sufficient strength.
Let b O be the radius of
the interior helix, or that
of the newel, and a O that
of the exterior helix; it is
usual to make the projection
a b, of the fillet, equal to the
thickness ad, measured in
the direction of the axis;
and for facility of execution,
the dimensions of the chan-
nel are made equal to those
of the fillet, that is to say,
c'bis made equal to ad; in
which case, the helical in-
terval a a' will be equal to
2 ad= 2 ab, when there is
but a single fillet. Should
there be two fillets, which
2
Fig. 269.
intermediate
helix should be
Bern daire small ;
0
are often employed to increase the helical interval without
changing the size of the newel, and therefore of r; then
will the helical interval be 4 ab. Considerations affecting
the union of sufficient strength with least friction, have
suggested this general rule in regard to the dimensions of
the fillet, viz.: make the projection ab equal to one third
of the radius 0b of the newel, or
ab = 30b.
proportion of the
different parts of
the screw;
rule;
This will give
Ob = 3ab;
radius of the
newel;

436
NATURAL PHILOSOPHY.
and
Ob + ab = r = 3ab + ab ab:
=
and because h
1
2 ab,
radius of
intermediate
helix;
r = 4 h;
work of friction;
its final value;
which substituted for r, in the expression for the friction,
gives
f. Qh
1 + 2.4
49
-
2
22
and making =
7
7
to which it is very nearly equal, the
expression reduces to
122
fx Qh
11 - ƒ
example;
result.
Endless screw;
To apply this to a particular example, let the screw be
made of wrought iron, and the nut of brass, and suppose
an unguent of tallow, in which case f= 0.103, see Table
III, § 212; hence the value of the friction becomes
1.152 × Qh;
which, substituted in Eq. (175), gives
2 R. F Qh + 1.152 Qh
11
2.152 Qh;
whence we see, that friction occasions a loss of work
greater than the whole work performed by the resistance.
§ 241.-The endless screw is employed to transmit a
very slow motion, and, at the same time, to overcome con-
siderable resistance. It is a short
screw, with
with square
square fillet,

MECHANICS OF SOLIDS.
437
and so supported as to revolve freely about its axis, with use and
no motion of translation. It is usually turned by means
of a crank. The fillet pass-
es between teeth on the
circumference of a wheel
of which the axis is per-
pendicular to that of the
screw. The resistance Q
is applied to the circum-
ference of the arbor of the
wheel. The rubbing faces
of the teeth, instead of be-
ing parallel to the axis of
the wheel, are slightly in-
clined to that line, so as
to make them parallel to
the surface of the fillet
when the latter is brought
in contact with the teeth.
aid)
Fig. 270.
A
dtons
nd of alza
od to attaq
Q
description:
surface of teeth
inclined to axis
of motion;
A rotary motion being communicated to the screw, its fillet
presses against the teeth of the wheel; and as the screw
can have no longitudinal motion, the wheel must turn
about its axis. As the teeth are withdrawn towards one operation and
end of the screw, others are interposed towards the other reason for the
end, and thus an endless motion may be kept up; hence
the name of the machine.
A plane through the
axis of the screw and per-
pendicular to that of the
wheel, will cut from the
rubbing surfaces of the
fillet and teeth a profile;
and if we confine our-
selves to what takes place
in this plane during the
motion, we shall find that
the circumstances will be
L
B
Fig. 271.
name;
section by a plane
through the axis
of the screw
perpendicular to
the axis of the
wheel;

438
NATURAL PHILOSOPHY.
the same as those of two
circumstances of wheels acting upon one
action same as
those of two
wheels with
teeth;
another through the inter-
vention of teeth; for, as
the screw turns about its
axis to bring different
parts of the fillet in this
cutting plane, the section
ab will move in the direc-
tion from A to B, driving
the section be of the tooth
before it.
C
A
B
Fig. 271.
Let Q, be the force applied at b in the direction AB,
which is tangent to the circumference whose centre is on
the axis of the wheel, and whose radius is Cb= R, and
which will sustain the resistance Q in equilibrio: then de-
noting by N the resultant of Q, and Q, by r the radius
of the arbor, and by r, that of the trunnion, will
quantity of work;
value of the
power;
friction;
Q, R, S,
=
Qrs, + fNr,s,;
in which s, is the arc described at the unit's distance from
the axis of the wheel.
Dividing by R,S,,
Q = Q1/2 + √ N/
(176).
Find, by the process explained in § 235, Eqs. (157) to (160),
the value of Q, and N. The pressure upon the tooth at b
will thus be known, being equal to Q. This pressure pro-
duces a friction upon the teeth of which the value is
f. Q
n + n'
nn'
T=
(
+
n
wherein n denotes the number of teeth on the wheel whose
centre is C, and n' the number on the other.
But the cir

MECHANICS OF SOLIDS.
439
cumference of this latter wheel being a right line, is in-
finite as well as the number of its teeth; hence
1
n'
=
= 0;
reciprocal of the
number of teeth
on section of
screw;
and the foregoing becomes
fr. Q₁
which must be added to
to turn the wheel and to
fillet of the screw.
無
​n
Q, to obtain the force necessary
obtain the total pressure on the
This sum, which is
Q₁ + f = · 2, = 2, (1 + ƒ
n
n
being substituted for Q in Eq. (175), will give
2 = RF = Q(1+ƒ) h +ƒQ(1+5)/2+
or
Th² + 4272
n/ 2xr-fh'
p2-
=
2 ~ RF Q, =) [h
1.
value of the
friction.
total pressure on
the fillet;
= 2 (1 + √ = ) [½ + √ b² + 1 +² (177), quantity of work
f.
2r-fh-
In the discussion of the screw, no reference has been
of power;
and collars
neglected.
made to the friction on the pivots and collars by which friction on pivots
the screw is kept in position. It will always be easy to
find this, in any particular case, by the rules for finding
the friction upon pivots, sockets, and shoulders or rings,
explained in § 223.

440
NATURAL PHILOSOPHY.
spied, losdw othe
dw rothel ald
XXI.
The lever;
fulcrum;
levers divided
into different
orders;
first order;
second;
third;
THE LEVER.
8 242.-The Lever is a
solid bar AB, of any form,
supported by a fixed point
O, about which it may freely
turn, called the fulcrum.
Sometimes it is supported
upon trunnions, and fre-
quently upon a knife-edge.
Levers have been divided
into three different classes,
called orders.
examples of
different orders of
levers.
A
Q
In levers of the first
order, the power F and re-
sistance Q are applied on
opposite sides of the ful-
crum 0; in levers of the
second order, the resistance
Qis applied to some point q
between the fulcrum O and
point of application of the
power F; and in the third
order of levers, the power
F is applied between the
fulcrum O and point of
application of the resist-
ance Q.
Q
F
The common shears fur-
nish an example of a pair
of levers of the first order;
the nut-crackers of the sec-
0
Fig. 272.
B
O
0 (1)
(3)
(2)
F
0
0

MECHANICS OF SOLIDS.
441
ond; and fire-tongs
x9
of the third. In all
orders, the conditions
of equilibrium are the
same.
F
which the
§ 243.-When the lever is supported upon a point, the Equilibrium of
equilibrium requires that the resultant of the power and the lever in
resistance shall pass through this point in order to be fulcrum is a
destroyed by its reaction; to have a resultant, the power
and resistance must lie in the same plane, and as the re-
sultant will also be in
this plane, the
power,
resistance, and ful-
crum, must be in the
same. If the result-
ant pass through the
fulcrum, its moment
taken in reference
thereto must be zero,
which requires that
the moment of the
power shall be equal
to that of the resist-
ance. That is, when
A
Fig. 273.
C
F
R
B
point;
power, resistance,
and fulcrum in
same plane;
a lever AB is in equilibrio and solicited by the power F moment of power
and resistance Q, O being the fulcrum, if we draw from equal to that of
this latter point Om and On, perpendicular respectively
to the direction of the power and resistance, then will
Fx Om = Q x On.
resistance;
If the lever turn upon trunnions, then will the moment of when lever is
power F, be equal to that of the resistance increased
the
by the moment of the friction on the trunnion.
ting the radius of the latter by r, then will
Designa-
supported on
trunnions;

442
NATURAL PHILOSOPHY.
moment of power
equal to that of
resistance, plus
FX0m
=
Qx On + fN.r;
that of friction; in which N is the resultant of F and Q.
Multiplying both members by s,, we have
work of the
power.
Use and
lever;
Fx Om x 8, = Q × On x s, + fNr.s;
that is to say, the quantity of work of the power F, must be
equal to that of the resistance Q, increased by the quantity of
work of the friction.
8244.-The lever is not intended to produce a con-
advantages of the tinuous rotation, but is usually employed to move a heavy
burden or great resistance through a short distance during
each separate effort of the power.
relation of power
to resistance on
an edge for a
fulcrura;
It is not, therefore, always necessary to make it turn
about trunnions which generally operate to disadvantage;
since these, to afford
sufficient resistance,
must be large, which
increases the term
fNrs, or the quan-
tity of work absorbed
by friction. If the
lever be laid upon a
simple knife-edge, r
becomes zero, and the
foregoing equation be-
comes
Q
Fig. 274.
0
FX Om xs, = Q X On x 8,
272
making the quantity of work of the power equal to that
of the resistance. The advantage of this machine, the
usually the lever most simple of all, is, that it transmits without loss, the
transmits without work of the power to the resistance. But this is not all,
resistance; a simple change in the point of support or fulcrum, which
loss the power to

MECHANICS OF SOLIDS.
443
may be made at pleasure, gives the means of establishing
any
desired relation between the power and resistance.
If, for example, the point of support O is placed so that
the distance On is one thousandth part of Om, then will
F=
Q
;
1000
diminution of
whence we see that with a very small power we may hold to effect a given
in equilibrio an enormous resistance; but as the quantity purpose, a
of work of the resistance must equal that of the power, power increases
the path described by the point of application of the latter
must increase in the same proportion.
its path;
To give an idea of the time necessary to raise a heavy
burden through a moderate height with the lever, suppose
the weight to be raised is 2000000 pounds, and that it is
to be elevated five feet. The quantity of work will be
2000000 lbs. X 5 ft. = 10000000 lbs. Supposing a man to
act by his weight = 150 lbs. at the end of a lever, he would example to
10000000
have to describe a path equal in length to 150
66666 feet, nearly. If in each second of time he move the
point of the lever at which he applies his weight, through
a distance of 0.2 ft., he will require
66666
0.2
=333333
seconds nearly, = 92.6 hours nearly, = 9.26 days, suppo-
sing the man to labor 10 hours a day: in fact a man left
to his individual efforts would never accomplish such a
task.
Jolly
illustrate this;
This example shows us that the lever is only useful for practical use of
momentary efforts, and when the burden, being considera-
ble, is to be moved through a very small distance.

444
NATURAL PHILOSOPHY.
Atwood's
machine;
objects of the
machine;
XXII.
ATWOOD'S MACHINE.
$245.-We shall terminate this branch of our subject
with a discussion of an instrument whose object is an ex-
perimental verification of the laws of constant forces. This
instrument is the invention of Atwood, an English philoso-
pher, and bears his name. Before proceeding to describe
it, let us first find the circumstances of motion under the
general case of which the machine in question is but a
particular instance. For this purpose, let A B and AD be
two inclined planes
having a common
altitude AE; H
and H', two wheels
of different diame-
the general case ters mounted upon
of which this
machine is a
particular
example;
the same arbor, to
which they are
firmly attached,
and of which the
axis is supported
upon trunnions par-
allel to the com-
W
Fig. 275.
H
H
E
C
B
E
mon intersection of
the two planes; W
W
and W' two weights supported upon the inclined planes
by means of cords c and c' wound, the first about the
one body ascends Wheel H and the second about the wheel H', the cords be-
while the other ing parallel to the inclined planes. Now if the weight
W be made sufficiently heavy, it will overcome all
opposition to motion and slide down the plane AB, while
descends;

MECHANICS OF SOLIDS.
445
the weight W' must
from its connection
move up the plane.
AD. It is required
Foo Fig. 276.
A
to find the circum-
טי.
stances of motion.
Denote the angle
which the planes
B
E
AB and AD make
respectively with the vertical A E, by o and o'; the radius
of the wheel H by R, that of H' by R', and that of the
trunnion by r.
The pressure of Wupon the plane A B
we have seen, is
W sin ;
that of W' on the plane AD is
W'. sin o';
Bas
and the friction on the planes A B and AD will be, re-
spectively,
f W. sin o,
and fW' sin o'.
The stiffness of the cord c', which alone opposes the mo-
tion since the cord c unwinds, is, § 229,
K+I. (0)
d
2 R'
to investigate the
circumstances of
motion;
in which d, represents d², dan, or d in Eqs. (127) to (130),
inclusive, according to the cord or rope used, and (0)
the tension of the cord c'. This latter is equal to the
component of W' parallel to the plane AD W'cos q',
increased by the friction due to its normal component
components of
the weights
normal to the
planes;
friction due to
these pressures;
stiffness of cord
which winds;

446
NATURAL PHILOSOPHY.
tension of the
=fW' sin q'; that is to say,
cord that winds; (2) = W' cos o' + f W' sin o' = W' (cos o' + f sin o');
its stiffness;
which, substituted in the expression above, for the stiffness
of the cord c', gives
d,
K+ I. W' (cos q' + f. sin o')
2 R'
length of paths in
direction of
weights;
At the instant motion be-
gins, let the centres of grav-
ity of W and W' be at G'
and G, respectively, and in
any subsequent instant at
G" and G; denote the dis-
tance G'G" by x, and G, G
by x', then will x and x' be
the paths described by the
Fig. 276.
A
11
G
B
E
centres of gravity parallel to the planes in the interval;
and
x cos, and
x' cos q',
quantity of work;
quantity of work;
total quantity;
will be the corresponding distances in the direction of the
weights.
The quantity of work performed by W will be
Wx cos,
and that performed by W' in the same time,
- W'x' cos q',
and the total quantity of work of both will be
Wx cos - W' x' cos p'.

MECHANICS OF SOLIDS.
447
The quantity of work absorbed by friction on the plane
AB is
f. W. x sin o,
and that absorbed by friction on the plane A D is
f. W'x' sin q',
and the total quantity absorbed by friction will be, sup-
posing the unit of friction the same on both planes,
f(Wx sin + W'x' sin q').
The quantity of work absorbed by the stiffness of the cord
c' will be
d
K+I. W' (cos o' + f sin o')
2 R'
work absorbed by
friction on one
plane;
that on the other;
work absorbed by
all the frictions;
x'.
work absorbed by
stiffness of cord;
The work consumed by friction on the trunnions will be
f'N.r.s,;
in which N is the resultant of the tensions of the cords c
and c'; in other words, is the diagonal of a parallelogram,
of which the contiguous sides have
work absorbed by
friction on
trunnions;
components of
W cos of W sin o, and W' cos o' +W' sin q', the pressure on
for their values, and + ' for their inclination to each
S
other.
is the arc described at the unit's distance from
the axis.
The work absorbed by the inertia of the wheels and
which is the same thing, half the living force of
arbor, or,
trunnions;

448
NATURAL PHILOSOPHY.
work absorbed
by the inertia of
wheels and
arbor;
the wheels and arbor will, § 159, Eq. (60)", be pa
VI
2
=
gVI
2 g.
a A
in which V₁ is the angular velocity, and I the moment of
inertia.
Denote by V the velocity of the body whose weight is
W, and by V' that of the body whose weight is W'; the
living force of the first will be
living force of
one body;
that of the other;
and that of the second,
Wy2
g
w' y²
g
;
doy and the quantity of action in the two bodies, will be
quantity of action
in the two
bodies;
W V² + W' V2
2 g
The quantity of work of the weights produces the
living force of the bodies, that of the wheels and arbor,
as well as the work of friction and that of the stiffness of
cordage; hence
work of the
weights equal to
the living forces
of moving parts
and the work of
friction and
stiffness;
WV+W V12
2g
+f(Wx sin + W'x' sin p')
Wx cos p-W' x' cos p' =
+d, x'
K+IW' (cos '+f sin o')
2 R'
+f' Nrs, +
VI
2
(178).
The variables in this equation, for the same inclination of
the planes, are V, V', V₁, x, x', and s; but these, by the

MECHANICS OF SOLIDS.
449
nature of the system, are connected by the following rela-
tions, viz.:
V: V:: R: R'. V':
=
VR'
R
.
(179),
x R'
x:x:: R: R'. x' =
(180),
R
relation between
the angular
velocities and
1: Rs, : x
90
x
=
.. s₁
•
.
R
(181),
spaces;
V
R: V: 1: V₁ .. V₁
=
(182).
R
These values of V1, V', x', and s,, being substituted in
Eq. (178), will give
R2
R2
2g
R
Wx cos - Wa
cos p' =
R
+d, x
R
R
+f(Wa sin +Wxsin p')
R K+IW' (cos d'+f sin p')
R
2 R'
ya I
+f'. Nr+g
R2 g
and solving this equation with respect to V³,
equation (178) in
different terms;
R
W. cos &-W'.
.
cos p
R
R
-f(W.sin &+W.
R
sin p')
value for the
square of the
R K+I.W'.(cos +fsin p')
velocity;
29
R/2
W+W.
R2
I
R2
R²+9⋅ R²-d₁⋅ R
-f. N
R
2 R'
The coefficient of x containing no variable, we find
that the space described by the body on the plane A B,
29

450
NATURAL PHILOSOPHY.
varied;
varies as the square of its velocity. Hence the motion is,
motion uniformly § 68, Eq. (8), uniformly varied; and the coefficient of x is
twice the velocity which the force producing this motion
is capable of generating in a unit of time.
Making
R'
W.cos - W'.
•
cos o'
R
R
value of the
velocity
-f(W sin p + W'
sin p')
R
g
A =
generated
in a unit of time;
W+W
R2
R2
I
+9R2
R K+I.W'. (cos +fsin o')
-d
R
2 R'
-f. N.
A
(182)
R
oviy in (611)
the foregoing equation may be written
square of the
velocity;
√² = x.2 A
(183).
space described;
value of velocity;
Since the motion is uniformly varied, if T denote the time
of describing the space x, then will Eq. (7) become
x = } AT²
(184);
writing A for v1, and x for s: substituting this for x
above, we find
or
V² = A² T²,
VAT
(185).
Eqs. (183), (184), and (185), give the laws of uni-
laws of uniformly formly varied motion, or, as it is usually expressed, the
varied motion; laws of constant forces. These laws are, 1st. The velocities
are to each other as the times in which the force produces them;
2d. The spaces described, are to each other as the squares of
the velocities acquired in describing them; or as the squares of
the times in which they are described.

MECHANICS OF SOLIDS.
451
m
Any device that will
make the time in which the
motion takes place com-
paratively great, while the
velocity acquired shall be
small, will enable us to
verify these laws from ob- m
servation. For this pur-
pose, A must be small. By
reference to Eq. (182), we
find that A may be dimin-
ished at pleasure by increas-
ing the angle o, or decreasing
o'; this will increase the
effect of friction, which op-
poses, while it will diminish
the component of W, which
aids the motion. Or A may
be diminished, by diminish-
ing the angles and q',
the difference between the
weights W and W' and that
between R and R'. Owing
to the uncertainty of friction
it is better to accomplish the
object by the latter method,
and this Atwood has done.
His machine consists es-
sentially of a fixed pulley
H, over which passes a
passes a cord
having attached to each ex-
tremity a basin s, for the
reception of weights; a ver-
tical graduated scaler of
equal parts, say inches, to
measure spaces; and a pen-
dulum clock h which beats
OIG
A
Fig. 277.
n
d
h
C
RICHARDSON.SC.
k
H
2
m
772
S
S

452
NATURAL PHILOSOPHY.
m
A
OIG
m
n
O
RICHARDSON.SC.
f
Fig. 277.
m
k
H
m
seconds, to mark the time.
The basins are short cylin-
ders of brass, having a wire
e coincident with the axis
and projecting some three
or four inches beyond the
upper bases; the cord is at-
tached to the ends of these
wires. The weights are either
circular plates m, or bars n,
of greater or less thickness,
depending upon the purpose
for which they are employed.
Both are perforated at the
centre, and a channel is cut
from the hole to the margin
to permit the cord f to enter,
that the weights may be
dropped upon the basins.
The scale piece r is pro-
vided with three sliding
stages, two of which a and
a' are rings, and the third c
is plane. The rings, whose
diameters are less than the
length of the bar-weights,
serve to take the latter from
a descending, or to add them
to an ascending basin. The
office of the plane stage, is
to arrest the motion of a
descending basin.
A fourth and revolving
stage o, connected by an arm
d with an arbor k, in front,
is used to support the basin

MECHANICS OF SOLIDS.
453
adjusting the
bearing the greater load opposite the zero point of the device for
scale. The arbor is also connected by means of a second basin to the zero
arm with the escapement-wheel of the clock. This stage of scale;
may
be thrown from under the basin when the seconds'
hand reaches a particular point on the dial plate; thus
causing the motion to begin at a particular instant, and
from the zero of the scale.
If we examine the value of A, we shall find that for
Atwood's machine, and o' are both zero, and therefore
sin ❤
1=
0;
sin q' = 0;
COS O
=
= 1;
cos q':
11
1:
reduction of
general equation
to the case of
Atwood's
machine;
moreover R is equal to R', and hence
R'
R
11
1.

The cord is very fine, and usually made of raw silk but
slightly twisted, so that the term involving the stiffness of omissions;
cordage has no appreciable value, and may be neglected.
The arbor of the pulley or wheel rests upon circumferences
of four other wheels of large radii compared with the radii
of their trunnions, after the manner explained in § 228, so
that the term involving the friction on the trunnions may
also be neglected without appreciable error.
Making the foregoing substitutions and omissions in the
value for A, we find
W
-
W'
A = g.
I
g
W+W' + 9 R2
The circumference of the wheel has the same velocity
as the points of the cord, and therefore the same as the
basins. Designate by M", the mass which if concentrated
in the circumference of the wheel would have a moment
corresponding
value of the
general
coefficient;

454
NATURAL PHILOSOPHY.
moment of inertia
of the wheel;
of inertia equal to that of the wheel, then
whence
M" R2 = I;
I
M" =
R2
velocity
generated in
unit of time;
and this, substituted above, gives
W W'
—
A = 9 W + W' + 9 M"
= g
WW'
W+W' W""
in which W" denotes the weight of the mass M".
This value of A, substituted in Eqs. (184) and (185),
gives
space;
velocity;
the weight W";
x=
WW'
W+WW"
g T. (186),
V
Ꮽ Ꭲ
W W'
-
W+W' + W 9 T.. (187).
Before proceeding to verify the laws expressed by these
equations, it will be necessary to determine the constant
experimental weight W". For this purpose load the machine by placing
determination of the same number of circular weights in each basin; then
add a bar-weight to the basin, which moves in front of the
scale. The basins being of the same weight, the difference
W-W' will be the weight of the bar; the sum W+W',
will be the sum of the weights of the basins, increased by
that of the circular weights added, and that of the bar, all
of which are known. Now place the basin which carries
the bar at the zero of the scale, by means of the revolving
stage; set the clock in motion, and, supposing the bar to
commence its descent at a particular beat of the clock,

MECHANICS OF SOLIDS.
455
coincidence of
note whether the bar is taken off by the upper ring stage,
coincidently with any subsequent beat of the clock; if it
is, then the distance of the ring below the zero of the scale
being substituted for x, and the number of seconds elapsed the experiment
from the beginning of motion till the bar is removed, be- repeated till
ing substituted for T'in Eq. (186), will enable us to find W", clock beat with
since all the other quantities in that equation are known. is obtained;
If the removal of the bar and the beat of the clock be not
coincident, the ring stage must be shifted, and the experi-
ment repeated till the coincidence is obtained.
Example. Let each basin weigh 11 units, and suppose
removal of bar
cons
14 units of weight to be placed in each basin, and a bar example;
weighing 1 unit to be added to the basin in front of the
scale, then will
WW' = 1,
W + W' = 51;
making g = 32 feet = 384 inches; g=192 inches. Sub-
stituting these values in Eq. (186), we find
data;
1
X=
192 X T²;
corresponding
51 + W"
value of space;
whence
192
W"
=
1
T2 51.
Now supposing the bar to be removed at the end of the
third second, and that we find x, or the space described by
the bar to be 27 inches, then will
value of W";
W"
=
192
27
X (3)² - 51 = 64 51=
-
13;
numerical value
of W";
that is to say, the moment of inertia of the wheel will have

456
NATURAL PHILOSOPHY.
conclusion;
space for the
the same effect to resist motion as the inertia of thirteen
units of weight placed in the basins.
Substituting this value for W" in Eqs. (186) and (187),
they become
particular
machine;
velocity in the
same;
experimental
verification;
times;
spaces:
1g T².. (188),
g T.
. (189);
X=
V
=
W-W'
W+W' + 13
W-W'
W + W' + 13 9
and, loading the machine as before,
X=
X 192 X T² =
3 T2,
V = X 384 x T = 6 T
Making Tequal to
1, 2, 3, 4, &c. seconds;
the corresponding spaces will be
3,
12, 27, 48, &c. inches;
velocities;
verification;
and the corresponding velocities,
6, 12, 18, 24, &c. inches.
Place the basin with the bar-weight at the zero of the
scale, and connect with the clock; adjust the ring so as
to remove the bar when its basin reaches the 3 inch
mark, and place the plane stage at the 9 inch mark
3+6. The clock being put in motion, the bar will
strike the ring at the first beat of the clock after it begins
to descend, and its basin will strike the plane stage at the
second beat. The bar being removed, there will be no
11

MECHANICS OF SOLIDS.
457
after the bar is
removed;
excess of weight in either basin, and the motion will be- motion uniform
come uniform, there being no reason why it should be
accelerated rather than retarded. To show that the motion
will be uniform, repeat the experiment, placing the plane
stage first at 1 foot 3 inches, then at 1 foot 9, then at
2 feet 3 inches, and so on, adding 6 inches each time,
and it will be found that the basin will be arrested at the its proof;
third, fourth, fifth, &c., beats of the clock after its motion
begins; thus showing that the spaces described are pro-
portional to the times, which is the characteristic of uni-
form motion. Next adjust the ring so as to remove the
bar when its basin reaches the 12 inch or 1 foot mark, and
place the plane stage at the 2 feet mark, it will be repetition of the
found that the bar will strike the ring at the second beat experiment;
after its motion begins, and that the scale will be arrested
at the third beat. That the motion is uniform after the
removal of the bar may be shown, as before, by repeating
the experiment, and adding 12 inches each time to the
space to be described after the bar is arrested. In the
same way all the other results may be verified.
If a bar-weight be placed upon the second ring, and the illustration of
latter be so adjusted that the ascending basin shall take retarded motion;
it
up at the moment the bar on the descending basin is
removed, the motion will become retarded, and we shall
have the case of a body projected vertically upward from
rest with a velocity equal to that of the basins. The
plane stage being placed at a distance below the ring
which takes off the descending bar, equal to that through
which the latter has descended, it will be found that the all the laws
basin will just reach this stage at the instant the motion which regulate
destroyed by the action of the ascending bar. All the
laws which regulate the fall of heavy bodies
verified by means of Atwood's instrument.
may be
the fall of heavy
bodies may be
verified by this
machine.

458
NATURAL PHILOSOPHY.
gnied
XXIII.
Impact of bodies;
direct impact;
impact;
IMPACT OF BODIES.
§ 246. When a body in motion comes into collision
with another, either at rest or in motion, an impact is
said to arise.
We have seen, § 204, that the action and reaction
which take place between two bodies, when pressed to-
gether, are exerted along the same right line, perpendicu-
lar to the surfaces of both, at their common point of
contact.
When the motions of the centres of gravity of the two
bodies are parallel to this normal before collision, the im-
pact is said to be direct.
When this normal passes through the centres of grav-
ity of two bodies which come
into collision, and the mo-
direct and central tions of these centres are
along that line, the impact is
said to be direct and central.
When the motion of the
centre of gravity of one of
the bodies is along the com-
mon normal, and the normal
does not pass through the
centre of gravity of the oth-
er, the impact is said to be
eccentric impact; direct and eccentric.
direct and
When the path described
by the centre of gravity of
one of the bodies, makes an
oblique impact; angle with this normal, the
impact is said to be oblique.
Fig. 278.
G
G
Fig. 279.
पै
G
Fig. 280.
G
G

MECHANICS OF SOLIDS.
459
collision;
Bodies resist, by their inertia, all effort to change circumstances of
either the quantity or the direction of their motion. figure during the
When, therefore, two bodies come into collision, each will
experience a pressure from the reaction of the other; and
as all bodies are more or less compressible, this pressure
will produce a change in the figure of both; the change
of figure will increase till the instant the bodies cease to
approach each other, when it will have attained its
maximum, and the bodies will have the same velocity.
The molecular spring of each will now act to restore the
former figures, the bodies will repel each other, and finally
separate.
the impact;
In the impact of bodies, three periods must therefore three periods of
be distinguished, viz.: 1st., that occupied by the process
of compression; 2d., that during which the greatest com-
pression exists, and in which it is obvious the bodies have
the same velocity; 3d., that occupied by the process, as
far as it extends, of restoring the figures. We are also
carefully to distinguish the force of restitution from the force force of
of distortion; the latter denoting the reciprocal action ex-
erted between the bodies in the first, and the former that
exerted in the third period.
The greater or less capacity of the molecular springs
of a body to restore to it the figure of which it has been
deprived by the application of some extraneous force
when the latter ceases to act, is called its elasticity.
The ratio of the force of distortion to the force of resti-
tution, is the measure of a body's elasticity. This ratio is
sometimes called the coefficient of elasticity.
two forces are equal, the ratio is unity, and the body is
restitution and of
distortion;
elasticity defined;
When these coefficient of
elasticity;
said to be perfectly elastic; when the ratio is zero, the body perfect elasticity;
is said to be non-elastic. There are no bodies that satisfy non-elastic.
these extreme conditions, all being more or less elastic,
but none perfectly so.
§ 247.-Suppose two bodies A and B to move in the
same direction, the body A to overtake B, and the impact

460
NATURAL PHILOSOPHY.
Direct impact of to be direct. The forces of distortion and of restitution,
two bodies;
notation;
equality of action
and reaction;
arising as they do from the reciprocal action of the bodies.
upon each other, are real pressures, measurable in pounds,
and are capable of generating in each body, in a given
time, a certain quantity of motion. Denote the intensity
of this force, at any instant of the impact, by F; the
small velocity lost by
the body A, in the
short time during
which F may be re-
garded as constant,
by v; and the small
velocity gained by B,
in the same time, by
the action of the same
Fig. 281.
A
B
force, by v'; also denote the mass of A by M, and that of
B by M'; then will F, which may be called indifferently
the action of one body or the reaction of the other, be
measured by Mv, or M' v'; and, because of the equality
of action and reaction,
Mv = M'v'.
That is to say, the quantity of motion lost or gained by one
of the bodies, in any small time, is equal to that gained
or lost by the other; and if we take the sum of all the
quantities of motion lost or gained by each of the bodies,
we shall have the whole quantity of motion gained or lost
by the one, equal to that gained or lost by the other.
Denoting the entire gain or loss of velocity of the body A
by V, that of the body B by V,,, we shall have
gain and loss of
motion equal;
MV
=
M' V
But the force Facts in opposite directions upon the two
bodies, and hence, if we give the positive sign to the
velocity generated in one body, that of the other must be

MECHANICS OF SOLIDS.
461
negative; that is, if V, be counted positive, V,, must be
negative, which will make
MV, = - M'V
=
forces producing
these act in
opposite
or
directions;
MV, + M' V,, = 0.
system constant.
That is to say, the algebraic sum, or the whole quantity of quantity of
motion lost and gained will be zero; and in every stage of the motion of the
impact the quantity of motion in the entire system will, there-
fore, be the same as before the impact began.
common velocity
at moment of
compression;
§ 248.-At the instant the bodies have ceased to ap- To find the
proach each other, they will have attained their greatest
compression, and, considering their condition before the greatest
retrocession begins under the action of the molecular
springs, it is obvious that they may be regarded as a single
body, having a common velocity. Denote this velocity by
U; also denote the velocity of the body A, before the
impact, by V; that of the body B, before the impact, by
V', the masses being M and M' as before. The whole
quantity of motion before the impact will be
MV + M' V',
and that at the instant of greatest compression will be
(MM) U. But these, by the last article, must be
equal, or
(M + M') U = MV + M' V';
whence
U
=
MV + M'V'
M+M'
(190).
the value of this
velocity.
That is to say, when two bodies moving in the same

462
NATURAL PHILOSOPHY.
expressed in
words;
direction have a direct impact, the common velocity, at the
instant of greatest compression, is equal to the sum of the quan-
tities of motion before the impact, divided by the sum of the
masses.
If the bodies moved in opposite directions, either V or
V' would be negative, say V', and
value when
bodies move in
opposite
directions. Many
Velocity gained
U
=
MVM'V'
M+M'
(191).
§ 249. The velocity lost by the body A, up to the
instant of greatest compression, is obviously equal to
VU,
up to greatest and that gained by the body B is equal to
compression;
U - V';
the force of distortion will, therefore, be measured by
M(V - U),
force of
distortion;
or by
force of
restitution;
M' (U- V').
Denote by V, the velocity which A loses by the force
of restitution; and by V, that which B gains by the
action of the same force; the force of restitution will be
measured by
MV, or M' Vi
and if e denote the coefficient of elasticity, then, from the
definition
coefficient of
elasticity;
MV
M(VU)
=
e

MECHANICS OF SOLIDS.
463
M'V
M' (U — V')
=e;
whence
coefficient of
elasticity;
V₁ = e(VU)... (192),
V₁ = e (UV)
(193).
velocities lost
and gained;
That is to say, the velocity which A loses by the force of
restitution, is equal to the coefficient of elasticity, into the the same
velocity which it lost by the force of distortion; and the velocity expressed in
gained by B by the same force, is equal to that which it gained
by the force of distortion, into the coefficient of elasticity.
The total loss of velocity which A will experience by
the impact will be
VU+e(V - U);
and the entire gain of B will be
UV'+e(U - V').
Denote by v the velocity retained by A, and by v' that
which B has after the impact; then, since the velocity
retained by A, must be equal to that which it had before
the impact, diminished by its loss,
v = V − V + U-e (VU) = (1 + e) U - eV;
-
and as B must, after the impact, have its primitive
velocity increased by its gain,
adt gode bus
v' = V' + U-V'+e(UV) = (1 + e) U - eV';
and substituting for U its value in Eq. (190), we have
v = (1 + e)
MV+M'V'
M+M'
words;
loss of velocity of
the impinging
body.
gain of the other;
velocity of the
- eV. (194), impinging body
after the impact;

