ELEMENTS OF NATURAL PHILOSOPHY. BY W. Hi. C. BARTLETT, LL. D., PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE UNITED STATES MILITARY ACADEMY AT WEST POINT. SECTION I. ME C HANIC S. NEW YORK: PUBLISHED BY A. S. BARNES AND COMPANY, NO. 51 JOIN-STREET. CINCINNATI: H. W. DERBY AND COMPANY. 1850 Entered, according to Act of Congress, in the year One Thousand Eight Hundred and Fifty, BY W. H. C. BARTLETT, In tle Clerk's Office of the District Court of the United States for the Southern District of New York. STEREOTYPED BY BILLIN & BROTHERS, No. 10 NORTH WILLIAM STREET, NEW YORK. F. C. GUTIERREZ, Printer, No. 51 JOHN STREET, OOR. OF DUTCH 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 comprehension of its contents, upon those which are to follow. In its preparation, constant reference was made to the admirable labors of M. PONCELLET, 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. CONTENT S. INTRODUCTION. Page Objects of Physical Science.....................9.................... 9 Physics of Ponderable Bodies........................................... 12 Primary Properties of Bodies....................................... 13 Secondary Properties....................................... 14 Force................................... 19 PART FIRST. MECIAN'ICS OF SOLIDS. I. Space, Time, Motion, and Fe................................ a Fore29 II. Action of Foiees, Equilibrium, Work................................ 42 III. Valried M~otion............................................... 67 IV. Forces whose Directions meet in a Point........................... 102 V. Forces whose Directions are Parallel. 123 6 CONTENTS. Page VI. Centre of Gravity of Bodies....................................... 134 VII. Motion of Translation of a Body or System of Bodies................... 150 VIII. Equilibrium of a System of Heavy Bodies................................. 162 Ix. Equilibrium of Several Forces, Virtual Velocities, and Motion of a Solid Body.. 167 X. Motion and Equilibrium of a Body about an Axis.......................... 180 Central Forces....................................................... 200 XII. Motion of the Heavenly Bodies...................... 230 XIII. Pendulum........................................ 241 Balistic Pendulum................................................... 266 XIV. Funicular Machine........... 271 XV. Action of Bodies resting upon each other, and upon Inclined Planes........... 299 XVI. Friction and Adhesion........................... a............... 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 XVII. Page The Wedge.................... 345 XVIII. Stiffness of Cordage.................... 379 Table of Rigidity of Cordage........................................... 382 XIX. Wheel and Pulley.......................... 387 Wheel and Axle............ 406 XX. Screw... 428 XXI. Lever............. 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................. 493 Table of Tenacities... e............................... 508 8 CONTENTS. V. Page Equilibrium of Floating Bodies....................................... 510 VI. Specific Gravity............... 517 Table of the Densities of Water at different Temperatures...... 522 Table of Specific Gravities......................... 531 VII. Compressible Fluids................................................. 536 VIII. Air-Pump...... 540 IX. Weight and Pressure of the Atmosphere...... 550!Mariotte's Law................................................. 556 XI. Law of the Pressure, Density, and Temperature........................ 559 XII. Barometer.................................. 556 Table for Finding Height by the Barometer.............................. 578 Water-Pumps..........6..... 582 XIV. Siphon........................................ 595 XV. Motion of Fluids.............................. 598 XVI. Motion of Gases and Vapors..................................... 606 Table of Aperture Coefficients............................ 618 XVII. Discharge of Fluid through Pipes........................ 622 ELEMENTS OF NATURAL PHILOSOP Y. INTRODUCTION. THE term nature is employed to signify the assemblage Nature. of all the bodies of the universe; it includes whatever 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 are in operation to produce them; and to investigate Physical sclence. nature with reference to these changes and their causes, is the object of Physical Science. All bodies may be distributed into three classes, viz.: Classification of unorganized or inanimate, organized or animated, and the bodies. heavenly bodies or primary organizations. The unorganized or inanimate bodies, as minerals, Inanimate water, air, form the lowest class, and are, so to speak, bodies, no (definite period, 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. The organized or animated bodies, are more or less Animated bodies, perfect individuals, possessing organs adapted to the per- orbansYitality formance of certain appropriate functions. In consequence of an innate principle peculiar to them, know as vitality, bodies of this class are constantly appropriating to themselves unorganized matter, changing its properties, and 10 NATURAL PHILOSOPHY. deriving, by means of this process, an increase of bulk. Reproduction, They also possess the faculty of reproduction. They and limited duration. retain only for a limited time the vital principle, and, when life is extinct, they sink into the class of inanimate Animal and bodies. The animal and vegetable kingdoms include all vegetable the species of this class on our earth. Celestialbodies; 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 olrgans- on the grandest scale. Their constituent parts may be continents, ocean, compared to the organs possessed by bodies of the second atmosphere. 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. Earth existed The earth supports and nourishes both the vegetable landbanimalants and animal world, and the researches of Geology have 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 animals pprt nd called into being before any of the organized bodies which vegetables. 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. Naturaatural P7ilosophy, or Physics, treats of the general hilochangesophy, 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; Ch/emistry, 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 substance of the body is altered and remodelled; and lastly, it detects and classifies the laws by which chemical changes are regulated. NIatural ffistory, is that branch of physical science Natural Historywhich treats of organized bodies; it comprises three anatomy, 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. 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 constitultion and physical condition, their mutual influences and actions on each other, and generally, seeks to explain the causes of the celestial phenomena. 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 ysicallaws. 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. Natural philosophy is a science of observation and ex- Natural periment, for by these two modes we deduce the varied philosophy, a science of information we have acquired about bodies; by the observation and former we notice any changes that transpire in the condi- t. tion or relations of any body as they spontaneously arise 12 NATURAL PHILOSOPHY. without interference on our part; whereas, in the performance of an experiment, we purposely alter the natural arrangement of things to bring about some particular condition that we desire. To accomplish this, we make use Apparatus; of appliances called 2philosoplhical or chemical cpparatus, the pheics.ental 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 experiment. means of freezing mixtures or evaporation, we cause water to freeze, we are then said to perform an experiment. These experiments are next subjected to calculation, by which are deduced what are sometimes called the laws re. of nature, or the rules that like causes will invariably produce 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 explanations or hypotheses. Hypotheses and A hypothesis gains in probability the more nearly it probatbilth of accords with the ordinary course of nature, the more numerous the experiments on which it is founded, and the more simple tlhe explanation it offers of the phenomena for which it is intended to account. PHYSICS OF PONDERABLE BODIES. Physical ~ 1.-The physical p2roperties of bodies are those exspeties.; the ternal signs by which their existence is made evident to our minds; the senses constitute the medium through which this knowledge is communicated. All the senses All our senses, however, are not equally made use of not equally for this purpose; we are generally guided in our decisions employed by the evidence of sight and touch. till sight alone is by the evidence of sight and touch. Still sight alone is INTRODUCTION. 1 frequently incompetent, as there are bodies which- cannot be perceived by that sense, as, for example, a11 colorless gases; again, some of the objects of sight are not substantial, 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. The properties of bodies may be divided into primary Primary and or principcl, and secondary or accessory. The former, are p'odpiarysof 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 existence, but become known to us by investigation and experience. PRIMARY PROPERTIES. ~ 2.-The primary properties of all bodies are extension and imnpenetrability. Extension is that property in consequence of which Extension; every body occupies a certain limited space. It is the lerngth, breadth, and thickness. condition of the mathematical idea of a body; by it, the volume or size of the occupied space, as well as its boundary, or figure, is determined. The extension of bodies is expressed by three dimensions, length, breadth, and thickness. The computations from these data, follow geometrical 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 impenetrble. this property is common to all bodies of whatever nature. 14 NATURAL PHIILOSOPHY. If a hollow cylinder into which a piston fits accurately, be filled with water, the piston cannot be thrust into the Experiment. water, thus showing it to be impenetrable. Invert a glass tumbler in any liquid, the air, unable to escape, will prevent 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 illustrate this fact: In one mouth of a twonecked bottle insert a funnel Fig. 1. a, and in the other a siphon b, the longer leg of which is im- \ a mersed in a glass of water. Experiment. NOW let water be poured into the funnel a, and it will be seen that in proportion as this water descends into the vessel F, the air makes its escape through the tube b, as is proved by the ascent of the bubbles in the water in the tumbler. SECONDARY PROPERTIES. Secondary The secondary properties of bodies are comnpressibility, properties. expansibility, porosity, cdivisibility, and elasticity. ~ 3.-Compressibility is that property of bodies by Compressibility, virtue of which they may be made to occupy a smaller expansibility. 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. INTRODUCTION. 15 Both changes are produced in all bodies, as we shall Change of presently see, by change of temperature; many bodies temperature, pressure, may also be reduced in bulk by pressure, percussion, &c. percussion. ~ 4.-Since all bodies admit of compression and expansion, it follows of necessity, that there must be interstices between their nminutest particles; and that property of a body by which its constituent elements do not completely fill the space within its exterior boundary, but leaves holes or pores between them, is called Porosity. porosity. The pores of one body are often filled with Pores filledwith some other body, and the pores of this with a third, as in other bodies. 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. all bodies and all In many cases the pores are visible to the naked eye; Visible and in otlhers they are only seen by the aid of the microscope, invisible pores. and when so minute as to elude the power of this instrument, 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. ~ 5.-The divisibility of bodies is that property in Divisibility. consequence of which, by various mechanical means, such as beating, pounding, grinding, &c., we can reduce 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- Smallnessofsome nished with numerous instances among natural objects, naturalobjects. 16 NATURAL PHILOSOPHY. whose existence can only be detected by means of the most acute senses, assisted by the most powerful artificial aids; the size of such objects can only be calculated approximately. Mechanical Mechanical subdivisions for purposes connected with the artbdsis 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 property of bodies. Divisibility of Some metals, particularly gold and silver, are suscepgold. tible of a very great divisibility. In the common gold lace, the silver thread of which it is composed is covered with gold so attenuated, that the quantity contained in a foot of the thread weighs less than 6 of a grain. An inch of such thread will therefore contain ~ 2 of a grain of gold; and if the inch be divided into 100 equal parts, each of which would be distinctly visible to the eye, the quantity of, the precious metal in each of such pieces would be -7I000 of a grain. One of these particles examined 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. Divisibility of Dyes are likewise susceptible of an incredible divisidyes. bility. 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 subdivided 62 millions of times. 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. threads. The single threads of the silkworm are also of an extreme fineness. 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. 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 thousands 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, investigations. in Bohemia, consist almost entirely of them. By microscopic 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 187 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 dimensions of these organic forms. In cases where our finest instruments are unaile to Divisibility render us the least aid in estimating the minuteness of detected by smelling. 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 P IIILOSO PHY. 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 astonished at their number and minuteness. Oil of lavender. In like manner a single drop of oil of lavender evaporated in a spoon over a spirit-lamp, fills a large room with its fragrance for a length of time. ~ 6. —Ilasticity 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 Elasticity, its very unequal tenacity. The measure of a body's elasticity, measure. is the ratio obtained by dividing the capacity of restitution 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 R 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 Examples of highly elastic bodies; viz., glass, tempered steel, ivory, elastic bodies. whalebone, &c. Let an ivory ball fall on a marble slab smeared with Experiment with some coloring matter. The point struck by the ball ivory. 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 considerable elasticity; this is particularly the case with meltedl Elasticity of some metals, more evidently sometimes than in their solid state. The following experiment illustrates this fact with regard to antimony and bismuth. Place a little antimony and bismuth on a piece of mJeltedbismuth charcoal, so that the mass when melted shall be about andantimoy. 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. cannonballs. FORCE. ~ 7.-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. Of the actual nature of forces we are ignorant; we Ignorant of their know of their existence only by the effects they produce, natue; existence 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, ~ 8.-We shall find, though not always upon superattractions, and ficial inspection, that the approaching and receding of repulsions. bodies or of their component parts, when this takes place apparently of their own accord, are but the results produced by the various forces that come under our notice. In other words, that the universally operating forces are those of attraction and of repulsion. Atomical action; ~ 9.-Experience proves that these universal forces are gravitation. 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 aggregate; the former distinguished as Atomical and Molecudar action, the latter as the Attraction of gravitation. Force of cohesion ~ 10. —Molecular forces and the force of gravitation, dissolution 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. Inertia, ~ 1. —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 condition 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 Known by amount of change. We derive our knowledge of this experience; principle solely from experience; it is found to be compassive in character. mon to all b6dies; 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 Actionequalto place, equal and directly opposed to each other. reaction. ~ 12.-Molecular action chiefly determines the forms Forms of bodies of bodies. All bodies are regarded as collections or determined by molecular action aggregates of minute elements, called atoms, and are formed by the attractive and repulsive forces acting upon them at immeasurably small distances. Several hypotheses have been proposed to explain the Constitution of constitution of a body, and the mode of its formation. bodies; 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 Firstpostulate. atoms. 2. These atoms are endowed with attractive and repul- Second postulat, 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, if a attract b, b must attract a with precisely the same force. 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 sensible contact is observed, and which do not exceed the Fifth postulate. 22 NATURAL PHILOSOPHY. 1000th or 1500th part of an inch, there are many alternations 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. sixth postulate. 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. Molecule, Two or more atoms may be so situated, in respect patile, body. to position and distance, as to constitute a molecule. Two or more molecules may constitute a particle. The particles constitute a body. Add inertia. 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. Exponential Boscovich represents his law of atomical action by c've. what may be called an exponential curve. Let the disFig. 2. c Br Dz.mI tance of two atoms be estimated on the line CA C, 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. WVe can represent this intensity by the length of the line i, perpendicular to A q, and can express the direction of the force, namely, from i to A, Attractive because it is attractive, by placing i I above the axis A C. ordinates above. Should the atom be at mn, and be repelled by A, we can express the intensity of repulsion by n n, and its direc- Repulsive Lion from rn towards G by placing min n below the axis. ordinates bow This may be supposed for every point on the axis, and a curve drawn through the extremities of all the perpendicular ordinates. This will be the exponential curve or scale of force. As there are supposed a great many alternations of curveonopposite sides of axis. attractions and repulsions, the curve must consist of many brancles lying on opposite sides of the axis, and must therefore cross it at C', 1D', C", D", &c., and at G. All these are supposed to be contained within a very small fraction of an inch. Beyond this distance, which terminates at G, the force Force of is always'attractive, and is called the force of grctvitation, 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 becomes infinite, when the distance from A is zero; whence the branch C' Q,, has the perpendicular axis, A y, 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 disturbed 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 indiffelene 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 N sATU R AL PHILOSO P Y. Limits of If the atom be at U', or C", &c., and be moved ever so cohesion. little towards A, it will be repelled, and when the disturbing cause is removed, will fly back; if moved from A, it Fig. 3. ___ D_'m 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 Permanent one of these points will, with that at A, constitute a molecule. 