464
NATURAL PHILOSOPHY.
velocity of the
other after the
impact;
in words;
v' = (1 + e)
MV + M' V'
M+M'
-
eV'. . (195).
Thus, the velocity of either body after impact, is equal to the
coefficient of elasticity increased by unity, multiplied into the
common velocity at the instant of greatest compression, and
this product diminished by the product of the coefficient of
elasticity into the velocity of the body before impact.
If the body B move to meet the body A, its velocity
will be negative, and the above reduce to
v = (1 + e)
MV-M'V'
M+M'
- eV . . (196),
when the bodies
meet.
MV-M'V'
v' = (1 + e).
+ eV'. . (197).
M+M'
When one of the
bodies is at rest;
coefficient of
elasticity;
its value when
the masses are
equal;
§ 250.-If the body B be at rest when the body A
impinges against it, then will V' be zero, and
v = (1 + e)
MV
M+M'
-
- .
- eV . . (198),
v' = (1 + e)
MV
M+M'
(199).
From the last equation we find
e=
v' (M + M') - 1. . . (200);
MV
- .
and when the masses of the bodies are equal, or M = M',
e=
2 v'
T
- 1 ..
(201);
which suggests a very easy method of finding the co-
efficient of elasticity of any solid body. For this purpose,

MECHANICS OF SOLIDS.
465
determination of
turn a pair of spherical balls of the same weight from the experimental
body whose coefficient of elasticity is to be found; suspend the coefficient of
them by silken strings, so that when the latter are vertical elasticity;
the balls shall just touch each other, be upon the same
level, and have their centres opposite the zeros of two cir-
cular graduated arcs whose
centres of curvature are at
the points of suspension.
The body A being drawn
back to any given degree
upon its scale and abandon-
ed, will descend and impinge
against the body B with a
velocity due to a height
equal to the versed sine of
the arc which it describes
before the impact; the body
B will ascend on the oppo-
radius
Fig. 282.
65432
description of
instrument, and
23 456
mode of using it;
A..
B
site are to a height due to the velocity with which it
leaves A; this height will be the versed sine of the arc
described by B before it begins to descend again. The
arcs being known, their versed sines are easily computed
from the properties of the circles. Denoting these versed
sines by h and h', then will
V = √2gh,
velocity of
impinging body
and that of the
body struck;
v' =
√2gh';
which, substituted in the value of e, gives
h'
e = 2
h
√ - 1
1.
(202).
coefficient of
elasticity;
odiopor nodeo
Example. Two ivory balls of equal weights, and there-
fore of equal masses, were made to collide in the manner
30

466
NATURAL PHILOSOPHY.
above described. One descended through an arc of 20
example of two degrees, and the other ascended through an arc of 18
degrees and 30 minutes; required the value of e.
By tables of natural sines and cosines, we find
ivory balls;
nat. cos 20° =
0.9396926;
versed sin 20° = 10.9396926
=
0.0603074;
height of fall of
the colliding
body;
and denoting the radius of the circular scale by R, we
have
h
=
0.0603074 R.
Again,
nat. cos 18° 30'
=
0.9483236;
versed sin 18° 30'
10.9483236
=
0.0516764;
height due to the
velocity of the
h'
0.0516764 R;
body struck;
and
numerical value
of the coefficient;
e=2
0.0516764.R
0.0603074.R
0.0516764
-1=21
-1=0.85138;
0.0603074
1500
example of the
whence we conclude that the coefficient of elasticity of
the specimen of ivory employed, is about 0.85; that of
glass will be found to be about 0.93, and that of steel
about 0.56.
Example. Two ivory balls, whose masses are repre-
sented by 6 and 4, move in the same direction with
collision of ivory velocities of 10 and 7 feet a second respectively. What
is the velocity of each after impact? The conditions
balls;
of the question require that the larger mass 6 shall over-
take the smaller mass 4, because the former has the greater
velocity. Hence

MECHANICS OF SOLIDS.
467
M = 6;
V = 10;
e= 0.85.
given data;
M' = 4;
V' = 7;
These data, in Eqs. (194) and (195), give
v = 1.85
60 + 28
10
—
0.85 × 10
=
ft.
7.78,
velocities after
impact;
60 + 28
ft.
v' = 1.85
—
0.85 x 7 = 10.33.
10
Example. Let the same balls move in opposite direc- another example.
tions so as to meet, each with the same velocity as before.
The same data, substituted in Eqs. (196) and (197), give
1.85
60 28
10
-
0.35
0.85 X 10 =
-
ft.
2.58,
—
60 28
v' = 1.85
+ 0.85 X 7
ft.
11.87.
10
§ 251.-Now suppose the bodies A and B to move, Oblique impact;
the first with a velocity V in the direction from E towards
F, and the second with a
velocity V' in the direction
from C towards D; and let
the collision take place at H.
Through the point H, draw
the common normal HN,
and resolve each of the ve-
locities V and V' into two
components, one in the di-
rection of the normal and
the other in the direction of
the tangent plane at H. For
this purpose designate the an-
gle FON by 9, and DO, N
N
Fig. 283.
F
D
B
description and
notation;
0
H
O
A

468
NATURAL PHILOSOPHY.
normal
velocities;
tangential
velocities
by '; the components in the direction of the normal,
will be
V cos o,
and V' cos q';
and those parallel to the tangent plane, will be
V sin o,
and
V' sin o'.
If the bodies were animated by these last velocities
alone, they would not collide, but would in general move
by one another without exerting any pressure; and hence
the impact will be wholly due to the components in the
direction of the normal; but these acting along the same
line perpendicular to the surfaces at their common point
of contact, will give rise to a direct impact, and denoting
the velocities of the bodies A and B after impact by v and
v', and the angles which their directions make with the
normal by and e', respectively, we shall have, from Eqs.
(194) and (195),
components of
v cos 0 = (1 + e).
MV cos+M' V' cos o'
M+M
- eV cos p... (203),
velocity in
direction of the
normal after
impact;
v' cos 0' = (1 + e)
MV cos + M' V' cos p'
M+M'
-e V' cos p'.. (204).
tangential
components of
velocity after the
impact;
Moreover, because the effects of the impact arising from
the components of the velocities in the direction of the
normal will be wholly in that direction, the components
of the velocities of each body before and after the impact
at right angles to the normal, will be the same, and hence
v sin
==
V sin o
(205),
v'sin 6' =
V' sin p' ..
(206).
Squaring Eqs. (203) and (205), adding, extracting the

MECHANICS OF SOLIDS.
469
square root, and reducing by the relation, (0) or hat
cos² + sin² = 1,
we find
guisd
velocity of the
[(1+e)
MV cos+M'V'. cos p'
M+M'
-eVcos ]2+V2 sin2 p.. (207); impinging body
after the impact;
and treating Eqs. (204) and (206) in the same way,
velocity of the
d = √ [(1+0)
MV cos &+M'V' cos '
M+M
-e V' cos p']+V2 sin2 p'.. (208). body struck after
the impact;
Again, dividing Eq. (205) by Eq. (203), we have
tan8=
(1+e):
V sin o
MV cos+M'V' cos o' - eV cos
M+M'
and, dividing Eq. (206) by (204),
tan '=
(1+e)
V' sin o'
ybod
direction of the
(209); first body's
motion;
Bod
(210).
that of the
second;
MV cos + M'V' cos o'-e' cos o'
M+M'
The Eqs. (207) and (208) will make known the velocities,
and (209) and (210) will give the directions in which the
bodies will move, after the impact.
Now suppose the body B at rest, and its mass so great suppose one body
that the mass of A is insignificant in comparison, then very large and at
will V' be zero, M' may be written for M+M', and
M
M
rest;
will be a fraction so small that all the terms into reductions;
which it enters as a factor may be neglected. Applying
these considerations to Eq. (207), we find
v = √ √ e cos² + sin²;
velocity of the
impinging body
after impact;

470
NATURAL PHILOSOPHY.
and to Eq. (209),
direction of the
impinging body's
tan ❤
tan =
motion after
e
impact;
(211).
The tangent of being negative, shows that the angle
NHK, which the direction
Chamini of A's motion makes with
graphical
the normal NN' after the
impact, is greater than 90
degrees; in other words,
illustration of this that the body A is driven
result;
ak pooded
body will not
rebound when
non-elastic;
back or reflected from B.
This explains why it is that
a cannon-ball, stone, or oth-
er body thrown obliquely
against the surface of the
earth, will rebound several
times before it comes to rest.
Fig. 284.
K
N
H
N
A
B
E
If the bodies be non-elastic, or, which is the same thing,
if e be zero, the tangent of becomes infinite; that is to
say, the body A will move along the tangent plane, or if
the body B were reduced at the place of impact to a smooth
plane, the body A would move along this plane.
If the body were perfectly elastic, or if e were equal to
unity, which expresses this condition, then would Eq.
(211) become
tan =
-
tan.
(212);
in perfectly
elastic bodies the
angle of
which means that the angle NHF=EHN' becomes equal
to KHN'. The angle EHN' is called the angle of inci
dence, the angle KHN', commonly, the angle of reflection.
incidence equal Whence we see, that when a perfectly elastic body is
thrown against a smooth, hard, and fixed plane, the angle
of incidence will be equal to the angle of reflection.
to angle of
reflection;
Barb
Φ
If the angles and ' be zero, then will cos = 1, cos
o' = 1, sin q = 0, and sin o' = 0, and Eqs. (207) and (208)

MECHANICS OF SOLIDS.
471
become
v = (1 + e)
MV + M'V'
M+M'
-
- e V,
v' = (1 + e)
MVM'V'
M+M'
-
- e V';
the same as Eqs. (194) and (195); and passing to the
limits, non-elasticity on the one hand and perfect elasticity
on the other, we have, in the first case, e = 0, and
case of direct
impact;
0 =
MV+M' V'
M+M'
(213),
bodies
non-elastic;
11
MV + M'V'
M+M'
(214);
and in the second, e = 1, consequently
v=2
MVM' V'
M+M'
-
V.. (215),
v' = 2
MVM' V'
M+M'
- V'. . (216).
$252.-The equations which have just been deduced,
bodies perfectly
elastic.
are sufficient to make known the circumstances of motion Oblique and
in a
of the centres of gravity of the colliding bodies, for we
have
seen, § 146, that whenever a body is acted upon
direction normal to its surface, its centre of gravity will
move as though the force were applied directly to that
point. But we have also seen, in the same article,
eccentric impact;

472
NATURAL PHILOSOPHY.
that when the direction of
in the eccentric the force does not pass
impact the bodies through the centre of
will rotate;
loss of velocity of
grav-
ity, which is the case in
the eccentric impact, the
body will also have a ro-
tary motion.
Employing the same no-
tation as before, and sub-
tracting Eq. (203) from the
identical equation,
V cos o = V cos ?,
we find
Fig. 285.
F
D
N
T
H
A
B
σ
one body in
direction of
normal;
V cos o
—
v cos 8 = (1 + e) M' (V cos o
-
V' cos
cos q');
M+M'
motion lost in
that direction;
the first member is the loss of velocity of the body A in
the direction of the normal, during the impact; and mul-
tiplying both members by the mass of A = M, we have, for
the quantity of motion lost in the direction of the normal,
-
M(V cos
-
v cos 8) = (1 + e)
MM' (V cos
M+ M'
V' cos')
construction;
If the force of which either member of this equation meas-
ures the intensity, and of which the direction coincides
with the normal, does not pass through the centre of grav-
ity, it will give rise to rotary motion. From the centre of
gravity G', of the body B, let fall the perpendicular G'C'
upon the normal, and denote its length by b; also denote
the angular velocity of the body B by s,, and its moment
of inertia with reference to an axis through the centre of
gravity, and perpendicular to the plane of the normal and
centre of gravity, by I; then, because the angular velocity

MECHANICS OF SOLIDS.
473
is equal to the moment of the impressed force divided by
the moment of inertia, Eq. (64),
$
s₁ = (1 + e) b
MM' V cos o
—
X
V' cos o'
I₁
angular velocity
.
.
(217). of one of the
M+M'
Also let fall from the centre of gravity G of the body A,
the perpendicular G Cupon the normal, and call its length
a. Since the reaction of the body B, which is equal to the
action of A, does not pass through the centre of gravity
of the latter, it will communicate a rotary motion; and,
denoting the angular velocity of A by s,, we shall have,
MM'
8,, = (1 + e) α M + M¹ X
a
V cos - V' cos o'
I'
bodies;
.
(218);
angular velocity
of the other;
in which I' is the moment of inertia of the body A, in
reference to an axis through its centre of gravity and
perpendicular to the plane containing this point and the
normal.
taken of friction;
In what precedes, no reference is made to friction, but thus far no
it is obvious that this principle cannot be wholly dis- account has been
regarded; for the bodies acting upon each other in the
direction of the normal with a pressure of which the
measure is
(1 + e) ·
MM'
M+M'
Φ
(V cos - V' cos o');
this
pressure will give rise to friction, whose intensity is
measured by
f(1 + e)
MM'
M+M'
(V cos - V' cos q');
UM
measure of the
friction;
and this acting in the direction of the tangential com-
ponents of the velocities will accelerate the one and retard

474
NATURAL PHILOSOPHY.
tangential force
to overcome
friction;
limits within
which friction
may act
to produce
rotation;
quantity of
the other. Let U, denote the tangential velocity lost by
the body A; then, the force exerted to overcome the
friction will be measured by
MU.
NE
D
Fig. 285.
F
Now if the tangential velocities be equal, it is obvious that
the bodies will move together in the direction of the tangent,
MU, will be zero, the friction will not be called into action,
and the bodies will not rotate from friction. If the tan-
gential velocities differ by a quantity that will make MU,
equal to the friction, then
will the whole of the latter
be exerted to produce ro-
tation. If the tangential
velocities be such as to
give to MU, any value
between these limits, a
part only of friction will be
exerted, and this part alone
will determine the rotation.
If the difference of the tan-
gential velocities be such
as to make MU, greater
than the friction, the bodies
will slide along each other
and rotate at the same
time; the latter motion
being due to the entire
B
T
0
A
H
G
T
0
E
friction, and the former to the excess of MU, over the
value of this force.
Denote by n, the ratio of the friction to MU,, then will
tangential motion MU, nf (1 + e)
1
lost;
MM'
M+M'
(V cos o - V' cos p').
Let fall from the centres of gravity of the two bodies the
perpendiculars GT and G' T', upon the tangent TT",

MECHANICS OF SOLIDS.
475
denote the length of the first by a, and that of the second
by b,. Then will the angular velocity of the body B,
produced by friction, be
MM'
nf (1 + e) b,
M+M'
—
V cos V' cos o'
I₁
angular velocity
of one body due
to friction;
and that of the body A,
nf (1 + e) a,
MM'
M+M'
V cos - V' cos q'.
I'
whence the whole angular velocities of the two bodies
angular velocity
of the other, due
to friction;
will become
MM'
8, = (1 + e) M+M'
V cos p
- V' cos p'
(b+nfb,),
I
whole angular
velocity of the
bodies;
8,1 = (1+e).
MM
M+M
V cos &
-
V' cos d'
·(a+nfa,).
I'
If the bodies be spherical and homogeneous, the nor-
mal will always pass through the centre of gravity, b and a
will reduce to zero, and the rotation will be due to friction
Φ
alone. If the impact be direct, then and o' will be zero, particular cases
there will be no tangential components of the velocities of figure.
MU, and consequently n will reduce to zero, and the
rotation will be due to the eccentricity of the impact.

boegoe bit to tadi
PART SECOND.
MECHANICS OF FLUIDS.
I.
Condition of all
upon the
a solid;
INTRODUCTORY REMARKS.
§ 253. We have seen, § 13, that the physical condition
bodies depends of every body depends upon the relation subsisting among
molecular forces; its molecular forces. When the attractions prevail greatly
over the repulsions, the particles are held firmly together,
and the body is called a solid. In proportion as the differ-
ence between these two sets of forces becomes less, the body
is softer, and its figure yields more readily to external
pressure. When these forces are equal, the particles will
yield to the slightest force, the body will, under the ac-
tion of its own weight, and the resistance of the sides of a
vessel into which it is placed, readily take the figure of the
latter, and is called a liquid. Finally, when the repulsive
exceed the attractive forces, the elements of the body tend
to separate from each other, and require either the applica-
tion of some extraneous force or to be confined in a closed
vessel to keep them together; the body is then called a
a gas or vapor; gas or vapor, according to the greater or less pertinacity
with which the repulsive retain their ascendency over the
attractive forces. In the vast range of relation among the
molecular forces, from that which distinguishes a solid to
a liquid;

MECHANICS OF FLUIDS.
477
and vapors run
into each other.
that which determines a gas or vapor, bodies are found in solids, liquids,
all possible conditions solids run imperceptibly into
liquids, and liquids into gases. Hence all classification of
bodies founded on their physical properties alone, must, of
necessity, be arbitrary.
8254.-Any body whose elementary particles admit of Definitions, &c.;
motion among each other, is called a fluid-such as water, a fluid;
wine, mercury, the air, and, in general, liquids, gases, and
vapors; all of which are distinguished from solids by the
great mobility of their particles among themselves. This
distinguishing property exists in different degrees in dif-
ferent liquids-it is greatest in the ethers and alcohol; it
is less in water and wine; it is still less in the oils, the
sirups, greases, and melted metals, that flow with difficulty,
and rope when poured into the air. Such fluids are said
to be viscous, or to possess viscosity.
approach so closely both a solid and
difficult to assign it a place among either class of these
bodies, as paste, putty, and the like.
Finally, a body may viscous fluids;
liquid, as to make it
paste; putty.
incompressible;
§ 255.--Fluids are divided in mechanics into two Classification of
classes, viz.: compressible and incompressible. The term in- fluids;
compressible cannot, in strictness of propriety, be applied compressible and
to any body in nature, all being more or less compressible;
but the enormous power required to change, in any sensible
degree, the volumes of liquids, seems to justify the term,
when applied to them in a restricted sense. The gases and
vapors are highly compressible. All liquids will, there- liquids
fore, be regarded as incompressible; the gases and vapors
as compressible.
incompressible;
gases and vapors
compressible
$256.-There are many fluids that readily pass from
the compressible to the incompressible class, when sub-
jected to moderate increase of pressure, and reduction of
temperature. These are called vapors, and are such as arise vapors;
from the application of heat to liquids, particularly when

478
NATURAL PHILOSOPHY.
vapors
distinguished
from mists and
clouds;
gases
distinguished
from vapors:
confined in closed vessels, as in the instance of steam in
boilers. Vapors are generally invisible, and must not be
confounded with the mists and clouds which are often seen
suspended above the surface of the earth, and which are
nothing more than water, in the form of small vesicles
filled with air, and supported by the buoyant action of the
atmosphere. Others of the compressible fluids are more
permanent, requiring very great pressure and reduction of
temperature to bring them to a liquid form. All such fluids
are called gases. The most familiar instance of this class
of bodies is the atmosphere which surrounds us on every
side and in which we live. It envelops the entire earth,
reaches far beyond the tops of our highest mountains, and
pervades every depth from which it is not excluded by the
presence of solids or liquids. It is even found in the pores
of these bodies. It plays a most important part in all
natural phenomena, and is ever at work to influence the
motions and to modify the results of machinery. It is
its composition; essentially composed of oxygen and nitrogen, in a state of
mechanical mixture. The former is a supporter of com-
bustion, and, with the various forms of carbon, is one of
the principal agents employed in the development of me-
chanical power.
atmosphere;
mechanical use
of oxygen;
proof of the
existence of
Vapors and gases;
The existence of air, gases, and vapors, is proved by a
multitude of facts. Contained in a flexible and imperme
able envelope, they resist pressure like solid bodies. The
gas in an inverted glass vessel plunged into water, will not
yield its place to the liquid, unless some avenue of escape
is provided for it. Those winds, hurricanes, and tornadoes
which uproot trees, overturn houses, and devastate entire
districts, are but air in motion. Air opposes, by its inertia,
the motion of solid bodies through it, and this opposition
is called its resistance. Finally, we know that wind is
used as a moter. employed as a moter to turn windmills and to give motion
to ships of the largest kind.
atmospheric
resistance;
§ 257.-Many bodies take, successively, the solid, liquid,

MECHANICS OF FLUIDS.
479
or vaporous state, according to the heat to which they are
subjected. Water, for instance, is solid in the state of ice change of state;
and snow, liquid in its ordinary condition, and vapor when
heated in a closed vessel. The process by which a body
passes from a solid to a liquid state, is called liquefaction or liquefaction;
fusion; from a liquid to a state of vapor, vaporization or vaporization;
volatilization; that by which a vapor returns to a liquid, condensation ;
condensation; and a liquid to a solid, solidification or congela- solidification.
tion. Some bodies appear to take but two of these states,
while others constantly present themselves only under one,
which is the case with the infusible solids and permanent
gases, including among the latter, the atmospheric air; but
the number of these bodies is constantly diminishing in
the progress of physical science.
considering the
subject;
hydrodynamics;
$258. The subject of the mechanics of fluids, is usual- Mode of
ly divided, as before remarked, into hydrostatics and hydro-
dynamics, the former treating of the equilibrium of fluids, hydrostatics;
and the latter of their motions; and not unfrequently the
compressible fluids are discussed under a separate head
called pneumatics. In the present instance, these divisions pneumatics;
will not be adhered to, as it is believed the whole subject
may be presented in a manner more connected and per-
spicuous by disregarding them. And in the discussions
which are to follow, the fluid will be considered as with-
out viscosity; that is to say, the particles will be supposed
to have the utmost freedom of motion among each other.
Such a fluid is said to be perfect. The results deduced perfect fluid.
upon the hypothesis of perfect fluidity will, of course,
require modification when applied to fluids possessing
sensible viscosity. The nature and extent of these modi-
fications can be known only from experiments.

480
NATURAL PHILOSOPHY.
II.
MECHANICAL PRINCIPLES OF FLUIDS.
Level surface;
when at rest;
normal to the
resultant of the
forces which act
$259. From the nature of a fluid, it is obvious that
when a force is applied to any one of its particles, the latter
must move in the direction of the force, unless prevented
by the reaction of the surrounding particles; but these
being equally free, can only react to prevent motion, by
being supported or acted upon by opposing forces. From
this arises a general law, viz.: that when a fluid is in
equilibrio, its free surface is always normal to the re-
sultants of the forces which solicit each of its surface
particles. For if the result-
ant O F of the forces which
act upon any one of these
particles O were oblique to
the surface AB, this result-
ant might be resolved into
upon the surface two components, one OF
particles;
level surface
defined;
normal, and the other OF"
tangent to the surface; the
Fig. 286.
F
F
B
former would be destroyed by the reaction of the fluid
mass supposed in equilibrio, while the latter would move
the particle along the surface, and with the greater facility
in proportion as similar components tend to move the
particles to which they are applied in the same direction.
Hence the supposition of an oblique resultant is inconsist
ent with the equilibrium. This free surface which every
fluid in equilibrio presents in a direction normal to the
resultant of the forces which act upon each of its surface
particles, is called a level surface. Hence every heavy

MECHANICS OF FLUIDS.
481
fluid upon the earth's surface in a state of repose, presents
heavy fluids;
its upper or free surface normal to the direction of the level surface of
force of gravity. If the earth did not rotate about an
axis PP', thus giving rise to a centrifugal force, every
such surface would be a portion of the surface of a sphere,
having its centre at the centre of the earth; but its cen-
trifugal force MC, combined
with the weight MG of each
element, giving rise to a re-
sultant MN slightly oblique
to the direction of the weight,
every free surface is in strict-
ness a portion of the surface
of a spheroid of revolution,
flattened at the poles and
protuberant at the equator.
Fig. 287.
P
M...C
GN
figure of the level
surface of heavy
fluids;
P
The great size of the earth, and the limited field that
may be brought under observation at the same instant,
will scarcely permit us however to distinguish any visible visible portions
portion of fluid surface from a plane. Instance, the
Instance, the sensibly plane;
ponds, lakes, ocean. The same is true of the atmosphere.
This fluid being elastic, its elements tend to recede from
each other and from the earth's surface; in proportion as
it expands, the repulsive action becomes less; the weight
of the elements tends to draw them towards the earth;
at the upper surface of the atmosphere these opposing case of the
forces, which act towards and from the centre of the earth, atmosphere.
become equal, and the further retrocession of the particles
is impossible. The atmosphere would, under the opera-
tion of these causes alone, come to a state of rest, and
present an exterior boundary similar to that of the earth.
$260.-Let the vessel ABDC contain a heavy fluid,
or a fluid acted on only by its own weight; the upper
surface RS will, from what we have seen, be horizontal
when at rest; and it is obvious that this position of the
surface will not be disturbed, or in the least altered, if the
31

482
NATURAL PHILOSOPHY.
portion of the fluid indicated by the shaded parts of the
second figure were to become solid, leaving the fluid
portions ET, FH,
HG, communicating
freely with each oth-
the
er; that is to say,
surfaces at E, F, and
G, of the communi-
cating fluid would be
upon the same level.
Fig. 288.
A
B
R
S
D
Whence we conclude,
A homogeneous that a heavy fluid, as
water or mercury,
heavy fluid in
vessels
communicating poured into several ves-
freely will stand
in all at the
same level;
sels which communi-
cate freely with each
RE F
GS
experimental
illustration;
other, will, when in
equilibrio, have its upper surface in all the vessels on the
same level. This important fact is easily illustrated by
experiment. A is a
vessel at the bottom
of which is a horizon-
tal tube connecting
freely with the vessels
B and C, and having
a stop-cock D inter-
posed, so that the con-
nection may be inter-
rupted or established
at pleasure. Fill A
with water, the stop-
cock being closed.
When the water in A
is at rest, open the
A
Fig. 289.
D
E
B
cock D; the water will descend in A and ascend in B and
C till it comes to the same level in all.
If the vessel C be broken off at E, the water will over-

MECHANICS OF FLUIDS.
483
flow at this point till it sinks in the vessels A and B to
the level of E.
Fig. 290.
transfer of water
reservoirs from
remote points;
To the operation of this principle we are indebted for this principle
the transfer of water from remote locations to artificial determines thes
reservoirs for the supply of cities and towns. Springs to artificial
also owe their existence to it. The greater part of the
solid crust of the earth consists of various strata ranged
one above another; many of these are of a loose and po-
rous nature and are penetrated with clefts, whilst others
are more dense and free from flaws. Through the former
of these, rains and melted
snows find their way to
the latter, where their fur-
ther progress is for a time
checked, till the water
accumulates in sufficient
quantity to force its way
through the sides of hills
and mountains, and often
at points of considerable
elevation. When the
harder and impervious strata form the outer crust of
mountain ranges, they often force the water to take an
oblique underground course through porous strata, that
extend to considerable
depth and reach to re-
mote districts. Here,
if a channel be provi-
ded for the water by
boring through the hard
crust which confines it, it
will spout forth or over-
flow, in its effort to gain
the level of its source
in the distant mountain.
This constitutes an Arte-
Fig. 291.
it is also the
cause of springs;
and is the cause
of the discharge
on from Artesian
wells.
sian well, a name derived

484
NATURAL PHILOSOPHY.
Principle of
equal pressures;
a heavy fluid;
several vessels
from the French province Artois, where, according to
account, this mode of obtaining water was first practised.
工
​Fig. 292.
A.
M
§ 261. From the principle of fluid level, it is easy to
pass to that of equal pres-
sure. Suppose a vessel,
ABDCFE, in which the
branches EF and BDC
have a free communica-
tion with the part A B;
then if water, mercury,
wine, or any other fluid, be
poured in either at E, A,
or C, and the whole be
suffered to come to rest,
the surface at IK of the
communicating; fluid in the part AB, at
L in the branch EF, and
at M in the branch BD C,
will be upon the same
level.
P
.
B
D
G
H
Through the point N, taken at pleasure below the sur-
face of the fluid, conceive a horizontal plane to be passed.
It is obvious that the weight of the fluid contained in the
vessel below PNQ can contribute nothing to the support
of the columns LP, IO, and MQ, since this weight acts
downward; and the equilibrium would obtain if the fluid
several columns contained in the part of the vessel below PNQ were
without weight. This fluid may therefore be regarded as
of unequal
weights
other;
supporting each solely a means of communication between the columns
LP, IO, and MQ, in such manner that it will transmit
the pressure resulting from the weight of the columns
LP and MQ to support the weight of IO, and recipro-
cally. If now, instead of the columns LP, IO, and MQ
of the fluid, pistons were applied to the surfaces at P, NO,
and Q, and were separately urged downward by pressures
respectively equal to the weights of these columns, the

MECHANICS OF FLUIDS.
485
equilibrium would manifestly obtain in like manner. Or
replaced by
if a pressure equal to that arising from the column MQ weight of
be applied to the surface Q, while the columns LP and columns of fluid
IO remain, the equilibrium will still subsist, and this, pressures upon
whatever be the directions and sinuosities at D, F, &c. pistons;
The weight W of the column QM is measured by b.h.d.g;
in which b is the area of the base at Q, h the height QM,
d the density of the fluid, and g the force of gravity. The
weight W' of the column IO is measured by b'.h.d.g, in
which b' is the area of the base NO, the other quantities
being the same as before. Dividing the latter by the
former, we find
W'
=
b'.h.d.g b'
b
W b.h.d.g
=
(219);
hence, the weights are to each other as the bases b' and b.
Now these weights act in the same direction, and are
unequal; they cannot, therefore, maintain each other in
equilibrio, unless the pressure arising from the column
10 were transmitted by the fluid down the vessel NB, up
the sinuous vessel BDQ to Q, and there diminished in
the ratio of the base NO to that at Q. In like manner,
the
pressure from the column MQ must be transmitted by
the fluid down the tube QDH, up the vessel BN to the
base NO, and there increased in the proportion of the base
at Q to that at N.
ratio of the
weights of
columns of equal
altitudes;
equilibrio when
the areas of the
pistons;
That is, the forces applied to two pistons in a vessel filled forces on two
with fluid, will be in equilibrio when their intensities are di- pistons are in
rectly proportional to the areas of the pistons to which they are proportional to
respectively applied. If the areas b and b' of the pistons
become equal, the forces will be equal, and this, whatever
be the actual dimensions of the pistons. Whence we con-
clude, that the force impressed upon a fluid, is transmitted by
it equally in all directions; and that every surface exposed to pressure
the fluid will receive a pressure which is directly proportional equally in all
to its extent. Moreover, this pressure will be perpendicular directions;
to the surface, for if it were oblique, it might be replaced
transmitted
996

486
NATURAL PHILOSOPHY.
normal to the
surface;
pressure always by its two components, one normal, the other parallel to
the surface; the former would be destroyed by the resist
ance of the surface, while the latter would give motion to
the fluid, which is contrary to the supposition that the
fluid is in equilibrio.
From Eq. (219) we find
value of the
pressure
transmitted;
1st. rule;
W' = W
•
b'
b
(220);
whence we have this rule for finding the amount of pres-
sure transmitted to any surface, viz.: Multiply the intensity
of the pressing force into the ratio obtained by dividing the area
Soll to which the pressure is transmitted, by that to which the force
is directly applied. Making b= 1, W will be the pressure
upon the unit of surface, and Eq. (220) becomes
pressure
transmitted
when the
pressure is
applied to a unit
of surface;
W' = W.b'.
(221);
2d. rule;
anatomical
siphon;
whence we have this second rule for finding the pressure
transmitted to any given surface, viz.: Multiply the intensity
of the force applied to the unit of surface by the area of the
surface to which the pressure is transmitted.
Fig. 293.
The truth of these deductions is finely illustrated by
the Anatomical Siphon. A short
cylindrical vessel A, made of
metal, and open at one end, is
connected with an upright glass
tube fh, say half an inch in diame-
illustration by the ter, open at the top. The vessel
is filled with water, and closed
by tying over it a bladder, on
which a plate of wood or metal
is laid to receive weights W'.
Water is now poured down the
glass tube fh; the water in A,
with its superincumbent weights
W', will be raised by the pressure
13

MECHANICS OF FLUIDS.
487
487
arising from the weight of that portion of the fluid in
the glass tube above the level of the bladder. Let this
difference of level be 50 inches, then will the volume, in illustrated by a
cubic feet, of the pressing water, be
in.
T R2 X 50
1728
in.
3.1416 x (0.25)2 × 50
=
c. ft.
= 0.00568.
1728
Now one cubic foot of water weighs sixty-two and a
half pounds, whence the weight of the pressing column or
W becomes, in pounds,
numerical
example;
lbs.
W: =
62.5 X 0.00568
1
The area of a section of the glass tube is
lb.
0.355.
in.
b = R2
=
3.1416 x (0.25)²
=
0.196;
or, in square feet,
b =
0.196
144
=
0.00136, nearly.
weight of the
pressing column;
area of a section
of the tube;
Let the diameter of the vessel A be one foot then will
b' = 3.1416 X. (0.50)2
=
ft.
0.7854;
and these values of W, b, and b', substituted in Eq. (220),
diameter of the
larger vessel;
give
W' = 0.355
0.7854
0.00136
lbs.
204.8, nearly;
weight sustained;
that is to
say, the trifling weight of three tenths of a pound
sustains in equilibrio a weight of more than two hundred hydrostatic
and four pounds; a result usually denominated the hydro- paradox;
static paradox.

488
NATURAL PHILOSOPHY.
if the bladder
were removed
If the bladder were removed, and the vessel extended
on a level with the fluid in the
rise in the larger tube, the water would rise in it to that height, when it
the water would upward to the line ed,
vessel;
would come to rest.
The volume of the added water, in
cubic feet would be
in.
50
X 0.7854
=
3.272;
12
and allowing 62 pounds to each cubic foot, the weight
of distilled water at 60° Fah. gives
lbs.
verification.
3.272 X 621
=
204.5, nearly,
as before.
III.
WORK OF THE POWER AND OF THE RESISTANCE.
§ 262.-It follows from Eq. (220), that a given power
Multiplication of may be multiplied at pleasure by this principle of equal
transmission of pressure. It will be sufficient for this
transmission of purpose, to provide a strong
power by the
principle of equal
pressure;
vessel for the reception of a
fluid, and to connect with it
a pair of pistons whose sur-
faces bear to each other any
desired ratio; the power F
being applied to the smaller
piston b will be transmitted
to the larger b' and made to
hold in equilibrio or over-
come almost any given re-
sistance R applied to the
latter. But we are not, there-
Fig. 294.
F
E R
D
W
H
B

MECHANICS OF FLUIDS.
489
however;
fore, to infer that there is any gain in the quantity of work no work gained,
performed, for if we multiply Eq. (220) by the distance
HI=s' through which the larger piston may have been
moved by the pressure transmitted to it, we have, by
writing R for W', and F for W,
Rs' F.
=
·
s' b'
b
(222).
work of the
resistance;
The product s'b', being the area of the larger piston into
the distance HI, is the measure of the volume of fluid
which has passed into the chamber CE, by the action of
the power Fupon the smaller piston; and if we regard the
water as incompressible, this must be equal to the volume
of fluid which has been pressed out of the chamber A B.
Supposing the smaller piston to have been depressed to I',
and denoting the distance H'I' by s, this latter volume
will be measured by sb, and, therefore, from what has just
been remarked,
s'b' = sb;
whence
s' b'
8=
which, substituted above, gives
=
Rs' Fs.
volumes of the
fluid equal;
(223).
work of power
equal to that of
resistance;
The first member of this equation is the work performed
by the resistance, the second that performed by the power,
whence we conclude, that in hydraulic machines depending conclusion;
upon the transmission of pressure, as in other machines, the
work of the power is equal to that of the resistance.
viscosity;
If the friction of the pistons against the sides of their friction and
respective chambers and the viscosity of the fluid be taken
into the account, the work of these must be added to the

490
NATURAL PHILOSOPHY.
the hydraulic
machine enables
perform what
it could not
without it;
term Rs', which would make the effective quantity of
work, measured by Rs', actually less than the work of the
a feeble power to power. What then is gained? The answer is the same
as before, viz.: the machine gives to a feeble power the
ability to perform, by a succession of efforts, an amount
of work which it could not accomplish by a single one.
It would be quite within the physical capabilities of an
individual to raise to the summit of a wall a ton of bricks,
by taking a few bricks at a time, whereas an effort to ele-
vate the whole at once by his unassisted strength would
prove an utter failure. And this is true of all kinds of
general principle machinery; whenever a given amount of work is accom-
plished by the application of a diminished power, the space
through which the latter is exerted must be proportionally
increased.
of all machines;
this principle
employed to
prove that of
equal pressure;
Had this principle, together with the incompressibility
of the fluid, been assumed at the outset, it would have
been an easy matter to deduce Eq. (220), and therefore the
principle of the equal transmission of pressure; for, the
volume of the fluid remaining the same, we should have
s' b'
=
sb,
and the quantity of work of the power and resistance
being equal, gives
Rs' =
Fs;
dividing the first of these equations by the second, we find
b'
=
b
F
pressures are
proportional to
the surfaces.
whence
FR: b: b';
that is to say, the pressures are directly proportional to
the areas of the pistons to which they are applied, when

MECHANICS OF FLUIDS.
491
there is an equilibrium, or when the pistons have a uniform
motion.
§ 263. One of the most interesting and important
applications of the principle of equal transmission of
pressure is exhibited by the Hydraulic or Bramah's Press. Hydraulic press;
The main features of this machine are the following: A
large and small
metallic cylinder
A and a, are
made to commu-
nicate freely with
each other by a
duct-piper. Wa-
ter stands in both
of the cylinders,
and each is provi-
ded with a strong
piston. The pis-
ton S of the larger
cylinder carries a
strong head-plate
R
W
Fig. 295.
T
cb
d
description, and
mode of applying
the power;
P, which works in a frame, so as to move directly towards
or from a plate R which is stationary. The substance to
be pressed is placed between these two plates. The piston
in the small tube a is worked by a lever c d, of the second
order, having its fulcrum at c, the piston-rod being attached
at b, while power is applied at d. The pressure exerted by
the smaller piston on the water is transmitted by the latter
to the piston S.
Let the diameter of the cylinder a be half an inch, that its power
of the larger 200 inches, then will
illustrated by an
example;
b'
b
=
(200)2
(2
=
160000;
and suppose the distance cd to be equal to 50 inches, and

492
NATURAL PHILOSOPHY.
data;
cb to be one inch, and let a man throw his weight, say
150 pounds, on the point d; then from the property of
the lever will the force F, applied to the smaller piston, be
given by the proportion
in.
in.
lbs.
:: 150 : F;
1 : 50 ::
power applied at
smaller piston;
whence
lbs.
F = 150 X 50
=
7500.
b'
b
in Eq. (222), and
value of
resistance;
path of the power
at smaller piston;
Substituting these values for F and
omitting the common factor s', we find
lbs.
R = 7500 X 160000 =
lbs.
1200000000;
thus an effort equal in intensity to a weight of one hundred
and fifty pounds applied at d, is capable of holding in
equilibrio a power, or of maintaining in uniform motion a
body subjected to a constant resistance, equal to one
billion two hundred million pounds.
Dividing both members of Eq. (223) by F, we find
S=
R.s'
F
Banely
substituting the above values for R and F, and suppose the
piston-head to have been raised through the distance of
one foot, we have
its numerical
value for one foot
of path of the
resistance;
S=
1200000000
7500
= 160000;
and because the power applied at d must pass over 50
times this distance, we find
ft.
160000 × 50 = 8000000,

MECHANICS OF FLUIDS.
493
or
8000000
5280
miles.
=
1515,
path of the
power;
for the distance described by the power to compress the
resistance one foot, or to raise a weight equivalent to the
resistance through that height. The hydraulic press is
used in the arts to press paper, cloth, hay, to uproot trees,
to test the strength of ropes, chains, building materials,
and guns; and two were recently employed with success
to raise, through a vertical height of more than one
hundred feet, the great iron viaduct-tube, weighing up-
ward of eighteen hundred tons, over the Menai Straits.
uses of the
hydraulic press
IV.
PRESSURE OF HEAVY FLUIDS.
264.-Let us now examine the pressure which a Pressure of heavy
heavy fluid exerts on the base of a vessel in which it is
contained. For this
purpose, let ABDC
be a vessel containing
a heavy fluid, as wa-
ter, in equilibrio. The
upper surface AB of
the fluid will be hori-
zontal. Conceive a
will be
horizontal plane G H
to be passed, and sup-
Fig. 296.
0
B
A 0
I
G
H
fluids:
the horizontal
pose the fluid below this plane, or that contained in the fluid below
the portion GCDH, to be devoid of weight; then it is stratum devoid
obvious, from our previous principles, that the weight of of weight;
any slender vertical column, as EI, will exert a pressure

494
NATURAL PHILOSOPHY.
at I, which is distributed equally in all directions through
the fluid GCDH, and that this pressure acts equally
upward to oppose the
descent of the other
each elementary columns which stand
column
sustaining all the vertically over the
others;
plane GH; the col-
umn EI alone keeps,
therefore, in equi-
librio all the other
columns of the mass
AGHB; conse-
A 0
G
Fig. 296.
E
0 B
D
H
quently, the mass G CDH, being still supposed without
weight, there will result no pressure upon the base CD,
except that which arises from the weight of a single fila-
ment EI, which being transmitted equally to all the
points of the base CD, the pressure on the latter will be
given by Eq. (220); that is, by
pressure upon
the base;
b'
W' = W
b
610
weight of the
pressing filament;
in which W is the weight of the column EI, b the area
of its base, b' the area of the base CD, and W' the pressure
which it sustains.
Denoting the height of the column EI by h, its weight
W will be given by
W=h.b.D.g;
in which D denotes the density of the fluid, and g the force
of gravity.
Substituting this above for W, we find
pressure upon
the base;
W'=h.b'. D.g
(224).
If now the plane GH be depressed so as to leave all
the heavy fluid above it, this plane will coincide with

MECHANICS OF FLUIDS.
495
the bottom, I will come to I', and I will become the ver-
tical height EI' of the surface of the fluid above the
base.
column which
measures this
pressure;
But the product b'h is obviously the volume COO' D weight of the
of the fluid contained in a right cylinder or prism having
for its base, the base of the vessel; D.b'. h is the mass of this
cylinder or prism, and D. b'. h.g is its weight. Whence
we conclude, that the pressure exerted by a heavy fluid
upon the horizontal base of a vessel containing it, is equal
to the weight of a column of this fluid, whose base is the base
of the vessel, and whose altitude is equal to the depth of this
base below the surface of the fluid.
independent of
In this measure for the pressure on the base of a vessel
containing a heavy fluid, there is nothing at all relating pressure
to the figure or actual volume of the vessel, and we are, figure of vessel
hence, to infer that this pressure is wholly independent of and quantity of
both, and will always be the same whenever the area of
the base and altitude.
of the fluid are the
same. The right cyl-
inder, inverted and
erect truncated cones,
having equal inferior
bases B, B, B, and
the same altitude h,
will, when filled, con-
tain very different
volumes of fluid, yet
the bases will all
experience the same
amount of pressure
from the weight of
the fluid, if it be the
same in kind, or of
the same density.
The experimental
verification of this
Fig. 297.
B
B
B
F
H
D
Fig. 298.
F"
F
E
H
H
the pressing
fluid;
illustration:
right cylinder,
truncated cone,
both erect and
inverted;

496
NATURAL PHILOSOPHY.
experimental
verification of
this fact;
apparent paradox is easy. A DCB is a glass tube, of
which the ends are open and bent upward; the end
B is furnished with
a brass ferrule upon
which a screw is cut
for the reception of a
mate-screw H around
the bottom of the ves-
sels F, F', and F",
also open at both
description of the ends. On the end A
apparatus for the is a sliding ring of
purpose;
details of the
experiment;
deductions;
At
metal or wood.
E is a short wire that
may be moved up and
down, and is held in
any desired position by friction.
F
Fig. 298.
F F"
E
H
H
H
B
묘
​G
D
C
Pour mercury in either end of the bent tube till it rises
to any
desired level, say that of the dotted line; next, screw
either of the vessels, say F, on its place at B, and fill it with
water. The water passing freely through to the surface of
the mercury will press upon the latter by its weight and
force it up the end A. When both fluids come to rest,
move the ring on the end A to a level with the mercury to
mark its place, and press the wire E down to the surface
of the water to determine its height. Now draw off the
water by the stop-cock G, remove the vessel F and replace
it by F'', and fill with water as before; when the level
of the water reaches the end of the wire E, the mercury
will be found to have reached the ring on the end A.
The experiment being repeated with the slender vessel
F", not even half as thick as the tube AD CB, the mer-
cury will again be found at the ring. In all these ex-
periments, the base pressed is the same, being a section
of the bent tube at the level of the mercury; and the
altitude is the same, being the difference of level of the
mercury in the end B and lower extremity of the wire E

MECHANICS OF FLUIDS.
497
when the mercury in the end A stands at the level of the
ring. The quantities of water employed in the three cases conclusion.
are very different, and yet the pressures exerted by their
weights are the same.
against inclined
§ 265. The pressure of a heavy fluid upon a horizon- Pressure of a
tal plane, enables us to pass to that on a plane inclined heavy fluid
under any angle whatever to the horizon, and thence to surfaces;
the pressure on a curved surface.
Let ABDC be a vessel
with plane or curved sides,
and filled with a heavy fluid;
suppose GH and G'H' to
be two horizontal planes in-
definitely near each other.
The layer of fluid between
these planes may be consid-
Fig. 299.
A
E
B
I
H
H
ered as without weight, and as transmitting the pressure
of the superincumbent fluid to the surface of the vessel
with which this layer is in contact; and the pressure
upon this surface will be the same as though it were in
either of the two planes in question. Designating the ex-
tent of this elementary surface by b', and the depth EI by
h', the measure of this pressure will be
D.g.b'.h';
in which D and g denote respectively the density of the
fluid and force of gravity. In like manner, the pressure
upon any other elementary portions b", b'", b'", &c., of
the surface at distances h", h'", and h""", &c., respectively,
below the upper surface of the fluid, will be
D.g.b".h", D.g.b"".h"", &c.;
and the pressure upon the entire surface will obviously be
the sum of these; or, if the total pressure be denoted by
pressure upon an
elementary
inclined surface;
similar pressures;
32

498
NATURAL PHILOSOPHY.
total pressure
upon the entire
surface;
value of this
pressure in
weight;
P, then will
P = Dg(b'h' + b'' h" + b'"' h'" + &c.).
But if we take the upper surface of the fluid as a plane
of reference, and denote by b the entire area of which
b', b', &c., are the elements, and of which the distance of
the centre of gravity from this plane of reference is h, then,
from the principle of the centre of gravity, will
bh= b'h' + b" h" + b"" h"" + &c;
which, substituted above, gives
PD.g.b.h.
.. (225);
expressed in
words;
example first;
that is to say, the pressure exerted by a heavy fluid against
the surface of any vessel in which it is contained, is measured
by the weight of a column of the fluid having for its base the
surface pressed, and for its altitude the depth of the centre of
gravity of this surface below the upper level of the fluid.
Example 1st. Required the pressure against the inner
surface of a cubical vessel filled with water, one of its
faces being horizontal. Call the
edge of the cube a, the area of each
face will be a², the distance of the
centre of gravity of each vertical
face below the upper surface will be
a, and that of the lower face a;
whence, the principle of the centre
of gravity gives,
distance of centre
of gravity below
the surface;
surface pressed;
Again,
h
=
4a² x a + a² x a
5 a²
b = 5a²;
=
340
Fig. 300.
a.

MECHANICS OF FLUIDS.
499
and these, substituted in Eq. (225), give
P = D.g.b.h = D.g. 3 a³.
Now Dg x 13=Dg, is the weight of a cubic foot of water
value of the
pressure;
= 62.5 lbs. whence
lbs.
P = 62.5 X 3 a³.
Make a 7 feet, then will
in pounds;
lbs.
its numerical
P = 62.5 × 3 × (7)3 = 27562.5.
value
The weight of the water in the vessel is 62.5 a³, yet the
pressure is 62.5 x 3 a³, whence we see that the outward
pressure to break the vessel, is three times the weight of conclusion;
the fluid.
Example 2d. Let the vessel be a
sphere filled with mercury, and let
its radius be R. Its centre of grav-
ity is at the centre, and therefore
below the upper surface at the dis-
tance R. The surface of the sphere
being equal to that of four of its
great circles, we have
Fig. 301.
example second;
whence
b = 4% R²;
surface pressed;
b.h = 4R3;
volume whose
weight is equal
to the pressure;
and, Eq. (225),
P = 4.D.g. R³.
The quantity Dgx 13- Dg, is the weight of a cubic foot
whole pressure;

500
NATURAL PHILOSOPHY.
pressure in
pounds;
its numerical
value;
of mercury = 843.75 lbs., and therefore, substituting the
value of 3.1416,
=
lbs.
P = 4 x 3.1416 x 843.75. R³.
Now suppose the radius of the sphere to be two feet, then
will R³= 8, and
lbs.
lbs.
P = 4 x 3.1416 x 843.75 x 8 = 84822.4.
The volume of the sphere is R3; and the weight of the
contained mercury will therefore be R³g D= W. Di-
viding the whole pressure by this, we find
ratio of weight of
pressing fluid to
pressure;
example third;
value of
pressure;
P
1=
3;
W
whence the outward pressure is three times the weight of
the fluid.
Example 3d. Let the vessel be a cylinder, of which the
radius r of the base is 2, and altitude 1, 6 feet. Then will
b.hrl(r+7)= 3.1416 x 2 x 6 x 8;
which, substituted in Eq. (225),
P = 301.5936 x Dg,
and
weight of
pressing fluid;
W
= 3.1416 x 22 x 6 x Dg = 75.398 x Dg;
whence,
ratio of weight to
pressure.
P
1=1
W
301.5936 x Dg = 4;
75.3984. Dg

MECHANICS OF FLUIDS.
501
that is, the pressure against the vessel is four times the
weight of the fluid.
§ 266.-Although the pressure of a heavy fluid de- Centre of
pends upon the position of the centre of gravity of the pressure;
surface pressed, yet the resultant of all the elementary
pressures passes through a different point, the position of
which for a plane surface
may be thus found. Let
EIF be any plane, and MN
the intersection of this plane
produced with the upper
surface of the fluid which
presses against it. Denote
the area of any elementary
portion n of the plane EIF
by b'; and let m be the pro-
jection of its place upon
the
upper surface of the fluid;
draw m M perpendicular to
E
Fig. 302.
M
712
N
MN, and join n with M by the right line n M, the latter
will also be perpendicular to MN, and the angle n Mm
will measure the inclination of the plane EIF to the sur-
face of the fluid. Denote this angle by 9, the distance
mn by h', and Mn by r'; then will
The
h' = 'sin q.
pressure of the fluid upon the element n will, Eq.
(225), be
=
D.g.b'.h' Dqb'r' sino;
and its moment, in reference to the line MN as an axis,
Dg b' sin o;
and for any other elements of which b", b", &c., denote the
geometrical
representation
and notation;
distance of an
elementary
pressed surface
below the fluid
surface;
pressure upon
this element;
its moment;

502
NATURAL PHILOSOPHY.
moments of the
elementary
pressures:
areas, we have, in like manner, aiaga
Dgb" sin
Dgb" sin o,
&c.,
&c.
depth of centre
of gravity of the
whole area
pressed;
Denoting by h the depth of the centre of gravity of the
area EIF below the surface of the fluid, and by r the dis-
tance of that point from the line MN, we shall have
h = r sin o;
whole pressure;
moment of the
entire pressure;
distance of the
and, for the total pressure upon EIF,
P = D.g.b.h = Dgbr sino,
in which b denotes the area of EIF; and if x denote the
distance of the point of application of this pressure from
the line MN, its moment will be
Dgbr sino.x.
But the moment of the entire pressure must be equal to
the sum of the moments of the partial pressures, and hence
Dgbrx sin o = Dg sin (b'+b² + b'''''² + &c.);
whence
Φ
point of total
pressure from the
x=
axis;
b' 212 + 6' 7'1² + b''''''² + &c... (226).
br

MECHANICS OF FLUIDS.
503
The numerator of the second member, is the moment interpretation of
of inertia of the plane EIF; the denominator is the the last equation;
product of the area of the plane itself by the distance of
its centre of gravity from the axis, and as a similar ex-
pression would result if the
pressures were referred to any
other line in the plane EIF
as an axis, it follows from
$184, Eq. (86), that the result-
ant pressure passes through
the centre of percussion of
the surface pressed. This
point is called the centre of
pressure. It is that point in
the surface to which, if a single
force be applied in a direction
contrary and equal to the total
E
Fig. 302.d
M
N
no F
centre of
pressure;
defined;
coincident with
pressure exerted upon it, the surface will remain in equi- centre of
librio.
percussion.
principles;
$267. The principles which have now been explained, Application of
are of high practical importance. It is not only interest- the preceding
ing, but necessary, often to know the precise amount of
pressure exerted by fluids against the sides of vessels and
obstacles exposed to their action, to enable us so to adjust
the dimensions of the latter as to give them sufficient
strength to resist. Reservoirs in which considerable
quantities of water are collected and retained till needed
for purposes of irrigation, the supply of cities and towns,
or to drive machinery; dykes to keep the sea and lakes
from inundating low districts; artificial embankments objects to which
constructed along the shores of rivers to protect the they are
adjacent country in times of freshets; boilers in which are
pent up elastic vapors in a high state of tension, to be
worked off at pleasure to propel boats and cars, and to
give motion to machinery generally, are examples.
applicable;

504
NATURAL PHILOSOPHY.
thickness of the
sustaining wall of
a reservoir;
Let ABCD be a section or
profile of the wall of a reser-
voir, MN the upper surface of
the water, and EE' the bottom.
Denote the length of the wall
by 1, the depth NE of the water
against its face, supposed verti-
cal, by d; then will the surface
pressed be measured by ld; the
distance of the centre of gravity
of this surface from the upper
level of the water will be d,
whence the whole pressure will be
pressure against
the face;
D.g.l.d2
2
Fig. 303.
A
B
M
N
372
TU
0
G
E
E
C
D
may slide;
in which D is the density of the water, and g the force of
gravity. The inner surface of the wall being vertical, this
pressure is exerted in a horizontal direction, and must be
resisted by the wall. Now the wall, if it move at all, may
either slide along its base D C, or turn about the horizontal
suppose the wall edge passing through C. First, let us suppose it slides.
Denote the depth of the face AD by d', the mean thick-
ness m n by t; then will the weight of the wall be
weight of the
wall;
friction on the
ground;
D'.g.l.d'. t;
and, denoting the coefficient of friction between the wall
and earth by f, the whole friction will be
f. D'.g.l.d'. t,
in which D' is the mean density of the wall; and the
condition of stability will be satisfied as long as we have
condition of
stability;
Dglda
2
=
=fD' gld't;

MECHANICS OF FLUIDS.
505
from which we find
t
=
D'
d2
Drx afd.
2fd'
The density of water is usually taken as unity, and on
ordinary earth, the value of f, for masonry, does not vary
much from, whence
value of mean
thickness;
t =
3d2
2 D' d'
value of the
thickness in
ordinary cases;
The thickness is the only unknown quantity, since d and
d' must result from the capacity of the reservoir.
the front line of
If the wall tend to turn about the edge C, then must suppose the wall
the moment of its weight be equal to the moment of the may rotate about
pressure when both are taken in reference to that line. its base;
Let G be the centre of gravity of the profile ABCD,
and denote the distance CO of its projection upon the
base of the wall from C, by r. Then, from the assumed
figure of the profile, we shall have
r
<= Ng or r = nt,
t
in which n is known; and the moment of the weight of
the wall will be
D'.g.l.d'. t. n.
The centre of pressure O', being that of a rectangle of
which the side through N is horizontal, is at a distance
below N equal to of NE, or from the bottom point E
equal tod; and adding the distance ED denoted by
a, the moment of the pressure, in reference to C, will be
Dgld (d+ a);
2
ratio of lever arm
of the wall to its
thickness;
moment of the
weight of wall;
moment of the
fluid pressure;

506
NATURAL PHILOSOPHY.
condition of
stability;
thickness of the
wall;
and, to insure stability, we must have but didY
D'gld' t2n =
2
Dgld (d+a);
whence
t =
1
D
d² (d + 3 a)
.
d'
6n D'
If the water come to the bottom of the wall, and the
reservoir be full, then will
and
a = 0,
d = d',
1
D
t = d.
6 n
D'
thickness of
water-pipes,
boilers, &c.;
Next, let A B C be a sec-
tion of a cylindrical water-
pipe or boiler perpendicular
to the axis, the inner surface
of which is subjected to a
pressure of p pounds on each
superficial unit. Denote by
R the radius of the interior
circle, and by 7 the length of
the pipe or boiler parallel to
surface pressed; the axis; then will the sur-
face pressed be measured by
A
whole pressure;
2T RI,
and the whole pressure, by
2 Rlp.
Fig. 304.
R
B

MECHANICS OF FLUIDS.
507
If, in virtue of this pressure, the pipe stretches so that its suppose the pipe
interior radius becomes R+r, it is obvious that the small to stretch;
distance will denote the path described by the whole
pressure, and its quantity of work will be
2 Rlpr.
The interior circumference before the application of the
pressure was 2 R, and afterward, 2 (R+r); the differ-
ence of which, or
2 (R)- 2 x R = 2xr,
is obviously the distance through which the resisting
molecular forces of the material of which the pipe or
boiler is made, have acted during the stretching process.
Denote the resistance which the material of the pipe or
boiler is capable of opposing, without losing its elasticity,
to a stretching force on a section of one superficial unit,
by B; the length of the pipe or boiler by 7; and its thick-
ness by t. The intensity of the force which a section
parallel to the axis is capable of resisting will be Blt, and
its quantity of work
Blt X 2 TV.
But by virtue of the principle of the transmission of work,
this must be equal to the work of the pressure, and we
have
quantity of work;
path of the
resisting
molecular action;
the quantity of
work of this
force;
2 Bltr =
2 Rlpr;
whence
Rp
t =
B
The value of p is easily estimated in the case of water
in a pipe, by the rules just given. In the case of steam in
condition of
stability;
thickness.

508
NATURAL PHILOSOPHY.
a boiler, it may with equal ease be found by rules to be
given presently. The value of B is readily obtained from
the following table giving the results of experiments on
the strength of materials:-
TABLE.
THE TENACITIES OF DIFFERENT SUBSTANCES, AND THE RESISTANCES
WHICH THEY OPPOSE TO DIRECT COMPRESSION.
SUBSTANCES EXPERIMENTED ON.
Tenacity in
Tons per
Square Inch.
Name of Ex-
perimenter.
Crushing
Force in Tons
per Sq. Inch.
Name of Ex-
perimenter.
Wrought iron, in wire from
inch in diameter -
1-20th to 1-30th of an
in wire, 1-10th of an inch 36 to 43
in bars, Russian (mean)
English (mean)
60 to 91
Lamé
Telford
27
Lamé
25
hammered
30
Brunel
rolled in sheets, and cut
lengthwise -
it }
14
Mitis
18
211
25
ditto, cut crosswise
in chains, oval links 6 in.)
clear, iron 14 in. dia.
ditto, Brunton's, with
stay across link
Cast iron, quality No. 1 -
Steel, cast
-
6 to 7
6 to 8
Hodgkinson 38 to 41 Hodgkinson
37 to 48
51 to 65
—
Brown
Barlow
2
3*
6 to 92
44
Mitis
cast and tilted
60
Rennie
blistered and hammered
591
shear
57
raw-
50
Mitis
Damascus -
31
ditto, once refined
36
ditto, twice refined
44
Copper, cast
81
Rennie
hammered
15
95
52
Rennie
46
sheet
21
Kingston
wire
271
Platinum wire -
17
Guyton
Silver, cast
18
wire
17
14
794862
Gold, cast
wire
Brass, yellow (fine)
Gun metal (hard)
Tin, cast
7
* The strongest quality of cast iron, is a Scotch iron known as the Devon Hot Blast,
No. 3: its tenacity is 93 tons per square inch, and its resistance to compression 65 tons.
The experiments of Major Wade on the gun iron at West Point Foundry, and at Boston,
give results as high as 10 to 16 tons, and on small cast bars, as high as 17 tons.-See
Ordnance Manual, 1850, p. 402.
Rennie
73
16

MECHANICS OF FLUIDS.
509
SUBSTANCES EXPERIMENTED ON.
TABLE-continued.
Tenacity in
Tons per
Square Inch.
Name of Ex-
perimenter.
Crushing
Force in Tons
per Sq. Inch.
Name of Ex-
perimenter.
Tin wire
Lead, cast
3
Rennie
4-5ths
31
Rennie
Marble (white)
Givry
Portland
milled sheet
wire
Stone, slate (Welsh)
4
It
I.I
Tredgold
Guyton
5.7
1.4
I
+
1.6
Brick, pale red
red -
Craigleith freestone
Bramley Fall sandstone
Cornish granite - -
Peterhead ditto
Limestone (compact blk)
Purbeck
Aberdeen granite
Hammersmith (pavior's)
2.4
2.7
2.8
3.7
4
4
5
.13
ditto
(burnt) -
.8
I
1.4
58.
.56
Chalk
.22
Plaster of Paris
.03
Glass, plate
4.
Bone (ox)
2.2
Hemp fibres glued together
41
Strips of paper glued together
13
Wood, Box, spec. gravity .862
9
Barlow
Ash -
-
.6
8
Teak
.9
7
Beech
-
.7
Oak
.92
5
1.7
Ditto
.77
Fir
.6
Pear
.646
Mahogany
.637
1112
Elm
6
Pine, American
6
Deal, white
6
.57
.73
.86
5885
|||
In the result just obtained for the value of t, no atten-
tion has been paid to the pressure upon the ends of the
boiler or pipe, but these are usually made thick enough to
throw the chances of breaking altogether upon the cylin-
drical portion of the surface.

510
NATURAL PHILOSOPHY.
Equilibrium of
floating bodies;
a body wholly
immersed in a
fluid;
first result;
V.
EQUILIBRIUM OF FLOATING BODIES.
§ 268. The rules for finding the pressure against the
sides of vessels are equally applicable to the determination
of the pressure on the surfaces of bodies, however sub-
jected to the action of a homogeneous heavy fluid. But
when it is the question to ascertain the circumstances that
determine a heavy body to be in equilibrio or in motion,
when immersed in a heavy fluid, it is usual to employ the
results deduced from the following considerations.
Suppose a vessel A to contain
any heavy fluid in a state of rest.
All parts of the fluid being in equi-
librio, it is obvious that this state
will in no respect be altered by sup-
posing any portion B to become
solid without changing its density.
This solid is entirely immersed in
the fluid, with which it has the same
density, and is in equilibrio. Now
this solid is urged downward by its
weight, which passes through its
Fig. 305.
B
A
K
centre of gravity. This weight can only be in equilibrio
with a single force when the latter is directed vertically
upward through the centre of gravity of the body, which
centre coincides with that of the fluid converted into a solid,
or that of the displaced fluid. But the only forces that
act upon the solid besides its weight, are the pressures of
the surrounding fluid; whence we conclude that
1st. The pressures upon the surface of a body entirely im-
mersed in a fluid, have a single resultant, and that this result
ant is directed vertically upward.

MECHANICS OF FLUIDS.
511
2d. The resultant of all the pressures is equal, in intensity, second result;
to the weight of the displaced fluid.
3d. The line of direction of the resultant, passes through third result;
the centre of gravity of the displaced fluid.
4th. The horizontal pressures destroy each other.
Again, if without altering the volume of this solid, we
give it an additional quantity of matter, it is obvious that
the weight of this latter will cause it to descend, that is,
sink to the bottom of the vessel. Or if, without altering
its volume, we conceive a portion of matter taken from its
interior, the equilibrium will again be destroyed, the weight
of the solid will be diminished by that of the subducted
matter, the resultant of the pressures will prevail, and the
body will rise to the surface, through which it will con-
tinue to ascend, till the weight of the fluid displaced by
the part immersed, is equal to that of the entire body.
fourth result;
float, according
greater or less
In the first case, the density of the body will be in-
creased, containing a greater quantity of matter under the an immersed
same volume, and in the second the density will be dimin- body will sink or
ished;
and as the density of the original body was the as its density is
same as that of the fluid, we see that when the density of than that of the
an immersed body is greater than that of the fluid, it will sink fluid;
to the bottom; when less, it will rise to the surface, and float.
It follows, also, from what has been said above, that
when a body is immersed in a fluid, it will lose a portion of
its weight equal to that of the
displaced fluid. This is beau-
tifully illustrated by what
is usually called the "cylin-
der and bucket" experiment.
Place a hollow cylinder a,
in one of the scales of a
balance; suspend to this
scale a second cylinder b, of
solid metal, exactly fitting
the former, and in the oppo-
site scale put a weight c, that
Fig. 306.
the body will lose
a portion of its
weight equal to
that of the
displaced fluid;

512
NATURAL PHILOSOPHY.
cylinder and
bucket
experiment;
weight of the
immersed solid
vessel;
shall restore the equilibrium
of the balance. Now im-
merse the cylinder b in a
vessel W of water, the scale
of the weight c will de-
scend; fill the cylinder a
with water taken from the
vessel W, the beam of the
balance will return to its
horizontal position.
The weight lost by the
solid is transmitted through
Fig. 306.
the fluid to the vessel, in the same way that the weight of a
person in bed is transmitted through the latter to the bed-
transmitted to the stead, and thence to the floor. This is proved, experiment-
ally, thus: Place a tumbler of water in one of the scales
A of a balance, bring the beam to a horizontal position by
means of the empty hollow cylinder a of the last experi-
ment and a weight c; sus-
experimental
proof;
this principle
used to find
weight of ships,
&c.;
pend the solid cylinder b by
means of a thread from a
detached ring R, and depress
it till it is wholly immersed
into the water of the tum-
bler; the scale A will fall;
fill the cylinder a with water
of the same temperature and
density as that in the tum-
bler; the equilibrium will
be restored.
This important principle,
OR
A
Fig. 263.
which determines the circumstances under which a body
will rest upon a fluid, is frequently employed to ascertain
the weights of large floating masses, such as ships, boats,
and the like, which are entirely beyond the capacity of our
ordinary weighing machines. For this purpose the vol-
ume, in cubic feet, of the immersed part is computed from

MECHANICS OF FLUIDS.
513
cargo;
the known figure and dimensions of the body, and this is weight of a ship's
multiplied by the known weight of a cubic foot of water,
which is 62.5 pounds avoirdupois; the product is the
weight of the floating body, in pounds. By taking in this
way the difference of weights of a ship, with and without
her cargo, the weight of the latter may be ascertained.
a fluid;
The upward action by which an immersed body appa-
rently loses a portion of its weight, is called the buoyant buoyant effort of
effort of the fluid; and as the line of direction of this effort
passes through the centre of gravity of the displaced fluid,
this point is called the centre of buoyancy. The vertical centre of
line through the centre of buoyancy, is called the line of buoyancy;
support. The weight of a body acting at its centre of grav- line of support;
ity downward, and the buoyant effort at the centre of
buoyancy upward, the body can only be in equilibrio
when the line joining these centres is vertical, for it is only
then that the forces are directly opposed. When the line
joining the centre of buoyancy and the centre of gravity
of the floating body is vertical, it is called the line of rest. line of rest;
When the equilibrium exists, it may be stable, unstable, stable, unstable,
or indifferent. If stable, the body will not overturn when and indifferent
careened; if unstable, it will; if indifferent, the body will
retain
any position in which it may be placed.
Let MQN repre-
sent a section of any
body, as a boat at
rest upon the water,
of which the upper
surface is AB, called
the plane of floatation.
When this plane is
produced through the
boat, it will divide her
into two partial vol-
umes, the lower of
A
M
Fig. 308.
N
equilibrium;
M
N
plane of
floatation;
M
which being supposed for an instant to consist of water,
would weigh as much as the entire boat and her load, and
33

514
NATURAL PHILOSOPHY.
planes;
agitea to gl this whatever be her position, whether careened or erect.
ogs Whence it follows, that if a series of planes M' N', M" N",
&c., be passed, making the volumes M'QN', M" QN",
series of cutting &C., respectively equal to MQN, these planes will, each
in its turn, come to coincide with the plane of floatation,
whenever the boat, in the process of careening, takes a
suitable position. But these planes may be regarded as so
to do good many tangent planes to a curved surface abc, which may
be conceived as invariably connected with the boat. Now
the effect, as regards the careening motion, will be the same
10 as though this surface were the boundary of a physical
axis which is made
Dink
pasyond
hogge to on to roll back and forth
oscillations of the on the plane of float-
youd
Fig. 308.
boat;
ation, regarded as a
physical surface, after
M
G
the manner of the
pendulum axis on A
M
α
its supporting plane,
N'
N
oldaber
during an oscillation.
When the boat has od ods
position of the
line of rest
a position of equi-
during the
equilibrium;
metacentre;
librium, the line of
M
support and of rest coincide, and are normal to this
surface at its lowest point c. As the boat careens, the
line of support, being always vertical, will still be normal
to this axis surface at its lowest point, being that in which
it is tangent to the plane of floatation; hence each of these
normal lines must in turn become a line of support. If
two normals a 0 and a'O, which lie in the same plane, be
drawn at tangential points answering to two consecutive
positions of the boat, these normals will intersect at some
point 0, which point will, obviously, be the momentary
centre of rotation, when the plane of floatation coincides
with M" N". When one of these normals coincides with
the line of rest, the point O is called the metacentre, being
the point of intersection of the line of rest, with an adjacent

MECHANICS OF FLUIDS.
515
line of support. But we have seen that the equilibrium of defined;
a heavy body which may turn about a fixed point, will be
stable or unstable, according as the centre of gravity during
a slight departure from a position of equilibrium is com-
pelled by the connection to ascend or descend; and it is
obvious that, in the present case, the centre of gravity will
ascend or descend on making a slight derangement of the
line joining the centres of buoyancy and of gravity from
the line of rest, according as the centre of gravity is below
or above the metacentre. Whence we see, that the equilib-
rium will be stable when the centre of gravity is below the meta- the nature of the
centre, unstable when the relative positions of these points are
reversed, and indifferent when these centres coincide, for then a the relative
slight derangement will cause no motion in the centre of gravity. centres;
It is also obvious that the stability of the equilibrium will
be the greater, in proportion as the centre of gravity of the
floating body be at a greater distance below the centre of
buoyancy. It is for this reason that ships sent to sea object of ship-
without cargoes are provided with ballast of stone, sand,
or other heavy mat-
ter, to diminish the
chances of upsetting.
The buoyant effort
of water is used to
great advantage in
raising heavy sunken
masses. For this pur-
pose it is usual to
connect two or more
C
A
Fig. 309.
D
B
equilibrium
determined by
positions of the
ballast;
buoyant effort
used to raise
sunken masses;
boats A and B, by means of a substantial cross-beam; to
fill them nearly full of water, that they may sink as low as
possible, and while in this condition to attach the body to
be raised to the cross-beam by means of a taught chain or a common mode
rope, and then to pump the water from the boats; the ten-
sion upon the chain will be equal to the weight of the
water pumped from the boats. If it is the question to
raise a sunken boat, one of the most effective means is to
of employing this
principle.