3permanent 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 Positions of limit of cohesion. The points D', D", &c., are also posiindifference. tions of indifference, in which the atom will be neither attracted nor, repelled by that at A, but they differ from UG, 0', C", &c., in this, that an atom being ever so little removed from one of them has no disposition to return Limits of to it again; these points are called limits of dissolution. dissolution. An atom situated in one of them cannot, therefore, constitute, with that at A, a permanent molecule, but the slightest disturbance will destroy it. BMolecules of It is easy to infer, from what has been said, how three, differentorders; four, &c., atoms may combine to form molecules of different orders of complexity, and how these again may be arranged so as by their action upon each other to form particles. 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 Attractionof atom in a spherical solid body is attracted towards the phericalmasses. centre by a force directly proportional to its distance from that point. The pressure towards the centre will, therefore, 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 elasticity. 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 body. their positions of rest by virtue of their inertia. It is only necessary therefore to suppose, that the Nebular heavenly bodies have been formed by the gravitation of hypothesis. 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 effets of motion. incandescent and self-luminous character of the sun. The'Incandescence same principle furnishes an explanation of the internal and luminosity of the sun. 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 thesun the other bodies of our system in regard to heat and light, greater because of simply because of his vastly greater size. 26 NATURAL PHILOSOP HY. 1g. —The molecular forces are the effective causes Effects of 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. P.:easure of This force is measured by the resistance it offers to cohesion. mechanical separation of the parts of bodies from eacb other. Three states of On the degree of this force, the three states or aggregation. gregate forms called solid, liquid, and gaseous depend. These different states of matter result from certain definite relations under which the molecular attraction and repulsion 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 proSolid, gas, liquid. cluces 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 liqidc form. Let A represent the attraction and R the repulsion, then the three aggregate forms may be expressed by the following formllula: A > R solid, Formulu. A < R gas, A= liquid. These three forms or conditions of matter may, for the most part, be readily distinguished by certain external External peculiarities; there are, however, especially between solids peculiarities of and liquids, so many imperceptible degrees of approximabodies; subject to change. tion, that it is sometimes difficult to decide where the one form ends and the other begins. It is further an ascertained fact that many bodies, (perhaps all,) as for instance water, are capable of assuming all three forms of aggregation. INTRODUCTION. 27 Thus, supposing that the relative intensity of the Change of molecular forces determines these three forms of matter, it molecular action in same body. 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 aeriform bodies come severally under our notice. ~ 14.-The molecular forces may so act upon the atoms c;otion of of dissimilar bodies as to cause a new combination or moec;ilarfolces between union of their atoms. This may also produce a separation dissimiiar 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 afIinity. Chemical affinity. ~15.-Beyond the last limit of gravitation, atoms Attraction of attract each other: hence all the atoms of one body attract bodies of sensible magnitude. 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- force erate motion when the bodies are free to move. Gravity, then, is a property common to all terrestrial Cravity common bodies, since they constantly exhibit a tendency to ap- to all bodies. Its consequences. proach the earth and its centre. In consequence of this 28 NATURAL PHILOSOPHY. 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 terrestrial bodies, and the cause of many phenomena, of which a fuller explanation will be given presently. SECTION I., MECHANICS. ~ 16.-That branch of Natural Philosophy which treats Mechanics, of the action of forces on bodies, is called lMechanics. MeStatics, chanics is usually divided into Statics, which treats of the mutual destruction of forces when applied to solid bodies; Hydrostatics, Hydrostatics, when applied to fluids; Dynamics, which treats Dyvnamics, of the motions of solid bodies; and Hydrodynazmics which Hydrodynamics. investigates the motions of fluids. Statics and Dynamnics Mechanics of will be treated together, under the general head, Mifesolids, chzanics of Solids, as will also EHydrostatics and Iydroand of fluids. dynamics, under the head, lfechanics of Fluids. PART FIRST. MECHAN I CS 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 tine, 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. The Instruments usually employed in measuring time Time are clocks, chronometers, and common watches, which are instruments. too well known to need a description in a work like this. The smallest division of time indicated by these timepieces is the second, of which there are 60 in a minute, 30 NATURAL PHIILOSOPHY. Performance of 3600 in an hour, and 86400 in a day; and chronometers, chronometer 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. Thus the number of hours, minutes, or seconds, between 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 distance. Time represented Time may be repby lines. resented by lines, by laying off upon a A -_ i2_ —— __ ~7 given right line A B, the equal distances firom 0 to 1, 1 to 2, 2 to 3, &c., each one of these equal distances representing the unit of time. Rest; ~ 19.-A body is in a state of absolute rest when it continues in the same place or position in space. There is relative. perhaps no body absolutely at rest; our earth being, without cessation, in motion about the sun, nothing connected 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. Example of A body, for example, which continues in the same place relative resti 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. Motion, like rest, ~ 20.-A body is in motion when it occupies succesisrelative. 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. It is continuous. 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 undlerstand that of a certain point connected with the body. Thus, the path of a ball, is that of its centre, &c. ~ 21.-The motion of a body is curvidinear or recti- Curvilinear and linecLr, according as the path described is a curve or mctilea right line. When the motion is curvilinear, we may consider it as taking place upon a polygon, of which Directionof the sides are very small and sensibly coincide with the body's motiton 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 Uniformmotion. 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 muniform. If otherwise, the motion is said to be varied. It is acceleratec 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 varn- 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 obvious that the space described in any given time will 32 NATURAL PHILOSOPHY. 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. Relation of space Denote by S the length of space described during to the time. the time T; s the length of the space described in the small portion of time t, then, from what precedes, we have: T:: s: t S s,.(1). a constant ratio. ~ 24.-Since in uniform motion, the spaces are proportional to the times employed in describing them, the Velocity velocity may be measured by the space described in any measured by the time whatever, for example in a second, minute, an hour, space described in any unit of &c. Thus we say the velocity is 2 feet a second, or 120 time. feet a minute, or 7200 feet an hour, or 2 of a foot in - of a second, &c; all of which amounts to the same thing, since the ratio of the space to the time is not changed. Rule for finding When a body describes uniformly a certain space in velocity. a given number of units of time, as the second, for example, which is usually takenr 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 V=..... (2). Example. Exam ple: The space described in I minute and 5 seconds or 65s being 260 feet, the space described in 1S, or the velocity, is given thus: MECHIANICS OF SOLIDS. S _ 260f V T 65 Reciprocally, if the velocity be multiplied by the number of units of time, the space will result. ~ 25.-It frequently happens in practice that the ve- Periodical locity is not constant, although the spaces described at the motion end of certain equal intervals are equal. Such for instance is the case in all periodical movements of which the different 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 — carriage and trian, are examples of this; the spaces described in eidean pedestrian. certain intervals, are often the same, while the motion is sometimes accelerated and sometimes retarded. ~ 26.-Conceive a table consisting of two vertical Relation of space columns, in one of which are arranged the numbers ex- epresented 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 0 B; assume any linear dimensioni as an inch, to repre-mot"' 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 0 t4 representing an interval of time given by the table; upon a perpendicular to OB at the e point t4, lay off a distance t4 e representing the distance passed over by the body in the time 0 t4. Do Io _II.I the same for the other times and corresponding spaces of the table, and we obtain the points el, e2, e3, &c., 34 NATURAL PHIILOSOPHIY. 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. 0, 0 t2, 0 t3, &c., are the abscisses, and el,, t2 e2, tSe, &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 described 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. Inluniform In uniform motion the spaces increase in the direct motion. ratio of the times, and the ordinates el, t2 e2, 13 e3) &c., are therefore proportional to the abscisses 0 4, Ot2, O t3, &c.; hence the curve becomes Fig. 6. a right line. Let the axis OB, of timnes, be divided into e4 any number of equal and very small parts; through J~ the points of division draw o B the ordinates or spaces, and 2.? 4 through the extremities of the ordinates draw the lines el b2, e2 b3, e3 b4, &c., parallel to the axis of times, we shall thus form a series of small right-angled triangles O01 el, e b.e2, &c., similar to the triangle 0 t4 e4, and because e3 b4 = t3 t4, we have t4e4: Ot4:: b4e4 t3 t4, whence t4e4 b4 e4 0 t4 t3 4 Relationofspaces but b4 e4 is the space s, described in the small time t3 t4 = t, to the times. and t4 e4 the space S described in the time 0 t4 -= T and the above may be written iECIHANICS OF SOLIDS. ~5 S S and making t= 1, s becomes the measure of the velocity V, and we have Tr- So Velocity equal to — T the ratio of the space to the time. the same as before, equation (1). Or, 0 t4 may be taken as the unit of time, in which case, t4 e4 becomes the velocity V; and we have Y _ S Same for any V t' space and time, In varied motion, the spaces not being proportional to Varied motiont the times, the line 0 e, el e e2e3, &c., is not straight, and the small spaces e2 b2, e3 b3, &c., described in Fig. 6. the elementary times,t t2, t12 1, &c., are not equal. The velocity must, therefore, vary e Acclratd a * 6 as Accelerated at every instant. For motion, represented the case represented e ometrically by the figure, the mo- I tion is accelerated, because the spaces e2 b2, e3'b3, &c., described in the equal elementary times, continually increase. Now let it be supposed that at the point e3 the motion ceases to be accelerated, and Motionceasesto that it becomes uniform with the velocity which the beccelerated; body had at this instant. The law of the motion afterward will be represented by the right line e3 m, the pro- becomes longation of e3 e4, and since, at the instant we are uniform considering, the body describes a space equal to e4 b4 in the elementary time e3b4 = t3 t4 it will, in virtue of 36 NATURAL PHILOSOPHY. its uniform motion, describe in a unit of time a space equal to mun, obtained by laying off from the Fig. 6. point e3, on e3 b4 produced, a distance e3 n equal to the unit of e Measure of the time. But the space,. j | inaint at any described in a unit of instant; e. time, at a constant e, 4 rate, is the measure...6 2 of the velocity corre-. 7a... ~ — spending to the point e3, or at the end of the time 0 t3. From the figure we obtain e4b4: e3:: mu e~8n; or making e4b4=8,8 e3 b4=t, mn= e3n=1, we have s' t'' V' 1; equal to the ratio whence of the element of the space, to the S element of the - time. as before. If we suppose the element of time t3 t4 sufficiently small, the line e3 e4 will coincide with the curve to which Tangent line; e3 in will become a tangent at the point e3. This tangent will give the being constructed geometrically, will give, in the manner velocity. 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 0 t3. MECH]ANICS OF SOLIDS. 37 Periodical motion, such as has been defined in ~ 25, will be rep- Fig. 7. resented by a waved line EE, &c, whose undulations are regu- G t larly disposed about the right representation of line eq, e 3, e, &c., which repre- periodical -~~J ~ ~ motion. sents the law of unyforn motion. It may be important to remark that the curves which have 2 just been described, and which connect the lengths of the spaces and the times, in any kind of motion, must not be con- Distinction founded with the actual path described by the body. betweenthe line giving the law of In this last, the tangent simply gives the direction of the the motion, and motion; and to obtain the velocity, the elementary por- desribed bythe 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 of itself the motion which it may have received. A body Fig. 8. at rest will forever A remain so unless disturbed by something 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 cot chng 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. If a billiard-ball, thrown upon the table, seem to Apparent exception 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 removed, will the motion itself remain unchanged. It results, from what has been said, that when a body is put in motion and abandoned to itself, Consequences of its inertia will cause ilertia. it to move in a straight line and preserve its rate of motion unchanged. If, from any extraneous cause the body is made to describe a curve A B, 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. IJorces; Weight ~ 28. —A force has been defined to be that which bid heat. 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.:itlustration. A body exposed to any source of heat, expands, its particles recede from each other, and thus the state of the body is changed.:'orces produce ~ 29. —Forces produce various effects according to cirvarious effects. curnstances. They sometimes leave a body at rest, by destroying 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 produced gradually, sometimes abruptly, but however produced they require some definite time, and are effected by con- Theseeffects tinuous degrees. If a body is sometimes seen to change require definite 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 duration imperceptible to our senses, yet some definite portion 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, a canon. or a sheet of paper when freely suspended, with a rapidity so great Fi. 1o 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 c — its ball to the same point as though it were on its carriage, which proves 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 are not. 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, corresponding to these effects, have passed in succession through their different degrees from the beginning to tthe ending. Forces which give motion to bodies are called motive Motiveforces; forces; they are accelerating when they accelerate the accelerating and retarding. motion at each instant, and retarding when they retard it. ~ 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 offorces obtained friom experience. 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. Forces are real weight, and thus forces are to us real pressures. Pressure piessures; may be strong or it may be feeble; it therefore has 9nagnitude, and may be expressed in numbers by assuming a unit of force. certain pressure as unity, which may easily be clone if we Equal forces. Can find pressures that are equal to each other. 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 TV, suspended from the extremity of a thread; the thread will as- Fig. 11. Forcesmeasured sume a vertical direction, and an effort will As by weights. 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 0B( Double, triple, weight of the body. A double, triple, &c., &c., force. force, will support two, three, &c., bodies, similar to the first, suspended one above ig. 12. another on the same thread; taking one of these forces, that, for instance, which supports -g,th of a cubic foot of distilled water at the temperature of 60~ Fahrenheit, and Unitofforcea 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 support. Forces compared ~ 31.-Weights are measured and compared by means bythebalance. f 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 whatever 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 conseqluently the magnitudes of forces. Springs, among others, in Use of spring supposing they preserve unim- Fig. 13. balanceto measure forces. 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 Veriflcation of the graduations if the elasticity of elasticity. the spring shall be found to have undergone a change since its construction. ~ 32.-It is known from observation that the action Variationin force of the force of gravity diminishes as the bodies upon ofgavinty small 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 -; 1 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. 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 ofthe limits of therefore distant about 4000 miles from the place of ordinary bodies. 4' 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 inappreciForce of gravity able to common instruments. It hence follows, that, constant, and acts within ordinary limits, the force of gravity may be regarded in parallel directions. as constanrt, and acting in pa.rallel directions. II. ACTION OF FORCES, EQUILIBRIUM, WORK. Action of exterior ~ 33.