516
NATURAL PHILOSOPHY.
Level strata in
heterogeneous
fluids;
mixture of
different fluids
having no
force empty and water-tight barrels between her deck and
hull.
§ 269.-We have just seen that when a body is im-
mersed in a fluid, it loses a portion of its weight equal to
that of the displaced fluid, and that it will sink or rise to
the surface, depending upon its relative density. This is
universally true whatever be the size and number of the
bodies immersed. If, therefore, one fluid be poured into
another for which it has no affinity, as oil into water, it
will sink to the bottom or rise to the surface and float,
according as its density is greater or less than that of the
fluid into which it is poured. The elements of the lighter
fluid will act as so many immersed bodies till they reach
the surface of the heavier fluid, where, being freed from the
buoyant action of the latter, they will arrange themselves,
affinity for each under the efforts of their own weight, into a stratum of
which the upper surface will, like that of the fluid below it,
be perpendicular to the direction of the force of gravity.
What is here said of two, is equally applicable to three,
four, or any number of fluids of different densities mixed
together; whence we conclude, that such fluids will come to
rest only after arranging themselves into LEVEL STRATA in the
strata; the most order of their densities; the most dense being at the bottom and
the least dense at the top. This is confirmed by daily obser-
vation, and may be easily illustrated by pouring mercury,
water, and oil, into a common tumbler. The mercury
will come to rest at the bottom, the oil at the top, the
upper surfaces of all being level.
other;
will form level
dense lowest ;
from the
properties of the
The same conclusion follows from the consideration,
the same results that these fluids when mixed constitute a heavy system,
which, we have seen, can only come to a state of stable
centre of gravity; equilibrium when its centre of gravity is at the lowest
point, a condition only fulfilled by the arrangement, in
respect to density, just described.
If the elements of one fluid have an affinity for those
of another, this affinity will, when the fluids come into con-

MECHANICS OF FLUIDS.
517
fluids have an
tact, counteract the buoyant action of the heavier fluid, they will not
and the lighter will be held in a state of mixture.
stance wine and water, water and alcohol, brandy
water, and the like.
In- obtain when the
and affinity for each
other.
VI.
SPECIFIC GRAVITY.
defined;
8270.-The specific gravity of a body, is the weight Specific gravity
of so much of the body, as would be contained under a
unit of volume.
It is measured by the quotient arising from dividing
the weight of the body by the weight of an equal volume
of some other substance, assumed as a standard; for the its measure;
ratio of the weights of equal volumes of two bodies being
always the same, if the unit of volume of each be taken,
and one of the bodies become the standard, its weight will
become the unit of weight.
The term density denotes the degree of proximity density;
among the particles of a body. Thus, of two bodies, that
will have the greater density which contains, under an
equal volume, the greater number of particles. The force
of gravity acts, within moderate limits, equally upon all illustration;
elements of matter. The weight of a substance is, there-
fore, directly proportional to its density, and the ratio of
the weights of equal volumes of two bodies is equal to the
ratio of their densities. Denote the weight of the first
by W, its density by D, its volume by V, and the force
of gravity by g, then will Eq. (26)'
W = g.D.V;
and denoting the like elements of the other body by W,
measure for the
weight of a body;

518
NATURAL PHILOSOPHY.
Jon the yo D, and V, we have one Jeyond
ed gode aldo
ved ebloh
weight of a
second body;dio
ratio of the
weights;
same when the
blok ed
doll out bas
Ban yboard Jodool W, g. D,. V
=
Dividing the first by the second,
W
W
=
gDV
g D, V
=
DV
D, V
and making the volumes equal,
volumes are glow
equal;
W
D
=
W
D
.... (227).
Now suppose the body whose weight is W, to be assumed
as the standard both for specific gravity and density, then
will D, be unity, and
specific gravity; todet
S
abe
W
=
=
D..
(228);
W
in which S denotes the specific gravity of the body whose
specific gravity density is D; and from which we see, that when specific
and density
expressed by
same number
for same
standard.
Choice of a
standard;
gravities and densities are referred to the same substance
as a standard, the numbers which express the one will
also express
express the other.
§ 271.-Bodies present themselves under every variety
of condition-gaseous, liquid, and solid; and in
every kind
of shape and of all sizes. The determination of their specific
gravity, in every instance, depends upon our ability to find
the weight of an equal volume of the standard. When a
solid is immersed in a fluid, it loses a portion of its weight
equal to that of the displaced fluid. The volume of the
boda to biste body and that of the displaced fluid are equal. Hence the
weight of the body in vacuo, divided by its loss of weight
when immersed, will give the ratio of the weights of equal

MECHANICS OF FLUIDS.
519
ased) lo gitar
ofioece
volumes of the body and fluid; and if the latter be taken
as the standard, and the loss of weight occupies the de-
nominator, this ratio becomes the measure of the specific
gravity of the body immersed. For this reason, and in
view of the consideration that it may be obtained pure at
all times and places, water is assumed as the general stand- water assumed as
ard of specific gravities and densities for all bodies. the standard for
Sometimes the gases and vapors are referred to atmo- and density;
spheric air, but the specific gravity of the latter being
known as referred to water, it is very easy, as we shall gases sometimes
presently see, to pass from the numbers which relate to
one standard to those that refer to the other.
very the latter being
specific gravities
referred to
atmospheric air.
of water;
al boamite
retow
§ 272.-But water, like all other substances, changes its varying density
density with its temperature, and, in consequence, is not
an invariable standard. It is hence necessary either to
employ it at a constant temperature, or to have the means
of reducing the specific gravities, as determined by it at
different temperatures, to what they would have been if
taken at a fixed or standard temperature. The former is
generally impracticable; the latter is easy.
standard
temperature;
Let D denote the density of any solid, and S'its specific reduction to a
gravity, as determined at a standard temperature corre-
sponding to which the density of the water is D. Then,
Eq. (227),
Fred Latonah ad
D
S=
=
Ꭰ "
specific gravity at
one temperature;
Again, if S' denote the specific gravity of the same body,
as indicated by the water when at a temperature different
from the standard, and corresponding to which it has a
density D, then will
whatis abnols
D
same at another
S
=
Du
vd
bolously inferotemperature;
Dividing the first of these equations by the second, we

520
NATURAL PHILOSOPHY.
have 1
ratio of these
Specific gravities;
ni bac
whence homine
S'
ESTER
Du
D
D₁,
bod flo
S = S'.
(229);
D
specific gravity
reduced to a
standard;
and if the density D, be taken as unity,
S = S'. Du
(230).
expressed in
words;
at different
temperatures;
That is to say, the specific gravity of a body as determined at
the standard temperature of the water, is equal to its specific
gravity determined at any other temperature, multiplied by the
density of the water corresponding to this temperature, the
density at the standard temperature being regarded as unity.
To make this rule practicable, it becomes necessary to
find the relative densities of water at different temperatures.
For this purpose, take any metal, say silver, that easily
resists the chemical action of water, and whose rate of ex-
pansion for each degree of Fahr. thermometer is accurately
density of water known from experiment; give it the form of a slender
cylinder, that it may readily conform to the temperature
of the water when immersed. Let the length of the cylin-
der at the temperature of 32° Fah. be denoted by 7, and
the radius of its base by ml; its volume at this tem-
perature will be,
volume of a
slender cylinder;
its expansion;
π m² 12 x 1 = m² 13.
Let nl be the amount of expansion in length for each
degree of the thermometer above 32°. Then, for a tem-
perature denoted by t, will the whole expansion in length
be
nl x (t - 32°),

MECHANICS OF FLUIDS.
521
and the entire length of the cylinder will become
l + nl (t - 32°) = 7 [1 + n (t - 32°)];
which, substituted for 7 in the first expression, will give
the volume for the temperature t equal to
π m² 13 [1 + n (t - 32°)]³.
its increased
length;
its increased
volume;
The cylinder is now
weighed in vacuo and
in the water, at dif-
ferent temperatures,
varying from
upward, through any
32°
desirable range, say
to one hundred de-
grees. The temper-
ature at each pro-
cess being substituted
above, gives the vol-
ume of the displaced
fluid; the weight of
the displaced fluid is
known from the loss
of weight of the cyl-
inder. Dividing this
weight by the vol-
ume, gives the weight
of the unit of volume
Fig. 310.
of the water at the temperature t. It was found by Stampfer,
experimental
determination of
the density of
water at different
temperatures;
that the weight of the unit of volume is greatest when the greatest density
temperature is 38.75 Fahrenheit's scale. Taking the den- at 38.75;
sity of water at this temperature as unity, and dividing the
weight of the unit of volume at each of the other tem-
peratures by the weight of the unit of volume at this,
38.75, the following table will result:-

522
NATURAL PHILOSOPHY.
Bailyo ode to dignat
TABLE
odt Ene
OF THE DENSITIES AND VOLUMES OF WATER AT DIFFERENT DEGREES
OF HEAT, (ACCORDING TO STAMPFER,) FOR EVERY 21 DEGREES OF
FAHRENHEIT'S SCALE.
dos doidw
(Jahrbuch des Polytechnischen Institutes in Wein, Bd. 16. S. 70.)
t
Temperature.
DI
Density.
V
Diff.
Diff.
Volume.
O
32.00
0.999887
1.000113
34.25
0.999950
63
1.000050
63
36.50
0.999988
38
I.000012
38
38.75
1.000000
12
1.000000
12
41.00
0.999988
12
1.000012
12
43.25
0.999952
35
1.000047
電話​: 35
45.50
0.999894
58
1.000106
59
47.75
0.999813
81
1.000187
81
50.00
0.999711
102
1.000289
102
52.25
0.999587
124
1.000413
124
54.50
0.999442
145
1.000558
145
56.75
0.999278
164
1.000723
165
59.00
0.999095
183
1.000906
183
61.25
0.998893
202
1.001108
202
63.50
0.998673
220
1.001329
221
65.75
0.998435
238
1.001567
238
68.00
0.998180
255
1.001822
255
70.25
0.997909
271
1.002095
273
72.50
0.997622
287
1.002384
289
74.75
0.997320
302
1.002687
303
77.00
0.997003
317
1.003005
318
79.25
0.996673
330
1.003338
333
81.50
0.996329
344
1.003685
347
83.75
0.995971
358
1.004045
360
86.00
0.995601
370
1.004418
373
88.25
0.995219
382
1.004804
386
90.50
0.994825
394
1.005202
398
92.75
0.994420
405
1.005612
410
95.00
0.994004
416
1.006032
420
97.25
0.993579
425
1.006462
430
99.50
0.993145
434
1.006902
440
With this table it is easy to find the specific gravity by
means of water at any temperature. Suppose, for example,
the specific gravity S' in Eq. (230), had been found at the
temperature of 59°, then would D,, in that equation, be
0.999095, and the specific gravity of the body referred to
water at its greatest density, would be given by
SS' X 0.999095.

MECHANICS OF FLUIDS.
523
volumes of the
The column under the head V, will enable us to determine relation of
how much the volume of any mass of water, at a tempera- same amount of
ture t, exceeds that of the same mass at its maximum den- fluid at different
sity. For this purpose, we have but to multiply the volume temperatures.
at the maximum density by the tabular number corre-
sponding to the given temperature.
nui olodw
specific gravity of
$273. Before proceeding to the practical methods of Instruments used
finding the specific gravity of bodies, and to the variations to find the
in the processes rendered necessary by the peculiarities of a body;
the different substances, it will be necessary to give some
idea of the best instruments employed for this purpose.
These are the Hydrostatic Balance and Nicholson's Hy-
drometer.
The first is simi-
lar in principle and
form to the common
balance. It is provi-
ded with numerous
weights, extending
through a wide range,
from a small fraction
of a grain to several
ounces. Attached to
the under surface of
one of the basins is
a small hook, from
Fig. 311.
lood
roubled
taplow
hydrostatic
30杯
​oddiy balance;
which may be sus-
pended any body bywyd bile
preda
means of a thin plat-
inum wire, horse-hair, or any other delicate thread that mode of attaching
will neither absorb nor yield to the chemical action of the the body;
fluid in which it may be desirable to immerse it.
hydrometer;
Nicholson's Hydrometer consists of a hollow métallic ball Nicholson's
A, through the centre of which passes a metallic wire,
prolonged in both directions beyond the surface, and sup-
porting at either end a basin B and B'. The concavities

524
NATURAL PHILOSOPHY.
conditions the
instrument must
satisfy.
of these basins are turned in the
same direction, and the basin B' is
made so heavy that when the in-
description, and strument is placed in water the
stem CC' shall be vertical, and a
weight of 500 grains being placed
in the basin B, the whole instru-
ment will sink till the upper surface
of distilled water, at the standard
temperature, comes to a point C
marked on the upper stem near its
middle. This instrument is pro-
Process for
finding specific
gravity of a solid
heavier than
waterby the
balance;
Fig. 312.
C
B
B'
A
vided with weights similar to those of the Hydrostatic
Balance.
§ 274.-(1). If the body be solid, insoluble in water, and
will sink in that fluid, attach it, by means of a hair, to the
hook of the basin of the hydrostatic balance; counterpoise
it by placing weights in the opposite scale; now immerse
the body in water, and restore the equilibrium by placing
weights in the basin above the body, and note the tem-
perature of the water. Divide the weights in the basin to
which the body is not attached by those in the basin to
which it is, and multiply the quotient by the density cor-
responding to the temperature of the water, as given by
the table; the result will be the specific gravity.
Thus denote the specific gravity by S, the density of
the water by D,,, the weight in the first case by W, and
that in the scale above the solid by w, then will
specific gravity;
S = D₁, X
W
พ
(2). If the body be insoluble, but will not sink in water, as
when the body is would be the case with most varieties of wood, wax,
and
the like, attach to it some body, as a metal, whose weight
in the air and loss of weight in the water are previously
lighter than
water;

MECHANICS OF FLUIDS.
525
described;
found. Then proceed exactly as in the case before, to find
the weights which will counterpoise the compound in air process
and restore the equilibrium of the balance when it is im-
mersed in the water. From the weight of the compound
in air, subtract that of the heavier body in air; from the
loss of weight of the compound in water, subtract that of
the heavier body; divide the first difference by the second,
and multiply by the density of the water answering to its
temperature, and the result will be the specific gravity
of the lighter body.
Example.
A piece of wax and copper in air
Lost on immersion in water
Copper in air
Loss of copper in water
Then
W+W' - W' = 438
grs.
= 438 =W+W', example;
= 95.8 = w + w',
=
388
=
W',
=
44.2 = w'.
388
=
507 = W,
w + w'
-
w'
=
95.8
44.2=
51.6 =
W.
the case of wax;
Temperature of water 43.25,
D,, = 0.999952,
W
S = D₁, X
11
0.999952 X
พ
50
51.6
0.968.
specific gravity
of wax;
(3). If the body readily dissolve in water, as many of
the salts, sugar, &c., find its apparent specific gravity in
some liquid in which it is insoluble, and multiply this
apparent specific gravity by the density or specific gravity when the body
of the liquid referred to water at its maximum density as
a standard; the product will be the true specific gravity. fluid;
If it be inconvenient to provide a liquid in which the
solid is insoluble, saturate the water with the substance
is soluble in
the standard

526
NATURAL PHILOSOPHY.
saturate the
liquid with the
and find the apparent specific gravity with the water thus
saturated. Multiply this apparent specific gravity by the
body and proceed density of the saturated fluid, and the product will be the
specific gravity referred to the standard. This is a com-
mon method of finding the specific gravity of gunpowder,
the water being saturated with nitre.
as before;
a liquid;
(4). If the body be a liquid, select some solid that
when the body is will resist its chemical action, as a massive piece of glass
suspended from fine platinum wire; weigh it in air, then
in water, and finally in the liquid; the differences between
the first weight and each of the latter, will give the
weights of equal volumes of water and the liquid. Divide
the weight of the liquid by that of the water, and the
quotient will be the specific gravity of the liquid, pro-
vided the temperature of water be at the standard. If the
water have not the standard temperature, multiply this
apparent specific gravity by the tabular density of the
water corresponding to the actual temperature.
rule;
Example.
example;
grs.
Loss of glass in water at 41°, 150 = w',
66
66
sulphuric acid, 277.5
= w₁
specific gravity
of sulphuric acid;
S=
277.5
=
X 0.999988
= 1.85.
150
(5.) If the body be a gas or vapor, provide a large glass
flask-shaped vessel, weigh it when filled with the gas;
when the body is withdraw the gas, which may be done by means to be ex-
a gas or vapor; plained presently, fill with water, and weigh again; finally,
withdraw the water and exclude the air, and weigh again.
This last weight subtracted from the first will give the
weight of the gas that filled the vessel, and subtracted
from the second will give the weight of an equal volume
of water; divide the weight of the gas by that of the
water, and multiply by the tabular density of the water
process;

MECHANICS OF FLUIDS.
527
answering to the actual temperature of the latter; the result
will be the specific gravity of the gas.
The atmosphere in which all these operations must be influence of the
performed, varies at different times, even during the same atmosphere;
day, in respect to temperature, the weight of its column
which presses upon the earth, and the quantity of moisture
or aqueous vapor it contains. That is to say, its density
depends upon the state of the thermometer, barometer,
and hygrometer. On all these accounts corrections must temperature;
be made, before the specific gravity of atmospheric air, or pressure;
that of any gas exposed to its pressure, can be accurately
determined. The principles according to which these cor-
rections are made, will be discussed when we come to treat moisture;
of the properties of elastic fluids.
Nicholson's
hydrometer for
To find the specific gravity of a solid by means of
Nicholson's Hydrometer, place the instrument in water, mode of using
and add weights to the upper basin till it sinks to the mark
on the upper stem; remove the weights and place the solid solids;
in the upper basin, and add weights till the hydrometer
sinks to the same point; the difference between the first
weights and those added with the body, will give the
weight of the latter in air. Take the body from the upper
basin, leaving the weights behind, and place it in the lower
basin; add weights to the upper basin till the instrument
sinks to the same point as before, the last added weights
will be the weight of the water displaced by the body;
divide the weight in air by the weight of the displaced
water, and multiply the quotient by the tabular density of
the water answering to its actual temperature; the result
will be the specific gravity of the solid.
To find the specific gravity of a fluid by this instru-
ment, immerse it in water as before, and by weights in the also for fluids;
upper basin sink it to the mark on the upper stem; add
the weights in the basin to the weight of the instrument,
the sum will be the weight of the displaced water. Place
the instrument in the fluid whose specific gravity is to be
found, and add weights in the upper basin till it sinks to

528
NATURAL PHILOSOPHY.
The scale
areometer;
description;
the mark as before; add these weights to the weight of the
instrument, the sum will be the weight of an equal vol-
ume of the fluid; divide this weight by the weight of the
water, and multiply by the tabular density corresponding to
the temperature of the water, the result will be the spe-
cific gravity.
Fig. 313.
C
§ 275.-Besides the hydrometer of Nicholson, which
requires the use of weights, there is another form of this
instrument which is employed solely in the determination
of the specific gravities of liquids, and its indications are
given by means of a scale of equal parts. It is called
the Scale-Areometer. It consists, gen-
erally, of a glass vial-shaped vessel
A, terminating at one end in a long
slender neck C, to receive the scale,
and at the other in a small globe B,
filled with some heavy substance, as
lead or mercury, to keep it upright
when immersed in a fluid. The appli-
cation and use of the scale depend
upon this, that a body floating on the
surface of different liquids, will sink
deeper and deeper, in proportion as
the principle of the density of the fluid approaches
that of the body; for when the body
is at rest its weight and that of the
displaced fluid must be equal. Deno-
ting the volume of the instrument by
V, that of the displaced fluid by V',
the density of the instrument by D, and that of the fluid
by D', we must always have
this instrument;
conditions of
equilibrium;
g VD = g V' D';
in which g denotes the force of gravity, the first member
the weight of the instrument, and the second that of the

MECHANICS OF FLUIDS.
529
085
displaced fluid. Dividing both members by D' V, and
omitting the common factor g, we have
D
=
D' T
ed Janm
How Food
bart to his
ratio of densities
equal to that of
the volumes
inversely;
In which, if the densities be equal, the volumes must be
equal; if the density D' of the fluid be greater than D, or
that of the solid, the volume V of the solid must be greater
than V', or that of the displaced fluid; and in proportion
as D' increases in respect to D, will V' diminish in respect
to V, that is, the solid will rise higher and higher out
of the fluid in proportion as the density of the latter is in-
creased, and the reverse. The neck C of the vessel should
be of the same diameter throughout. To establish the
scale, the instrument is placed in distilled water at the
standard temperature, and when at rest the place of the construction of
surface of the water on the neck is marked and numbered
1; the instrument is then placed in some heavy solution
of salt, whose specific gravity is accurately known by
means of the Hydrostatic Balance, and when at rest the
place on the neck of the fluid surface is again marked and
characterized by its appropriate number. The same pro-
cess being repeated for rectified alcohol, will give another
point towards the opposite extreme of the scale, which
may be completed by graduation.
the scale;
To use this instrument, it will be sufficient to immerse use;
it in a fluid and take the number on the scale which coin-
cides with the surface.
instrument;
To bring into view the circumstances which determine
the sensibility both of the Scale-Areometer and Nicholson's
Hydrometer, let s denote the specific gravity of the fluid, sensibility of the
the volume of the vial, 7 the length of the immersed
portion of the narrow neck, r its semi-diameter, and w the
total weight of the instrument. Then will r², denote the
area of a section of the neck, and 27, the volume of fluid
displaced by the immersed part of the neck. The weight,
34

530
NATURAL PHILOSOPHY.
weight of fluid
displaced
condition of the
equilibrium;
therefore, of the whole fluid displaced by the vial and
neck will be
Sc + Sπp²l;
but this must be equal to the weight of the instrument,
whence
T
w = s (c + π p² 1),
from which we deduce
specific gravity;
length of neck
immersed;
s=
พ
c + π r² l'
w
-
SC
7 =
(231).
length immersed
for second fluid;
difference of
specific gravity;
inference;
sensibility of
Nicholson's
hydrometer;
Now, immersing the instrument in a second fluid whose
specific gravity is s', the neck will sink through a distance
7', and from the last equation we have
w-s' c
1' = =
T 12 s'
subtracting this equation from that above and reducing,
we find
-
27' =
พ
s s'
The difference 7-7' is the distance between two points on
the scale which indicates the difference s'-s of specific
gravities, and this we see becomes longer, and the instru
ment more sensible, therefore, in proportion as w is made
greater and less. Whence we conclude that the Are-
ometer is the more valuable in proportion as the vial por-
tion is made larger and the neck smaller.
If the specific gravity of the fluid remain the same,
which is the case with Nicholson's Hydrometer, and it
becomes a question to know the effect of a small weight

MECHANICS OF FLUIDS.
531
added to the instrument, denote this weight by w', then
will Eq. (231) become
7' =
w + w' - sc
π p² s
subtracting from this Eq. (231), we find
w'
-
7' 7 =
odendal A
=
π p² S
From which we see that the narrower the upper stem of
Nicholson's instrument, the greater its sensibility.
TABLE
OF THE SPECIFIC GRAVITIES OF SOME OF THE MOST IMPORTANT BODIES.
[The density of distilled water is reckoned in this Table at its maximum 384° F. 1.000.]
Name of the Body.
Specific Gravity.
I. SOLID BODIES.
(1) METALS.
Antimony (of the laboratory)
4.2
Brass
7.6
4.7
8.8
Bronze for cannon, according to Lieut. Matzka
8.414
8.974
Ditto, mean
8.758
Copper, molten
7.788
-
8.726
Ditto, hammered
8.878
Ditto, wire-drawn
Gold, molten
Ditto, hammered
Iron, wrought
Ditto, cast, a mean
Ditto, gray
Ditto, white
8.78
19.238
19.361
7.207
7.251
7.2
7.5
-
-
8.9
19.253
19.6
7.788
Ditto for cannon, a mean
Lead, pure molten
7.21
11.3303
7.30
Ditto, flattened -
11.388
Platinum, native
16.0
18.94
Ditto, molten
20.855
Ditto, hammered and wire-drawn
21.25
Quicksilver, at 32° Fahr. -
13.568
13.598
Silver, pure molten
10.474
Ditto, hammered
10.51
-
10.622
Steel, cast
7.919
Ditto, wrought
7.840
Ditto, much hardened
7.818
Ditto, slightly-
7.833
Ditto, hammered
Tin, chemically pure
7.291
7.299
-
7.475
Ditto, Bohemian and Saxon
7.312

532
NATURAL PHILOSOPHY.
TABLE-continued.
Name of the Body.
Specific Gravity.
Tin, English
7.291
Zine, molten
6.861
-
7.215
Ditto, rolled
7.191
(2) BUILDING STONES.
Alabaster
Basalt
2.7
2.8
-
3.0
Dolerite
2.72
Gneiss
Granite
Hornblende
Limestone, various kinds
Phonolite-
Porphyry
Quartz
Sandstone, various kinds, a mean
Stones for building
Syenite
Trachyte
Brick
(3) WOODS.
Alder
Ash
Aspen
Birch
Box
Elm
Fir
Hornbeam
Horse-chestnut
Larch
Lime
Maple
Oak
Ditto, another specimen
Pine, Pinus Abies Picea
Ditto, Pinus Sylvestris
Poplar (Italian)
Willow
Ditto, white
(4) VARIOUS SOLID BODIES.
3.1
2.93
2.5
2.9
2.5
2.66
2.9
3.1
2.64
2.72
2.51
2.69
2.4
2.6
2.56
2.75
2.2
2.5
1.66
2.62
2.5
3.
2.4
2.6
1.41
1.86
Fresh-felled.
Dry.
0.8571
0.5001
0.9036
0.6440
0.7654
0.4302
0.9012
0.6274
0.9822
0.5907
0.9476
0.5474
0.8941
0.5550
0.9452
0.7695
0.8614
0.5749
0.9206
0.4735
0.8170
0.4390
0.9036
0.6592
1.0494
0.6777
1.0754
0.7075
0.8699
0.4716
0.9121
0.5502
0.7634
0.3931
0.7155
0.5289
0.9859
0.4873
Charcoal, of cork
0.1
Ditto, soft wood
0.28
-0.44
Ditto, oak
1.573
Coal
1.232
1.510
Coke
1.865
Earth, common
1.48
rough sand
1.92
rough earth, with gravel
2.02
moist sand
2.05
gravelly soil
2.07
clay
2.15
clay or loam, with gravel
2.48

MECHANICS OF FLUIDS.
533
MARUTAN
TABLE-continued.
Name of the Body.
Specific Gravity.
Flint, dark
2.542
Ditto, white
2.741
Gunpowder, loosely filled in
coarse powder
0.886
ontrold
musket ditto
0.992
gay siberiby
Ditto, slightly shaken down
musket-powder-
Ditto, solid
Ice
Lime, unslacked
Resin, common
Rock-salt-
Ditto, crystallized
Slate-pencil
Sulphur
Tallow
Turpentine
Wax, white
Ditto, yellow
Ditto, shoemaker's
Saltpetre, melted
Degerby
iqiy H
2.563
0.9268
1.089d bododgson
2.2579819 de masie
2.745 Biss Rundglue
1.900
1.8
1.92
0.942 wom
0.991
0.965 of Rasio
1.069
2.248
0.916
-
1.842
-
2.24
1.99
to oldal a to s
0.969
0.897
II. LIQUIDS.
Acid, acetic
1.063
Ditto, muriatic
1.211
Ditto, nitric, concentrated
1.521
1.522
Ditto, sulphuric, English
1.845
Ditto, concentrated (Nordh.)
1.860
Alcohol, free from water
0.792
Ditto, common
0.824
0.79
Ammoniac, liquid
0.875
Aquafortis, double
1.300
Ditto, single
I.200
Beer-
1.023
1.034
Ether, acetic
0.866
Ditto, muriatic
0.845
-
0.874
Ditto, nitric
0.886
Ditto, sulphuric
0.715
Oil, linseed
0.928
0.953
Ditto, olive
0.915
Ditto, whale
Ditto, turpentine
Quicksilver
Water, distilled -
Ditto, rain
0.792
0.891
0.923
13.568
13.598
1.000
Ditto, sea-
Wine
III. GASES.
Water
Temp. 38° F.
1.
1.0013
1.0265
0.992
-
-
1.028
1.038
Barometer
30 In.
1
Temp. 34°.
Atmospheric air =770=
0.00130
1.0000
Carbonic acid gas
0.00198
1.5240
Carbonic oxide gas
0.00126
0.9569
Carbureted hydrogen, a maximum
0.00127
0.9784
Ditto, from coals
0.00039
0.3000
0.00085
0.5596

534
NATURAL PHILOSOPHY.
TABLE-Continued.
Name of the Body.
Specific Gravity.
Barometer
Water=1.
30 In.
Temp. 381° F. Temp. 34°.
Chlorine
0.00321
2.4700
Hydriodic gas
0.00577
4.4430
Hydrogen
0.0000895
0.0688
Muriatic acid gas
Hydrosulphuric acid gas
0.00155
1.1912
0.00162
1.2474
Nitrogen
0.00127
0.9760
Oxygen
0.00143
1.1026
Phosphureted hydrogen gas
0.00113
0.8700
Steam at 212° Fahr.
0.00082
0.6235
Sulphurous acid gas
0.00292
2.2470
use of a table of
The knowledge of the specific gravities or densities of
specific gravities; different substances is of great importance, not only for
weight of any
body;
weight of a cubic
foot of distilled
water at
maximum
density;
scientific purposes, but also for its application to many of
the useful arts. This knowledge enables us to solve such
problems as the following, viz. :-
1st. The weight of any substance may be calculated,
if its volume and specific gravity be known.
2d. The volume of any body may be deduced from its
specific gravity and weight. Thus we have always
W = gDV;
in which 9 is the force of gravity, D the density, V the
volume, and W the weight, of which the unit of measure
is the weight of a unit of volume of water at its maxi-
mum density.
Making D and V equal to unity, this equation becomes
W₁ = 9;
but if the density be one, the substance must be water at
38.75 Fahr. The weight of a cubic foot of water at
60° is 62.5 lbs., and, therefore, at 38.75, it is
lbs.
62.5
0.99914
=
lbs.
62.556;

MECHANICS OF FLUIDS.
535
whence, if the volume be expressed in cubic feet,
lbs.
W = DV..
62.556 x D V. . . (232),
in which W is expressed in pounds; and if the unit of
volume be a cubic inch,
volume in cubic
feet;
weight of a body
in pounds,
volume being in
cubic feet;
W
=
62.556
1728
Also
DV = 0.036201 DV,.. (233).
weight in
pounds,
volume in cubic
inches;
V =
lbs.
W.
62.556. D
volume in cubic
•
(234),
feet;
W.
V₁ =
•
(235).
lbs.
volume in cubic
inches;
0.036201. D
Example 1st. Required the weight of a block of dry example first;
fir, containing 50 cubic inches. The specific gravity or
density of dry fir is 0.555, and V = 50; substituting these
values in Eq. (233),
lbs.
W
=
0.036201 x 0.555 X 50
1.00457.
weight of 50
cubic inches of
fir;
Example 2d. How many cubic inches are there in a example second;
12-pound cannon-ball? Here W is 12 pounds, the mean
specific gravity of cast iron is 7.251, which, in Eq. (235),
give
12
in.
V₁ =
45.6.
volume of a 12-
pound cannon-
0.036201 X 7.251
ball.