-WVhen a force acts against a point in the surface forces on bodies; 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 rewhen some of the motest particles of the body. If these latter particles are particlesare fixed. fixed or prevented by obstacles from moving, the result will be a compression and change of figure throughout the WTihen none of the body. If, on the contrary, these extreme particles are particlesarefixed. free they will advance, and motion will be communicated 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 cannot be generatedt, 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 ~ 34.-As the molecular springs cannot be compressed and contrary to action. 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. Two Fig. 14. CA, Illustration. weighing springs attached to the extremities of a thread or bar, will indicate the same degree of tension, and in contrary directions when made to act upon' each other through the intervention of the thread or bar. ~ 35.-In every case, the action of a force is trans- Point of mitted through a body to the ultimate point of resistance, application take at any point in by a series of equal and contrary actions and reactions lineofdirection. 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. ~ 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 NAT URAL PHILOSOPHY. regarded asrigid which are usually employed to transmit the action of and issextensible; ibe; 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 ~ 37. —We have just seen that when a force acts upon by means of forces; 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 proportional 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 Fig. 15. thread, the thread will stretch and even break if the action be'too violent, and this will the more probably happen in proportion as the body is more massive. conduct of a If a body be suspended by means spring when under the action of a vertical cord, and a weighing of inertia; spring be interposed in the line of traction, the graduated scale of the spring will indicate the weight of the: body when the latter is at rest; but if we suddenly elevate the upper end of 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 cohaon of become uniform, the spring will resume and preserve motion; 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 climinishedl, the M3[ECHANICS OF SOLIDS. 45 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 ispdiingte the motions of a body, and the energy of its force of changes in motion, inertia, which acts against or with a power exerted in the direction of the motion, according as the velocity is increased or diminished. ~ 38.-The effect of every force depends, 1st, upon its stfect of a force; point of acpplicction; that is, the point to which it is point of application; line 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- intensity. 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 A B; from A, lay off upon Graphical the direction in representation of a force; which the force acts, a distance AP, containing as many linear units, say inches, as there are pounds in the -. —intensity of the 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 A Q, 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 bysymbol geometrical figures. lI46 NATURAL PHILOSOPHY. Equilibrium of ~ 40. —Vhen the forces applied to any body balance, or forces; mutually destroy each other, so as to leave the body in the same state as before their application, these forces are said statical and to be in equilibrio. The equilibrium may be statical or dynlamical. 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 Illustration — motion. Two men pulling with equal strength at the optwo men. 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. No case of In reality there is no case of absolute statical equiabsolute rest. 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 necesRepose not sary to that of an equilibrium, which only requires the lecessary to mutual destruction of all the forces which act at the same equilibrium. instant upon a body. Forces in ~ 41.-When a body, subjected to the action of several equilibrio; 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 chanesth mot.io equilibrio; but if we take into account the force of inertia of the different particles of the body, and introduce among Effect of inertia the extraneous forces one equal to it and capable of preon equilibrium of forces. venting the modification of the motion, there will again be an equilibrium among all the extraneous forces. A horse Illustration- which draws a carriage along a road, destroys at each horse and instant all resistances which are opposed to his action; if carriage. 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 conme into action and add to the other resistances 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 overcome all the resistances, or to maintain the equilibrium. Thus inertia stands always ready to maintain an equi- Inertia alw-ays librium among forces of whatever nature; and hence the ready to establish an equilibriunm 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 Lupon by a force, it will move in the direction of this force, while the force itself is maintained in equilibrio by the inertia developed during the yielding of the point. Action and reaction are equal and contrary. ~ 42. —When an equilibrium exists among several Reaction equal forces, as 0, P, Q, &c., one of them, as 0, may be con- and contraryto action. sidered as preventing the effect of all the others. If, then, we conceive a force R, equal and directly Fig. 17. opposed to 0, at the same point of appli- R cation C, this force will destroy of itself the force 0, and will therefore produce the same effect upon the body as 0 ~e Resultant of the forces P, Q, &c., Rsfotrces, taken together. This components of a force. force R is called the resultant of the forces, P, Q, &c., and these latter the conmponents of the force R. Reciprocally, if to the resultant R of several forces P, Q, &c., an equal force 0, be immediately opposed, there will be an equilibrium between this force and the several 48 NATURAL PHILOSOPHY. Resultant and forces P, Q, &c.: hence, the resultant is a single force components 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. Resultant of ~ 43.-When several forces act along the same straight several forces line and in the same direction, their joint effect will obacting along the same line. 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. Mechanical wolk ~ 44.-The most simple case of equilibrium, is that in of forces. 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. Roesisacean Mechanical work not only supposes a resistance overreproduced. 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 mnst 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 resistance 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, Workincreases at each instant, increases with the intensity of the effort and andhpath the length of the path described by its point of appticatwon in described by the the direction of the efort. point of *edrcinof tefor.application. ~ 45.-Let us suppose a constant resistance and, there- Measure of the fore, a constant effort which is equal and directly opposed work when the resistance is 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 PHILOSOPHIY. path described by the point of ap2plication of the resistance. Illustration. 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 described by the point of action, we shall have Equation of the quantity of work. To represent this geometriGeometrical cally, assume any linear unit, representation of Fig. 18. thequantity of aS the inch, to represent 1 work. pound, and the same to represent the unit of linear length; lay off from 0 on the indefinite right line 0 B, the distance 0 el, equal to the length ~ e B of path described by the point of action, and at el, the perpendicular el r1, containing as many inches as the constant effort contains pounds; then will the number of square inches in the rectangle 0 el rl r, express the quantity of work. Work when the ~ 46.-If the resistance, or the equal effort which devariable. 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 tMECHANICS 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, will be measured by the sum of all these elementary products. Draw the curve r, rz, r2, r,, &c., of which the abscisses Oe, 0 el, 0 e2, 0e3, &c., shall represent the spaces described by the point of action of the resistance up to certain Fig. 19. given successive in-. stants of time, and n"'.' of which the ordi-,. nates e r, ne r, e2 r2, e3 r3, &c., shall rep-' resent the corre- e i rz;{is,' i':' Represented by sponding resistan- S geometry. ces. Let e el, e e,.:e2e3, &c., be the equal and very 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 constant for each one, that is, by the products ee Xer, e, e2 X el rl, e2 e3 X e2 r2, these elementary portions of work are represented respectively by the elementary areas e r sl e1, e rl s2 e2, e2r23 e3, &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. of division el, e2, e3, &c., by diminishing the distances eel, ele2, e e3, &c., it is obvious that the sum of the rectangles will not sensibly differ from the area included by the curve rrz1'r2...r7, the whole path e e7 described by the point of action, and the two ordinates er and e7 r7 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 an area. 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 any even number of parts, and erect the ordinates at the points of division, and at the extremities; number the Fig. 20. ordinates in the order of the natural Rule for finding numbers; add tothie area. gether the extreme ordinates, increase 4: e e e B this sum by four times that of the even 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 elementary areas, by drawing ordinates, as in the last figure, M:ECHANICS OF SOLIDS. 53 and to regard each of the elementary figures, q, e2 r2 rl, e2 e3 r3 r2, &c., as trapezoids; and it is obvious that the Demonstrationof the rule. error of this supposition will be less, in proportion as the number of trapezoids between given limits is greater. Fig. 21. Take the first two trapezoids of the Al preceding figure, and divide the distance el e3 into three equal parts, and at the points of division, erect the ordinates n n, mnl n; the area computed l - -- from the three trapezoids el m n r1, m m1 nl n, mn e3 r3 11, will be more accurate than if computed from the two el e2 r2 r1, e2 e3 r3 r2. The area by the three trapezoids is e r mn u m n+ - m ml em ml n1 + e3 r3 2 2 2 But by construction, el m = mm = m1 e3 -e e3 = -el e2, and the above may be written, el e2 (el rl + 2 mn + 2 ml n + e3r3), but in the trapezoid mm nm n, 2mnn + 2mlnl = 4e2r2, very nearly; whence the area becomes 3 ee2 (erl + 4e2r2 + e3r3); the area of the next two trapezoids in order, of the preceding figure, will be el e2 (e3r3 + 4e4r4 + e5r ); 54 NATURAL PHILOSOPHY. and similar expressions for each succeeding pair of trapezoids. Taking the sum of these, and we have the whole area bounded by the curve, its extreme ordinates, and the axis of abscisses; or Algebraic expression of the Q = 3 ele2 [elr1 + 4 e2r2 + 2e3r3 + 4 e4r4 + 2 e5r5 + 4e6r6 + e7 r7]; rule. whence the rule. ~ 47.-When the value of the mechanical work of a variable resistance for any distance passed over by the Iean resistance; point of action, is found by the method just explained, if this value be divided by the distance, the quotient will equalto the entire be a mean resistance, or the constant effort which, exerted work divided by through the entire path, will produce the same quantity the entire path. 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. Examples of ~ 48.-When a motive force is employed to bend a mechanical mechanical spring, it will develop, at each instant, an effort which is greater in proportion as its point of action describes, that of a force in the direction of the effort, a greater path; an effort belding 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., We have already taken as an example the work produced by a constant force in drawing a body over a horizontal plane, and above we have taken the work which arises from the action of a variable force in bending a spring; of thedraftof a the reasoning applied to these is applicable to all kinds horseo of work employed in the arts. Does a horse pull upon of t he effort of water from a well; a man, the shaft of a mortar mill; a man draw water from a well; 4ECE0IANICS OF SOLIDS. 55 an artificer saw, plane, file, polish; a turner fashion his Of the materials in the lathe; the quantity of work performed manipuilatins of an artificer, 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 portions of work, if the resistance is variable. ~ 49.-In seeking to appreciate different kinds of work, Distinction to be we must be careful not to confound that which is really observed inwork. 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 measurements. 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. ~ 50.-To show the complication incident to certain Complication kinds of mechanical work, take, for example, the work inherent in certain kinds of 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 opposed 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 Theworkreduced manner already described. The work of the operator and measured as before. may be reduced to this, by supposing the file placed upon a level surface, loaded with a given weight, and the 56 NATURAL PHIILOSOPHY. operator or motive power only employed in drawing it uniformly in the direction of its length. What nus ose 51. —In general, then, we must henceforth understand understood by by mechanical work, that which results from the simple mechanical work of a force; 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 work of the same things of the resistance which it overcomes at each resistance. instant, in each particular case of its application. ~ 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 Invariable of inertia. The work in this case obviously increases as standard by which to estimate weight and vertical height increase, and is measured the quantity of by the product of the two, agreeably to what is said in work; ~ 45 and ~ 46; here the unit of work, is the unit of weight raised through a unit of height. 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, utility of this as, for instance, the quantity of work necessary to grind standard. 1, 2, or 3 pounds of corn, which is the old standard of MECHANICS OF SOLIDS. 57 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 millers; 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 consideration of the effort, and the path described by its point of action in the direction of the effort. It will remain to be found how many pounds of corn Means of this unit of work is capable of grinding, how many square comparing 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 distinguished from the object accomplished, this latter being but its effect. SMEATON calls it mechanical power; CARNOT, moment Different names of activity; MONGE and HACHETTE, dynamic effect; Cou- given to mechanical LOMB, NAVIER, and others, quantity of action; and this work; last expression is now generally adopted. It will hereafter be employed, and will always signify the quantity of work-mechanical work. Sometimes the mechanical work has been called quan- sometimes called tity of motion, and sometimes living force, both of which are quantity of but simple effects of mechanical work upon a body free force. to move. We shall explain, in the proper place, the meaning 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 eth useful but we have seen that this quantiy of result is propor. 58 NATURAL PHILOSOPHY. tional to the quantity of mechanical work necessary to its production, and hence mechanical work or quantity of action is what pays in forces. To express the ~ 54.-When a motive force acts with a constant effort, continued work conumbersd wo; and its point of action moves uniformly during any considerable 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 comlparing 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 work in unit 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 duration of the effort. 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. The path Ordinarily, we take for the length of path, that which is described in a described in one second, this latter being taken as the unit second is usually taken; 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 coincidence, 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 causthe consequences ing them, as we shall see further on, to confound -the of this. qGantity of work or of action with the uanctity of motion, although their measures are in fact very different. All units In the same way that the unit of time is arbitrary, so arbitrary; 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 pound; unit of distance. We shall take for the unit of effort 1 pound, distance, one foot; and for the unit of distance 1 foot, so that the unit of MECHANICS OF SOLIDS. 59 work will be, as before, the effort one pound exerted through unit of work, the product of these; a distance of one foot. Suppose, for example, that the effort 75 pounds is exerted through the distance 4 feet, then will 4 X 75 = 300 units of work, 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'fe'ence to time.. expressed thus, 3001b. f.; and is read, 300 pounds raised through 1 foot. And this has no reference to the time in which the work is performed. ~ 55.-Mechanicians long felt the necessity of some Differentunitsof well defined unit by which to express the work performed, work proposed, 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 likelihood 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-powver 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 trough 1 lfoot in 1 second; through 1 foot in 1 minute, or 1,980,000 pounds raised 60 NATURAL PHILOSOPHY. through 1 foot in 1 hour; all of which amount to the same thing. WVhen, then, we are told that a machine or engine is of 30-horse power, or has a power equal to 30, for inExample. 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. Error of ~ 56.-We can now appreciate the error we should greatest effort commit, if, in estimating the productive power of a motive alone; 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 example, in estimating the productive effort of a man, we only consider the greatest burden he is capable of supporting 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 this effort may clear from the consideration that we may in all such cases xed obacle; by replace the moter 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 error of quantity of action from the path described by the point of considering the action, without taking into account the effort exerted at path alone. h 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 producing no useful effect; he overcomes no resistance external to himself, which it can be an object to destroy. MECHANICS OF SOLIDS. 61 In a word, the productive effect of every motive force is Productive measured, at each instant, by the product of the effort ablitebte into the path described in the direction of the effort; so product of the effort into the that, if either the effort or path be zero, the quantity of path. action will also be zero. ~ 57.-It must be remarked, however, that, since all Alwayssome bodies are more or less extensible and compressible, a fworkbteven; 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 resistances 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- Fig. 22. scribed, the curve O rl r r2... r6, of which the f~ abscisses O el, e e2, &c., represent the spaces de- r its value represented scribed by the point of r/ o, geometrically. action in each successive instant of time in the direction of the force, and the ordinates, the corresponding pressures or 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 e2e3, is the 62 NATURAL PILOSOPiHY, area of the trapezoid e2e3 r3r2, 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 e6 r6, which denotes the maximum resistance, and the axis of abscisses. If, then, it should happen that the body or obstacle is either compressed or extended by any appreciable quantity as 0 e6, which is the path described by the point of action, and the. greatest resistance e6 r6 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 most cases be reference to their capacity to resist all change of figure; neglected; so that when well chosen and judiciously disposed in combinations, 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. 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, comespecially when pelled to describe. The force in this case will only action and the compress or stretch the body uselessly, without adding to motion are at 1 right angles to or subtracting from the work in the direction of the each other, 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 ~ 60.-These considerations are important, as they Motive forces prove, in general, that forces may work without produ- mayi ork without useful 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 compressed 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 tornsmitting the 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. ~ 61.-We also see that if the action of the force or Lossgreater in moter, or the resistance occasioned by the work, undergo proportion as tho moter, or force changes. 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. ~ 62. -The shock of bodies develops considerable Still greater in pressure, and produces sensible changes of figure;he the case of shocks. quantity of action destroyed or generated will, therefore, always be appreciable. On this account it becomes indispensable, 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 NATURAL PHILOSOPHY. stiff and elastic from the use of very stiff and very elastic materials in the materials. construction of those pieces which are employed to receive and transmit the action of forces, and to regulate the motions they produce. ~ 64.-To obtain a clear idea how the molecular Elastic bodies springs of a body may develop or restore a certain restore, in quantity of mechanical work, we have but to consider expanding, the work absorbed what takes place at the instant when a body begins to in being resume, progressively, its primitive figure after it has been compressed. 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. Loss of work If, on the contrary, the body be not perfectly elastic, when the bodies it will not return to its former figure; the pressures will are not perfectly elastic. be less during the process of restitution, there will be a loss of space described by the point of action, and, consequently, less work performed than in the first change of figure, there will be a certain quantity of action lost. There are scarcely any perfectly elastic bodies except Examples of the gases and vapors, and these must be confined in a elastic bodies; 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 future use; forces are employed to compress or bend them, their use. in which state they are retained by mechanical contrivances 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, communicates it to something else, and so on to the ultimate object to be attained. The balistas, catapultas, and bows Examplesof the ancients, throwing arrows, stones, and other missiles blitas, bows, air-gun; 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 alternations of heating and cooling. No one is ignorant of steamand 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. ~ 65.- -Weight also affords the means of storing up Weight asa mechanical work, and of rendering it available when means of storing mechanical 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 Elevationof the ocean the form of vapor, ascends to elevated regions water by heat. 5 66 NATURAL PHILOSOPHY. 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 to break, &c., not destroy, in a word, the natural affinity of bodies. The reproduced. 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 Springs, like animals, and combustibles which give spndings, animals, heat, have this peculiarity, viz.: they are very portable, combustibles. and may be even used as a motive power for vehicles. Thus carriages have been put in motion by springs attached, 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 Nourishment work which they have received. Finally, animals, and and fuel heat even, the primitive source of all the mechanical representatives of mechanical work employed in the arts, require a certain expense in work. nourishment and fuel which, according to the beautiful theory of Leibig, are the same in principle. This nourishment and fuel become, therefore, the representatives of a certain amount of mechanical work, so that it is really impossible to create a motive force, without having previously incurred an equivalent expenditure. Inertia a source ~ 66.-Thus far we have only examined the work of of reproduction forces when employed to overcome the weight of bodies, of mechanical work. 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 MECHANICS OF SOLIDS. 6-l artificer must overcome the inertia of the matter of which his tool is made; the draft-horse, that of the carriage, and of the load it bears, &c. [But inclependently of this, it is very important to be able to estimate the quantity of work which a body will absorb in acquiring 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. Besides, 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 rilGe- Exnmpeesram, common hammer, &c.; the inertia of bodies is thus pi-."""eild made, like weight, elasticity, &c., to restore the quantity of ha),,me. 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. III. VARIED MOTION. ~ 67.-We will begin with the most simple case of Varieddmotion; varied motion, viz: that in mwhich a body is pressed by a'~~"/ for 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 words, the velocity will be proportional to the time 68 NATURAL PHILOSOPHY. r:iformly varied, elapsed since the commencement of motion. This is acelerated, and called uniformly varied motion in general; which becomes retarded. uniformly accelerated or uniformly retarded, according as the force increases or diminishes the velocity of the body. Uniformly 68. —First, take the case of uniformly accelerated acceleated; motion, and recall to mind that the velocity acquired at any instant is, ~ 25, 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. Let 0 be the point of starting. Draw the Fig. 23. line 0 vl v2... v6, of which the abscisses 0 4, 0 t, &c., represent the time Ail elapsed from the origin V., graphical or beginning of the mo- -..e'esentat~ tion, and of which the ordinates motion. t,,... t, ------ represent the velocities acquired at the end of the times 0 4, 0 t,... 0 t6. Since in uniformly varied motion, the velocities 4 v1, t2 V2,... t v6 are proportional to the times 0 4O, 0 t2,... 0 t6, the line O v v v3...,V is a right line, which passes through the point 0 from which the body takes its departure; for at this point, the velocity and time are zero together, at the instant of starting. The: distances 0t, 44t2, t2 t3 &c., being equal, if through the points vl, v2,,... v,, lines be drawn parallel to the axis 0 B of times, there will be formed a series of right-angled triangles, 0 4 vl, vl b2 v2,... v5 b6 v6, all equal to each other. The sides t4 vi, vA2 b, va b3,... v3 b6, 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 corresponding intervals of time 0 t, v b2, v2 b3,....v5 b, are equal. The successive intervals of time 0 1, t1 t2 t2, t3, &c., being Path represented supposed very small, we may regard the body as moving byethagl ho uniformly during any one of them as t3 t4 or its equal velocity into time. v3 b4, and with the velocity t3 v3 acquired at its commencement. 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 superficial units, and, in this sense, the distance passed over by the body in the time t3 t4, will have for its measure the product of this elementary portion of time by the velocity t3 r, or the area of the rectangle t3t4b4v3: for another interval t4 t, the path described will have for the measure of its length, the area t4 t5 b5 4, and so on; so that the total length of path described by the body during the time 0 t6, will be the sum of all the partial rectangles t1 t2bsvl, t2 t3 b3 Vi.... t5 t6 b6 v5; which sum will not differ sensibly from the area of the triangle 0 t6 v6, when the points of division tl, t2,... t, 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 uniformly varied motion. Since the area of the triangle 0 t6 v6, has for its measure, Laws of the half of its base into its altitude, and as the base niformlyv'i ried,motion. into the altitude, or the entire rectangle, represents the length of path described in the time Ot6, with a constant velocity t6 v6, acquired at the end of this time, it follows, 1st. In uniformly accelerated motion, the path describedFirst law. at the end of any time, is half that which the body would TOf NATURAL PHILOSOPHY. 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 Ot3, 0 rt, are represented by the triangles 0 t3 v3, O t5 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, Second taw. 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. hirid aw, 3dcl. That these paths are to each other, as the squares of the velocities acquired at the end of the corresponding tJmies. o,,lulato AVWhen in uniformly accelerated motion, the velocity ~,.ollplti.te t-r Z, acquired at the end of a given time 0 t5, say one of this totion. second, taken as the unit of time, is given, the law of the motion or the right line v6, which represents it, is completely determined, and we may compute the velocity and space which correspond to any other time. Denote by el and v%, the length of path and velocity which correspond to the first second, and by S and X, the path and velocity corresponding to any other time, as T; w-e have by the first law, Space'in tuti a el - V1 X I - (4) -re-latin1 of' space, - VT. ()); thue, and velocity; and by the second law, el: S:: ls X Is TX T e 1s T2; whence, 8pace in (any inej X= el X T'... (6); MECHANICS OF SOLIDS. 71 and replacing el by its value, Eq. (4), 8-= v1 2,...... (7). From the third law, Space in any time; el, or S 12 whence V = 2 vS..(8). Velocity due to any space. By the definition of uniformly varied motion, we have, v'l V:: is T; whence V = vlT..... (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 all acquired velocity; motion, passes through the point of departure 0. But if the body have already a velocity 0 vo, acquired previously, this right line Fig. 24. will pass through vo, the vX extremity of the ordinate V4/ which represents the ve- V3/ locity of the body at the instant from which the time is reckoned. The velocity Ov0, It...Vo t tAd t initial velocity. is called the initial velocity. By drawing v0 t'5, parallel to O B, we see that the I! i velocity t3 v3, which corre- sponds to the time Ot3, is o5 0 f B composed of two parts, viz. t t'3, and t'3 v3; the first is equal to the initial velocity Ovo, and the second to the 72 NATURAL PHILOSOPHY. Formulas to velocity which the body would acquire in the time vo t1', computee of equalto 0 t3, under the action of the constant force, had it circumstances of the motion; moved from the point v0 with no initial velocity, as in the preceding case; for the line v0 v5 gives, in reference to the line v t'5, 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 v0 v5, in relation to v0 t'5 or its parallel 0 t5, 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 O t4. This path will contain as many linear units as the trapezoidal area 0 t4 v4 v0 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 O vo, and represented by the rectangle 0 t4 t'4 v0, and that described in virtue of the constant force and represented by the triangle vot'4 v4. But, denoting by a the initial velocity, and by T the time, we have for the measure of the rectangle a T; and for the measure of the triangle, Eq. (7), ~ V1 1; and if we denote by S the total length of path actually described by the body, we have valueo S = aT + ~vlT2.... (10): space; 2 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), valueoft Vt a + th.. velocity. lV a + vx 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 retrded motion; it, the motion becomes uniformly retarded, and the line v0 v4 gives the law of the motion. By drawing v0 t'5 parallel to Ot5, we see that the velocity Fig. 25. V3 t3, which corresponds to the time 0 t3, is nothing else than the initial velocity Ovo dimin- ished by the velocity t'3 V3, graphical which the body would acquire representation. under the action of the constant force at the end of the, time 0 t3 had it moved from 0 1 4 rest. The length of path described is now represented by the trapezoidal area O t3 v3 v0; and is equal to that which would be uniformly described in the same time, with the initial velocity 0 vr, diminished by that which would be described in the same time, if moved from rest under the action of the constant Arce, by a motion uniformly accelerated; that is to say, the length of path is represented by the rectangle 0 t t'3 vo diminished by the triangle vo v3 t'3. The equations (10) and (11), which appertain to uni- Formulas to formly accelerated motion, become, therefore, applicable compute the, 7 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 = a IT - 2 v1 T2,.. (12), Value of space; V = a - v.... (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 PHILOSOPHIY. velocity; we have only to make VI= 0, and equation (13) becomes a - v T = 0O whence Time required to destroy all a = (14); body's velocity. V1 from which we conclude that the time required for a constant 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 The path a2 described during... (15); the destruction of, 2 V its velocity; 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 is destroyed, the return along the path already described, and pass in body will return; 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 a, 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 MECHANICS OF SOLIDS. 75 this latter motion, while equations (14) and (15) are to the former. But from equation (8) we have V-= V2vl,, and substituting for S its value given by equation (15) we get and have at its / a2 previous - V - = a; -paositions the 2 v, same velocity as before. that is to say, the velocity V; which the body has acquired 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. ~ 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. 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 motion; 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. Experiment 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 velocity and extent of surface; 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. 76 NATURAL PHILOSOPHY. 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 on the fall of bodies; the action of their weight. In permitting bodies to fall through the air, from the same height, it is bodies which observed that those which weigh most under weigh most and the same volume, or those which present the Fig 26. have least P surface,ffall most 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 in vacno all fall equally fast, and therefore will reach the bodies fall equally fast; bottom at the same instant if they start together. This is called the guinea and feather experiment, from the fact that a 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 ncdiscriminately 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. gravity acts on We may easily assure ourselves that the force of e interior andes gravity acts on the interior as well as on the exterior exterior particles r 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 The weight of a body, is the resultant of all the actions between the weight of a body of the force of gravity upon its elementary particles; we and the force of must be careful, therefore, not to confound the weight with gravity, the force of gravity itself, which is, in fact, only the elementary force impressed upon each particle. MECHANICS OF SOLIDS. 77 ~ 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 fainth,apidyr; 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. 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 experiment, a constant accelerating force, acting with an equal intensity at each instant whatever be the velocity acquired. Atwood, an English philosopher, in resuming the experiments of Galileo, with greatly improved means, obtained the same results. Laws of the 0 72.-Hence, when a body falls from rest through a motion offalling bodies; 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. Although the force of gravity, may, without sensible Force of gravity error, be regarded as constant at the same locality, it yet latitudeiththe 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. body in the first second of its fall from rest in vacuo, will be given by the following formula, viz: space a body feet. will describe el = 16.0904 - 0.04105 cos. 2 J.. (16), under its action in first second; in which el is the space, and d 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 el, Eq. (4), as given by the above equation, we have, velocity it can feet. generate in one g = 32.1808 - 0.0821 cos. 2 -.. (17); second; 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 vl. 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 Hf = 2V VT.. (18), formulas which H 2, (19), relate to the fall of bodies in vacuo; V2 = 2 g H.. (20), - = T. * ***(21), in which, for all ordinary cases we may take g = 32.1808 feet... (22). application to Suppose we are required to find the velocity acquired examples; 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), 32.1808 = 21808 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. 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 surface of the fluence if the falling body be very dense, as iron, lead, body andheight &c.; or if the surface of the body be small; or if the be small; 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 to,,heheag indicated by a watch beating tenths or fifths of seconds, depth ofwells. 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 Suppose a body to fall through a height of 80 feet, then will Y = VX 32.1808 X 80 = 71.75 feet. This proposition is of frequent occurrence in practical mechanics. Velocity due to a The quantity V is called, the velocity due to a.given given height; height due to a height H; and,H the height due to a given velocity V. given velocity. 80 NATURAL PHILOSOPHY. A body thrown ~ 73. —When a body, as the ball from a gun, for vertically 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 uniformly 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. WVhen it returns to its point of departure, its velocity will be the same as it was at starting. Denote by EL, the greatest height the body will attain; and V; the primitive or initial velocity; then will, equations (20) and (21), greatest height 172 to which it will H.. (23), ascend; 2g time required to V reach its greatest T = -.