536
NATURAL PHILOSOPHY.
VII.
COMPRESSIBLE FLUIDS.
Peculiarities of
§ 276.-The properties of liquids which have now been
gases and vapors; considered, are common to all fluids. But gases and
vapors have, in addition, properties peculiar to themselves
which we now proceed to consider.
oldeo mi
contract and
Gases and vapors differ mostly from liquids, in the
readiness with which they yield a portion of their volume
expand according and contract into smaller spaces when subjected to an
to pressure.
aldes
conditions of
rest;
no marked
variety of
fluidity:
augmentation of external pressure, and diffuse themselves
in all directions when this pressure is withdrawn. These
distinguishing properties are due to the repulsive forces or
molecular springs by which the particles are urged to
separate from each other, and which make it impossible
for compressible fluids, that are also highly elastic, ever
to be at rest, unless these forces are opposed by the reaction
of inclosing surfaces, as the sides of vessels, or the appli-
cation of some other antagonistic forces acting inwardly,
as in the case of the earth's attraction upon our atmosphere.
Besides these essential peculiarities, there are other
characteristics that distinguish compressible fluids, usually
denominated aeriform bodies, from the other forms of
aggregation. Between solids and liquids, a gradation is
observable, and in the degree of fluidity of the latter, a
strongly marked variety obtains-as in tar, oil, water,
ether, and the like; but between compressible and incom-
pressible fluids, these connecting links are less obvious.
Again, as a general rule, gases are highly transparent,
for most part colorless, and therefore invisible, and are
small density; distinguished from all other bodies by their small degree
of density and consequent low specific gravity.
usually
transparent;

MECHANICS OF FLUIDS.
537
The atmosphere, as being the most important of the
aeriform bodies, may be taken as the representative of the atmosphere the
whole class, as regards their mechanical properties. It is type of the class;
to this class of bodies, what water is to liquids. It exists
all over the earth, and its ever-active agency in the pro-
duction of phenomena, makes it not less interesting than
important to determine the laws of its equilibrium and
motion.
and elasticity
shown;
(1) The compressibility and elasticity are easily shown compressibility
by inclosing the air in a bag of some impervious substance,
as india-rubber, and pressing it with the hand; the hand
will experience a resistance, while the volume of the con-
fined air will diminish: on removing the hand, the bag
will be distended by the elasticity
of the air, and restored to its for-
mer dimensions. Air-pillows and
cushions, in common use, are famil-
iar illustrations.
(2) A is a two-necked bottle con-
taining some liquid, as water, B an
inflated bladder, or india-rubber
bag, attached by the neck to one of
the mouths. A glass tube a b, open
at both ends, is fitted air-tight to
the other mouth, its lower end a
reaching nearly to the bottom of
the bottle. On compressing with
the hand, the air in the bladder or
bag, the liquid will be seen to
mount up the tube.
A
(3) Hero's Ball.-A hollow globe
from which the external air can
be excluded by turning a cock b,
contains a tube that reaches nearly
to the bottom, and fits in the neck by
a screw. Fill the vessel about half
full of water, screw in the tube cd,
Fig. 314.
india-rubber bag;
B
Fig. 315.
india-rubber bag
connected with a
two-necked
bottle;
Hero's ball;
a

538
NATURAL PHILOSOPHY.
breathe through c, and close the
stop-cock b; the breath will ascend
through the water, mingle with the
air in the space a, take from it a
portion of its volume and thus in-
crease its elasticity, which, reacting
upon the surface of the water, will
force the latter up the tube cd on
turning the cock b. On this princi-
Fig. 315.
principle of the ple depend the operations of the air-chamber in fire-
engines and similar machines.
fire-engine;
(4) Hero's Fountain.-In this apparatus, also, the com-
pression of air and consequent increase of elasticity, are
manifested in producing a water-jet.
Two vessels a and g are united by
a tube t, open at both ends, extend-
ing from the upper surface of the
lower vessel to near the top of the
Hero's fountain; other. A pipe c d, provided with a
stop-cock b, screws into the top of
the vessel a, and extends nearly to
its bottom, as in Hero's Ball. Upon
the top of this vessel is a basin no,
from the bottom of which a pipe ef,
open at both ends, passes clear
through, nearly to the bottom of
the vessel g. The tube cd, being
unscrewed, is removed, and after
pouring water into the vessel a till
its surface comes nearly to the up-
per end of the tube t, the pipe cd is
replaced, and the stop-cock b closed.
description;
n
Fig. 316.
T
a
mode of action; Water is now poured into the basin no; this will descend
through the tube e f into the vessel g, and expel a portion
of its air by forcing it up the tube t into the vessel a;
there, finding no means of escape, it will be compressed,
and its increased elasticity made to act upon the water

MECHANICS OF FLUIDS.
539
rest;
precisely as in the case of the Ball. The water will con- conditions of
tinue to descend through the tube ef from the basin, till
the increasing elasticity of the air becomes equal to the
pressure arising from a head of water equal to the differ-
ence between the level of the water in the basin and that
in the lower vessel, when the flow will cease, and every
thing will come to rest. In this condition of things turn
the cock b, and the water will spout through the tube c d.
The fluid in the upper vessel being thus ejected, there will conditions to
be room for more air; this will pass from the lower vessel cause the flow;
through the tube t, and the water will again descend from
the basin to the vessel g. The water discharged by the
jet falls into the basin n o, and is ready, in its turn, to pass
down the tube ef. A constant flow is thus maintained as
long as the fluid in the vessel a remains above the bottom
of the tube cd.
(5) The Cartesian Devil.-This is a well-known figure,
constructed so as to float in a glass vessel of water, above Cartesian devil;
the surface of which a portion of air is confined in such
manner, that if this air be com-
pressed, the figure will descend, and
rise again when the compression
ceases. It is thus contrived: In
the middle of the figure a is a small
capillary tube b, through which so
much water is admitted into the in-
terior of the body as to make its
mean density a little less than that
of the water in which it is to float.
Being thus adjusted, the figure is
immersed in a wide-mouthed glass
vessel, over which a piece of blad-
der or sheet of india-rubber is then
stretched to confine the air over the
fluid. The finger being now pressed
upon the bladder or india-rubber,
Fig. 317.
a
d
description;
b
motion of the
the air will be compressed, the increased elasticity thus pro- figure;

540
NATURAL PHILOSOPHY.
duced will be exerted upon the water, which will be forced
by it through the tube b, the mean density of the figure
will be increased, and it will sink to the bottom; on re-
explanation of moving the finger, the air above the water as well as that
in the figure, being relieved from the pressure, expands, the
water is forced back through the tube b into the vessel
again, and the figure will rise to the surface in consequence
of diminished mean density.
the motion.
VIII.
THE AIR-PUMP.
Air-pump, or
air-syringe;
the receiver;
the barrel and
piston;
$277. Seeing that the air expands and tends to diffuse
itself in all directions when the surrounding pressure is
lessened, it may be rarefied and brought to almost any de-
gree of tenuity. This is accomplished by an instrument
called the Air-Pump or Exhausting Syringe, one of the
most important pieces of apparatus used by the natural
philosopher. It will be best understood by describing one
of the simplest kind. It consists, essentially, of
1st. A Receiver R, or chamber from which the exterior
air is excluded, that the air within may be rarefied. This
is commonly a bell-shaped glass vessel, with ground edge,
over which a small quantity of grease is smeared, that no
air may pass through any remaining inequalities on its
surface, and a ground glass plate mn imbedded in a metallic
table, on which it stands.
2d. A Barrel B, or chamber into which the air in the
receiver is to expand itself. It is a hollow cylinder of
metal or glass, connected with the receiver R by the com-
munication ofg. An air-tight piston P is made to move
back and forth in the barrel by means of the handle a.

MECHANICS OF FLUIDS.
541
m
R
h
Fig. 318.
B
P
9
graphical
representation;
3d. A Stop-cock h, by means of which the communica-
tion between the barrel and receiver is established or cut
off at pleasure. This cock is a conical piece of metal fitting stop-cock, or
air-tight into an aperture just at the lower end of the bar- valve;
rel, and is pierced in two directions; one of the perfora-
tions runs transversely through, as shown in the first figure,
and when in this position the communication between
the barrel and receiver is
established; the second
perforation passes in the
direction of the axis from
the smaller end, and as it
approaches the first, in-
clines sideways, and runs
out at right angles to it,
as indicated in the second
figure. In this position
Fig. 319.
h
P
description of
stop-cock;
of the cock, the communication between the receiver and
barrel is cut off, whilst that with the external air is opened.
Now, suppose the piston at the bottom of the barrel,
and the communication between the barrel and the receiver mode of action;
established; draw the piston back, the air in the receiver
will rush out, in the direction indicated by the arrow-head,
through the communication ofg, into the vacant space
within the barrel. The air which now occupies both the

512
NATURAL PHILOSOPHY.
mode of
>peration;
to find the degree
of exhaustion;
ratio of the
densities;
barrel and receiver is less dense than when it occupied the
receiver alone. Turn the cock a quarter round, the com-
munication between the receiver and barrel is cut off, and
that between the latter and the open air is established;
push the piston to the bottom of the barrel again, the air
within the barrel will be delivered into the external air.
Turn the cock a quarter back, the communication between
the barrel and receiver is restored; and the same operation
as before being repeated, a certain quantity of air will be
transferred from the receiver to the exterior space at each
double stroke of the piston.
To find the degree of exhaustion after any number of
double strokes of the piston, denote by D the density of
the air in the receiver before the operation begins, being the
same as that of the external air; by r the capacity of the
receiver, by b that of the barrel, and by p that of the pipe.
At the beginning of the operation, the piston is at the
bottom of the barrel, and the internal air occupies the re-
ceiver and pipe; when the piston is withdrawn to the
opposite end of the barrel, this same air expands and
occupies the receiver, pipe, and barrel; and as the density
of the same body is inversely proportional to the space
it occupies, we shall have
r+p+br+p :: D: x;
in which x denotes the density of the air after the piston
is drawn back the first time. From this proportion, we
find
first diminished
density;
x = D.
r + p
r+p+b
The cock being turned a quarter round, the piston pushed
back to the bottom of the barrel, and the cock again
turned to open the communication with the receiver, the
operation is repeated upon the air whose density is x,
and

MECHANICS OF FLUIDS.
543
we have
r+p+br+p :: D
r+p
r+p+ b
x; ratio of densities;
in which x' is the density after the second backward mo-
tion of the piston, or after the second double stroke; and
we find
201
x' = D
r p
\r+ p + b
( + + 2)²;
and if n denote the number of double strokes of the piston,
and xn the corresponding density of the remaining air,
Xn
then will
Xn = D.
=
67
r+ p
+p+ b
n
second
diminished;
the nth
diminished
density;
From which it is obvious, that although the density of
the air will become less and less at every double stroke,
yet it can never be reduced to nothing, however great n
may be; in other words, the air cannot be wholly removed
from the receiver by the air-pump. The exhaustion will
go on rapidly in proportion as the barrel is large as com-
pared with the receiver and pipe, and after a few double the air can never
strokes, the rarefaction will be sufficient for all practical exhausted from
purposes. Suppose, for example, the receiver to contain the receiver;
19 units of volume, the pipe 1, and the barrel 10; then
be wholly
will
r + p
r+p+b
1=
3180
20
=
2013
3:
and suppose 4 double strokes of the piston; then will illustration;
n = 4, and
7)" = (3)
=
r+p
\r+p+b
16
81
=
0.197, nearly;
density after 4th
double stroke;

544
NATURAL PHILOSOPHY.
rarefaction by
best pumps;
gauges;
objects, and
construction;
scale of the
gauge, and
position;
first inventor;
that is, after 4 double strokes, the density of the remaining
air will be but about two tenths of the original density.
With the best machines, the air may be rarefied from four
to six hundred times.
Fig. 320.
The degree of rarefaction is indicated in a very simple
manner by what are called gauges. These not only indicate
the condition of the air in the receiver, but also warn the
operator of any leakage that may take place either at the
edge of the receiver or in the joints of the instrument.
The mode in which the gauge acts, will
be better understood when we come to
discuss the barometer; it will be suffi-
cient here simply to indicate its con-
struction. In its more perfect form, it
consists of a glass tube, about 60 inches
long, bent in the middle till the straight
portions are parallel to each other; one
end is closed and the branch termina-
ting in this end is filled with mercury.
A scale of equal parts is placed between
the branches, having its zero at a point
midway from the top to the bottom, the
numbers of the scale increasing in both directions. It is
placed so that the branches of the tube shall be vertical,
with its ends upward, and inclosed in an inverted glass
vessel, which communicates with the receiver of the air-
pump.
Repeated attempts have been made to bring the air-
pump to still higher degrees of perfection since the time
of OTTO VON GUERICKE, burgomaster of Magdeburg, who
first invented this machine in 1560. Self-acting valves,
opening and shutting by the elastic force of the air, have
improvements; been used instead of cocks. Two barrels have been given
to the air-pump instead of one, so that an uninterrupted
and more rapid rarefaction of the air is brought about,
the piston in one barrel being made to ascend as that of
the other descends. The most serious defect in the air

MECHANICS OF FLUIDS.
545
defect of the
older pumps;
pump was, that the atmospheric air could not be entirely the most serious
ejected from the barrel, but remained between the piston
and the bottom of the barrel. This intervening space is
filled with air of the ordinary density at each descent of
the piston; when the cock is turned, and the communica-
tion re-established with the receiver, this portion of air
forces its way in and diminishes the degree of rarefaction
Fig. 321.
K
RICHARDSON.N.Y
N
H
I
B
A
M
section of one of
the most
approved
pumps;
already attained. If the air in the receiver is so far rare-
fied, that one stroke of the piston will raise only such a
quantity as equals the air contained in this space, it is plain
that no further exhaustion can be effected by continuing
to pump.
This limit to rarefaction will be arrived at the limits to
the defect above;
sooner, in proportion as the space below the piston is rarefaction due to
larger; whence one chief point in the improvements has
35

546
NATURAL PHILOSOPHY.
been to diminish this space as much as possible. AB is a
highly polished cylinder of glass, which serves as the bar-
description of the rel of the pump; within it the piston works perfectly air-
improved pump; tight. The piston consists of washers of leather soaked in
oil, or of cork covered with a leather cap, and tied together
about the lower end C of the piston rod by means of
two parallel metal plates. The piston-rod Cb, which is
Fig. 321.
section of the
pump;
K
I
RICHARDSON.N.Y
TH
B
M
use of oil;
shape of the
lower end of
piston-rod;
toothed, is elevated and depressed by means of a cog-wheel
that is turned by the handle M. If a thin film of oil be
poured on the upper surface of the piston the friction will
be lessened, and the whole will be rendered more perfectly
air-tight. To diminish to the utmost the space between
the bottom of the barrel and the piston-rod, the form of a
truncated cone is given to the latter, so that its extremity

MECHANICS OF FLUIDS.
547
may be brought as nearly as possible into absolute contact
with the cock E; this space is therefore rendered indefi-
nitely small, the oozing of the oil down the barrel con-
tributing still further to lessen it. The exchange-cock E exchange-cock;
has the double bore already described, and is turned by a
short lever, to which motion is communicated by the rod
cd. The communication GH is carried to the two plates communication;
I and K, on one or both of which receivers may be placed;
the two cocks N and O below these plates, serve to cut off cut-off cock
the rarefied air within the receivers when it is desired to
leave them for any length of time. The cock O is also an
exchange-cock, so as to admit the external air into the cock to readmit
receivers.
the air;
this kind of
pump.
Pumps thus constructed have advantages over such as advantages of
work with valves, in that they last longer, exhaust better,
and may be employed as condensers when suitable receivers
are provided, by merely reversing the operations of the
exchange valve during the motion of the piston.
§ 278.-The following are some of the most interesting
experiments performed with the aid of an air-pump, show-
ing the expansive force of the atmosphere, and also the
relations between air of ordinary density and that which
is highly rarefied:-
1st. Under a receiver place a bladder tied tightly about Experiments with
the neck and partly filled with air; exhaust the air in the air-pump;
receiver, and that confined within the bladder will gradu- first experiment;
ally distend, proving experimentally the expansive force
of atmospheric air. When the air is readmitted into the
receiver, the bladder will resume its former dimensions.
An analogous appearance will be exhibited if a jar, over
which some india-rubber has been tied, be placed beneath second;
a receiver, and the air be then exhausted.
2d. The expansive force of our atmosphere is further
shown if a long-necked flask, or retort, be inverted so that
its mouth shall be below the surface of some water con-
tained in a vessel, and the whole be placed under the

548
NATURAL PHILOSOPHY.
showing also the
expansion of air;
Fig. 322.
receiver of an air-pump; when the
air within the receiver is rarefied,
that which was contained in the
bulb, expanding, escapes through
the water; and on readmitting the
atmosphere the water will rise and
occupy the
space vacated by the air.
3d. The transfer of a fluid from
Let there be a fluid in the flask A.
contains a glass tube fitted air-tight into it, and reach-
one flask to another.
The neck of this flask
third, illustrating ing almost to the bottom; the tube being bent twice
the same
principle;
fourth.
Atmospheric
resistance
illustrated;
Fig. 323.
Ъ
at right angles, the other end
passes freely through the neck of a
second bottle B. Place this appa-
ratus under the receiver of an air-
pump, and exhaust; the fluid will
mount up from the bottle A and
pass through the tube over into the
bottle B. Readmit the air, the fluid will pass back again.
4th. Place Hero's ball under the receiver when half
filled with water, and exhaust; the expansion of the air
within will send the water up through the tube in a jet.
§ 279.-When a piece of metal and a feather are aban-
doned to their own weight in the air, they fall with very
different velocities. The cause is the great disparity in
the extent of surfaces exposed to the resistance of the air
as compared with the weights.
Let a and b be two wheels re-
sembling the arms of a windmill,
with this difference only, that the
vanes of b shall strike the air with
their broad faces, whilst those of a
shall cut it edgewise; each has a
separate axis on which it revolves.
By means of a mechanical contri-
vance a rapid rotary motion is com-
Fig. 324.
Ъ

MECHANICS OF FLUIDS.
549
municated to them.
In order that this may act under
use of the
instrument.
a receiver, a rod must be made to pass through an air- description and
tight leather stuffing-box e; at the end of the rod is a
curved arm d, which drives the wheels. If the rotation
take place in vacuo, the two wheels a and b will cease to
revolve simultaneously; whereas, if the motion take place
in the ordinary atmosphere, the resistance of the latter
will bring b to a stand long before a ceases to turn.
rarefaction on
sound;
§ 280.-The atmosphere is the ordinary medium through Effects of
which sound is transmitted to the ear. In proportion as
the air becomes more rarefied, the transmission of sound
through it becomes more feeble.
Fig. 325.
Under a receiver furnished with a leather stuffing-box,
place a bell whose clapper may be struck by a rod passing
through the box, taking care to place the bell on some soft
unelastic substance, to prevent its communicating sound to
the plate of the pump and thus to the external air. The
annexed figure represents such an
apparatus, which may, however, be
considerably varied: a is the bell,
b the clapper attached by a spring
to a thin plate of wood c, into
which the support of the bell is
screwed; g is a leather drum stuffed
with horse-hair, fitting into the
upper wooden plate c, and into a
lower plate d, by which the whole
apparatus is fastened down to the
plate of the pump; lastly, h is the
lever by which the clapper is agi-
T
d
tated. After about 10 strokes of the piston, the sound
becomes sensibly more feeble, and if the exhaustion be
continued long enough it will cease altogether.
instrument by
which this may
be illustrated;
Air is necessary to respiration. Place a bird beneath air is necessary to
the receiver of an air-pump; a few strokes of the piston will respiration;
cause it to make convulsive struggles, and death will soon

550
NATURAL PHILOSOPHY.
place a bird in
the receiver of a
pump and
exhaust;
ensue unless air be admitted. Warm-
blooded animals, as birds, die if
rarefaction be carried to a small
degree; cold-blooded animals, on
the contrary, endure a high degree
of rarefaction. Many birds ascend to
considerable heights in the atmo-
sphere, and it may be hence inferred
Fig. 326.
that the density of the air at these altitudes is greater than
that in the exhausted receiver of an air-pump.
Air is necessary to combustion. Introduce a taper into
a bell-shaped receiver full of atmospheric air, and observe
air is necessary to the time it will continue to burn. Light the taper again,
place it beneath the receiver and exhaust quickly, after it
has been replenished with fresh air; the flame will expire
much sooner than before.
combustion.
To the same cause it is owing that in vacuo no light is
produced by striking a flint and steel together.
has weight;
IX.
WEIGHT AND PRESSURE OF THE ATMOSPHERE.
§ 281. From the resistance which the atmosphere
opposes to the motion of bodies through it, we might
infer that it has weight as well as inertia. That it has
The atmosphere Weight is obvious from the fact that the atmosphere
incases, as it were, the whole earth: if it were destitute
of weight and subjected only to the repulsive action
among its own particles, it would recede further and
further and extend itself throughout space.
But the
existence of weight in the atmosphere may be shown
experimentally, thus:-

MECHANICS OF FLUIDS.
551
Take a flask of some two or three inches in diameter,
having an air-tight stop-cock. Suspend it from one end
of the balance-beam and as-
certain its weight when filled
with air. Exhaust the air,
by means of the air-pump,
and the flask will be found
lighter than before; readmit
the air, it will regain its for-
mer weight. Force into the
flask an additional quantity
of air, by means of the air-
Fig. 827.
pump, used as a condenser, and the weight will be found
to be increased.
Since the atmosphere has weight, it must exert a
experiment to
show this;
pressure upon all bodies in it. To illustrate the truth the air exerts a
of this, fill with mercury a glass tube, about 32 or 33 pressure upon all
inches long, and closed at
one end by an iron stop-cock.
Close the open end by press-
ing the finger against it, and
invert the tube in a basin of
mercury; remove the finger,
the mercury will not escape,
but remain apparently sus-
pended nearly 30 inches
above the level of the mer-
cury in the basin.
If we consider the cir-
cumstances attending this
experiment, it will be seen
Fig. 328.
bodies within it
experiment to
illustrate this;
that the tube containing the mercury forms with the
basin a system of communicating tubes, as in § 260. Now
the atmosphere rests on the mercury in the basin, and is
excluded by the glass from that in the tube, above which effect of
there is therefore a vacuum. Withdraw the atmosphere
from the surface of the mercury in the basin, and, by the atmosphere;
withdrawing the
external

552
NATURAL PHILOSOPHY.
an instrument
well suited to
exhibit the facts
law of equilibrium of fluids, the mercury will descend in
the tube till it comes to a level with that without; restore
the pressure of the atmo-
sphere, and the mercury in
the tube will again rise to
its former height. This is
This is
well illustrated by the fol-
of this ad words lowing device. R is a re-
experiment;
use;
ceiver closed air-tight at the
top by means of a metallic
plate; a is a tube filled with
mercury after the manner
just described, and termina-
ting at the open end in an
inverted vial-shaped vessel
-this tube passes air-tight
through the plate on the
receiver; b is a second tube
bent in the manner indicated
in the figure, and, like the
72
Fig. 329.
b
R
description and tube a, it terminates at one end in a vial-shaped vessel,
but is open at both ends; this tube communicates with
the receiver by passing through the metallic plate at top,
and thus a connection is established between the open air
and the interior of the receiver. Mercury being poured
of into the vial of the tube b, it will rise to the same level
on either side of the bend m, and the communication
between the interior of the receiver and exterior air
will be interrupted. The receiver being placed upon
the
plate of the air-pump and the air exhausted, the mercury
will descend in the tube a, and ascend in the tube b to-
wards the bend at the top; readmit the air into the
receiver, the mercury will rise in the tube a and fall in the
tube b.
inference from
this experiment;
From this we see, that the atmospheric air presses on
the mercury, and indeed upon the surfaces of all bodies
exposed to it, with a force sufficient to maintain the quick-

MECHANICS OF FLUIDS.
553
silver in the tube at a height of nearly 30 inches; whence,
the intensity of its pressure must be equal to the weight of a
column of mercury whose base is equal to that of the surface
pressed and whose altitude is about 30 inches. The force thus atmospheric
exerted is called the atmospheric pressure.
pressure;
The absolute amount of atmospheric pressure was first
discovered by Torricelli, a pupil of Galileo; the tubes
employed in the experiments are called, on this account,
Torricellian tubes, and the vacant space above the mercury Torricellian
in the tube is called, the Torricellian vacuum, to distinguish tubes;
it from that of a receiver, which is frequently called the vacuum;
Guerickian vacuum, from Otto von Guericke, who first
invented the air-pump.
The pressure of the atmosphere at the level of the sea
will support a column of mercury 30 inches high. Now,
if we suppose the bore of the tube to have a cross-section
Torricellian
pressure at the
of one square inch, the atmospheric pressure up the tube atmospheric
will be exerted upon this extent of surface, and will sup-level of the sea;
port 30 cubic inches of mercury. Each cubic inch of
mercury weighs 0.49 of a pound-say half a pound-from
which it is apparent that the surfaces of all bodies, at the level
of the sea, are subjected to an atmospheric pressure of fifteen
pounds to each square inch.
the surface of a
man;
The body of a man of ordinary stature has a surface of pressure upon
about 2000 square inches; whence, the whole pressure to
which he would be exposed, at the level of the sea, is
15 pounds x 2000 = 30000 pounds.
The pressure of the atmosphere, resulting as it does
from its weight, it is an easy matter to estimate the weight
of the entire atmosphere of the earth. It will be sufficient weight of the
to compute, from the known diameter of the earth, the
extent of its surface in square inches, and to multiply this
by fifteen; the product will be the weight in pounds.
entire
atmosphere;
When the height of the mercury in the Torricellian
tube is 30 inches, the atmospheric pressure will support in column of water
vacuo a column of water 34 feet, the specific gravity of supported by the
mercury being 13.6 referred to water as a standard. This pressure;
atmospheric

554
NATURAL PHILOSOPHY.
Magdeburg
hemispheres;
has been verified by Hanson and Sturm, who actually
performed the experiment at Leipzig.
The atmospheric pressure is exhibited in a most stri-
king way by means of the Magdeburg hemispheres. These
are two hollow hemispheres, of brass or copper, whose
edges fit air-tight, each hemisphere being furnished with a
strong ring or handle, one of them also having a tube with
stop-cock. Place the two hemi-
spheres together, connect them with
description and the communication-pipe of the air-
mode of using;
pump, exhaust the air, and turn the
stop-cock, and disconnect from the
pump. It will be found that great
force will be necessary to pull the
hemispheres asunder. If the diame-
ter of the hemispheres, as in the case
of those employed by Guericke, in
one of his experiments, were 2 feet,
the number of square inches in a
great circle would be
Fig. 330.
examples of
Guericke's
hemispheres;
3.1416 X
2412
2
=
452.39,
the forcing of
fluid through
pores of solids;
and the force, estimated in pounds to overcome the pres-
sure, would be
15 x 452.39 = 6785.85.
In the experiment referred to
above, there were successively from
14 to 30 horses harnessed to the
hemispheres, without effecting the
separation.
The pressure of the atmosphere
will force fluids through such solid
bodies as are porous. Let R be a
long receiver, provided with a tube
Fig. 331.
ъ
R

MECHANICS OF FLUIDS.
555
exhibit this;
and stop-cock C at one end, for the purpose of connecting
with the air-pump, and at the other a perforated metallic
plate a a, into which fits, air-tight, a wooden cup b, whose instrument to
pores are in the direction of the axis of the tube. This
cup being filled with mercury, and the air exhausted by
the air-pump, the mercury will fall in a fine shower down
the receiver. The tube below is made to enter the re-
ceiver, and to curve over at the top to prevent the mer-
cury from falling into the communication-pipe of the
pump.
The atmosphere presses not only
downward, but upward, and later-
ally in all directions. This is shown
by the following experiment: The
two hemispheres A and B, are con-
nected by a tube in such manner
that one of them may turn about a
joint C, while the other is stationary.
Place the hemisphere A upon the
plate of the air-pump, and upon B
Fig. 332.
A
lay a plane plate of glass or metal fitting it air-tight. Ex-
haust the air, and the hemisphere B may be turned in any
direction without its plate falling off. This equal pressure
of the atmosphere in all directions, is of great practical
utility, as we shall presently see when we come to speak
of siphons and water-pumps. To this pressure it is owing
that flies, and other insects, are
enabled to support themselves upon
smooth vertical walls, and in in-
verted positions upon the ceilings
of rooms, &c. The feet being flat
and flexible, are brought close a-
gainst the wall or ceiling so as to
exclude the air, the centre of the
Fig. 333.
foot is then drawn away, leaving the margin in contact;
a partial vacuum is thus formed, and the external pressure
of the air is sufficient to support the weight of the insect.
atmospheric
pressure is
exerted in every
direction;
exemplification
of this in the
adhesion of
insects to walls
and ceilings.

556
NATURAL PHILOSOPHY.
of polly
X.
Mariotte's law;
connecting the
pressure, density,
and elasticity;
the instrument
MARIOTTE'S LAW.
$282.-We have seen that the
atmosphere readily contracts into a
smaller volume when pressed exter-
nally, that it as readily regains its
former dimensions when the pres-
sure is removed, and that it is,
therefore, both compressible and
elastic. Let us now consider the
law which connects the pressure,
density, and elasticity. For this pur-
pose, procure a siphon-shaped tube
ABD, open at A, the end of the
longer branch, and hermetically
sealed at the end D of the shorter
branch. Place between the branch-
es, and parallel to them, a scale of
for compressing equal parts, say inches, having its
the air;
zero on the line o o.
Pour in, at the open end A, as
much quicksilver as will fill the
horizontal part of the tube, and
bring its upper surface to the zero
0-
Fig. 334.
B
D
line in both branches; a quantity of atmospheric air of
mode of using it; ordinary density will then be confined in the shorter
branch. The expansive action of this air, resisting, as it
does, the pressure of the external air, is measured by the
weight of a column of mercury, whose base is a section of
the tube and height 30 inches. Pour into the longer
branch an additional quantity of mercury; it will rise in

MECHANICS OF FLUIDS.
557
rationale of the
experiment;
20
the shorter branch, and cause the air above it to be com-
pressed into a smaller space, but the heights at which it
will stand in the two branches will be different. The
difference between these two heights, added to 30 inches, details and
will be the altitude of the column of mercury, whose
weight is just sufficient to resist the expansive action of
the confined air. Now it is found by trial, that when the
air in the shorter branch is compressed into half its primi-
tive volume, the difference of level of the mercury in the
two branches is just 30 inches, thus making the compress-
ing force double what it was before; that when it is
compressed into one third of its original volume, the dif
ference of level is 60 inches, thus trebling the pressure;
when compressed into one fourth, the difference of level is
90 inches, thus quadrupling the pressure, and so on.
Hence we see, that in compressing the same quantity of
air into smaller spaces, the volumes occupied by it are in- volumes are
versely proportional to the pressures.
This law holds equally when the
air, instead of being compressed, is per-
mitted to expand. Let ab be a glass
tube, about 33 inches long, one end a,
being fitted with an air-tight cock, and
the entire length of the tube being
graduated in inches. Open the cock a,
immerse the tube with its open end
downward into the vessel A, previously
half filled with mercury, which will, of
necessity, stand at an equal height within
and without the tube. Now close the
cock a, and so confine a portion of air
at its ordinary density within the tube
above the surface of the mercury.
Elevate the tube any distance what-
ever, taking care that its open end shall
be below the surface; the air will ex-
pand, and fill a larger portion of the
Fig. 335.
Q
A
2
46
S
30
12
14
16
18
20
22
inversely
proportional to
the pressures;
instrument for
expanding the
air;
24
26
23
description and
130
mode of using;

558
NATURAL PHILOSOPHY.
weight of the
suspended
column of
mercury plus
elastic force of
confined air,
equal to
atmospheric
pressure;
experiments
made at Paris;
expression of
Mariotte's law.
tube, though a column of mercury will still stand at an ele-
vation above the outer level, so that the weight of this
column, with the elastic force of the inclosed air, counter-
balances the natural pressure of the atmosphere. The pres-
sure therefore which the included air sustains, is equal to
the weight of a column of mercury 30 inches high, minus
that of the column supported in the tube. Let the
space full of air above the mercury in the closed tube
be 3 inches; lift up the tube so that this space shall
be 6 inches, the mercury will be found to stand in
the tube 15 inches above that in the outer vessel. Here
the volume of the air is doubled, and the pressure
upon it is 30-15= 15 one half of 30, what it was
before. Again raise the tube till the volume of air be-
comes 9 inches long, the mercury in the tube will be
found to stand 20 inches higher than in the outer vessel;
here the volume is three times its primitive volume, and
its pressure 30-20=10= one third of 30, its original
pressure; whence the law is manifest.
By experiments made at Paris, it has been found that
this law obtains when air is condensed 27 times, and rare-
fied 112 times. Other gases obey it also, till the pressure
becomes a few atmospheres less than that at which they
assume a liquid form.
The density of the same quantity of matter is inversely
proportional to the volume it occupies. If, therefore, P
be the pressure upon a unit of surface necessary to pro-
duce a density unity, p the pressure corresponding to a
density D, then, according to this law, will,
P = PD...
(236).
This law was investigated by Boyle and Mariotte, the
former in 1660, and the latter in 1668, and is now known
as Mariotte's law.

MECHANICS OF FLUIDS.
559
XI.
LAW OF THE PRESSURE, DENSITY, AND TEM-
PERATURE.
the pressure,
density, and
§ 283.-It is a universal law, of nature that heat ex- Law connecting
pands all bodies, and is ever active in producing changes
of density. We have now to consider the law of this temperature;
change in air.
It has been ascertained, experimentally, that air, sub-
jected to any constant pressure, will expand 0.00208th of rate of the air's
its volume at 32° Fahr., for each degree of the same scale expansion;
above this temperature; so that if V, be the volume of the
air at 32°, and V its volume at any other temperature t,
then will
V₁ [1 + (- 32°) 0.00208]... (237).
V = (t
If D, be the density at 32°, under a pressure p, and
D that at the temperature t, under the same pressure,
then, because the densities are inversely as the volumes,
will
V₁: V₁ [1 + (t - 32°) 0.00208] :: D: D,;
volume for any
temperature
under a constant
pressure;
whence
D
D =
.
1 + (t - 32°). 0.00208
(238).
If p, denote the pressure necessary to restore this air to
the density D, we shall have from Mariotte's law
D₁
1+(-32°) 0.00208
:D, p Pi
density at any
temperature
under a constant
pressure;

560
NATURAL PHILOSOPHY.
pressure to
produce at a
given
temperature a
density at 320
under a given
pressure;
weight of a
column of
mercury at to;
whence
P₁ = p[1 + (t - 32°) 0.00208] . . (239).
Again, let the pressure p be produced by the weight of
a column of mercury, having a base unity, and an altitude
h,,, taken at a given latitude, say that of 45°, in order that
the force of gravity may be constant. Denoting the den-
sity of the mercury by D,,, its weight will be
Dhug';
in which g' denotes the force of gravity at the latitude
of 45°.
Substituting this for p, in Eq. (236), we have
Dhg'= PD;
pressure to
produce a unit of
density at to;
same in different
form;
density at to
under a constant
pressure;
whence
P = D₁₁ 9';
Dhg'
D
and substituting the value of D, given in Eq. (238), this
becomes
P=
Dh9' [1 + (t - 32°) 0.00208].. (240).
D
From Eq. (236), we have
80300.0 D =
and substituting the value for P above, we get
D=
p D
Dikg [1 + (t – 32°) 0,00208]

MECHANICS OF FLUIDS.
561
Denote by h the height of the column of mercury at 32°,
necessary to produce upon a unit of surface the pressure
p, then will
p = Dhg';
which, substituted for p above, gives, after striking out the
common factors,
weight of a
column of
mercury at 32°
equal to the
constant
pressure;
D=
D, h
h, [1+ (t-32°) 0.00208]*
density at to
under a constant
Now, when h,, becomes 30 inches, then will D, take the
value given in the table of § 275 opposite the name of
the gas or vapor under consideration, and we have, for
the practical application of that table,
D = 1/4 x
h
1+ (t - 32°) 0.00208
.
pressure:
density of any gas
answering to a
given
(240)'; temperature and
in which D, is the tabular specific gravity or density, h
the height of the column of mercury expressed in inches,
and D the density of the gas pressing upon the mercury.
barometric
column;
Example. What is the density of atmospheric air, when
the barometer stands at 26 inches and thermometer at example to
42°? In this case, D, will be found in the table to be
0.0013, whence
illustrate the use
of this formula;
0.0013
D=
X
26
0.0011.
30 1+ (42° 32°) 0.00208
We are now prepared to understand how the values of
D, in the table just referred to, were obtained, and of
which no explanation has, thus far, been made.
tabular specific
gravity of gases,
It will be recollected that, when referred to the same to obtain the
standard, the numbers which express the specific gravities
of bodies also express their densities, and that the specific &c.;
36

562
NATURAL PHILOSOPHY.
of any body;
gravity of a body is the ratio obtained by dividing the
specific gravity weight of the body by that of an equal volume of the
standard substance. The gases and vapors are incessantly
changing their densities, on account of the varying pres-
sures and temperatures to which they are subjected.
Tabulated densities must, therefore, correspond to a
standard of temperature and of pressure. Thirty-two
degrees Fahrenheit's scale is adopted for the former; and
density of gases, the weight of a column of mercury, at the same tempera-
ture, having an altitude equal to thirty inches, and resting
upon a base whose area is a superficial unit, is taken for
the latter.
standard
temperature and
pressure for
&c.;
By a very simple transformation of Eq. (240)', we find
tabular value for
density;
D, =
30 x [1 + (t - 32°) 0.00208]
h
x D.
To make this formula applicable to any gas, it will only
be necessary to observe h, by means of a barometer in the
atmosphere; t, by a thermometer
in contact with the gas; and to find
D, corresponding to these quan-
tities, by the following process:
Provide a glass vessel A, whose
mouth may be closed by a stop-
cock B, air-tight, and of which
the bottom terminates in a long
vessel for finding narrow tube C, closed at the end.
the weights and Let the capacity of this vessel be
carefully ascertained by filling it
with water, and pouring this water
afterward into a graduated vessel;
also let the tubular portion C be
graduated and numbered by tenths,
hundredths, &c., so that the num-
bers shall increase towards the
volumes of gases;
A
B
Fig. 336.
10,8
0,9
C
smaller end, and express that portion of the entire capacity

MECHANICS OF FLUIDS.
563
of the vessel, regarded as unity, which is comprised be-
tween its mouth B and these numbers.
This being understood, denote the weight of this ves-
el by W.; that of a volume of air, or of the gas under
consideration, equal to the contents of the vessel, and notation;
under the pressure h and temperature t, by Wa; the buoy-bas
ant effort of the atmosphere, under the same pressure and
temperature, by e; and the weight required to
poise the vessel filled with gas by W₁, then will
Spe
W₁ = W₂+ Wa
-
e.
to counter-
To Mat
stow bonialno weight of vessel
(a).
filled with air;
Connect with the air-pump, and exhaust as far as conve-
nient; close the stop-cock, disconnect and weigh again, and
denote the weight necessary to counterpoise the vessel
with its rarefied air by W2, and we shall have
W₂ = W。 + Wa, − e; (5) .på gnibbs
in which Wa, denotes the weight of the rarefied air re-
maining in the vessel.
Subtracting this from the equation above, we find ban
—
W₁ W₂ = Wa-Wai
which is obviously the weight of the extracted air.
weight of vessel
filled with
rarefied air;
weight of the
extracted air;
Now immerse the vessel in water, mouth downward,
and open the stop-cock; the liquid will enter, and taking
care to keep its level on the inside and outside the same,
the water will come to rest at or near some one of the
graduated points on the tube. The air or gas within will
then have the same elasticity as the external atmosphere,
and the reading h of the barometer becomes applicable to
the gas. This graduated point will make known the volume of the air
volume V of air or gas extracted; and, knowing its extracted under
weight, that of a volume equal to the contents of the pressure;
whole vessel, which we have denoted by Wa, may be
the barometric

564
NATURAL PHILOSOPHY.
found from the proportion
V: W₁
-
W₁ W₂ :: 1 :
1: Wa;
whence
weight of the
vessel full of
air under the or
barometric
W₁ - W₂
V
=
Wa...
pressure;
(b).
Next fill the vessel with water, and weigh again; denote
the counterpoising weight by W3, and the weight of the
contained water by Wu, and we shall have
W₂ = W₂+ Ww - e;
and subtracting Eq. (a), we find
WW₁ = Ww - Wai
adding Eq. (b), we find
weight of the
W₁ - W2
vessel full of
water;
W3 − W₁ +
1
Ww;
V
ratio of the
weights of equal
volumes of water
and gas;
density of the
air;
and dividing Eq. (b) by this one, we get
W₁- W2
(W₂ — W₁) V + W₁ — W₂
Wa
11
Ww
Multiplying both members by the tabular density d of
water corresponding to the temperature of that employed,
and dividing both numerator and denominator of the first
member by W₁- W2, we finally get
21
d
Wa
=
W3
Wi
X d.
W
V+ 1
W₁- W2
But the second member is the specific gravity or density
D of air or gas, under the pressure h and temperature t

MECHANICS OF FLUIDS.
565
finding the
density or specific
Whence, to find the value of D, we have this rule, viz.:
Weigh the vessel full of the gas under consideration;
exhaust, and weigh a second time; find, by admitting process for
water, the volume of gas exhausted by the pump; fill
with water, and weigh a third time; then divide the dif- gravity of a gas.
ference between the last and first weights by the differ-
ence between the first and second; multiply this quotient
by the volume exhausted; increase this product by unity,
and divide the tabular density of water, corresponding to
its observed temperature, by this sum. The value of D,
thus found, and the observed values of h and t, being sub-
stituted in the value for D,, this latter may be found and
tabulated.
XII.
BAROMETER.
atmosphere at
§ 284.-The atmosphere being a heavy and elastic fluid, The barometer;
is compressed by its own weight. Its density cannot be
the same throughout, but diminishes as we approach its density and
upper limits where it is least, being greatest at the surface pressure of the
of the earth. If a vessel filled with air be closed at the different places;
base of a high mountain and afterward opened on its
summit, the air will rush out; and the vessel being closed
again on the summit and opened at the base of the moun-
tain, the air will rush in.
The evaporation which takes place from large bodies
of water, the activity of vegetable and animal life, as well
as vegetable decompositions, throw considerable quantities
of aqueous vapor, carbonic acid, and other foreign ingre- foreign
dients temporarily into the permanent portions of the ingredients in the
atmosphere. These, together with its ever-varying tem- change of
perature, keep the density and elastic force of the air in a
state of almost incessant change. These changes are indi-
air, and its
density;

566
NATURAL PHILOSOPHY.
barometer;
weather-glass;
wiverg
description of
the barometer;
column of
mercury in
equilibrio with
atmospheric
pressure;
common
mountain
barometer;
cated by the Barometer, an instrument employed to measure
the intensity of atmospheric pressure, and frequently called
a weather-glass, because of certain agreements found to ex-
ist between its indications and the state of the weather.
The barometer consists of a glass tube about thirty-four
or thirty-five inches long, open at one end, partly filled
with distilled mercury, and inverted in a small cistern also
containing mercury. A scale of equal parts is cut upon a
slip of metal, and placed against the tube to measure the
height of the mercurial column, the zero being on a level
with the surface of the mercury in the cistern. The elastic
force of the air acting freely upon the mercury in the cis-
tern, its pressure is transmitted to the interior of the tube,
and sustains a column of mercury whose weight it is just
sufficient to counterbalance. If the density and conse-
quent elastic force of the air be increased,
the column of mercury will rise till it
attain a corresponding increase of weight;
if, on the contrary, the density of the air
diminish, the column will fall till its di-
minished weight is sufficient to restore
the equilibrium.
In the Common Barometer, the tube
and its cistern are partly inclosed in a
metallic case, upon which the scale is
cut, the cistern, in this case, having a
flexible bottom of leather, against which
a plate a at the end of a screw b is made
to press, in order to elevate or depress
the mercury in the cistern to the zero of
the scale.
De Luc's Siphon Barometer consists
De Luc's siphon of a glass tube bent upward so as to form
barometer;
two unequal parallel legs: the longer is
hermetically sealed, and constitutes the
Torricellian tube; the shorter is open,
and on the surface of the quicksilver
Fig. 337.
30
29
0

MECHANICS OF FLUIDS.
567
the pressure of the atmosphere is exert-
ed. The difference between the levels
in the longer and shorter legs is the
barometric height. The most conve-
nient and practicable way of measuring
this difference, is to adjust a moveable
scale between the two legs, so that its
zero may be made to coincide with the
level of the mercury in the shorter leg.
31.
Fig. 338.
30.
n
23
C
Different contrivances have been
adopted to render the minute variations
in the atmospheric pressure, and conse-
quently in the height of the barometer,
more readily perceptible by enlarging
the divisions on the scale, all of which
devices tend to hinder the exact meas-
urement of the length of the column.
Of these we may name Morland's Diagonal, and Hook's
Wheel-Barometer, but especially Huygen's Double-Barom-
eter.
The essential properties of a good barometer are: width
of tube; purity of the mercury; accurate graduation of the
scale; and a good vernier.
moveable or
sliding scale;
different devices
for appreciating
slight changes of
barometric
column;
temperature;
Heat affects the density of mercury as well as that of
all other bodies. When its temperature is increased, it
expands; when diminished it contracts. The same atmo-
spheric pressure will sustain the same weight-in other
words, the same quantity of mercury; but the same quan-
tity of mercury will occupy different volumes, according effects of
to its temperature, and the same atmospheric pressure will,
hence, sustain a longer column when the temperature is
high than it will when the temperature is low. The indi-
cations of the barometer must, therefore, be reduced to
what they would have been, if taken at a standard or fixed
temperature, without which reduction they would be
utterly worthless.
From the experiments of Dulong and Petit, it is found

568
NATURAL PHILOSOPHY.
expansion of
mercury;
barometric
column reduced
to standard
temperature;
attached
thermometer;
example for
illustration;
data;
9990
that mercury expands 9th part of its volume for each
degree of Fahrenheit's scale by which its temperature is
increased, and that it contracts according to the same law
as its temperature is diminished. If, therefore, T denote
the standard temperature, and T" the temperature of ob-
servation; b the altitude which the barometer would have
at the standard temperature, and b' the observed altitude,
then will,
b = b' [1 + 2
b=b'
+7-2
T-T
9990
=b'[1+(T-7'') 0.0001001].. (241);
when T'' becomes T, b' will be equal to b.
A thermometer is usually attached to the barometer
tube for the purpose of observing the temperature of the
mercury.
Example. Observed the barometric column to stand at
29.81 inches, while its thermometer gave a temperature of
93°. What would have been the column under the same
pressure, had the temperature of the mercury been 32°?
Here we have
in.
b' = 29.81,
T'' = 93.00,
T 32.00,
=
and
T-T=
61.00;
in.
reduced column.
b =
Barometer used
to measure the
elasticity of
confined gases,
&c.;
=
in.
29.81 [1 - 61 × 0.0001001] 29.63.
§ 285.-The barometer may be used not only to meas
ure the pressure of the external air, but also to determine
the density and elasticity of pent-up gases and vapors,
and furnishes the most direct means of ascertaining

MECHANICS OF FLUIDS.
569
2
Fig. 339.
60
its use and
application;
the degree of rarefaction in the receiver of an air-pump.
When thus employed, it is called the barometer-gauge, barometer gauge;
In every case it will only be necessary to establish a
free connection between the cistern of the barometer and
the vessel containing the fluid whose elasticity is to
be indicated; the height of the mercury in the tube,
expressed in inches, reduced to a standard temperature,
and multiplied by the known weight
of a cubic inch of mercury at that
temperature, will give the pressure in
pounds on each square inch. In the
case of the steam in the boiler of an
engine, the upper end of the tube is
sometimes left open. The cistern A is
a steam-tight vessel, partly filled with
mercury, a is a tube communicating
with the boiler, and through which the
steam flows and presses upon the mer-
cury; the barometer tube bc, open at
top, reaches nearly to the bottom of the
vessel A, having attached to it a scale
whose zero coincides with the level of
the quicksilver. On the right is marked
a scale of inches, and on the left a scale
of atmospheres.
If a very high pressure were exerted,
one of several atmospheres, for example,
an apparatus thus constructed would re-
quire a tube of great length, in which
case Mariotte's manometer is considered
preferable. The tube being filled with
air and the upper end closed, the sur-
face of the mercury in both branches
will stand at the same level as long
as no steam is admitted. The steam
being admitted through d, presses on
the surface of the mercury a and forces
74
8254
34
2 4
45
1
30
15
Fig. 340.
Atm
20/10/22 CAR
d
C
scale of inches
and another of
atmospheres;
HH
Mariotte's
manometer;

570
NATURAL PHILOSOPHY.
its mode of
action.
Levelling by
means of the
barometer;
it up the branch bc, and the scale from
b to c marks the force of compression
in atmospheres. The greater width of
tube is given at a, in order that the
level of the mercury at this point may
not be materially affected by its ascent
up the branch bc, the point a being the
zero of the scale. dio
§ 286.-Another very important use
of the barometer, is to find the differ-
ence of level between two places on the
earth's surface, as the foot and top of a
hill or mountain.
Fig. 340.
Alm
1
d
Since the altitude of the barometer depends on the
pressure of the atmosphere, and as this force depends upon
the height of the pressing column, a shorter column will
exert a less pressure than a longer one. The quicksilver
in the barometer falls when the instrument is carried from
the foot to the top of a mountain, and rises again when
restored to its first position: if taken down the shaft of a
mine, the barometric column rises to a still greater height.
effect of change At the foot of the mountain the whole column of the
of place upon the
atmosphere, from its utmost limits, presses with its entire
height of the
weight on the mercury; at the top of the mountain this
weight is diminished by that of the intervening stratum
between the two stations, and a shorter column of mercury
will be sustained by it.
barometer;
effects of
irregularity of the
earth's surface;
It is well known that the surface of the earth is not
uniform, and does not, in consequence, sustain an equal
atmospheric pressure at its different points; whence
the mean altitude of the barometric column will vary
at different places. This furnishes one of the best and
most expeditious means of getting a profile of an ex-
tended section of the earth's surface, and makes the
barometer an instrument of great value in the hands of
the traveller in search of geographical information.

MECHANICS OF FLUIDS.
571
To find the relation which subsists between the alti-
tudes of two barometric columns, and the difference of
level of the places where they exist, conceive the atmo-
sphere to be divided into an indefinite number of elementary
horizontal strata of equal thickness, and so thin that the relation between
density from the top to the bottom of the same stratum
may without error be regarded as uniform, the density difference of level
varying from one stratum to another.
Then, commencing at any
elevated position O, above
the level of the sea, denote
by p the pressure exerted
upon the unit of surface by
the whole column of atmo-
sphere above this point. The
density of the stratum of air
A, immediately below this
Fig. 341.
0
B
C
D
HAGAE
point, will be due to this pressure; call this density D,
then, from Mariotte's law, Eq. (236), will
p = PD;
Exce
the barometric
columns and
in which P is the pressure necessary to produce, on a unit
of surface, a unit of density. From this equation, we
have
of the places;
elastic pressure;
D =
D =
The weight of so much of this stratum as stands upon a
unit of surface will be
density
corresponding;
g Dh = p.
gh
P
in which h denotes the indefinitely small height common
to all the strata, and g the force of gravity.
The pressure upon the unit of surface of the second
stratum B, will be p, transmitted through the first stratum,
weight of a small
column on unit
of surface;

572
NATURAL PHILOSOPHY.
pressure on unit
of surface of
second stratum;
augmented by the weight of this stratum found above;
and, denoting this pressure by p', we shall have
=
p' p + p
gh
P
= p(1 + 2h).
P
Denoting by D' the density of the second stratum B,
we have again by Mariotte's law
30
or
p' = PD',
p'
weight of second
stratum;
same under
another form;
pressure upon
unit of surface of
third stratum;
D' = P
and for the weight of this stratum upon a unit of surface,
gh
gh D'p'. P
and substituting the value of p', found above,
gh). gh.
gh D' = p(1+gh).
The pressure upon the unit of surface of the third stratum
C, will be the pressure p', transmitted through the second
stratum, increased by the weight found above for this same
stratum; hence, denoting this pressure by p", will
p" = p(1+)+p (1+)=(1+) (+)=(1+);
and in the same way will the pressure p'"', upon the fourth
stratum, be given by the equation
same for fourth
stratum;
p""
= p
P(1 + 9h),
P
and so on to the surface of the earth: and supposing n to
denote the number of strata between these limits, then will

MECHANICS OF FLUIDS.
573
P₂ = p(1 + 2h)";
Pn
P
in which p₂ denotes the pressure at the lowest station.
Developing the second member of this equation by the
binomial formula, and dividing by p, we have
ghn(n-1) g² h² n(n-1) (n-2) g³ h³
Pn1+n
P₂ = 1 + n %
p
p
1.2
P2
+
1.2.3
pressure upon
unit of surface
of nth stratum;
ratio of the upper
p3+ &c. and lower
pressure;
The strata being indefinitely thin, the number in any
definite altitude will be indefinitely great, and this being
the case in the above series, it is obvious that the numbers
1, 2, 3, &c., connected with n by the minus sign, may be
disregarded without sensibly impairing the result, which
will give
Pn
ngh
=1+
n² g² ha
+
n³ g3 h3
+
+ &c.
p
P
1.2 P2
1.2.3 P3
same reduced;
But the second member is equal to
ngh
e P
in which e= 2.7182818, the base of the Naperian system
of logarithms. Whence,
Pn
p
ngh
eP
But n being the number of strata, and h the common
height of each, nh will be equal to the difference of level
between the first and last points. Calling this 2, and
taking the Naperian logarithm of both members, we find,
after substituting 2,
log Pr
Pn
p
=
92
P
same under
different form;
Naperian
logarithm;

574
NATURAL PHILOSOPHY.
common
logarithm;
difference of
level;
and passing to the common logarithms
M. Log Pr
=
p
gz
P
doldw ni
in which M denotes the reciprocal of the modulus of the
common system; whence we have
MP
2
Log
Pn
g
p
ratio of pressures
in terms of
barometric
columns:
same reduced to
a standard
temperature T;
value of force of
gravity;
Denote by b, the height of the barometric column at
the lower station, where the pressure is pn, and by b that
at the upper station where the pressure
is P then will
Pn
p
=
bn
Ъ
and reducing the barometric column b to the temperature
of ba taken as the standard, we have, Eq. (241),
on as the standard, we
Pn
p
=
bn
b [1+ (TT) 0.0001001]'
in which T becomes the temperature of the mercury at
the lower, and T" that at the upper station. Moreover,
we have, Eq. (81)',
or
ft.
g= 32.1808 [1- 0.002551 cos 2 +1,
g=g' (10.002551 cos 2 +);
ft.
in which g'= 32.1808, the force of gravity in the latitude
of 45°.
Substituting these values of Pn,
g, and the value of P
p
given by Eq. (240), in the value for z above, and we find
value for
difference of
level;
2=
MD,,h,, 1+(-32) 0.00208
Y
X
1
D, 1- 0.002551 cos 2 × Log [× 1+ (77) 0.0001001].

MECHANICS OF FLUIDS.
575
In this it will be remembered that t denotes the tem-
In this it w
perature of the air; but this may not be, indeed scarcely
from difference of
ever is, the same at both stations, and thence arises a dif- difficulty arising
ficulty in applying the formula. But if we represent, for temperature of
a moment, the entire factor of the second member, into air at the two
which the factor involving t is multiplied, by X, then we
may write
≈ = [1 + (t − 32°) 0.00208] X.
stations;
difference of level
for constant
no temperature;
If the temperature of the lower station be denoted by t,
and this temperature be the same throughout to the upper
station, then will
-
2, = —
= [1 + (t, 32°) 0.00208] X.
And if the actual temperature of the upper station be
denoted by t', and this be supposed to extend to the lower
station, then would
elgand
≈' = [1+(32°) 0.00208]. X.
at
temperature
throughout the
same as at lower
station;
temperature
same as upper
station;
豆漿
​difference of
level, the true
Now if t, be greater than t', which is usually the case,
then will the barometric column, or b, at the upper station.
be greater than would result from the temperature t', since
the air being more expanded, a portion which is actually
below would pass above the upper station and press upon
the mercury in the cistern; and because b enters the de- mean value of
nominator of the value X, 2, would be too small. Again,
by supposing the temperature the same as that at the
upper station throughout, then would the air be more
condensed at the lower station, a portion of the air would
sink below the upper station that before was above it, and
would cease to act upon the mercurial column b, which
would, in consequence, become too small; and this would
make too great. Taking a mean between z, and z' as
the true value, we find
one;

576
NATURAL PHILOSOPHY.
true value for
difference of
level;
2=
2, + 2'
2
=
= (t t'
-
[1 + (+ 64°) 0.00208] X.
Replacing X by its value,
value for
difference of
level;
MD,,h, 1+(t,+t64°) 0.00208
z=
D
.
1-0.002551 cos 2 ¥
The factor
MD,,h
DO
guirlage
ba
1
0.0001001]-
× Log. [1+(-7) 0.0001001-
(T
we have seen,
is constant, and it
For this purpose,
only remains to determine its value.
measure with accuracy the difference of level between
two stations, one at the base and the other on the summit
of some lofty mountain, by means of a Theodolite, or
levelling instrument-this will give the value of 2; observe
the barometric column at both stations-this will give b
and b; take also the temperature of the mercury at the
two stations-this will give T and T''; and by a detached
find the value of thermometer in the shade, at both stations, find the values
the coefficient; of t, and t. These, and the latitude of the place, being
substituted in the formula, every thing is known except
the coefficient in question, which may, therefore, be found
by the solution of a simple equation. In this way, it is
found that
its value;
MD,,h
D
1
60345.51 English feet;
final value for
difference of
level;
which will finally give for z,
ft.
1+,+t64°) 0.00208
1
z=60345.51.
1-0.002551 cos 2 ¥
X × Log. [x
1+(T-T") 0.0001001-
0.0001001].
To find the difference of level between any two
stations, the latitude of the locality must be known; it
will then only be necessary to note the barometric
data for its use; columns, the temperature of the mercury, and that of
the air at the two stations, and to substitute these observed
elements in this formula.

MECHANICS OF FLUIDS.
577
Much labor is, however, saved by the use of a table labor saved by a
for the computation of these results, and we now proceed table;
to explain how it may be formed and used.
Make
60345.51 [1 + (t,+t' - 64) 0.00104] = A,
1
1 0.002551 cos 2+
-
1
1 + (T' — T') 0.0001
=
= B,
=
C.
mode of
computing one;

Then will
2 = AB Log.
.
Cbn
b
2 = AB. [Log. C+ Log. bn
-
and taking the logarithms of both members,
Log. b];
abbreviated
formula;
Log. 2 Log. 4+ Log. B+ Log. [Log. C+ Log. bn-Log. 6].. (242). its logarithm;
Making t,+t to vary from 40° to 162°, which will be variations of the
sufficient for all practical purposes, the logarithms of the temperature of
corresponding values of A, are entered in a column, under
air;
the head A, opposite the values t, + t', as an argument.
Causing the latitude to vary from 0° to 90°, the variations in
logarithms of the corresponding values of B are entered
in a column headed B, opposite the values of 4.
The value of T-T' being made, in like manner, to
vary from
30° to +30°, the logarithms of the cor-
responding values of C are entered under the head of C,
and opposite the values of T-T". In this way a table
latitude:
mercury;
is easily constructed. That subjoined, was computed by variations in
Samuel Howlet, Esq., from the formula of Mr. Francis temperature of
Baily, which is very nearly the same as that just described,
there being but a trifling difference in the coefficients.
37

578
NATURAL PHILOSOPHY.
TABLE FOR FINDING ALTITUDES
Detached Thermometer.
t+t'
A
t+t'
A
t+t
A
t+t
A
40 4.7689067 75 4.7859208 110
4.8022936 145 4.8180714
41
.7694021 76
.7863973 III
.8027525 146
.8185140
42
.7698971 77
.7868733 112
.8032109 147
.8189559
43 .7703911 78
.7873487 113
.8036687
148
.8193975
44
.7708851 79
.7878236 114
.8041261 149
.8198387
45
-7713785 80
.7882979 115.
.8045830 150
.8202794
46
7718711 81
.7887719 116
.8050395 151
.8207196
47
.7723633 82
.7892451 117
.8054953 152
.8211594
48
.7728548 83
.7897180 118
8059509 153
.8215988
49
.7733457 84
.7901903 119
.8064058 154
.8220377
50 .7738363 85
.7906621 120
.8068604 155
.8224761
51
-7743261 86
52 .7748153 87
53 .7753042 88
.7911335 121
.8073144 156
.8229141
.7916042 122
.8077680 157
.8233517
.7920745 123
.8082211 158
.8237888
54 7757925 89
.7925441 124
.8086737 159
.8242256
55
.7762802 90
.7930135 125
.8091258 160
.8246618
56
.7767674 91
.7934822
126
.8095776 161
.8250976
57
.7772540 92.
58 .7777400 93
.7939504 127
.8100287 162
.8255331
-7944182 128
.8104795 163
8259680
59
.7782256 94
.7948854 129
.8109298
164
.8264024
60
.7787105 95
.7953521 130
.8113796
165
.8268365
61
-7791949 96
7958184 131
.8118290 166
.8272701
62
.7796788 97
-7962841 132
.8122778 167
.8277034
63
.7801622 98 7967493 133
.8127263 168
.8281362
64
.7806450 99 -7972141 134
.8131742 169
.8285685
65
66
67
71
.7811272 100 7976784 135
.7816090 101 .7981421 136
.7820902 102 .7986054 137
68 .7825709 103
.7990681 138
69
.7830511 104 .7995303 139
70 .7835306 105
.7999921 140
.7840098 106. .8004533 141
72 .7844883 107.8009142 142
73
.7849664 108 .8013744 143
74 4.7854438 109 4.8018343 144
.8136216 170
.8290005
.8140688 171 .8294319
.8145153 172 .8298629
.8149614 173 .8302937
.8154070 174 .8307238
.8311536
.8158523 175
.8162970 176 .8315830
.8167413 177 .8320119
.8171852 178 .8324404
4.8176285
179
4.8328686

MECHANICS OF FLUIDS.
579
WITH THE BAROMETER.
Attached Thermometer.
Latitude.
F
B
T-T
C
C
00
0.0011689
+
3
.0011624
00
0.0000000
0.0000000
6
.0011433
.0000434
9.9999566
6
.0011117
.0000869
.9999131
12
0.0010679
3
.0001303
.9998697
15
.0010124
4
.0001737
.9998262
18
.0009459
5
A.0002171
8.9997828
21
.0008689
6
.0002605
-9997393
24
.0007825
27
.0006874
8
78
7
.0003039
.9996959
.0003473
.9996524
30
.0005848
9
.0003907
.9996090
33
.0004758
ΙΟ
.0004341
.9995655
36
.0003615
II
.0004775
.9995220
39
.0002433
12
.0005208
.9994785
42
.0001223
13
.0005642
.9994350
45
.0000000
14
.0006076
.9993916
48
9.9998775
15
.0006510
.9993481
49
.9998372
16
.0006943
.9993046
50
.9997967
17
.0007377
.9992611
51
.9997566
18
.0007810
9992176
52
.9997167
19
.0008244
.9991741
53
.9996772
20
.0008677
-9991305
54
.9996381
21
.0009111
9990870
55
.9995995
22
.0009544
.9990435
56
.9995613
23
.0009977
.9990000
57
.9995237
24
.0010411
.9989564
58
-9994866
25
.0010844
.9989129
59
.9994502
26
.0011277
.9988694
60
9994144
27
13.0011710
.9988258
63
.9993115
28
.0012143
9987823
66
200-9992161
29
.0012576
.9987387
69
.9991293
30
.0013009
.9986952
75
.9989852
31
0.0013442
9.9986516
388
81
.9988854
90
9.9988300
05580
NATURAL PHILOSOPHY.
rule for
computing
difference of level
with a
barometer;
Taking Eq. (242) in connection with this table, we have
this rule for finding the altitude of one station above
another, viz. :--
Take the logarithm of the barometric reading at the lower
station, to which add the number in the column headed Coppo-
site the observed value of T- T', and subtract from this sum
the logarithm of the barometric reading at the upper station;
take the logarithm of this difference, to which add the numbers
in the columns headed A and B, corresponding to the observed
values of t,+t' and ; the sum will be the logarithm of the
height in English feet.
Example. At the mountain of Guanaxuato, in Mexico,
M. Humboldt observed at the

Upper Station.
Lower Station.
6500
Detached thermometer, t'
=
70.4; t = 77.6.
example first;
Attached
T'= 70.4; T 77.6.
Barometric column,
b
=
23.66; b= 30.05.
What was the difference of level?
Here
0000000
observed data;
t, + t'
=
148;
T- T' = 7.2;
Latitude 21.
in.
To Log. 30.05
1.4778445
Add C for 7.2
=
0.0003165
1.4781610
in.
Sub. Log. 23.66
1.3740147
Log. of-
0.1041463 =
-
1.0176439
Add A for 148°06.
=
4.8193975
Add B for 21°
0.0008689
ft.
height of
Guanaxuato;
6885.1
3.8379103;
whence the mountain is 6885.1 feet high.
MECHANICS OF FLUIDS.
581
formula true only
podala srowb!
It will be remembered that the final Eq. (242) was de- barometric
duced on the supposition that each stratum of air pressed when there is no
that ea
with its entire weight on that below it, a condition which wind;
can only be fulfilled when the air is in equilibrio-that is to
say, when there is no wind. The barometer can, therefore,
only be used for levelling purposes in calm weather. More-
over, to insure accuracy, the observations at the two stations
whose difference of level is to be found, should be made
simultaneously, else the temperature of the air may change observations at
during the interval between them; but with a single in- both stations
strument this is impracticable, and we proceed thus, viz.: simultaneously
Take the barometric column, the reading of the attached
and detached thermometers, and time of day at one of the
stations, say the lower; then proceed to the upper station,
and take the same elements there; and at an equal interval or at equal
of time afterward, observe these elements at the lower intervals apart;
station again; reduce the mercurial columns at the lower
uns at the l
station to the same temperature by Eq. (241), take a mean
of these columns, and a mean of the temperatures of the
air at this station, and use these means as a single set of
observations made simultaneously with those at the higher
station.
should be made
Example. The following observations were made to de- example second;
termine the height of a hill near West Point, N. Y.
Upper Station.
Lower Station.
(1)
(2)
Detached thermometer, t' = 57;
t₁ = 56°
and 61.
แ
T"= 57.5; T = 56.5 and 63.
in.
Attached
Barometric column, b = 28.94; b₂ = 29.62 and 29.63.
in.
in.
First, to reduce 29.63 inches at 63°, to what it would
have been at 56.5. For this purpose, Eq. (241) gives
in.
observed data;

b(1+7-7" x 0.0001)=29.63 (16.5 x 0.0001)=29.611. reduction
582
NATURAL PHILOSOPHY.
Pad Then
Thenew (2)
sory bowong is to
Deal wood
reduced column; did not b
of si indt
Sant
29.6229.611
in.
bn
=
=
a oft no boob
29.6105, eli dile

2
edaned
temperature at
80M
lower station;
56+61
=
2
anallats ovet od
shams of bloede bandt
t, + t' = 58.5 + 57...
T-T" = 56.5 - 57.5
in.
ballilut ad loo man
ood now you
58.5, boa od vho
1=
115.5, Jaconetivosiz
=
-
1.
To Log. 29.6105
=
1.4714458ck ed
Add C for -1°
H
9.9999566
1.4714024
in.
computation;
Sub. Log. of 28.94
1.4614985

Log. of
0.0099039
1
-
Add A for 115.5 -
Add B for 41.4
height of the hill.
ft.
632.07
3.9958062
4.8048112
0.0001465
2.8007639;
whence the height of the hill is 632.07 English feet.

XIII.
Pumps:
Slow
PUMPS.
§ 287.-Any machine employed for raising water from
one level to a higher, in which the agency of atmospheric
pressure is employed, is called a Pump. There are various
MECHANICS OF FLUIDS.
583
kinds of pumps; the more common are the sucking, forcing, different kinds.
and lifting pumps.
Fig. 342.
§ 288.-The Sucking-Pump consists of a cylindrical Sucking-pump;
body or barrel B, from the lower end of which a tube D,
called the sucking-pipe, descends into the water contained
in a reservoir or well. In the interior of the barrel is a
moveable piston C, surrounded with leather to make it piston;
water-tight, yet capable of moving up and down freely.
The piston is perfora-
ted in the direction
of the bore of the
barrel, and the orifice
is covered by a valve
F called the piston-
valve, which opens
upward; a similar
valve E, called the
sleeping-valve, at the
bottom of the barrel,
covers the upper end
of the sucking-pipe.
Above the highest
point ever occupied
by the piston, a dis-
charge pipe P is in-
serted into the barrel;


P
piston-valve;
H
G
B
C
E
-A
D
I
L
sleeping-valve;
discharge-pipe;
the piston is worked
by means of a lever H, or other contrivance, attached to
the piston-rod G. The distance A A', between the highest
and lowest points of the piston, is called the play. To ex- play;
plain the action of this pump, let the piston be at its low-
est point A, the valves E and F closed by their own
weight, and the air within the pump of the same density
or elastic force as that on the exterior. The water of the
reservoir will stand at the same level LL both within and operation of the
without the sucking-pipe. Now suppose the piston raised pump;
584
NATURAL PHILOSOPHY.
to its highest point A', the air contained in the barrel and
sucking-pipe will tend by its elastic force to occupy the
action during the space which the piston leaves void, the valve E will, there-
ascent of the
piston;
of the fore, be forced open, and air will pass from the pipe to the
equilibrium;
barrel, its elasticity diminishing in proportion as it fills a
larger space. It will, therefore, exert a less pressure on the
water below it in the sucking-pipe than the exterior air does
on that in the reservoir, and the excess of pressure on the
part of the exterior air, will force the water up the pipe till
the weight of the suspended column, increased by the elastic
force of the internal air, becomes equal to the pressure of
the exterior air. When this takes place, the valve E will
close of its own weight; and if the piston be depressed,
the air contained between it and this valve, having its
density augmented as the piston is lowered, will at length
have its elasticity greater than that of the exterior air;
action during the this excess of elasticity will force open the valve F, and


descent of the
piston;
air enough will escape to reduce what is left to the same
density as that of the exterior air. The valve F will then
fall of its own weight; and if the piston be again elevated,
the water will rise still higher, for the same reason as
before. This operation of raising and depressing the
piston being repeated a few times, the water will at length
enter the barrel, through the valve F, and be delivered
from the discharge-pipe P. The valves E and F closing
after the water has passed them, the latter is prevented from
returning, and a cylinder of water equal to that through
few strokes of the which the piston is raised, will, at each upward motion, be
forced out, provided the discharge-pipe is large enough.
As the ascent of the water to the piston is produced by
the difference of pressure of the internal and external air,
it is plain that the lowest point to which the piston may
reach, should never have a greater altitude above the
greatest altitude water in the reservoir than that of the column of this
of lower limit of fluid which the atmospheric pressure may support, in
vacuo, at the place. We
the result of a

piston;
the play;
From a little reflection upon what has been said of the
MECHANICS OF FLUIDS.
585
Fig. 343.
94
A
depends the rise
of the water;
to find the
relation of the
play to the other
operations of this pump, it will appear that the rise of fact upon which
water, during each ascent of the piston after the first,
depends upon the expulsion of air through the piston-
valve during its previous descent. But air can only issue
through this valve when the air below it has a greater
density, and, therefore, greater elasticity, than the external
air; and if the piston may not descend low enough, for
want of sufficient play, to produce this degree of compres-
sion, the water must cease to rise, and the working of the
piston can have no other effect than alternately to com-
press and dilate the same air between it and the surface of
the water. To ascertain, therefore,
the relation which the play of the
piston should bear to the other
dimensions, in order to make the
pump effective, suppose the water
to have reached a stationary level
X, at some one ascent of the piston
to its highest point A', and that, in
its subsequent descent, the piston-
valve will not open, but the air
below it will be compressed only to
the same density with the external
air, when the piston reaches its
lowest point A. The piston may be
worked up and down indefinitely,
within these limits for the play, with-

A
dimensions;
hypothesis in
respect to rise of
water;
out moving the water. Denote the play of the piston by a;
the greatest height to which the piston may be raised above
the level of the water in the reservoir, by b, which may
also be regarded as the altitude of the discharge-pipe; the notation;
elevation of the point X, at which the water stops, above
the water in the reservoir, by x; the cross-section of the
interior of the barrel by B. The volume of the air volume of the
between the level X and A will be
confined air,
when the piston
is at its lowest
point;
Bx (b-x-a);
586
NATURAL PHILOSOPHY.

volume of same
air expanded
when piston is at
highest point;
weight of the
column of water
which the first
will support;
weight supported
by the second;
the volume of this same air, when the piston is raised to
A', provided the water does not move, will be
B(b-x).
Represent by h the greatest height to which water may be
supported in vacuo at that place. The weight of the
column of water which the elastic force of the air, when
occupying the space between the limits X and A, will
support in a tube, with a bore equal to that of the barrel,
is measured by
Bh.g.D;
in which D is the density of the water, and g the force of
gravity. The weight of the column which the elastic
force of this same air will support, when expanded between
the limits X and A', will be
Bh'.g.D;
in which h' denotes the height of this new column. But
from Mariotte's law we have
B(b-x-a): B(b-x) :: Bh'g D: Bhg D;
whence
ratio of the
heights;
condition of
equilibrium;
altitude of point
of stopping;
b-x
-
a
h'=h
.
b Xx
But there is an equilibrium between the pressure of the
external air and that of the rarefied air between the limits
X and A', when the latter is increased by the weight of
the column of water whose altitude is x. Whence, omit-
ting the common factors, B, D, and 91
x+h'=x+h.
b-x-a
b x
=h;
or, clearing the fraction and solving the equation in refer-
ence to x, we find
x = b b ± 1 √ b2-4ah
.. (243).