(24). height. U 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 Example; 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 EH= 2 X(150)2 = 350.28 feet, 2 x 32,1808 150 T 3180 - 4.658 seconds. 3271808 MECHANICS OF SOLIDS. 81 This is on the supposition that the air opposes no resist- effect of ance. The body will not ascend so high in the a::; and, atioPnheric moreover, will fall with less velocity than in vacuo. ~ 74.-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, required to impressing upon itself a certain velocity, or in overcoming impress upon a its inertia. Denote by W,, thle weight of the body, express- belocity; 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 position; this will measure the constant effort exerted upon the body during its descent through the height iI. 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 = 2gH; from which we have V2 2g and multiplying both members by W, WBH - W X V2T., (25). ~ 75.-Thus, the quantity of work developed by the work required to weight of a body to impress a certain degree of velocity irvelocity; upon itself, is equal to half the product obtained by multiplying 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, ~2 living force; is what mechanicians call the living force of the body tequal t ofuble whose weight is V. We see, therefore, that the quantity the quantity of action necessary of action expended by the weight of ca body, is half the living to prodceit; force imjpressed.;, or that the living force impressed, is doubl the quantity of action expended by the weight. half the living It is to be remarked, that when a body is thrown verforce lost or gained, equal to tically upward with a certain velocity, the quantity of the work that action of the weight, which is always measured by the overcomes the inertia. 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. This principle is, as we shall soon see, general, whatever 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. meaning of living ~ 76.-As the expression of living force, employed to force; designate the product, TVmay 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 not a force, but W. Et the result of a force's action; 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 dynramic effect of a force, or rather double this effect, since a dynallic effect., -. V2 = 2 W. g A body in motion, or a certain dynamic effect, may A body inmo tion indeed become, in its turn, a source of work; as, for Ybe a cause of work; 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 moter. 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, ~ 66, when it has been thus conquerel, 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 a.n more. than an elevated 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 sprlig, 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 W. y2 is a Force. It is the samne of men, animals in general, oranimals, of caloric, of water-courses, of wind, &c., &c.; these are Yot ctotc, th wimd, agents of work, or moters-not simple forces. 8I4 NATURAL PHIILOSOPHY. otiect of It is the object of mechanics, in its application to the,,:lianics ase arts of life, to study the different transformations or applied to the ar1tX. metamorphoses which the work of moters 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 remember, 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. ~ 77. —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 The mass of a quantity of matter it contains, or of its mass, for it is plain bodY; 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. Exforce of gravity perience shows that the velocity impressed by the force of ptellvtioitalto gravity, in one second of time, is directly proportional to,nosy impress in the intensity of this force, and that therefore the ratio one second. W must remain the same for all places, since the weight is also directly proportional to the force of gravity. Thus if W and WV', 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: W':: g g'; MECHANICS OF SOLIDS. whence g g''TV This invarible ratio W-, is taken, in mechanics, as the g measure of the mass of a body. Ordinarily the mass is Measureof tll represented by M, whence mass of a body; W g or 11W M= ifg. (26), measure of the in which TV 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. ~ 78. —By substituting the value of the weight, as given by equation (26), in the expression for the living force, we find TV Living force in - V72 = M V2. terms of the nlass g 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 Ai V, the quantity of motion of the body; and this it must be quantity of remarked is very different from the quantity of action or motion; of work. To understand what is meant by this new expression, denote the quantity of motion by Q, then will Q=_ - V= ifV. (27); 8 go NATURAL P II LOSOPHY. or, which is the same thing, i8, meaning; W::: V g. But Wr is the weight of the body, and g, the velocity which this weight can generate in this body, in one second itis a pressLre, of time; hence Q must designate either a weight or an like weight; equivalent effort, which can generate in the body, the velocity V;, in one second. Ve see also that the living force, IfV2, or M~ V V= Q V; living force is the product of this effort, by the velocity V, or by quantity of the path described uniformly by the body in a unit of motionl into the time in virtue of its acqcuired velocity. velocity. These observations show the distinction between the qucntity of notion of any body and its living force, and the identity between this latter and double the quantity of action. Use of the ~ 79.-It is principally to abridge and simplify the denominations mass and computations and reasonings, that the denominations mass q!iantity of and quantity of motion, are employed in mechanics; and miotion. they might easily be dispensed with. But as authors generally have used them, it becomes important to understand their precise significations. A force is ~ 80.-WVe have just seen that the force of gravity will eloorcity t impress upon a body, during one' second of time, velocities ean generateina which are constantly proportional to its intensity, or to tvel' timoni'neY the absolute weight of the body in each locality. But when constant. this property arises only from the fact, that the weight Wbien the force remains constant during the fall, so that the total velocity is tariable, it is at the end of the fall, is proportional to the equal degrees proportional to the small degree of velocity impressed at each instant. When the motive of velocity a force, instead of being constant, varies at each instant, it is imrted a t its itnstan no longer be measured by g-ven instant. obvious that its intensity can no longer be measured by 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 smcall degree of velocity which it communicates at a given instant. By observing what takes place at the surface of thO forces earth, and in our planetary system, it is found that the proportional to the small degrees motive forces or pressures are, in fact, proportional to the smnall of velocity they. can impress in a degrees of velocity which they impress upon the same body in very small 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. ~ 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 for, of inertia by the velocity which it can impress upon a body at any velocity impressed in a instant or epoch, during an indefinitely small interval smatime. of time, denoted by t; also, let V17 be the pressure exerted by the weight of a 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, F: VW:: v vI; Consequences of this law; whence F V * V. But from the first law of falling bodies V': g t isec.; whence v' = gt; 88 NATURAL PHILOSOPHY. therefore measure for the TV v v intensity of any F = -- -. -. (28). motive force; g t 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, impressed 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 inertiaexerted, given by Eq. (28), shows us that the force of inertia, which proportional to the product of s equal and contrary to F, is directly proportional to the mass into the mass, and to the velocity v which this mass receives during velocity the elementary time t. imparted; Let F' be the measure of a second force, which acts upon the mass A', impressing upon it in the same time t, the small velocity v', then will Fi' = AL v' t' which, with Eq. (28), gives relation of any F FI two motive forces,''. 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 F. t impressed in any V = -A' short time; M' MECHANICS OF SOLIDS. 89 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, increases with the intensity of by the mass. the force, and inversely as the mass, or weight. ~ 83.-If now we suppose, at any instant, the foice Measure ofineltia suddenly to cease to vary, and to continue to act upon the and of the equal and contrary 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 V': V Isec. t; whence V which, in Eq. (28), gives F =.......(29); 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 quantity of A7~~~ ~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. conit of time, when When the mass becomes the unit of mass, Eq. (29) becomes F-...... (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. measured by the and is always measured by the velocity it is capable of imnvelocity imprsed on a pressing on a unit of mass in a unit of time, acting with a impressed on a unit of mass in constant intensity. unit of time; And from Eq. (29), which gives, Vl=- M' is equal to the it appears that the acceleration or retardation due to the motive force divided by the force, is, in every case, nothing more than that portion mass. of the entire motive force which results from dividing the latter by the number of units in the mass acted on. Geometrical ~ 84.-Trace, according to the method described for illutration; uniformly varied motion, ~ 68, the curve vOv v2 v3, &c., which represents the law of the times and velocities; let t3 v3 and t4 V4 represent the, velocities which correspond to the 4 end of the times I 0 t3 and 0t4, or at the beginning ~ and end of the very small portion of time t3t4 = t. Draw through v3 the line vs b4, parallel to the axis 0 B of times, and produce it till v3mn= 1 second; this line will meet the ordinate t4 v4, and b4 v4 will be the small portion of velocity = v, impressed by the force, during the small portion of time t. Now if, at the instant corresponding to the end of the time 0 t3, the force become constant, it will subsequently impress upon the body equal MECIHANICS OF SOLIDS. 91 increments of velocity during the equal intervals of time t, and the curve v3 v4 v5 will become the straight line v3 n, tangent to the curve at the point V3. Drawing through m a line parallel to t4v4, the portion m n will represent the velocity E1 impressed in one second, and the two similar triangles, v3 b4 v4 and v3 m n, will give 3 b4 b v4 V V3;m a n; or t' v Ise c.:; whence V the value of the V1 =; velocity at any instant; 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 foundby the compute the value of the intensity of the force from the tangent line; equation, MXVi = -.1; the measure of Fv = M V1 =. vE; 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 intensity of the force F at each instant, we deduce from it Value of the accelerating force, the corresponding value of ao.al to motive force divided by mass. V; = ]r 92 NATURAL PHIILOSOPHY. Inclination of or of the inclination of the tangent v3n, or that of the tangent to the element of the curve of velocities to the axis 0 B of times. curve. The tangent of this inclination is given by m n v3 m and if the initial velocity 0 vo be given, nothing is easier Curve constracted than to construct the curve, of which the ordinates shall be by means of this tangent. 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. Wosk necessary ~ 86.-By the aid of what precedes, we may readily to impress a given compute the quantity of work which must be expended velocity; against a body, whose weight is W; by a force ], equal and contrary to the force of inertia, to impress upon it a certain velocity V; Fig. 28. or, more generally, to v6 augment or diminish its V4'4 velocity by a given quan- V2 tity.: The quantity of work: expended during any small interval of time t, has, for its measure, the 0 z product of the intensity of the force F, into the elementary portion of the path described by the body during this time. This small path is given by the area of the small rectangle V3 t3 t4 b4, whose base is the element t3 t4 = 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 F Vt, for each instant of time, or for each small increment b4v4 of velocity V. But from Eq. (28) we have F = X t; replacing F by this value, in the preceding expression, we have, for the elementary quantity of work, elementary Vv; ~quantity of work; 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 O w,, W1 w2, w2 w,3 &c., to represent the different increments of velocity during the different successive elementary Fig. 29. portions of time t, V7 which have elapsed v since the beginning of / geometrical V4/ 5; method of finding motion - increments VStll Aeeet - U/;ithe whole work. that will not be equal aI in the case of a vari- able force; then will 0 W1, O W2, O W3, &c., 0 Wi W4 represent the velocities of the body at the corresponding instants: lay off these same lengths upon the ordinates wl, wv v2, w3 v, &c., so that we shall have w1 V1 = OwI, wv2 = O= w, w3V3 = 0 W3, &c.; 94j: NATURAL PHItLOSOPHY. the series of points v,, v2, v3, &c., will lie on a right line, inclined to the axis 0 B, in an angle of 450. Consider now the velocity v3 w = w, for instance, of which the increnmeit W3W4 or v3 b4 4, Fg 2., The area of a is called v. The product representsthe Vv, will here be represent- 2V l represents the sutm of all the ed by the small rectangle v products Vv., V3 w3 V4 bN, or by the trape- v3/lb4i' zoid 3 3 W4 V4, to which it'' becomes sensibly equal when v the increment of velocity or /! 0 ew1 202'03 that of the time is very small. The sum sought, of all the partial products Vv, has for its measire the sum of all the corresponding elementary trapezoids, or the area comprised within the right line 0 v,, the axis O w,, and the ordinate w7 v7, 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. Work consumed ~ 87.-For example, if the body sets out from rest, and when the body's ie desire to find the sum of the products of Vv, correspondmotion is accelerated; ing to the acquired velocity w44 = V', this sum being represented by the area of the triangle 0 wu4 v4, we shall have - O 4 X %4v - 2 (W4Vq)2 - a 2; hence the quantity of work corresponding to the velocity F', and consumed by the inertia of the body whose mass equal to half the iS M,1; will be measured by ~X VT2, or by half the living living force force communicated from the beginning of the motion, ~ 76. communicated; This principle obtains, therefore, for any kind of motion, or for a motive force different from the force of gravity. For another velocity, W7 7 = V", the consumption of work will be in like manner measured by 1 FV"2, 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 2orresponding to the trapezoid W4 w7 v7 v4. But f V'2 and work consumed in any interval, M V'2 are the living forces at the beginning and end of equalto halfthe the interval of time during which we are considering the difierence of living force at work of the motive force; the expression above is, there- beginning and fore, one half the increment of living forcer or half the living end. 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 mreasure, hayf 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 when the motion developed by the resistance F, (equal and contrary to the is entarded; force of inertia now become a power,) during the time necessary, to reduce the velocity from V' to V"', would have for its measure, 1 ( If VI2 - -;I I"2) or half the living force destroyed or lost. equal to half the Thus, the diminution of the living force of a body difference of living force at between any two given instants, supposes that a quantity the beginning and end of interval. 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 cugmentction supposes, on the part of a power, a consumption of work equal to the half of this augmentation. ~ Inertia serves to transform work ~ 89.-We now clearly perceive how the inertia of into living force, and living force a body, serves to transform work into living force, and into action; 96 NATURAL PHILOSOPHY. living force into work; or, to use the expressions employed, ~ 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. examples in the The mechanic arts offer a multitude of instances in mechanic arts; 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 exampleofthe from the reservoir, is changed into a certain quantity of grist-mill; work; this is communicated to the wheels of the mill, and these latter transmit it to the millstones which pulverize the corn. The air confined in the reservoir of an air-gun, represents the value of the mechanical work the air-gun; expended by a certain moter in compressing it; on opening 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, Fig. 30. developed by the the action of the spring, becomes equal ball against a to half the living 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. ~ 90. —If, then, the spring be perfectly elastic, the Perfectly elastic velocity communicated to the ball, will be precisely equal bodies restore all the living force to that impressed upon it by the air-gun in a contrary lostduringan direction. Thus, in the example before us, the quan- impact. 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 impressed 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. ~ 91.-HIence, 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 always lost in the bodies are perfectly elastic, and as the vast majority are, collision of bodies to a considerable degree, deficient in this quality, the notperfectly 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 therefore 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 remarked, to avoid all shocks in the motion of machinery. ~ 92. —We also see, from what precedes, that it is as The work restored can impossible for the force of a spring to develop, in un- neverexceed bending, a living force greater than that consumed in thatconsumedin creating the bending it, as for the force of gravity, ~ 65, to give catigthe 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. 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 accomplish, 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. What talres place ~ 93.-We have shown, by examples, how the quantity in periodical of work or of action may be transformed alternately into motion; 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 when the examples of this last kind. When the velocity of a body increased, inertia augments, it is a sign that some portion of the moter's opposes the work acts in the direction of the body's motion, and inforce; creases its living force by a quantity double this portion when the of work; the other portion being absorbed by resistances; diminishes, if, on the contrary, the velocity of the body diminish, inertiaaidsthe notwithstanding the power may be exerted in the direcforce. force. 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 moter by a quantity equal to half the living force thus expended, and so on, according to the number of alternations. BIECHANICS OF SOLIDS. 