MECHANICS OF FLUIDS.
587
stoppage;
When x has a real value, the water will cease to rise, condition of
but x will be real as long as b² is greater than 4 ah. If, on
the contrary, 4 ah is greater than b2, the value of x will
be imaginary, and the water cannot cease to rise, and the
pump will always be effective when its dimensions satisfy
this condition, viz.:-
or
4ah b²,
a4h
condition of
incessant flow;
the piston;
that is to say, the play of the piston must be greater than the rule for play of
square of the altitude of the upper limit of the play of the piston
above the surface of the water in the reservoir, divided by four
times the height to which the atmospheric pressure at the place,
where the pump is used, will support water in vacuo. This
last height is easily found by means of the barometer.
We have but to notice the altitude of the barometer at the value of h
the place, and multiply its column, reduced to feet, by
131, this being the specific gravity of mercury referred to
water as a standard, and the product will give the value
of h in feet.

found by the
barometer;
Example. Required the least play of the piston in a
sucking-pump intended to raise water through a height
of 13 feet, at a place where the barometer stands at 28 example;
inches.
Here
b = 13, and b² = 169.
Barometer,
in.
28
2.333 feet.
12
ft.
h = 2.333 x 13.5
= 31.5 feet.
Play
b2
=a> =
4 h
169
ft.
1
4 x 31.5
1.341+;
that is, the play of the piston must be greater than one
and one third of a foot.
data;

resulting limit for
play;
588
NATURAL PHILOSOPHY.

of the moter in
the sucking-
pump;
The quantity of work performed by
the motor during the delivery of water
quantity of work through the discharge-pipe P, is easily
computed. Suppose the piston to have
any position, as M, and to be moving
upward, the water being at the level
LL in the reservoir, and at P in the
pump. The pressure upon the up-
per surface of the piston will be equal
to the entire atmospheric pressure de-
noted by A, increased by the weight of
the column of water MP', whose height
is c', and whose base is the area B of
the piston; that is, the pressure upon
the top of the piston will be
pressure on top
of piston;
pressure on the
under surface of
piston;
pressure to be
overcome by the
power;
A+ Be' g D,
Fig. 344.

P
IN
P
M
I
gied

in which g and D are the force of gravity and density of
the water, respectively. Again, the pressure upon the un-
der surface of the piston is equal to the atmospheric pres-
sure A, transmitted through the water in the reservoir and
up the suspended column, diminished by the weight of the
column of water NM below the piston, and whose base is
B and altitude c; that is, the pressure from below will be
A- Beg D,
and the difference of these pressures will be

A+ Be'g D - (A - Beg D) = Bg D (c+c');
but, employing the notation of the sucking-pump just
described,
c+c= b;
whence the foregoing expression becomes
weight to be
overcome;
Bb.g.D;
MECHANICS OF FLUIDS.
589

which is obviously the weight of a column of the fluid
whose base is the area of the piston and altitude the
height of the discharge-pipe above the level of the water
in the reservoir. And adding to this the effort necessary to which friction
to overcome the friction of the parts of the pump when in must be added;
motion, denoted by o, we shall have the resistance which
the force F, applied to the piston-rod, must overcome to
produce any useful effect; that is,

F:
F= BbgD+ 9.
value of the
motive force;
Denote the play of the piston by p, and the number of its
double strokes, from the beginning of the flow through
the discharge-pipe till any quantity Q is delivered, by n;pantom
the quantity of work will, by omitting the effort necessary
to depress the piston, be
Fnpnp [Bb.gD+q];
or estimating the volume in cubic feet, in which case p
and b must be expressed in linear feet and B in square
feet, and substituting for g D its value 62.5 pounds, we
finally have for the quantity of work necessary to deliver
a number of cubic feet of water Q=Bnp,
Fnpnp [62.5. Bb + ].. (244);
in which must be expressed in pounds, and may be
determined either by experiment in each particular pump,
or computed by the rules already given.
quantity of work;

quantity requisite
to deliver a given
number of cubic
feet;
It is apparent that the action of the sucking-pump
must be very irregular, and that it is only during the
ascent of the piston that it produces any useful effect; sucking-pump
during the descent of the piston, the force is scarcely irregular in its
exerted at all, not more than is necessary to overcome
the friction.
action.
§ 289.-What is usually called the lifting-pump, does Lifting-pump;
not differ much from the sucking-pump just described,
590
NATURAL PHILOSOPHY.
barrel and pipe
reversed in this
pump;
except that the barrel and
positions of the sleeping-valve E are placed
at the bottom of the pipe, and
some distance below the sur-
face of the water L L in the
reservoir; the piston may or
may not be below this same
surface when at the lowest
point of its play. The pis-
ton and sleeping valves open
upward. Supposing the pis-
ton at its lowest point, it
mode of action; will, when raised, lift the
column of water above it,
the result of
several strokes
of the piston;
and the pressure of the ex-
ternal air, together with the
H
L
Fig. 345.

R
CARI

head of fluid in the reservoir above the level of the sleep-
ing-valve, will force the latter open, the water will flow
into the barrel and follow the piston. When the piston
reaches the upper limit of its play, the sleeping-valve will
close and prevent the return of the water above it. The
piston being depressed, its valves F will open and the
water will flow through them till the piston reaches its
lowest point. The same operation being repeated a few
times, a column of water will be lifted to the mouth of the
discharge-pipe P, after which every elevation of the piston
will deliver a volume of the fluid equal to that of a cyl-
inder whose base is the area of the piston and whose
altitude is equal to its play.
As the water on the same level within and without the
pump will be in equilibrio, it is plain that the resistance
to be overcome by the power, will be the friction of the
rubbing surfaces of the pump, augmented by the weight of
a column of fluid whose base is the area of the piston, and
work estimated altitude, the difference of level between the surface of the
water in the reservoir and the discharge-pipe. Hence the
quantity of work is estimated by the same rule, Eq. (244).
by the same rule
as for sucking-
pump;

MECHANICS OF FLUIDS.
591
If we omit for a moment the consideration of friction, and
take but a single elevation of the piston after the water
has reached the discharge-pipe, n will equal one, o will be
zero, and that equation reduces to
Fp=62.5 Bp x b;
but 62.5 x Bp is the quantity of fluid discharged at each
double stroke of the piston, and b being the elevation
of the discharge-pipe above the water in the reservoir,
we see that, the work will be the same as though that
amount of fluid had actually been lifted through this
vertical height, which, indeed, is the useful effect of the
pump for every double stroke.
$290. The for-
cing-pump is a fur-
ther modification of
the simple sucking-
pump. The barrel
B and sleeping-valve
E are placed upon
the top of the suck-
ing-pipe M.
The
piston F is without
perforation and valve,
and the water, after
being forced into the
barrel by the atmo-
spheric pressure with-
out, as in the sucking-
pump, is driven by
the depression of the
piston through a lat-
eral pipe H into an
air-vessel N, at the
bottom of which is
B
2
Fig. 346.
E
H
M
I
P
K
work for one
elevation of
piston;
measure of the
useful effect.
Forcing-pump;
description;
N
action of the
piston and
sleeping-valve;
air-vessel;
592
NATURAL PHILOSOPHY.
second
a second sleeping-
sleeping valve; valve E', opening,
like the first, upward.
Fig. 346.

P
Through the top of
the air-vessel a dis-
discharge-pipe; charge-pipe K passes,
Coup air-tight, nearly to the
action of the
air-vessel, second
valve;
in forcing-pump;
bottom. The water
when forced into the
air-vessel by the de-
scent of the piston,
rises above the lower
end of this pipe, con-
fines and compresses
the air, and this, re-
acting by its elastici-
ty, forces the water
up the pipe, while the
valve E' is closed
by its own weight
and the pressure from
above, as soon as the
B
H
E
M
I
I
K
N
piston reaches the lower limit of its play. A few
strokes of the piston will, in general, be sufficient to raise
water in the pipe K to any desired height, the only limit
being that determined by the power at command and the
strength of the pump.
During the ascent of the piston, the valve E' is closed
and E is open; the pressure upon the upper surface of
quantity of work the piston is that exerted by the entire atmosphere; the
pressure upon the lower surface is that of the entire
atmosphere transmitted from the surface of the reservoir
through the fluid up the pump, diminished by the weight
of the column of water whose base is the area of the
piston and altitude the height of the piston above the
surface of the water in the reservoir; hence the resistance
to be overcome by the power will be the difference of


MECHANICS OF FLUIDS.
593
overcome by the
power;
these pressures, which is obviously the weight of this resistance to be
column of water. Denote the area of the piston by B, its
height above the water of the reservoir at one instant by
y, and the weight of a unit of volume of the fluid by w,
then will the resistance to be overcome at this point of the
ascent be
w.B.y;
and denoting the indefinitely small space described by the
piston from this position by s, the elementary quantity of
work will be
its measure;
w By.s.
In like manner, denoting by y', y', y", &c., the different
heights of the piston, and by s', s", s'", &c., the correspond-
ing elementary spaces described by it, the elementary
quantities of work of the power will be
w By's', w By"s", w By"" s'", &c.;
and the whole quantity of work during the entire ascent,
will be
w [Bys + By's' + By"s" + By""s"" + &c.];
but Bs is the volume of a horizontal stratum of the fluid
in the barrel, and Bs Xy is the product of this volume
into the distance of its centre of gravity from the surface
of the fluid in the reservoir; and the same of the others.
Hence, if y, denote the height of the centre of gravity of
the play p of the piston, in other words, of its middle
point, then will
Bpy, Bys + By's' + By"s" + &c.;
=
=
elementary
quantity of work;
same for different
positions of
piston;
work during one
entire ascent;
equivalent
expression for
the same;
and
w. Bp.y,
38

594
NATURAL PHILOSOPHY.
will measure the quantity of work of the motor during one
ascent of the piston. During the descent of the piston,
the valve E is closed and E' open, and as the columns
of the fluid in the barrel and discharge-pipe, below the
horizontal plane of the lower surface of the piston, will
maintain each other in equilibrio, the resistance to be
work during one overcome by the power will, obviously, be the weight of a
column of fluid whose base is the area of the piston, and
altitude, the difference of level between the piston and
point of delivery P; and denoting by z, the distance of
the central point of the play below the point P, we shall
find, by exactly the same process,
descent; a
its measure;
work during one
double stroke;
w Bp:
for the quantity of work of the motor during the de-
scent of the piston; and hence the quantity of work
during an entire double stroke will be the sum of these,
or
+
w Bp (y + z).
But y, z, is the height of the point of delivery P above
the surface of the water in the reservoir, and denoting this,
as before, by b, we have
same;
w Bpb;
work for any
number of double
strokes:
motion made
regular in
forcing-pump;
and calling the number of double strokes n, and the whole
quantity of work Q, we finally have
Q = nw. Bpb
(245).
b
If we make 2,=y,, or b=2y,, which will give y
=
2'
the quantity of work during the ascent will be equal to
that during the descent, and thus, in the forcing-pump, the
work may be equalized and the motion made in some

MECHANICS OF FLUIDS.
595
d) to astoillboes
lifting and
degree regular. In the lifting and sucking pumps, the
motor has, during the ascent of the piston, to overcome
the weight of the entire column whose base is equal to the it is very
area of the piston, and altitude the difference of level be- irregular in the
tween the water in the reservoir and point of delivery, and sucking-pumps
being wholly relieved from this load during the descent,
when the load is thrown upon the sleeping-valve and its
box, the work becomes exceedingly variable, and the
motion irregular.
nottaneique
XIV.
THE SIPHON.
§ 291.-The siphon is a bent tube of unequal branches, siphon;
open at both ends, and is used to convey a liquid from a
higher to a lower level, over
an intermediate point higher
than either; and although
its discussion more naturally
appertains to the motion of
fluids, its analogy with the
pumps, renders a descrip-
tion of it here proper. The
siphon having its parallel
branches vertical and plun-
ged into two liquids whose
upper surfaces are at LM
and L'M', the fluid will stand
L
Fig. 347.
0
M
sts audiom
description; de
L M
mode of using;
at the same level both within and without each branch
of the tube when a vent or small opening is made at 0.
If the air be withdrawn from the siphon through this
vent, the water will rise in the branches by the atmo-
spheric pressure without, and when the two columns
oda

19596
NATURAL PHILOSOPHY.
flow;
explanation;
conditions of the unite and the vent is stopped, the liquid will flow from
the reservoir A to A', as long as the level L'M' is below
LM, and the end of the shorter branch of the siphon is
below the surface of the liquid in the reservoir A.
b. The cause of this apparent paradox will be manifest
from the following consideration, viz.: The atmospheric
pressures upon the surfaces L M and L'M', tend to force
the liquid up the two branches of the tube. When the
siphon is filled with the liquid, each of these pressures is
counteracted in part by the weight of the fluid column in
the branch of the siphon that dips into the fluid upon
which the pressure is exert-
ed. The atmospheric pres-
sures are very nearly the
same for a difference of level
of several feet, by reason of
the slight density of air. The
weights of the suspended
columns of water will, for
the same difference of level,
differ considerably, in conse-
quence of the greater density
of the liquid. The atmo-
motion due to
the excess of
pressure up the
Fig. 347.
0
L
M
Sha
spheric pressure opposed to the longer column will there-
fore be more diminished than that opposed to the shorter,
shorter branch; thus leaving an excess of pressure at the end of the shorter
branch, which will produce the motion. Thus, denote by
A the intensity of the atmospheric pressure upon a surface
a equal to that of a cross-section of the bore of the siphon;
by h the difference of level between the surface L M and
the bend O of the siphon; by h' the difference of level be-
tween the same point 0 and the level L' M'; by D the
density of the liquid; and by g the force of gravity: then
will the pressure, which tends to force the fluid up the
branch which dips below LM, be
pressure up the
shorter branch;
A-ah Dg;

MECHANICS OF FLUIDS.
597
and that which tends to force the fluid up the branch
immersed in the other reservoir, be
A-ah' Dg;
and subtracting the second from the first, we find
aDg (h'- h),
for the actual intensity of the force which urges the fluid
within the siphon, in a direction from the upper to the
lower reservoir.
Denote by the entire length of the siphon. It is
obvious that this will be the distance over which any one
stratum will move, while subjected to the action of the
above force, and that the quantity of action will be meas-
ured by
a Dg (h' - h) l.
The mass moved will be all the fluid in the siphon which
is measured by a lD; and if we denote the velocity by V,
we shall have, for the living force of the moving mass,
pressure up the
longer branch;
pressure which
determines the
flow;
quantity of action
in passing a
siphon full from
the upper to
lower reservoir;
al D.V²;
કરવી.
and because the quantity of action is equal to half the
living force, we find
a Dg (h' - h) 1 = a DIV
l=
2
whence
V = √2g (h' - h);
from which it appears, that the velocity with which the liquid
will flow through the siphon, is equal to the square root of
twice the force of gravity, into the difference of level of the fluid
living force;
velocity of the
flow;

598
NATURAL PHILOSOPHY.
flow will cease
when the water
in the two reservoirs. When the fluid in the reservoirs
comes to the same level, the flow will cease, since, in that
in the reservoirs case, h' - h = 0.
comes to same
level;
practical
application of
the siphon;
The siphon may be employed to great advantage to
drain canals, ponds, marshes, and the like. For this pur-
pose, it may be made flexible by constructing it of leather,
well saturated with grease, like the common hose, and fur-
nished with internal hoops to prevent its collapsing by the
pressure of the external air.
It is thrown into the water
to be drained, and filled;
when, the ends being plug-
ged up, it is placed across
the ridge or bank over which
mode of using it the water is to be conveyed;
for draining
purposes;
the plugs are then removed,
the flow will take place, and
thus the atmosphere will be
Fig. 348.
0
made literally to press the water from one basin to another,
over an intermediate ridge.
It is obvious that the difference of level between the
greatest elevation bottom of the basin to be drained and the highest point 0,
over which the water is to be conveyed, should never
exceed the height to which water may be supported in
vacuo by the atmospheric pressure at the place.
over which the
water may be
raised.
XV.
Motion of fluids;
MOTION OF FLUIDS.
$292.-The purpose now is to discuss the laws which
govern the motion of fluids; and we shall begin with
those that relate to liquids. Suppose ABDC to be
any vessel containing a heavy fluid whose upper level is

MECHANICS OF FLUIDS.
599
A B. If a small opening ab be
made in the vertical side of the
vessel, the pressure from within will
urge the fluid out, and this pres-
sure being greater as we descend
to a greater distance from the upper
surface AB, the fluid will flow with
a greater velocity and in greater
quantity during a given time, in
proportion as the opening is made
nearer the bottom. The quantity
A
Fig. 349.
B
C
D
flow of liquids
from vessels,
through
apertures:
of fluid discharged in a unit of time, as a second, is called
the expense. The liquid on leaving the vessel forms a expense;
continuous stream called the vein or jet, which takes the vein or jet;
form of the curve described by a body thrown per-
pendicularly from the side of the vessel with the velocity
which the fluid has at its exit, and afterward acted upon
by its own weight. This, we have seen, is a parabola. in shape, a
At every point of this parabola, the weight of the fluid parabola;
tends to alter its velocity, but at the orifice, the vo-
locity is determined solely by what takes place within
the vessel.
aspe
Fig. 850.
direction of the
vein determined
by the face of the
vessel.
If the orifice be in the horizontal bottom, as at a'b',
the jet will be vertical, and the liquid will flow down-
ward; if, as at d, the orifice be in
a horizontal face pressed vertically
upward, the jet will also be vertical,
and the liquid will ascend on leav-
ing the vessel. In general, when
the sides of the vessel are thin, the
direction of the vein will be per-
pendicular to the surface through
which the orifice is made.
d
bubow
orifices;
§ 293.-The interior surface of every vessel containing Motion through
a heavy fluid is subjected, as we have seen, to a pressure
therefrom, which depends upon the extent of surface and

600
NATURAL PHILOSOPHY.
to find the
velocity of a fluid
flowing freely
through an
orifice in a thin
plate;
the distance of its centre of gravity
below the upper level of the fluid.
At the moment an orifice a b is
made, the fluid at its mouth is urged
by this pressure to leave the vessel,
the neighboring particles crowd to-
wards the opening, describing paths
which converge towards and lead
through it. This movement is soon
propagated in some modified degree
Fig. 351.
A
B
to all parts of the fluid, and speedily each point of space
within the vessel becomes distinguished by the constant
velocity which every particle of the fluid mass that passes
through it will there possess. It is from this instant, when
permanent flow; the motion of the fluid becomes permanent, that we are to
consider the flow.
equal volumes
If the fluid be incompressible, it is obvious that the
same volume will flow through each horizontal section of
the vessel above the orifice in the same time, and that this
volume must be equal to that dis-
different sections charged through the orifice. De-
note by A the area of the section
NB of the interior of the vessel, at
flow through the
in same time;
data;
Fig. 352.
N
B
B
Q
G
G
0
M SM
the upper surface of the fluid; by a
the area of the orifice MO; by s
the distance through which the up-
per stratum NB descends in any
indefinitely small portion of time;
and by S the distance O O' through
which the stratum at the mouth of
orifice passes in the same time. The volume of the fluid
which flows through the section NB in this time will be
measured by As; and that through the orifice, by a S;
and because these must be equal, we have
equal volumes;
whence
=
As a S;

MECHANICS OF FLUIDS.
601
S
α
S
Α
But because the distances s and S are described in the
same time, they will be proportional respectively to the
velocities of the strata which describe them; and denoting
the velocity of the stratum at the upper surface by v, and
that of the stratum at the orifice by V, we have
ratio of spaces
and areas of
sections;
S
8 = 7
S
ย
Ti
which, substituted above, gives
=
V
α
That is to say, the velocities of the strata are inversely
proportional to the areas of the sections through which
they flow, and from which we obtain
a
v = V.
A
ratio of spaces
and velocities;
ratio of velocities
and areas;
(246).
velocity through
any section;
Again, since the flow is permanent, it is obvious that
the living force of the fluid mass N'B' MQ must always
be the same. Denote this by L, and let w represent
the weight of the fluid mass in NBB' N', equal to that
in MM'0' 0; then will the living force of the mass
NBMQ be
W
L+
v2,
g
and that of the mass B'N' QM' O' O be
L + 20 y²;
g
and subtracting the first from the second, we find for the
difference of living force of the same mass NBMQ, and
living force of
the interior fluid;
that of a portion
within and that
at the jet;

602
NATURAL PHILOSOPHY.
difference of
living force of
the same
mass;
B'N' QM M'O' 0, moving with the velocities v and V
respectively, the expression
10
9
(V² — v²).
work of the
weight of the
entire fluid;
work at the end
The quantity of work performed
by the weight of this same mass
in the interval between its oc-
cupying the space NBMQ, and
B'N' QM' O' O, is, as we have
seen, equal to this weight multi-
plied by the vertical distance
through which its centre of gravity
may have descended in the interval.
Let G be the centre of gravity
N
N
Fig. 352.
B
B
G-
G
0
Q
MSM
of the whole mass when in the position NBMQ, and
G" when it occupies the space B'N' QM' O' O. Denote
the vertical distance of G' below the upper surface NB
by h', that of G" below the same surface by h", and the
weight of the entire fluid by W, then will the quantity of
work of this weight be
W(h"-h') Wh" - Wh';
=
and calling the distance of the centre of gravity of the
mass MM'O' O below the upper surface, h""; that of the
centre of gravity of the mass N'B' MQ below the same
surface, 7; and denoting the weight of this latter mass by
W'; we have, from the principles of the centre of gravity,
of any short
interval;
that at the
beginning;
work during the
interval;
Wh"
=
W'l+wh"",
Wh' W'1+ws;
=
in whichs denotes the distance of the centre of gravity
of the mass NBB' N' below the surface 'N B; whence
Wh" - Wh' w (h'"'- 1 s);
=

MECHANICS OF FLUIDS.
603
but h'"-s is the vertical distance between the centres of
gravity of the masses NBB'N' and M M' O' O, and when
these masses are considered as elementary, this distance
becomes the depth of the centre of gravity of the orifice
below the upper level of the fluid. Denote this distance
by h, and the quantity of work of the weight of the fluid
while the stratum NB is passing to N'B', and the stratum
MO to M' O', becomes
wh.
If the upper surface be subjected to any pressure, as that
of a piston or the atmosphere, then will the quantity
of work due to this pressure be
pAs;
in which p denotes the pressure exerted upon the unit of
surface. If, moreover, the fluid at the orifice be also sub-
jected to a like pressure inward, this pressure would be
transmitted to the lower face of the stratum whose area is
A, and its work would be measured by
p'As;
and taking the difference, we have, for the effective work
of these pressures,
the same;
elementary work
from external
pressure above;
elementary work
from external
pressure below;
(p' - p) A s.
Now As Dg=w, from which
W
As =
Dgi
and, substituting this above, we have
—
W
(p' - p) As = (p' - P) Dg'
whence the whole quantity of work due to the weight of
effective work of
external
pressures;
volume of the
stratum;
effective work;

604
NATURAL PHILOSOPHY.
total effective
work;
quantity of work
equal half gain
of living force;
the fluid and the pressures at the upper surface and the
orifice, becomes
-
พ
wh + (P' - P) Dg'
Dging
and because the difference of the living force at the begin-
ning and end of any interval, is equal to twice the quantity
of action in this interval, we have
p' -
Dg
1/2 (V - v) = 2 w (2 + 2);
310
or, dividing out the common factor, multiplying by g,
and substituting for v its value, given in Eq. (246), we
have
p'-
V² - V². 2² = 2 (gh + " ");
A2
D
velocity of egress
through the
orifice;
whence
2gh +
p'
D
V:
V =
(247).
a²
1.
42
same when the
pressures at top
and orifice are
the same;
If p and p' denote the atmospheric pressures upon
the unit of surface, they become equal when the altitude
of the fluid above the orifice is not very great, in which
case
2gh
V =
1-
A2
.
(248);
and if the area of the orifice be very small as compared
with that of the upper surface of the fluid, the fraction
a²
A2
will be so small, that it may, without sensible error,
be
omitted; in which case, the fluid at the surface will be at

MECHANICS OF FLUIDS.
605
comparative rest while it flows through the orifice, and
V = √2gh;
velocity of egress
through a very
small orifice;
that is to say, when a liquid is flowing through a small
orifice in the side or bottom of a large vessel, its velocity is
equal to the square root of twice the force of gravity multiplied rule;
by the depth of the centre of gravity of the orifice below the
upper surface of the fluid.
a heavy body in
It is apparent from the form of the above expression,
that this velocity is the same as that acquired by a heavy velocity same as
body while falling, in vacuo, from a state of rest, through that acquired by
the distance of the orifice below the fluid level. The failing through
distance h is called, in the case of discharging fluids, the the depth of
generating load.
If a be equal to A, that is, if the bottom of the vessel
be removed, then will, Eq. (246),
v = V.
The space described uniformly by the stratum of fluid
at the orifice in a unit of time being V, the expense,
estimated in volume, will be
orifice;
a V;
and in weight,
a VD g.
So that, if t denote the time of flow, expressed in seconds;
Q the quantity in volume, and Q' the quantity in weight
discharged, then will
Q = a Vt
expense in
volume:
in weight;
(249),
quantity in
volume in a
given time;
quantity in
Q' = a VDgt
in which Dg is the weight of the unit of volume.
(250); weight in a given
time;

606
NATURAL PHILOSOPHY.
example;
Example. The upper surface of the water, which is
15 feet above the centre of gravity of the orifice, is pressed
with an intensity equal to 20 pounds upon the square
foot; the area of the orifice being 0.02 of a foot. What
is the velocity of egress, and what the expense? Here, the
atmospheric pressure upon the piston and at the orifice
being the same,
=
p'-p 20 pounds,
D = 1,
data;
h
=
15,
g = 32 nearly;
and neglecting the small fraction
a2
A29
we find, from Eq.
(247),
velocity of
egress;
=
quantity in
volume in one
second;
quantity in
pounds in one
second.
V = √ 30 X 32 +40 31.6 feet;
and for one second,
Q = 0.02 x 31.6
=
lbs.
X
=
0.632 cubic feet,
Q' = 62.5 × 0.632 = 39.5 pounds.
XVI.
MOTION OF GASES AND VAPORS.
§ 294. In the preceding case, we have supposed,
1st, that the volume of the fluid which escapes through
Motion of gases the orifice, is equal to that which passes, during the
same time, through any interior horizontal section of the
and vapors;

MECHANICS OF FLUIDS.
607
vessel; 2d, that the density in all parts of the vessel both the volume
remains the same: both of which suppositions are sensibly through different
true for liquids, but are not so in the cases of gases and density vary;
vapors.
sections and
least at orifice;
When fluids of this latter class are confined and sub-
jected to any compressing action, as that of a piston, and
are permitted to escape through an orifice at which the
resistance of external pressure is too feeble to retain them, density greatest
the density, tending as it always does to conform to Mari- at piston and
otte's law, will be greater at the piston where the pressure
is greatest, than at the place of egress where it is least.
Again, the motion being permanent, the same amount, in
weight, of gas will flow through
any section A'B' of the vessel as
through the orifice ab; but the den-
sities at these places being different,
the volumes of these equal weights
will also be different. In these par-
ticulars, the circumstances attending
the motion of gases and vapors dif-
fer from those of liquids.
To find the velocity of egress at
the orifice, we remark, that the
fluid is subjected, as in the case of
liquids, to the action, 1st, of its
own weight; 2d, to that of the
opposing pressures at the piston and
orifice; and 3d, to the additional
Fig. 353.
A
B
Α'
B
B
equal quantities,
by weight, will
flow through the
different sections
in the same time:
the volumes of
these will be
different;
the forces which
act: weight,
pressure from
the piston, and
molecular
repulsion;
weight;
action arising from the repulsions of the particles for each
other, this latter producing expansion whenever the pres-
sure from without will permit it. The quantity of work work of the
upon the stratum issuing through the orifice, due to the
weight of the fluid mass, is, as we have seen, measured by
wh; in which w denotes the weight of the stratum, and h
the height of the fluid above the orifice. To find the to find the work
work due to the pressures, denote the pressure upon a
unit of surface at the piston by p; that on the same
due to the
pressures;

608
NATURAL PHILOSOPHY.
fine
notation;
test extent of surface at the orifice by
p'; the area of the piston by A;
that of the orifice by a; the dis-
tance between any two consecutive
positions, as A B and A'B', of the
piston by s; the distance between
the two corresponding positions ab
gs and a'b' of the stratum at the
orifice by S. Then, because the
weights of the volumes A B B'A'
and abb'a' of the fluid are equal,
we have
A
Α'
Fig. 353.
the equal
weights;
volumes
inversely as
densities;
SEUR
=
As Dg a SD'g
B
B
(251);
in which D and D' denote the densities of the gas at the
piston and orifice, respectively, and g the force of gravity.
Whence
As
a S
=
D'
D
densities directly
as pressures;
But by Mariotte's law the densities are directly propor-
tional to the pressures, hence
D'
1=
D
བོ། ས་
relation of
volumes and
pressures;
which substituted above, gives
p'
pherd As
=
a S
p
(252).
loss or gain of
work due to
pressure;
Clearing the fraction and transposing, we find
p. As p'a S = 0.
But p A is the pressure on the whole extent of the piston,
and p As is, therefore, the whole work of this pressure;
also p'a is the pressure on the surface of the stratum of

MECHANICS OF FLUIDS.
609
fluid in the orifice, and p' a S is the quantity of work
of this pressure; and as these quantities of work are pro-
duced in the same time, we see that the loss or gain of this is zero;
work, due to these pressures, is zero.
molecular action;
the primitive
The quantity of work due to the molecular actions, to find quantity
arises in consequence of the expansion which takes place of work due to
when the gas passes from the pressure p, within the vessel
and near the piston, to the pressure p', at the mouth of
the orifice. The amount of this work is directly propor-
tional to the primitive volume expanded during the change it is directly
of pressure; if the primitive volume to be expanded be proportional to
doubled, tripled, or quadrupled, &c., the quantity of the volume to be
work will be doubled, tripled, quadrupled, &c. Hence,
taking a cubic foot of the gas under the pressure p, and
denoting the quantity of work due to the expansion, cor-
responding to a change from the pressure p to the pressure
p', by E, then will the work due to the expansion of the
volume A B B'A' to a b b' a', be measured by
A.s. E.
But since w denotes the weight of the gas in the volume
ABB'A', we have
expanded;
work of the
expansion from
the volume at the
piston to that at
orifice;
whence
and
w= Asg D;
w
As=
g D'
weight of the
stratum;
its volume;
A.s.E=
w.E
gD
;
whence the whole quantity of action or work due to the
weight and expansion of the fluid will be
work due to
expansion;
E
wh+w.
B = w(r + B).
E
work due to
weight and
expansion of the
D
stratum;
39

610
NATURAL PHILOSOPHY.
Denoting, as before, the velocity at the piston by v, and
that at the orifice by V, we have, from the principle of
aad living forces,
living force equal
ข
E
ni booob
out of sub from
to twice quality solo malie (v) = 2 w (+);
of action;
polson selenolog
จ
g
vº) (h
D
Rup odl
or
to debom alt
yoyong poor V2 - v² = 2gh+
of lanolis From Eq. (252) we have
avillating of
ed of ent
relation of
elementary
paths;
same as ratio of
velocities;
bas
2 E
D
(253).
Riming
bolqurbanp to bolqis boldnob
p'a boldweb od fw how
02 bolquh S
=
PA' Jo Joot
2.
109 polartigo put of pub low la viito
and the spaces s and S, being described in the same time,
they are to each other as the velocities v and V, hence
yd beriansen od do of A
or
velocity at
piston;
บ
V
=
p'a
PA'
estoph woonis pl
p'a
A
PA
v = V. ; oved ow
which substituted in Eq. (253) for v, we find
oonofw
2¹³
№³ (1 –
2 E
p² A2
=2gh+
D
Making
how
value for the
velocity of
egress;
edt of oub dow
the above gives
,2
1 - pa²
=
p² Aa
= K²,
slody od:
1
2 E
V =
2gh+
D
.. (254).

MECHANICS OF FLUIDS.
611
expansion of a
from one
pressure to
It remains to find the value of E. For greater sim- to find the work
plicity, let us take for the primitive volume of gas a unit due to the
or cubic foot; and suppose this unit of volume to be con- unit of volume,
tained in a tube, of which the area of the internal cross-
section is a unit of surface, or square foot, so that in its another;
primitive condition, under the pressure p, the length of
the tube it occupies will be the linear unit, one foot.
When the pressure is reduced to p', the volume becomes
dilated, and because the volume, and therefore the length,
since the base is supposed constant, is inversely as the
pressure, we have, calling the new length 1,
p
whence
p'l: 1;
ow Infot oft
why olis
p
1 =
ह
TO
new length of the
volume of gas;
The path described by the moveable face of the cubic
foot of the gas, during the expansion, will be
7-12-12-p'
p
offt Joere has
expansion during
the change;
Dividing this path into two equal parts, and adding one
of them to unity, the original length, we have
1+1 P-p_p+p
=
2 p'
2 p'
for the length of the fluid when its expansion is half
completed; and denoting the corresponding pressure by
P, we have, by Mariotte's law,
length when the
expansion is half
completed;
ovoda od
roq how
yd beques 1:
p+p'
2 p'
:: P, P;
whence
P₁ =
2 pp'
p+p"
corresponding
pressure;

612
NATURAL PHILOSOPHY.
the three
consecutive
If we now observe that the consecutive pressures are
ed of em P
2 pp'
p+p
and p';
pressures;
space described
by the pressure
and that the constant space passed over, during the inter-
val which separates the instants in which these pressures
are exerted, is
while its value is al volorid
changing from
the first to
second;
the computation of
p-p'.
2 p'
Fig. 354.
M
determination of
the work;
its value;
the total work be-
comes easy by the
rule given in § 46.
For this
take
purpose,
M'
M
4.
C
B
AC=CB=P-p'
and erect the perpendiculars
2 p''
sho yaibba han
AM
diag sids
=
P
CM' =
2 pp'
p+p"
BM" = p';
join the points M, M', and M"; the area A B M" M will
be the value of E: that is to say, the value of the quantity
of work performed by the gas during its expansion. But
this area is, by the rule just referred to, measured by
AC(AM+4 CM' + BM");
and, substituting the values above, we have

MECHANICS OF FLUIDS.
613
p-p'
p-P² (p +4.
2 p'
$8 of Janpe
133
-ups eds
2 pp'
od Hiw Y doidw
+p') = E;
p+p'
same in other
terms;
1
V=
K
2/ √√2gh
1
p-p' 8pp'
p
which, substituted for E, in Eq. (254), gives
value of velocity
V = = √2 g h + 3D 27 (P + 27p+p).. (255), in terms of
When the orifice is small, as compared with the area of
the piston, the fraction
pressures:
of Sagwedselle
Yajniti aa
Joeepy
doidw.
2
p' a2 based
mored live di
p² A2
fed bill add dolde
may be neglected, and K will become equal to unity.
Moreover, the term 2gh, in the case of gases, is scarcely
ever appreciable in practice; making these suppositions,
Eq. (255) becomes
no hook
1
1=
V=V 3D
p-p'
p'
(p +
8pp'
p+ p'
+p').. (256).
velocity in case
of small orifices;
((088)
The pressures p and p' are usually ascertained by means
For this
determine the
pressures;
of gauges, or manometers, as they are sometimes called, and use of gauges to
it will be convenient to express the velocity of egress in
terms of the indications of these instruments.
rms of
purpose, denote by h the height of a column of mercury
resting on a unit of surface, and whose weight is equal to
p, and by h' the same for the pressure p'; then, denoting
the density of the mercury by D,,, will
p=gh Dun and p' gh'D; g
1
which, substituted above, give
V=V9 3D
Vg. Du h
h-h'
h'
.
8hh'
(h +
velocity in terms
gobyl of the indications
h+h+h').. (257); of the
manometer;

614
NATURAL PHILOSOPHY.
todto at oms
expense;
Theo ban to ch
in which V will be expressed in feet, g being equal to 32
feet very nearly, and D,, equal to 13.5 nearly.
tion
The expense e, in volume, will be given by the equa-
ovly (168) paisol bouditadura doldw
e = a V
·
(258);
od and the quantity Q in volume, discharged in a given
time t,, expressed in seconds, will be known from
quantity
discharged in
volume;
To so oft diw bong a Vt,... (259);
in which a must be expressed in square feet. The density
D, it will be remembered, is that of the fluid in the vessel
near the piston, where the pressure is p; the density D',
which the fluid assumes on leaving the orifice, deter-
mined by the pressure p', and is connected with D, ac-
cording to Mariotte's law, by the relation
1970010 M
oldsharqge rove
gablem
DP'
D' =
h'
h
= D
D mooed (608) pil
density on
leaving the
orifice;
p
Viol
only Hence, the expense Q', in weight, will be given by
woods Hema
to
quantity in
weight
discharged in
unit of time;
20
quantity, in
weight, in time
t,;
example;
ang ni tor
vd benig
h'
Q' = D'gaV = DgaV
DgaV. (260);
and the quantity Q" in weight, discharged in the time t,,
aidi -10
norem to
Q"
stoomu" =
meios
of laupo at ddpiew on
od lliw di
Dga V
h
hibai od
V,
(261);
stone
in which a must be express-
ed in square feet, as above.
The density D is com-
puted by Eq. (240)'.
Example. The open gauge,
connected with a gasometer,
containing heavy carbureted
Bolbo od to hydrogen, shows a difference
dof level in the mercury of 8
odime to time a no guitest
100018 Fig. 355. bos
.

MECHANICS OF FLUIDS.
615
inches; the barometer in the air stands at 28 inches;
the thermometer of the gasometer, at 52°: required the conditions;
velocity with which the gas will flow into the open air,
and the volume and weight discharged through an ori-
fice 0.02 of a square foot of area in 20 minutes = 1200
seconds.
bot
Here, option dysouls guiwolt
begindbeth Bruft
Ades
h-h' = 8 inches = 0.666 feet,
at quianup
Adglow
找 ​To ne dguardi
h' = 28 66
=
2.333
data;
whence alfe of esse
trodtiw son
idaiw
h = 36
= 3.000
of
bus xtisos
Inolienoods
anofticoqqos
odtoysol oëls bastisol
9ad to no g = 32
aily frottorouds adt bris
t = 52°;
door bin D,, = 13.5 itu odt
32 follerng bus lapo dity solito
hollatut one anotibnos
and from Eq. (240)', after substituting the values of h and
t, above, and that of D,, in the table, page 533, for heavy
carbureted hydrogen, we find
di doid
0.00127
D=
X
36
30 1+ (5232) 0.00208
1199 & of
= 0.001465; density;
and these values, in Eq. (257), give itoni tiodt foods Hiw
ad
32.
13.5
dioft.
ft.
sonatges
aid time
0.666
8 X 3 X 2.333
3+2.333
gatvoll ausla bout doidwanao
adi toshnos of
+2.333)=668.02. velocity;
3 X 0.001462883X(3+
iw brodt
d moft
Substituting this and the numerical values of a and t, in
Eq. (259), we find
lever
odt toodard miss adt
Q = 0.02 x 668.02 x 1200 = 16032.00 cubic feet.
ada yd boeashorti od lliw Jodho
The quantity Dg, in Eq. (261), is the weight of a cubic
quantity in
volume;

616
NATURAL PHILOSOPHY.
quantity in
weight.
Vein.
theoretical
suppositions;
results of
experience;
foot of the gas, whose density in this case is 0.001465; and
b as a cubic foot of water weighs 62.5 pounds, the value of
Dg becomes 62.5 X 0.001465 0.0916, nearly; whence
noordt Jografai afgins saulov adt
lbs.
=
28
S
lbs.
Q"=
Q" 0.0916 x 0.02 x 668.02 X X 1200 = 1142.4.
36
§ 295.-A stream flowing through an orifice is called.
a vein. In estimating the quantity of fluid discharged
through an orifice, it is supposed, 1st, that the orifice is
very small, as compared with a section of the vessel at
the upper surface of the fluid; 2d, that there are neither
within nor without the vessel any causes to obstruct the
free and continuous flow; 3d, that the fluid has no vis-
cosity, and does not adhere to the sides of the vessel and
orifice; 4th, that the particles of the fluid reach the
upper surface with a common velocity, and also leave the
orifice with equal and parallel velocities. None of these
conditions are fulfilled in practice, and the theoretical dis-
charge must, therefore, differ from the actual. Experience
teaches that the former always exceeds the latter. If we
take water, for example, which is far the most important
ar the most impos
of the liquids in a practical point of view, we shall find
it to a certain degree viscous, and always exhibiting a
tendency to adhere to ununctuous surfaces with which it
may be brought in contact. When water flows through
an opening, the adhesion of its particles to the surface
will check their motion, and the viscosity of the fluid will
transmit this effect towards the interior of the vein; the
velocity will, therefore, be greatest at the axis of the
latter, and least on and near its surface; the inner particles
causes which tend thus flowing away from those without, the vein will
increase in length and diminish in thickness, till, at
a certain distance from the orifice, the velocity becomes
the same throughout the same cross-section, which usually
takes place at a short distance from the aperture. This
effect will be increased by the crowding of the particles,
arising from the convergence of the paths along which
to contract the
vein;

MECHANICS OF FLUIDS.
617
contraction;
eolite delt
they approach the aperture, every particle, which enters
near the edge, tending to pass obliquely across to the
opposite side. This diminution of the fluid vein is called
the veinal contraction. The quantity of fluid discharged veinal
must depend upon the degree of veinal contraction, and
the velocity of the particles at the section of greatest lait at
diminution; and any cause that will diminish the viscosity
and adhesion, and draw the particles in the direction of
the axis of the vein as they enter the aperture, will
increase the discharge.
exoflito odp
contraction;oo
mos
depends upon;
discharge
Experience shows that the greatest contraction takes
place at a distance from the vessel varying from a half to place of greatest
once the greatest dimension of the aperture, and that the
amount of contraction depends somewhat upon the shape
of the vessel about the orifice and the head of fluid. It is
further found by experiment, that if a tube of the same its amount
shape and size as the vein, from the side of the vessel to
the place of greatest contraction, be inserted into the
aperture, the actual discharge of fluid may be accurately
computed by Eq. (261), provided the smaller base of the
tube be substituted for the area of the aperture; and that, the actual
generally, without the use of the tube, the actual be
may obtained from
deduced from the theoretical discharge, as given by that the theoretical;
equation, by simply multiplying the theoretical discharge
into a coefficient whose numerical value depends upon the
size of the aperture and head of the fluid. Moreover,
all other circumstances being the same, it is ascertained
that this coefficient remains constant, whether the aper-
ture be circular, square, or oblong, which embrace all coefficient of
cases of practice, provided that in comparing rectangular
with circular orifices, we compare the smallest dimension
of the former with the diameter of the latter. The value
of this coefficient depends, therefore, when other circum- depends upon;
stances are the same, upon the smallest dimension of the
rectangular orifice, and upon the diameter of the circle,
in the case of circular orifices. But should other cir-
cumstances, such as the head of fluid, and the place of
discharge;
Sol sinaloffieno
ho

618
NATURAL PHILOSOPHY.
the orifice, in respect to the sides
Fig. 356.
ya yous
and bottom of the vessel, vary,
then will the coefficient also vary. au shisogo
When the flow takes place through
discharge thin plates, or through orificesab
whose lips are bevelled external- o
through orifices
in thin plates;
Sastors to a
coefficient
deduced from
experiments;
ly, the coefficient corresponding to
given heads and orifices, may be
found in the following table, pro-
vided the orifices be remo.e from gradben
the lateral faces of the vessel. This
table is deduced from the experi-
ds
orbe brie
ode/constroqz
leib Josly
ments of Captain Lesbros, of the French engineers, and
agrees with the previous experiments of Bossut, Miche-
lotti, and others.
tamoms all onse od
To sbie
don't
TABLE
odt en osie bid qede
ogadosib
mot bouteldy
lasbygoods od
COEFFICIENT VALUES, FOR THE DISCHARGE OF FLUIDS THROUGH THIN
PLATES, THE ORIFICES BEING REMOTE FROM THE LATERAL FACES OF
THE VESSEL.
Head of fluid
above the
centre of the
orifice, in feet.
bebivory (108) pued
Values of the coefficients for orifices whose smallest dimensions or
diameters are-
ft.
0.66
ft.
0.33
ft.
0.16
ft.
ft.
ft.
0.08
0.07
0.03
0.05
0.700
0.07
It to Head be
0.6270.660
0.696
0.13
0.618
0.20
0.592
table of
coefficients;
0.620
0.632 0.657 0.685
0.640
0.656
0.677
0.26
190.33
0.66
0.602
0.625
0.638 0.655 0.672
0.593
0.596
0.608 0.630
0.613
0.631
0.637 0.655 0.667
0.634 0.654 0.655
ogadoelb
1.00
Aling1.64 01
0.601 0.617
0.6020.617
3.28
0.605
5.00
0.603
6.65
32.75
0.602
0.600
mogu shangab
coefficients for
gas; and for
orifices not in the
table;
0.632 0.644 0.650
0.63000.640
0.628 0.633 0.632
0.620 0.621 0.618
0.615 0.610 0.610
0.600 0.600
In the instance of gas, the generating head is always greater than 6.65 ft., and the
coefficient 0.6, or 0.61, is taken in all cases.
For orifices larger than 0.66 ft., the coefficients are taken as for this dimenson; for
orifices smaller than 0.03 ft., the coefficients are the same as for this latter; finally, for
orifices between those of the table, we take coefficients whose values are a mean
between the latter, corresponding to the given head.
0.630
0.628
0.644
0.612 0.626
0.610 0.615
0.600 0.600
0.600

MECHANICS OF FLUIDS.
619
As the orifice approaches one NE
of the lateral faces of the reservoir,
the contraction on that side becomes
less and less, and will ultimately be-
come nothing, and the coefficient
will be greater than those of the
table. If the orifice be near two
of these faces, the contraction be-
comes nothing on two sides, and the
coefficient will be still greater.
Under these circumstances, we
have the following rules: Denote
by C the tabular, and by C' the
true coefficient corresponding to a
given aperture and head, then, if
the contraction be nothing on one
side, will
Fig. 357.
Fig. 358.
ods
to
組
​077989
d
when orifice is
near one lateral
nos
faced on
gav-coluls a
apitaton
near two lateral
faces;
C' = 1.03 C;.
conrodw
if nothing on two sides,
C'
=
1.06 C;
coefficient in the
first case;
nas to oulev
coefficient in the
second;
if nothing on three sides,
XIX
C' = 1.12 C;
and it must be borne in mind, that these results and those
of the table are applicable only when the fluid issues
through holes in thin plates, or through apertures so
bevelled externally that the particles may not be drawn
aside by molecular action along their tubular contour.
coefficient for no
contraction on
three sides.
through
leollowing
§ 296.-When the orifice is rectangular, and has no Discharge
upper limit, or is open at the top, it is called a sluice-way. sluice-ways;
It is usually a cut made in the edge of a reservoir, through

620
NATURAL PHILOSOPHY.
which the fluid may
sador Fig. 359.
A
flow when it rises in
A
Bod to
above a certain level.
ed
gi pnittto กา w
estimate of the
expense through
a sluice-way;
notation;
The expense is esti-
mated in this wise. doi
Denote by 7 the length d
of the horizontal side
TO
D
of the sluice-way; by h the head or distance BI, of the
centre of gravity of a transverse section of the flowing
fluid below the upper surface of the latter in the reservoir;
by H the height of the fluid above the sill C, of the sluice-
way; and by V the mean velocity: then, supposing the
sluice-way filled to the upper level of the fluid in the
reservoir, will
godt badd big astroge vig
h = =H, no no guidton of noitestimo
1=
Hiw obis
V² = 2gh = 2g X H = (2g H);
and text
value of mean
velocity;
booste
whence
w do ganiton T
V = 0.707 √2g H;
00.1
theoretical 000
expense;
practical
expense,
fils dons
and the theoretical expense will be
Asbie oonil ne quidfor
VX 1 X H = 0.707 √2gHxlx H.
But this is too great, and experience shows that it should
be multiplied by the coefficient 0.57 for all ordinary cases
of practice; that is to say, the true expense, denoted by
E, will be given by the equation,
E=0.57 X 0.737 XIX HX √2gH=0.403 1. H. √2g H... (262).
bus
The experiments of Dubuat, Bidone, Eytelwein, and
Lesbros, show that the coefficient 0.403 should be re-

MECHANICS OF FLUIDS.
621
value of the
coefficient;
duced to about 0.39 when I becomes equal to or greater
than 0.66 of a foot, and increased to 0.415 when H be-
comes less than 0.07 of a foot; but that it remains variation in the
sensibly the same, whatever be the total contraction or
position of the sluice-way in regard to the vertical sides
of the reservoir, provided I be measured from the level
of the upper surface of the sill to that of a point, as A,
in the surface of the fluid in the reservoir which has no
sensible velocity. When the sill is on a level with the
bottom of the reservoir, the velocity of the upper surface
is everywhere sensible, and the coefficient increases to
about 0.45. On the contrary, 0.403 is already too large
when the sluice-way is prolonged into a trough-like duct,
of slight inclination, wherein the fluid may have impressed
upon it a whirling or irregular motion by the roughness
of the surface.
plates;
coefficients;
The foregoing conclusions suppose that the fluid is discharge
discharged through orifices in thin plates, and that, du- through thick
ring the flow, the fluid particles are not drawn aside from
the converging paths, along which they tend to approach
the orifice, by the action of any extraneous cause. When
the discharge is through thick plates without bevel, or
through cylindrical tubes whose lengths are from two to
three times the smaller dimension of the orifice, the
expense is increased, the mean coefficient, in such cases,
augmenting, according to experiment, to about 0.815 for values of the
orifices of which the smaller dimension varies from 0.33
to 0.66 of a foot, under heads which give a coefficient
0.619 in the case of thin plates. The cause of this in-
crease is obvious. It is within the observation of every
that water will wet most surfaces not highly polished
or covered with an unctuous coating-in other words,
that there exists between the particles of the fluid and explanation;
those of solids an affinity which will cause the former to
spread themselves over the latter and adhere with con-
siderable pertinacity. This affinity becoming effective
between the inner surface of the tube and those particles
one,

622
NATURAL PHILOSOPHY.
effects of
molecular action.
of the fluid which enter the orifice near its edge, the latter
will not only be drawn aside from their converging direc-
ditions, but will take with them, by the force of viscosity,
the other particles, with which they are in sensible con-
tact. The fluid filaments leading through the tube will,
therefore, be more nearly parallel than in the case of
orifices through thin plates, the contraction of the vein
will be less, and the discharge consequently greater.
dieme
mottod
ode
Discharge of
fluids through
pipes;
bu XVII.
एम XVII.
nd notion
linnilont idylla to
baibidws -nogy
DISCHARGE OF FLUIDS THROUGH PIPES.
de ar biul et tert soyqueenofaeloco gaiogend ed.
We have considered the discharge of fluids through
thin and thick plates. It remains to discuss the discharge
through pipes. When the flow is through pipes whose
length does not exceed two or three times their diameter,
the quantity discharged in a given time is, as we have
seen, greater than through bevelled orifices of the same
size; but when the length is increased much beyond this
limit, the reverse is the case and, all other things being
orifices;20
less than through equal, the discharge will be less as the pipe is longer. The
same pipe may be of variable bore, that is to say, it may
have a greater cross-section at one point than at another;
in which case, the living force of any given portion of the
moving fluid cannot be constant throughout. When of
considerable length, pipes are rarely perfectly smooth, the
fluid particles cannot, therefore, flow through them in par-
allel filaments, but must be incessantly deflected from their
onward course into partial eddies formed by the small ir-
regularities of surface. Moreover, as the pipes increase in
length, will the surface exposed to fluid pressure increase,
and as the extent of surface, all other things being equal,
causes which
obstruct the
motion;

MECHANICS OF FLUIDS.
623
determines the amount of pressure, the friction, which de- friction.
pends upon the pressure, augments so as greatly to impede
the motion. We shall proceed to estimate the value of
these influences.
§ 297.-But first of all let us
compute the amount of living force
resulting from the shock of fluids,
flowing with different velocities.
Lydgalliy
A
For this purpose, let the fluid in and
the pipe LK flow with the velocity
V, and denote by M the mass which
flows into the vessel B C in a unit
of time; also let the velocity of the
fluid in the vessel BC be V', and its
mass M'; then will the correspond-
ing living force be
MV² + M' V²;
Fig. 360.
10
Loss of living
force arising from
the impact of
K
fluids;
w so
L
B
C
D
and supposing the fluid to be water, which we have re-
garded as unelastic, the common velocity after impact will
be obtained from either of the Eqs. (194) or (195), by
making e = 0; hence the common velocity denoted by v,
will be given by
living force
before the
impact;
v=
M.V+M'. V'
M+M'
;
common velocity
after the impact;
and the corresponding living force,
(M + M') v² =
M+M
M.V+ MV) x (M+ M')=
(MV+ M'V')2
M+M
corresponding
living force;
Бл
moved bee
and the loss of living force in a unit of time, denoted
by L,

624
NATURAL PHILOSOPHY.
loss of living
force
L = MV²+M'V² (MV + M'V')
M+M'
MM' (V- V').
M+M;
game;
and, dividing by M',
M(V - V')
L=
(263);
M
1+
M'
same when a
small mass flows
into a large
mass.
or when the mass M' is very great as compared to M,
L = M(V - V') ... (264).
§ 298.-It will be an easy matter now to estimate the
loss of living force, arising from a contraction of the vessel
or pipe through which the fluid
may be flowing. Let ABCD be
a vessel containing a heavy fluid,
cross-section of a of which AB is the upper level,
Loss of living
force from
contraction of
pipe;
hypothesis;
notation;
and issuing through an opening ab
in the bottom CD; and suppose
A' B' to be a diaphragm, pierced
by an opening a'b'. Denote by A"
the area of the section at A" B", by
a the area of the contraction at ab,
and by a' that of the contraction
at a'b'. The fluid, in passing
through the contraction a'b', im-
C
Fig. 361.
B
B
כי
pinges against that below the diaphragm A' B', and if
the opening a b is beyond the reach of the eddies formed
by this conflict, the velocity at either contraction may be
computed from that at the other.
Denote by V the velocity of the fluid as it passes the
contraction at a b, by V' that at the contraction a' b', and
by V" that at the section A" B", supposed beyond the
region of eddies; and let m represent the coefficient of the
expense at ab, and m' that at a'b': these coefficients

MECHANICS OF FLUIDS.
625
may
be found from the table. The expense at a b will be
m a V, that through the section A"B" will be A" V",
the different
sections;
and that through the contraction at a' b' will be m' a' V'; expense through
but as the same quantity of fluid must pass through the
sections of ab, A" B", and a'b', in the same time, we
have
ma V = A" V",
ma V = m'a' V';
whence
T":
11
mav
A
mav
=
;
m' a'
and the velocity with which the fluid through a' b'
impinges against that below the diaphragm, will be
1
1
velocities;
y' - y"
= ma
-
m'a'
Denoting by w the weight of fluid that passes a'b' in any
small portion of time, its loss of living force will be
10 (V' -
g
1
V₁ = 20 m² a² (m²α- 1). v;
g
•
m' a'
1
m' a'
---
1
A"
and denoting the factor ma (-) by K, the
relative velocity
of the impact;
quantity of work lost will be
W
K2 V2.
2 g
The work of the weight, during the same time, will be
wh, and the quantity of work remaining will be
work lost;
40

626
NATURAL PHILOSOPHY.
the work
remaining;
which is equal to
wh
-
พ
2g
K2 V2;
but this must be equal to half of the living force, hence
half the living
force;
velocity of egress
through a b.
whence we find
W
V2 = wh
-
g
W
2g
V =
2gh
1 + K2
K2 V2;
(265);
Loss of living
force in short
pipes;
notation;
and from which we see that the velocity will be less than
that due to the height A C, equal to h.
§ 299.-Let us apply this to the discharge of a fluid
through a short pipe, inserted into the orifice in the side
of a vessel. The fluid hav-
ing contracted to its mini-
mum dimensions at n, again
dilates, and fills the tube at
a'b'. Let V be the mean
hypothesis and velocity at a'b', where the
area of the cross-section of
the pipe is a. The fluid
particles moving in parallel
paths at a'b', the expense
will be a x V; while that
Fig. 362.
through a section at ab, where the velocity is V', and
cross-section a', will be ma' V', in which m is the co-
efficient corresponding to the area a'; and, as these must
be equal, we have
a V = ma' V';
velocity at the
entrance of pipe;
whence
V'
=
a
ma'
V:

MECHANICS OF FLUIDS.
627
and the loss of living force,
טי
1 x (V' - V² = 2 x V (-1).
g
g
ma'
The quantity of work of the weight, in the same time,
is wxh, and this, diminished by half the loss above,
must be equal to half the actual living force; and,
therefore,
W
29
V² = wh-
w
α
2.g
ma'
(-1);
or making - 1= K, we find
a
ma'
2gh
V=V1+ K³°
When the tube is cylindrical a = a', and
K:
=
1
m
— 1;
loss of living
force;
ogiq
velocity of egress
from the pipe;
when the contraction is complete in n, and the head
varies from 3 to 7 feet, it is found that m is equal to value of m
0.62 very nearly; whence.
1
K= - 1 = 0.613 very nearly,
0.62
and
To orod odi
1
V1+ K2
value of the
0.85;
constant;
whence
V = 0.85 V2 gh.
Experiments give the coefficient 0.82, but, in com-
final value for
velocity of
egress;

628
NATURAL PHILOSOPHY.
coefficient given puting the foregoing value, no account was taken of
by experiment a friction, which is an additional cause to diminish the
work of the weight wh.
little less.
Flow of fluids
through pipes of
any length;
§ 300.-When the velocity of a fluid is considerable,
and the length of the pipe through which it flows is great,
friction, which has thus far been neglected, becomes an
effective cause of obstruction, and can never be neglected
in estimating the circumstances which determine the
quantity discharged. The amount of friction depends,
as we have seen in the case of fluids, upon the pressure,
and this latter is determined by the extent of surface, and
the head which impresses the velocity, so that the length
of pipe and the velocity of flow, are the elements from
which friction is to be estimated.
Let abb'a' be
a pipe of uniform
bore throughout, con-
necting two reser-
Fig. 363.
B
H
case stated;
notation;
De-
voirs ACDB and
A' C' D' B', partly
filled with fluid, the
former to the level
AB, and the latter to
the level A'B'.
note by H the differ-
ence of level between
AB and A' B'; by a
the area of a cross-sec-
tion of the bore of the
D
B
pipe; by C the contour of this section; by L the length of
the pipe; and by V the constant velocity of the fluid flow-
ing through it. Experience shows, and the computations
of COULOMB, DE MEST, PRONY, EYTELWEIN, and NAVIER,
loss of work from teach us, that the loss of work occasioned by friction of
friction in pipes; pipes, in the time during which a weight of the fluid
denoted by w is discharged, is proportional to the value

MECHANICS OF FLUIDS.
629
of the expression
w LX CX V
9
α
and that this loss is a certain fraction n of this function, or
is equal to
n.
w LX CX V2
พ
-
g
α
If, therefore, there be neither contractions in the pipe,
nor sudden turns giving rise to shocks, the only loss of
work will be that measured by the above expression, and
that due to a diminution at the orifice ab, measured by
the expression
proportional to
this function;
the loss of work
from friction;
พ
V2
29
m
v. (1-1)² = 29
w
V² K³,
in which
1
m
—
1 = K;
btw sobota
and, because of the principle of fluid level, H is the only
distance through which w can act to produce work, we
work lost from
diminution at the
entrance of the
pipe;
have
V²=
2 g
w y² = wH-
พ
.V². K - n
w L. C.V
(266);
2g
g
α
whence
2g H
V =
(267),
C
velocity of
egress;
1+ K² + 2 n. L.
α
from which the velocity may be found.
The expense, denoted by Q, will be given by
Q = a V...
(268).
expense;

630
NATURAL PHILOSOPHY.
value of the
constant;
value of the
coefficient n, for
water;
and for gas;
Taking the value of m equal to 0.60, (see table,) we find
1+K 1.4444.
=
Experiment shows that, for water,
and for air or gas,
n=
0.0035;
n=
0.00324;
modification in
the formula for
gas;
and it is important to remark that, when the question
relates to the discharge of gas, we must make
H = 1 D 11
3D
h-h'
h'
.
(
h +
8hh'
h+h'
+
+ h²),
as indicated by Eqs. (254), (257), in the latter of which h
and h' denote the mercurial altitudes corresponding to the
interior and exterior pressures.
Denote by D the internal diameter of the pipe, then
will CD, and a =
C
α
T D2
4
=
so that
4 D
T
T D2
=
4
D
Substituting these different values and that of gravity,
Eq. (22), in the expression for the velocity, we have,
after dividing both terms of the fraction by 8n,
velocity in case
of water;
for water, ... V = 47.94
D. H
L+ 51.57. D
(269),
in case of air;
for air, . . . . . V = 49.83
D. H
L+ 55.72. D
(270);

MECHANICS OF FLUIDS.
631
in which all linear dimensions are expressed in English either formula
feet. The first formula may be employed even for gas, for gas
may be employed
because of the small difference between the values of n
for the two fluids, provided we employ the proper value
for H.
Finally, if the aperture
a'b' of final egress be small-
er than ab, or of less section
than V being the velocity
within the pipe, the expense
may still be deduced from a
gig Fig. 364.
slight modification of the value of the velocity, as given
by Eq. (267). For let V' denote the velocity of egress,
a' the area of the section at a' b', and m' its coefficient
of contraction, then will
a V = m' a' V';
when the
aperture of final
egress is smaller
than section of
pipe;
condition of
permanent flow;
whence
y'
=
a v
m'a'
SAT
and the living force of the fluid as it issues through a'b',
will be
thron 5.1
310
2
w
a²
=
X
max V²;
g
m12a12
which, being placed equal to the second member of Eq.
(266), will give
living force of the
discharging
fluid;
V =
2g H
a²
+K²+2nL.
σ
• •
(271). its velocity;
m12a12
α
When a' is very small as compared with a, the value
of m' is about 0.60. If the values of a and a' differ but

632
NATURAL PHILOSOPHY.
values of the
coefficient m';
example;
slightly, or if the pipe term-
inates at a'b' in a conical
tube, then will the value of
m' vary from 0.82 to 0.96.
Example. Let the height
Fig. 365.
of the reservoir above the point of delivery be 70 feet,
the diameter of the pipe 0.5 of a foot, and its length 1200
feet: required the quantity of water discharged in 24
hours. In this case,
data;
ft.
ft.
ft.
D = 0.5;
H = 70;
L:
=
1200;
which, in Eq. (269), give
velocity;
V = 47.94
0.5 X 70
= 8.102.
1200 + 51.57 X 0.5
The value of a, in Eq. (268), will be given by
area of the
section of pipe;
α = T
D2
4
0.25
= 3.1416 X
=
0.196;
4
expense.
which, in Eq. (268), gives
Q = a V = 0.196 × 8.102
=
1.6 nearly;
and this multiplied by the number of seconds in 24 hours,
equal to 86400, gives 138240 for the number of cubic feet
discharged in the given time.
(0
END OF MECHANICS.






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will not disappoint the reasonable expectations thus excited. They are elementary
works prepared by authors in every way capable of doing justice to their respective
undertakings, and who have evidently bestowed upon them the necessary time and
labor to adapt them to their purpose. We recommend them to teachers and
parents with confidence. If not introduced as class-books in the school, they may
be used to excellent advantage in general exercises, and occasional class exer-
cise, for which every teacher ought to provide himself with an ample store of
materials. The volumes may be had separately; and the one first named, in the
hands of a teacher of the younger classes, might furnish an inexhaustible fund of
amusement and instruction. Together, they would constitute a rich treasure to a
family of intelligent children, and impart a thirst for knowledge."-Vermont Chron.
"Of all the numerous works of this class that have been published, there are
none that have acquired a more thoroughly deserved and high reputation than this of
series. The Chambers of Edinburgh, well known as the careful and intelligent pub-
lishers of a vast number of works of much importance in the educational world,
are the fathers of this series of books, and the American editor has exercised an
unusual degree of judgment in their preparation for the use of schools as well as
private families in this country."-Philadelphia Bulletin.
"The titles furnish a key to the contents, and it is only necessary for us to say,
that the material of each volume is admirably worked up, presenting with sufficient
fulness and with much clearness of method the several subjects which are treated."
-Cincinnati Gazette.

UNIVERSITY OF MICHIGAN
3 9015 04843 3992
B 535359
A. S. BARNES & COMPANY'S PUBLICATIONS.
NATURAL AND EXPERIMENTAL PHILOSOPHY,
FOR SCHOOLS AND ACADEMIES.
BY R. G. PARKER, A. M.
Author of " Rhetorical Reader,"
,"" Exercises in English Composition,"
Outlines of History," etc., etc.
44
1. PARKER'S JUVENILE PHILOSOPHY.
11. PARKER'S FIRST LESSONS IN NATURAL PHILOSOPHY.
III. PARKER'S SCHOOL COMPENDIUM OF
PHILOSOPHY.
The use of School Apparatus for illustrating and exemplifying the principles
of Natural and Experimental Philosophy, has, within the last few years, become so o
general as to render necessary a work which should combine, in the same course of
instruction, the theory, with a full description of the apparatus necessary for illus-
tration and experiment. The work of Professor Parker, it is confidently believed,
fully incets that requirement. It is also very full in the general facts which it pre-
sents-clear and concise in its style and entirely scientific and natural in its
arrangement.
"This work is better adapted to the present state of natural science than any
other similar production with which we are acquainted." Wayne Co. Whig.
"This is a school-book of no mean pretensions and no ordinary value."-Albany
Spectator.
"We predict for this valuable and beautifully-printed work the utmost success."
Newark Daily Advertiser.
"The present volume strikes us as having very marked merit."-N. Y. Courier.
"It seems to me to have hit a happy medium between the too simple and the
too abstract."-B. A. Smith, Principal of Leicester Academy, Mass.
"I have no hesitation in saying that Parker's Natural Philosophy is the most
valuable elementary work I have seen."-Gilbert Langdon Hame, Proj. Nat. Phil.
N. Y. City.
"I am happy to say that Parker's Philosophy will be introduced and adopted in
Victoria College, at the commencement of the next collegiate year in autumn;
and I hope that will be but the commencement of the use of so valuable an ele-
mentary work in our schools in this country. The small work of Parker's (Parker's
First Lessons) was introduced the last term in a primary class of the institution
referred to, and that with great success. I intend to recommend its use shortly into o
the model school in this city, and the larger work to the students of the provincial
Normal School."-E. Ryerson, Superintendent of Public Instruction of Upper Canada.
"I have examined Parker's First Lessons and Compendium of Natural and Ex-
périmental Philosophy, and am much pleased with them. I have long feit dissatis-
faction with the Text-Books on this subject most in use in this section, and am
happy now to find books that I can recommend. I shall introduce them imine-
diately into my school."-Hiram Orcutt, Principal of Thetford Academy, Vermont.
"I have no hesitation in pronouncing it the best work on the subject now published.
We shall use it here, and I have already secured its adoption in some of the high-
schools and academies in our vicinity."-M. D. Leggett, Sup. Warren Public Schools.