99 ~ 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 intervals between 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- employed to 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 eot by velocity had increased incessantly, the inertia of the body half the living would have opposed the motive force without intermis- force acqu~ire.d r destroyed. 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 Fig. 31. by means of the second v/ figure employed in ~ 86, in observing that when v the velocity of the body 4 eontrical diminishes, after hav- v illstation; ing augmented during a certain time, so will the abscisses and ordinates of the right line 0 v7, which represent this velocity; the extreme ordinates w7 V7, after receding from the point 0, while the velocity is increasing, will, on the contrary, approach this point while 100 NATURAL PHILOSOPHIY. the velocity is diminishing, to keep the triangular area Ow7V7, constantly proportional to the quantity of work absorbed by the inertia, or to its equal, one half the living example of a force. A carriage travelling at a variable rate, sometimes caylige 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 diminished, 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 quantity of work consumed, so that nothing will be lost. In what is here said, it is understood, however, that no diminution of velocity results from opposition or holding back of the horses, for in that case, the moter would be converted into resistance. The same ~ 96.-The same reflections are applicable to the reflections apply weight of a carriage in ascending and descending a hill. to weight as well as to inertia. The quantity of work employed in overcoming the weight while ascending will be restored during the descent, provided the latter be not so steep as to cause the horses to hold back, by which a quantity of work would be consumed uselessly. And this consideration shows us one of the many advantages which results from giving gentle slopes to roads. When a force is ~ 97.-When a moter is employed to raise a burden,,mployed to raise through a vertical height, it takes the body from a state a weight, inertia retains nothing of of rest, and hence a quantity of work must be expended the mloter's work-'work; to overcome its inertia. Arrived at the desired height, the effort of the moter is relaxed to restore the body to a state of rest, and during this diminished action, a portion of the living force acquired is employed to destroy MECHANICS OF SOLIDS. 101. 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 arte inertiaoo an artificer's tool. 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 destroyed, 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. ~ 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 by inertia; 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. 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 plane; 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. 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 ~k; which the inertia of the bung opposes to the sudden motion 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 transforming quantity of action into living force, take the com- the common mon slinjg, from which a stone may be thrown with much sling 1.02 NATURAL PHILOSOPHY. greater velocity than from the naked hand. HIere, living force is accumulated in the stone, by whirling it through many accelerated turns about the hand before it is disthe common charged. The common top turns and runs along the shuiring top. ground, in virtue of the living force acquired during an accelerated unwinding of the string from the coils of which it is thrown. ~ 99. — Ve would recommend to the reader, to consider attentively these examples, as well as all others of like nature which his observation and memory may.tia sometimes furnish. They will aid his conceptions of the manner in,'ati'ee; which the inertia of bodies, like their weight and molecular spring, sometimes acts as a mere passive resistance, so:,etiles a real and sometimes as a real motive force, according to the luotive force. circumstances. 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 translation, 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 ~ 100.-Thus far we have only considered the effect of dlirections meet tuo point; 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 through its intervention. WVhen 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 preventsthe effects of all the produce, if the first did not exist. The body itself is in others; equilibrio, if the different forces applied to it, leave it at rest. This last kind of equilibrium can never be abso- noabsolute lute, because all. bodies connected with the earth partake equilibrium of bodies; 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 asi 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 dynamical 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 circumstances. 101. — It has already been, stated,. ~ 43, that when Resultantof several forces act along the same right line and in the severalforces; 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 diree- when acting tions, and along the same straight line, their resultant will along the same line, in same or 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 directions; 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 tensionofacord; its consecutive portions are urged to separate from each other, 104. NATURAL PHILOSOPHY. the effect of and this being the same throughout, the excess of the unequal forces acting uapon a sum of the forces which act in one direction over that of cord. 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 0 J and Parallelogram of 0 N situated paths; i the same Fig. 82. plane with the line A B, by drawing parallels to these, A... _ __A/ lines considered as axes, the place A giving 0 _! __ the two co-ordinates AA' 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 0 M and 0 N; are, therefore, described by the projections in the same time as the path relative or A B by the moving point. The first are called comptonent omponent or reatve paths in such and such directions. Prolong the co-ordinates of the points A and B, till the parallelogram A E BF is formed, and this principle will appear, viz.: the rectilineal path described by a point, may always be resolved into two relative or component paths, in any two resolution of any directions, and these component paths will be the sides of a path into component _ parallelogram, constructed upon the path described by the paths; 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 trueand relative A' B' and A"' B"' are described by its projections on velocities; the directions O and 0 NA 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 ulpon its two relative velocities, found from relative velocities, estimated in any given directions whatever. and the reverse. ~ 104.-If the motion be curvilinear, the rectilineal di- Relative paths in agonal A B can no longer represent, in general, the path varied motion; described. Nor, if the motion be varied, can its length measure the velocity, when the time of description is considerable. In such cases, conceive a given interval of time divided into a great number of small and Fig, 33, equal portions, and determine the relative zr. --—.. paths described during X - each, by the projec- -- tions of the moving --------------- geometrical point on the axes. I representation; Each pair of these I relative paths will de- 0 - r termine a parallelogram, of which the diagonal will be the corresponding elementary path described by the point itself. Any one of these diagonals, 106 NATURAL PHILOSOPIY. constructionof as A B, will sensibly coincide with an element of the the direction of the body's curve, and its prolongation A T will be tangent to the motion. curvilinear path. This tangent will determine the direction 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" 1, respectively, parallel to the directions 0 T" and 0 T'. ~ 105.-When the law of a body's motion in two directions is known, it is always possible by the preceding Roberval's method to draw a tangent to the path described. Take, method of constructing the for example, the ellipse: tangent; this curve is generated by fixing at two points F and F', called the 7? foci, the ends of a thread FA F', equal in length to a given line, lfl2f1', called the transverse axis, and moving the point of a pencil A to all positions in which it will keep the thread 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 A F' will diminish, and therefore the point A tends to describe equal relative paths, or will have equal relative results from the velocities, in the two directions A B and A F'. Hence, law which determinesthe taking upon FA produced, and upon A F', the equal path. portions A B and A B', and completing the parallelogram A B CB', 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 animated by two or more simultaneous velocities. To illus- Illustration of the coexistence of trate, let it be supposed that while a boat is crossing a cimultaneous river, a man walks from one side of the boat to the other, velocities; and that, starting from the point A, for example, he arrives at B at Fig. 85. the moment the boat reaches a position such /tN1Qlllll!.L that the point A shall X. be at A', and the point irj' B at B'. It is plain, that the man, though only conscious of having walked across the 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 simultaneous motions; 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 animated 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. tant of these two with the third, and the resultant of the three with the fourth. In fact, when a body has several resultant of simultaneous motions, the effect is several simultaneous the same as if the body had re- Fig. 36. velocities; ceived, one after the other, all the motions which it possesses at the same time. Hence, this rule, i viz.: The resultant of several simul- 5 / \ i taneous velocities is found by conrule; structing a polygon, of which the sides are equal and parallel to the / component velocities, and by joining, with a right line, the point of departure with] the extremity of the last side. This right line will represent the resultant required. illustration. Thus, let the point 0 have the simultaneous velocities 0 V, 0 V V', V, V'; from the extremity V of 0 I; draw Vm parallel, and equal to 0 VT; from m draw m ml parallel, and equal to 0 V"; from m' draw m' n" parallel, and equal to 0 V"', and join 0 with m"; the line 0 om" will be the resultant velocity. Independence of ~ 107.-The action of a force upon a body, whether at the actionof rest or in motion, is always the same, and impresses upon simultaneous forces; it the same degree of velocity. Let a body fall, for example, 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 downward by the action of some other force. For Fig. 3'. example, when a bombshell is thrown into the T 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 MECHANICS OF SOLIDS. 109 weight, and its velocity at any instant is the resultant HR, of the velocity Hi]Q, which it has received at the instant immediately preceding that we are considering, and the small velocity MIP, which its weight can impress upon it during the very short interval of time between these two instants. Thus, when two forces are applied to two forces the same body, they impress upon it, at each instant, and impress simultaneously, simultaneously, the same degree of velocity which each the same velocity would impress if acting alone. This degree of velocitya,s if acting separately. we have said, ~ 81, is, from the general law of nature, proportional to the intensities of the forces. ~ 108.-Accordingly, let a material point A be acted Parallelogram upon by the two forces P and Q, represented in intensity and o orces direction by the lines A B and A C respectively. These forces will impress simultaneously, and in their respective directions, the same degrees of velocity A m and A n, as though each acted separately. The resultant velocity will, ~ 107, be represented by the Fig. 88. diagonal A r of the parallelogram A m r n. 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 x forces in equilibrio. Take, upon the diagonal, the distance A D = X, 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 NATUIAL PHILOSOPHY. resultant A D, equal in intensity to X, are proportional to the velocities A wm, A n, and A r, which they can simultaneously produce, and, therefore, A D must be the diagonal of the parallelogram constructed upon the lines the resultant of A B and A C as sides. Whence results this important any two oblique principle, known under the denomination of pcaralelogram forces applied to a point; of forces, viz.: the resultant of any two forces a2pplied to the same point, is represented, in magnitude and direction, by represented the diagonal of a parallelogram, constructed upon the lines by pthaallelogram f which represent, in in tensity and direction, the twoforces. It must not be forgotten that a force is, in geometrical investigations 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 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 balancespring, to measure the efforts exerted in the directions C AL and Fig. 39. 0 B. Laying off upon the vertical through D C, and from the point C a distance CD A experimental equal to 15 inches, illustration of the and completing the parallelogram of forces. parallelogram by drawing D a and D b parallel respectively to CB and CA, we shall find C the number of inches in C a and C b to be the same as the number of pounds indicated by the balances A and B. ~ 109.-By the same principle that two forces, applied ~MECHANICS OF SOLIDS. 111 to the same point, may, without change of effect, be re- Resolutionofa placed by a single one, may a single force be replaced by fotce into two two others, acting in given directions. Let a given force, applied to the point O, be represented in direction and intensity by the line 0 r: its com- Fig. 40. ponents, in any two assumed directions, as O A and O B, are P A 4/" thus found. Through the point r, the extremity of O r, draw 91 r m and r n parallel, respec- 2 tively, to OB and OA; the 0 portions 0 qn and 0 n will represent the components required. Make Om = P; On = Q; Or = R; the angle trigonometrical A O B = p = rem B = 1800 - rn O. Then, in the tri- relation of resultant to its angle 0 r n, because 0 n = r n = P, we shall have two components; R2= p2 + Q2 + 2P Q cos p, or R = v'P2 + Q2 + 2P Q cos. (31); valueofresultant; and because the angle 0 r n is equal to the angle r 0 m, and sin r n 0 = sin A 0 B, we also have, from the same triangle, R: Q:sinp: sinr0m, R: P:: sin p: sin r On; whence, sinr m Qsin R its inclination to sin.. (32). its components. P sin p sin rOn = ~ 110.-We have heretofore supposed the resistance 112 NATURAL PHILOSOPHY. Quantity of work immediately opposed to the force destined to overcome when tathe not it. Let us now consider the case in which the resistance resistance is not immediately is exerted in any line of direction other than that of the foprsed to 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, A R represent a force applied to the point A, Fig. 41. which can only move / in the direction AB. - - Decompose this force, /'" which denote by R, into two components P and Q- the first per- p pendicular to A B, and the other in the direc- tion of that line, and, consequently, immediately 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 Q X A a, will measure the elementary quantity of work necessary to overcome the elementary quantity of resistance over the same path; such will be the measure of the effective quantity of work of the force R. Draw from the point a, a r perpendicular to A R; A r 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 a r and A Q R, which are similar. having a common angle A, and each a right angle, equal to the Aa Ar::Q product of the force into the whence, path, estimated in direction of force. a X Q =Ar X.. (33); 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. ~ Ill.-When a heavy body Quantity of work is compelled to move upon the ofthe weightof a body, moving on curve A B C, the elementary Fig. 42. a curve; 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 product W X b'c' estimated upon B the vertical line A D'. It is also the measure of the quantity of work expended in the direction of the curve. Adding together all the elementary D quantities of work by which the body is made to describe 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 A D' = H; or to W X H. This is also the measure of the quantity 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 ~ 88, the double of this last quantity of the weight inent is equal to the living force of the body; that is to say, direction of to the product curve. Wx V; g in which V denotes the velocity of the body in the direction of the curve, at the instant the work terminates; whence 8 114 NATURAL PHILOSOPHY. 2 WHE- =. V2 g or V2 = 2gH; The velocity 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 the height, and not onthepath same as though the body had fallen vertically through described. the same height. And we see, from this investigation, that the quantity of work which a moter 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 ~ 112. —It has just been shown, ~ 110, that the elequantityof work mentary quantity of work of a force, of which the point of two forces applied to a of application is moved in a direction different from that point; 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 remarked, that this component Fig. 43. is nothing more nor less R than the projection of the force on the direction of the motion. Accordingly, let us consider two forces, Q P and Q, applied to the point A, R their resultant, and a A the small path described by the point of ap- Q' A a R' p' when the plication. Let fall from the projections of points Q, P, and R, the perpendiculars Q Q', R R', and components fall on opposite sides P P', upon A a produced; the projection of the force of point of P will be A P', that of Q, A Q', and that of the resulapplication; taut R, A R'. MECIHANICS OF SOLIDS. 115 Now, A R' = AP'- R P' but A Q and R P, being equal and parallel, their projections A Q' and R' P' upon the same line, are equal, and hence A R' = AP' - Q,' and multiplying both members by the path A a, we have work of resultant equal to difference of AR' X Aa = AP' X Aa - A Q' x Aa; worlkof components; 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 A 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 when the Fig. 44. projections fall place so as to cause the R on same side; 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 equation would become AR' X Aa = AP' x Aa + A Q' X Aa, the work of and that the effective quantity of action of the compo- resultant equal to sum of that of nents A P' and A Q', would be the sum of the quantities components. 116 NATURAL PHILOSOPHtY. taken separately, and the equation may be written, generally, AR' x Aa = AP' X Aa i A Q' x Aa..(34). Thie work of Hence, the elementary quantity of work of the resultant resultant equal to the qalgebraic of tWO forces, applied to a point, is equal to the algebraic.u[a of the work Sum of the quantities of work of the two components. of its CotleoltS. 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. Motion about a fixed point. ~ 113.-The small space A a, may be described in different ways. If 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 arc of a circle, which we may regard as a small right line Fig. 45. perpendicular to A 0. From Q the point a, let fall the perpendiculars a b, a d, and ac, upon the directions of the forces T?. P, Q, and their resultant R; ])~ then will the ele'mentary quantities of work due to these forces be respectively P X A b, Q x A d, and R X A c; and from ~112. R x Ac = P X Ab ~:Q x Ad. From the point 0, about which the motion takes place, let fall the perpendiculars Op, Oq, and. 0 r, npon the directions of the forces P, Q, and R, respectively; the triangles A O p and A a b are similar, since each has a -right angle, and the angle A 0 p, of the first, is equal MECHANICS OF SOLIDS. 117 to the angle a A b of the second, the sides A 0 and Op being, respectively, perpendicular to the sides A a and A b; hence, Ab:Aa:: Op: AO;.:"w-hence, Aa Ab= Op X AO; and, in like manner, from the similar triangles A d a and 0 A q, we have Aa Ad= Oq. AO' and from the similar triangles A c a and A 0 r, Aa Ac = OrxAO these values, substituted in the above equation, give, after omitting the common factors, and making Or = r, Oq = q, and Op = p, Rr = Pxp~ Q xq... (35). The effective quantity of work which a force is capable Moment of a of performing, while its point of application is constrained force; to describe an elementary path A a, about a fixed centre 0, is called the moment of the force; the fixed point 0 is called the centre of moments; and the perpendiculars the centre of p, q, and r, the lever arms of the forces P, Q, and R, moments; lever arms; 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, Q q, and R r, multiplied each by the constant ratio A a; and if this A n;adifti 118 [ NATURAL PHILOSOPHY. the relative constant ratio be omitted, these products may be taken iasre of a 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 ilte' moment of from Eq. (35) we see that the moment of the resultant the resultanlt of two foces. of two forces, applied to a point, is ejqual 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 hlen the forces be applied to different points;Ue liote point;o C and B, it is evident that we may suppose two rigid bars, Fig. 46. CA and BA, 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- 0 ing invariably connected with the body, may be taken as the common point of application, without changing the effect of the forces. The resultant A R will be obtained by means of the diagonal of the parallelogram A PR Q, and the point D, where it meets the surface, may be taken as its point of application. If, now, the body be constrained to move around any point, as 0, the common point of application A, will describe the small arc of a circle, which may be regarded as a small right line, to t., Tl-omrellnt of be projected on the directions of the forces, as in the last tL...resultant ti article; and the same reasoning will show us, that in this sti!. equal to the allhr.lc sum of case also, the moment of the resultant is equal to the mb tos omeents of;th:e:opnoents, algebraic sum of the moments of the components. ~ 115.-The relations which have just been established between the quantities of work, and between the mo MECHANICS OF SOLIDS. 119 ments of forces and of their resultant, will always obtain These relations wherever the point 0 be taken, since its selection was eqally true, entirely arbitrary; but these relations were obtained by centreof moments be considering the motion of the point, common to the taken; 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 position of the point of application. The moment of the or wherever the force P, for example, will be the same whether it be sup- points ofn 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. ~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 theorem of the quantity of work; 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 y any point of of both combined. line of direction; First case. The body and the di- Fig. 47. Aa' BY' P First-inmotion rection A P, of 7j1!t \i of rotation; the force P, being of rotation; supposed to have 0 a motion of rotation about the point 0, any two points, as A and B of the 120 NATURAL PHILOSOPHY. line A P, will de- Fig. 4'7. scribe arcs which A, are proportional to their distance, OA i and /OB, from O; and we shall have Aa Bb AO O0' 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,:Px Ox A a P X Of X OA; or estimated by the motion of its point of application, supposed at B, will be measured by Bb PX OpX OB' Hence, the quantities of work are equal, being measured by the product of the intensity P, the length of the perAa Bb pendicular Op, and the equal factors A, and B' Second-in Second case. If the motion of translation; body only have a motion of translation, Fig. 48. any two points of ap- plication, as A and i B, will describe the equal and parallel paths A a and B b, which will be projected upon the direction A P, in the equal paths A a' and B b'; and the quantities of work in the two cases being P X A a' and P X B b', are equal to each other. Third case. Suppose the line of direction- A P of the MECHANICS OF SOLIDS. 121 force P, to take the position A1 B1, in virtue of the com- Third-when the bined motion of rotation and translation, and the points motion is of 7O fl5 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 combined; into a motion of rotation around the point 0, the centre of a circle, tangent to the two positions Fig. 49. of the line of direc- tion, supposed in- B < definitely near each other, and of trans- lation along the see- a iP ond position of this _ I line. By the first,' the points A and B 0 ) are carried in the arcs of circles to A1 and B1, and by the 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 b b', we shall have for the quantities of work, considered in reference to the motion of the points A and B, P x A a' and P x B b', respectively. But by projecting the points A1 and B, on the primitive direction, by the perpendiculars A1 Al' and B1 B1', we have A a' = A' a' - A1'A, Bb' = B1 b' - B, B; multiplying each equation by P, P X Aa' = P X Al' a' - P X A1' A, Px Bb' = Px B1'b' -- PxB1'B. Now P X A1' a', and P x B1' b', are the quantities of work, on the supposition of a simple motion of translation 122 NATURAL PHILOSOPHY. alone, in the direction A B, and these Fig. 49. have been shown, in the second case, to B be equal; whence, A1' a' B1' b'. m - a' P no matter where The products " the points of application be P X A1' A, \ taken on the / lines of direction; and P X B1' B, measure the quantities of work due to the motion of A and B, on the supposition of a simple motion of rotation about 0, which have been shown to be equal, in the first case; whence, A1' A =B B~'B; and consequently, Px Aa' = PX Bb'. the work of the Thus, the relation given in ~ 112, between the quantity esultant, isequal of work of the resultant of two forces, and the total quanto the algebraic sum of the tites of work of the components, subsists in all cases, quantities of whatever be the points of application, and whatever be work of the components. the nature of the motion. ~ 117.-Resuming Eq. (35), Rr = Pp i Qq, When the in which r, p, and q, denote the lengths of the lever armn resultantiszero of the resultant R and of the two components P and Q, or when its line of direction we see that the moment R r, of the resultant, can only passes througha reduce to zero when the moments of the components P fixed point, and Q are equal and have contrary signs. But the prod MECHANICS OF SOLIDS. 123 net R r, 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 considerations. 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 equilibrium about a fixed point, when the resultant of the forces which act upon the body, passes through this fixed point. V. OF FORCES WHOSE DIRECTIONS ARE PARALLEL. ~ 118.-It has been shown of two forces whose direc- Theorem of the tions intersect: 1st, that the line quantity of work, of direction of the resultant, will Fig. 60. moments equally intersect those of the compo- forces are nents in the same point; 2d, parallel. that the moment of the resul- | / taut is equal to the sum or dif- I r 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 position of the point of meeting O, and of its distance from the body or centre of moments, will not cease to be true when 124 NATURAL PHILOSOPHY. the point 0 is so far removed as to make the directions 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 direction 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 moments. Fig. 51. Resuming Eq. (31), and re- Q 2' 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 direction, the angle q will become zero, and we shall have Value of resultant when R = P2 + Q2 + 2 PQ = P + Q. the components act in same direction; Fig. 62. Again, revolving the directions / as before, till they become par- A'[ allel and the forces act in opposite directions, the angle q will equal 1800, and Eq. (31) reduces to i 8/~~~~~~ MECHANICS OF SOLIDS. 125 B =v /P2 + Q _2 p =P Q value of resultant 2P9=P- 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 sin R OA Q sin q, P sin sin R OB = in which, if we make q = 0, or 180~, we obtain sin BOA = 0, sin R 0 B = 0; 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 the resultleant of 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 The theorem of limits of the case in which the Fig. 68. moments true of parallel forces. directions of two forces P and Q, applied at the points A and P B of any body, meet in a point; assume any point as ], in the plane of the forces, and let fall the perpendiculars pKa, Kb. Denote by R, the intensity of the resultant, supposed to act along the line R c, 126 NATURAL PHILOSOPHY. then, from the principle of moments, will R x Kc = P x Ka Q x K b); the upper or lower sign Fig. 63. being taken, according as the forces tend to turn the body in the same or opposite directions about the point K. Relation of Replacing R by its resultallt to its value P i Q, the above two parallel components; becomes.ib | Ci ai (Pi Q) Ke =Px Ka K Q x-Kb; which, by an obvious reduction, becomes P (Kc-)- a) = ( Kb KF Kc); but Kc - Za = ca;:.Kb =F: Kc =: be; whence P x ac =: Q x b c or P Q:: be: ac; the distance of 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 prothe other component. portional to the forces. MECHANICS OF SOLIDS. 127 Fig. 54. ~ 120. -Let the parallel At 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 b a, then will Bc' = bc; c'a' = ca; and because Cc' is parallel to A a', the triangles Bc' C and Ba'BA, give the proportion, Bc' c'a' BC: A; whence P Q: BC AC; that is to say, the line of di- Fig. 55. rection of the resultant of any P Rule for position two parallel com2ponents, divides the line joining their C points qf application into pro2p9ortional to the intensities of the components. The above proportion gives by composition, Q $ PiQ: P B CI:hAC: BC, P~ Q Q:: BC~= A C: A C; 128 NATURAL PIIILOSOPHIY. or, replacing P:= Q by R, and B C ~: [ A C by the whole line BA, R: P:: AB: BC, 1R: Q:: AB: AC; relation of that is to say, the resultant of two parallel components is to ceoSmponttteither 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 comiponent. M'oments of ~ 121.-When two forces are parallel, their moments reference to an may not only be taken in reference to a point, but also in axis; reference to a. right line, supposed fixed. Thus, suppose the forces P, Q, and their resultant R, to act along the parallel Fig. 56. lines AP, BQ, and K r a c C R, respectively. l. Assume any line, as ML, at pleasure; conceive a plane/ Ld/ drawn through this / / line and perpendicnular to the plane of L the forces, and let KL' be the intersection of these planes. From the point ]; draw KL" perpendicular to the direction of the forces; then, regarding moments referred K as the centre of moments, will to a centre; R x KC' = P X KA'+ Q x KB'; whence MECHANICS OF SOLIDS. 129 KA.' KB' R= P X + Q x,. But from the similar triangles, KEA' A, KB' B, and KC' C, we have KA' KA KCG'- K C' KB' KB KO' =KC Ei~-C1 0'K C"' which, substituted in the above equation, gives, on clearing fractions, Rx KC= P x KA + Q x KB... (36). Dividing both members by R X K C, P KA Q KB From the points A, B and draw the lines KC a, B From the points A, B, and 6, draw the lines A a, B b, and Cc, perpendicular to the line KL. Also, resolve forces replaced each of the forces P, Q, and R, supposed applied at A, B, bYthoen; qC respectively, into two components, one parallel, and the other perpendicular, to the line KL; and let A P", B Q", and C6R" be the former, and A P', B Q', and CGR', the latter of these components. In the similar triangles P A P', R CR', and Q B Q', we have, denoting the components A P', CR', and B Q', by P', R', and Q', respectively, P P' 7J B" R =- R'; 9 130 NATURAL PRILOSOPIIY. and from the similar triangles KA a, K Cc, and KB b, KA Aa KC Cc' KB Bb KC Cc' which values, substituted in the foregoing equation, give, after clearing the fractions, moments of components R' x Cc = P' X Aa +' x Bb.. (37). perpendicular to the axis; The effective quantity of work per- Fig. 56. formed by each of ct c _ l the forces P, Q, and R, may be replaced c by the algebraic sum I,/,2' of the quantities of P I work performed by its components; but the effective quantimoments of the ties of work of the parallel components which are parallel to the line KL, will be zero, components; 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. moment of a The moment of a force in reference to a line, is the effecforce in reference tive quantity of work which the force is capable of perto 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 component 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. the axis o, moments. ~ 122.-Dividing Eq. (36) by K C, we find P. KA QKB and substituting the values of KA XKB K C and C' as given on the opposite page, we find, after clearing the fraction, R x Cc = P x Aa + Q x Bb; 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 distances of their of application from any given straight line, is equal to the points of sum of the products of the forces into the perpendicular application. omn a line, attd planse. distances of their respective points of application from the 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. 123.-Now let us suppose any number of parallel forces —for instance, five. Find the resultant of any two Resultant of,y of them; compound this resultant with the third force, nlmbe",of parallel Ibrccs; 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 rulerfio, fldig; algebraic sum of the moments of the components. Men pulling upon parallel ropes, horses drawing upon 132 NATURAL PHILOSOPHY. exwinples of their traces attached to whipple-trees, are examples of p.Malled forces. parallel forces. 124.-Suppose a body to be drawn in one direction'the work by any number of parallel forces P, Q, R, &c., and in the performed by the:,soaislt of' opposite direction, by the parallel forces P', Q', R', &c. ix!.allei forces; If the points of the body move in parallel lines, it is plain that the paths described by the points of Fig. r application will be equal to each R' 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 quantities of work performed by the P Q R forces P, Q, R, &c., diminished by the sum performed by the Oqual to tIhe forces P', Q', R', &c.; that is to say, it will be equivalent,l.roicsumof to t the product of the common path, multiplied into the the work of the;orplpciellts. 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 perpendicular 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 r = z-, Position of the resultant of whence parallel forces. K r = 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 ]; 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. ~ 126.-To illustrate the principle of parallel forces, Illustration of the let us take the example of the common steelyard, an Prnlfcipleofy parallel forces by instrument employed to ascertain the weight of different the steelyadl. substances. It consists of a bar MN, whiclt turns freely Fig. 68. about an axis C sus-A pended from a fixed d point; the substance Q to be weighed, is Q placed at one end A, while a constant 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 C= P x B, from which BC = Q x AC; NATURAL PHILOSOPHY. or, if P be taken equal to one pound, then will BC= QxA C. The scale of the If Q be taken suc- Fig. 58. steely-alrd costeelyard cessively equal to 1, A 2, 3, 4, &c. pounds, - then will the corresponding values of B C, become A C, 2 A C, 3 A C, 4 A C, &c. Thus, if a scale of equal parts be constructed on the longer arm, having its zero at the point C, and the constant distance between the consecutive divisions equal to A C; 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. CENTRE OF GRAVITY OF BODIES. Poilt of ~ 127.-Whatever be the angle which two parallel application of forces, P and Q, make resultant of arallel forces; with the line A B, join- Fig. 59. ing their points of ap- B plication, the intensity of the resultant R, and | the position of its point of application C, will always be the same, however the direction ": of the forces may revolve about their points 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 R = P i= Q, and the point C, by P x AC= Q x BC; 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 X, 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 0, the point of application of the resultant T through which the resultant will 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 parallel forces, there is one point through which the resultant will always pass. This point is called the centre qf parallel forces. the centre of parallel forces. ~ 128.-Every body is composed of an indefinite number of elementary heavy particles, which are the points of application of as many vertical or parallel forces. Their resultant is a force equal to their sum, and is called the weight of the body. The point of application of the weight Weightofabody; is obtained by combining the parallel forces in the manner before explained; this point will be the centre of the system, 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. particles, this point must remain invariable, while the forces, without ceasing to be parallel, revolve about the Twomethods of points of application. Instead of causing the forces to finding centre of gravity; 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- A body being suspended by means of a thread A C, by suspension; from the point A, will take such a position, that the effort exerted along the thread to support it, will be in equilibrio with the weight, and thus, when Fig. 60. 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 second line C' G, also passing through the centre of gravity, and whose intersection with the first, will determine the position of that point. 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 Fig. 61. body which satisfy this condition, give as many lines intersecting at secondmethod- the centre of gravity. The upper by poising; and lower points, in which any two ] of these lines pierce the surface, being known, and connected by rectilineal openings, these openings will MECHANICS OF SOLIDS. 137 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 planesnthrough the centre of 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. ~ 130.-This method becomes impracticable in the case Centre of gravity of very heavy bodies, of those which are fixed, or of such foputi,,; 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 distanceofcentre distance of its centre of gravity from the same plane. of ga'vityfroma plane; Hence, the distance of the centre of gravity from any plane, is equal to the sum of the products obtained by nzmultiplying the weight of each element of the body into its distance from this plane, divided by the whole weight of the body. Find the distance, given by this rule, from any three from three arbitrary planes, and the position of the centre of gravity vnmea 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 abridged inthe to be homogeneous, when any two of its parts have the homogeneous bodies. same weight under equal volumes. 138 NATTURAL PHILOSOPHY. ~ 131.-Experience shows us that a bar A B, of wood, Centre of gravity metal, or any other material, which is perfectly homoof regular and homogeneous geneous, will remain in bodies; equilibrio in a horizontal position, if suspended by Fig. 62. its middle point C; and hence the centre of gravity A v R of this bar is situated at the middle of its length. of a bar; The bar is also found to remain in equilibrio when placed in a vertical position, if suspended by the central point of its end; 4 and hence the centre of gravity is situated at the central point of its thickof a bar with ness. If the bar support equal spheres at at its ends equal spheres, the ends; 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 homofreguleneous of gravity of all regular and homogeneous bodies are at bodies, at the their centres of figure. And, hence, a right prism or centre of figure; right prism; cylinder has its centre of gravity at the middle of its circle, &c.; length, breadth, and thickness; a circle at its centre; and a right line at its middle point. centre of gravity By the centre of gravity of a surface, is understood of a surface; of a line. 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 Body very small as compared with its length. symmetrical in reference to a plane; ~ 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 terminated 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. eferencee tofa line; line of This line is called a line of symmetry. symmetry; A surface is symmetrical in reference to a line, when surface the latter cuts into two equal parts, all the perpendiculars symmetrical in 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 gravty rical bodies, is situated in their planes, or lines of symmetry. in planes and Consider, for example, a symmetry; curve having A B for its line of symmetry, and of Fig. 63. which we have found the centres of gravity G and G, of the two halves A illB and A it B. These two halves being turned about the line of symmetry till one is applied to the other, their 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 symmetrical upon a right line G G, perpendicular to the line A B. curve; Hence, if the curves be supposed concentrated at their respective centres of gravity, Ga becomes a right line, terminated by two material points whose common centre of gravity is at the middle point 0, on the line of symmetry. A similar reasoning may be applied to all bodies of symmetrical dimensions. The centre of gravity of a surface which has two axes centre of gravity of symmetry, is at the intersection of these axes. The wo axes f itofh transverse and conjugate axes of the ellipse, for ex- symmetry; ample, being axes of symmetry, cut each other at the 140 NATURAL PHILOSOPHY. case ofthe centre of gravity of the elliptical ellipse; surface. For the same reason, the Fig. 64. rectangle; 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 line of symmetry, its centre of gravity is on this line. A volumewithone right cylinder, with an elliptical base, has two planes of ymmety; symmetry, determined by the longer and shorter axes of the ellipse, its centre of gray- Fig. 65. ity is, therefore, on the line -. aG G, joining the centres of gravity of the bases, and at its middle point 0. sphere many Other bodies are divided symmetrically, in an infinity axes of of ways. Such, for example, is the sphere of which all symmetlpy. the planes of symmetry pass through the centre of figure; it is for this reason that this point is also its centre of gravity. Centre of gravity ~ 133.-If the regular homogeneous body contain of two within its boundary another homogeneous body of difhomogeneous bodies, one ferent density, the centre of gravity of the whole mass is within the other. 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 Fig. 66. body supposed homogeneous, to be concentrated at the centre of gravity 0, and subtracting this weight ~ w from the total weight W, we obtain a difference W- w, neglected in finding the point 0. Let O' be the centre of gravity of the volume corresponding to this MECHANICS OF SOLIDS. 141 difference; join 0 with O' by a right line, and divide this line at the point K, so that w x OK= (WI- w) KO'; the point K will be the common centre of gravity. ~ 134.-Whenever a body may be divided into parallel When the layers layers, and the centres of gravity of these are situated on theof a body havef 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 line 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 C D, Centre of gravity supposed to possess a of a parallelogram; small thickness, be divided by planes par- Fig. 67. allel to C:D, into an 3r indefinite number of strata or layers, the I _ centre of gravity of each one will be at A its middle point, and therefore on the line FE, joining the middle points of the opposite sides C D and A B; 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 IV, joining the middle points of the opposite sides C B and D A; it must, therefore, be at their intersection 0. 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 parnlopipdon gravity of their opposite faces. 142 NATURAL PHIILOSOPHY. ~ 136.-The triangle A B C, 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 triangle, 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 ( I c of gravity of the triangle will also be on the line A F, drawn from 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 a; 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 acngles to the middle point of the opposite side, and at a distance from this side equal to one third of the line. common centre This point is also the common centre of gravity of of gravity of three equal balls, whose centres of gravity are situated at three equal balls. 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 B D at the point G, so that B G shall be double G D. MECHANICS OF SOLIDS. 143 ~ 137.-To find the centre of gravity of any polygon, Centre of gravity as A B CD EF, draw from any one of the angles, as A, of a polygon. the diagonals A C, AD, A E, &c., and thus divide the polygon Fig. 69. into triangles. Find the centres 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 denote the areas of the triangles ABC and A CD by a and a', respectively; then will the centre of gravity of the area A B CD A, be found by the proportion a + a': a:: gg': g'G. In like manner, joining G and g" by a right line, and denoting the area of the triangle A D E by a", will the centre of gravity of the area A B C D E A be found from the proportion, a + a' + a": a" G:: g": G'; and so on to the last triangle; the quantities g' G, G GI, &c., being the only unknown quantities become known from the proportions. ~ 138. —A series of planes Fig.'o. parallel to the base D B C, of the triangular pyramid A pyramid divided into A B C D, will give rise to a layers parallel series of strata or layers per- tothebase; fectly similar to the base, and all their centres of gravity will be situated upon a right i line joining the centre of gravity of the base and the vertex, because they are all similarly situated to the base. 14.I-I NATURAL PHILOSOPHY. 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 its centre of the centre of gravity of the Mg t0 gravity found; 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. / i Let G' and G" be the centres of gravity of the triangu- I - lar faces A BD and B CD; / join these points with the opposite vertices by the right X lines A G" and C G', 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 CD are divided proportionally at the points G' and G", the line G' G" is parallel to A C, the triangles G G' G" and G A C are similar, and give the proportion, G" G GG":: A C 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 joining one of the angles with the centre of gravity of the opposite face, and at a distance fromn this face, equal to one fourth of the line. The common The same result may be obtained for the common ceof fourequlit centre of gravity of four equal balls, whose centres of balls. 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 S to 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 Fig. 71. through this line, and through the angles A, B, CU 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 \ c intersection of this line and plane. Hence, to find the centre of gravity of 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 ~ 140.-Since every polyhedron may be divided into of any triangular pyramids whose weights may be supposed to polyhedron. 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, parallelo10 146 NATURAL PHILOSOPHY. grams, pyramids, parallelo- Fig. 1/2. pipedons, polyhedrons, &c.; the partial the sum of the products E which result from multiplying the weight of each c into the distance of its centre of gravity from some assumed plane, or right line,! must be found, and this sum divided by the entire the sum of these weight of the body; the result will be the distance of the dntided by ghthe centre of gravity from the plane or line. Let it be required, for example, to determine the centre of gravity of any plane area Ca b Fd c; 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 be regarded as a small rectangle, and, supposinig its density uniform, its centre of gravity is at its middle point 0; denoting the density by D, and the force of gravity by g, one of the partial products will be ac + db ac+db ea + ec illustration; D X g X X = Dg X X 2 2 2 The other partial products being found in the same way, and their sum divided by the product of D g into the entire area Cc d Fb a 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 A B and A E, and distant from them, equal to the results obtained by the above process. when the force It iS to be remarked, that when the force of gravity g of gravity is constant and is constant and the density D is uniform throughout the density uniform; body, these quantities strike out, and leave the distance MECHIANICS OF SOLIDS. 14]7 of the centre of gravity from the line, or plane, equal to the martial the sum of the products arising from multiplying the ele- products in terms of the mentary volumes into the distances of their respective -olumes. 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 ordinary process. The screw, the curbs of stair-ways, surfaces of revolution generated by the rotation of a plane curve Use of the centre CD E about an axis A B, Fig. 73. of graity in computilng situated in its plane, are ex- A olumes and surfaces amples. Suppose, in the case C of a volume, the generating area CD E to be divided into - -.. — D small rectangles, of which the sides are parallel and perpendicular to the axis A B. Each z rectangle will generate around the axis an elementary ring, 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 at of the rectangle, multiplied by the mean circumference of the ring, 2 ~r 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, and hence we shall have for the value of the ring the expression 2 n r a. The volumes generated by the other rectangles, whose areas are a', a", a"', &c., will be 2 n r' a,' 2,r r" a", 2 r r"' a"', &c. And denoting by V the total volume generated, we shall have V = 2r (ar + a'r' + a"r" + a"'r"' + &c.); 148 NATURAL PHILOSOPHY. but the quantity within the brackets, is the sum of the products which result from multiplying the elementary volumes of the generating area CE), by the distances of their respective centres of Fig. S3. gravity from the line A B, A 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 ClED by A, the distance a of its centre of gravity from A B by R, we, therefore, have relation of volume to generatrix and path of centre of V = 2 RA A(38). gravity; If, instead of an area, we had considered a plane curve C E, the quantities a, a', a", &c., would represent the lengths of elementary portions of this curve, A Fig. 74. would represent its entire length, a 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 a generated by the motion of any plane, or surface generated by the motion of rule; 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 profile, 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 example-the find the volume generated by the ro- volumeofa Fig. "/5. cone; tation of the right-angled triangle A B C, about the side A B. The cen- tre of gravity G, being found by the / rule already explained, draw G D -. — $. perpendicular to A B. Then, in the i triangles E G D and EB a, we have CB GD: CE: G.E:: 3: 1; whence GD = ~ CB; and 2e GD = 5 B, which is the length of the path described by the centre of gravity. The area of the triangle is 2-AB x CB; whence the volume V becomes V-= 3 CB2 x AB, which is the usual measure of the volume of a cone. Example 2d. Let it be required to Fig.'76. example-the:D surface of a conic find the surface generated by the rota- frustum tion of the line C D, about A B. The centre of gravity of CD is at its middle point G; and G D', CA, and D B being perpendicular to A B, we have A.-D,' —GD' ='(AC+ BD); and for the path described by G, 2e GD7 = 2(AC+ BD) 2tGD, 2ir 2~ 1Sib NATURAL PHILOSOPHY. ancd hence, 2rA C + 2 r BD V= 2.CD; which is the usual measure for the convex surface of a conic frustum. example-the _Example 3d. Let it be required vaolume o r to find the volume of the curb of Fig. 71. stairway curb; 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 of the mean helix. excavation from The excavation taken from a a ditclhes; lditch, of which the profile is constant, may be estimated in the same way. In examples 1st and 2d, the. centre of gravity is supposed to l~e holds for have described an entire circumfeany portion of rence; but had it moved through only an eighth, tenth, or an entire,revolution. 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. VII. MOTION OF TRANSLATION OF A BODY OR SYSTEM OF BODIES. Motion of ~ 143.-A body, or system of bodies, is said to have a translation; 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 communicates 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 7P V the measure of f [f the inertia of an element; and the force of inertia f', of an element whose weight is p', by 1Y' V. f g t' 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 respectively move, and as these are supposed parallel, the forces of inertia are parallel, and give a resultant equal in intensity to their algebraic sum. Denoting the intensity of this resultant by F, we have V f+ + + &c. F=f+f +f" + &c. = (P + P + P +.) and replacing the sum of the partial weights by the entire weight P. and -by the entire mass Mof the system, we shall finally have VF = -.. (39) of that of the t * 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, ip', p", &c., to which they are respectively applied, and thus the point of application of F, will coincide 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 ineria inwords. 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, multilied iito the ratio which the small degree of velocity communicated bears to the time during which the velocity is impressed. And the total force of inertia has its point of application at the centre of gravity. This coincidence of the centre of inertia with the centre of gravity, results from the assumption that the The force of force of gravity is the same in its action upon the different gravity being parts of the system. Had it been otherwise, that is to say, constant, the centre of gravity had the force of gravity varied in intensity from one eleand of inertia coincide; ment to another, the centre of inertia, being always at the centre of mass, would be different from the centre of gravity. The intensity of the force of gravity being regarded as these centres the same within the limits of a body on the earth's surface, ensiblythe same the centre of inertia and of gravity may be regarded as in bodies on the earth. coinciding, and hence these terms will be used indiscriminately. ~ 144. —Let V represent the velocity of a body having a motion of translation, supposed uniform at any instant; Quantity of the quantity of motion of any one of its elements whose motion of a weight is p, is measured by body; PV g and of an element whose weight is p', Tp' 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 v= f.. (40). its measure. g 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. ~ 145. —When a certain degree of velocity v, is im- Motionofa pressed upon all the elements of a body during a very hdinctYne ofthe short interval of time t, we have seen that the total force motive force of inertia is given by, Eq. (39), passes throlgh gravity; For= Al x X V 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 Fig. 78. centre of gravity of the body, a motive force X, it will commu- x nicate to it a simple motion of translation. For this force X will be equal and directly opposed to the force of inertia ]1 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 intensities of these partial forces will be proportional to the respective weights of these elements. Denoting the masses of the elements by m, in', m", &c., we shall have, m f m',, ml f= F~z~, - 7 f i &c., iM ~ ~~~~ iMao 10b4 NATURAL PHILOSO P IY. The degree of velocity which each of these forces impresses upon the part on which it acts, will, ~ 82, be measured by f. t f'. t f".t, m',, &c.; or, replacing f, f' f", &c., by their values as given above, simply by the expression F.t AL' and as this measure is the same as that before deduced, Eq. (39), for the degree of velocity impressed on the centre will be of gravity by the force F, or its equal X, we see that, that of simple m n translation. 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 tle centre of gravity; and, reciprocally, if the line of direction of the force, or that of the resultant, in the case of severalforces, 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 A B, Fig. 79. Motion when the not passing through the centre fpathough othe of gravity G, it is easy to see m F centre of gravity; that the motion cannot be one of simple translation. For, if " i this latter motion obtained, the partial forces of inertia would have a resultant of which the 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 extremities of a physical line or bar A G, can produce an MECHANICS OF SOLIDS. 155 equilibrium, unless they act in the direction of the bar. willbethatof Hence, when a body receives the action of a force, of which tanslation and of rotation at the 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 perpendicular to the planes in which the other elements, for the instant, rotate, and let A B be its trace upon that one Fig. 80. position of the element having of these planes which e a motion of contains the point C.- -:;l, translation. and its rectilinear A path. Let mh be the projection of some one element m' upon this latter plane, and take C C1 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 m 2 equal and parallel to CC1; then would n1 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 m mn2, diminished or increased by the projection of m2 ma upon the line C C1 according to the direction of the rotation. Project the points ml, m2, and m3, upon A B, by the perpendiculars nm k1, m2 k2, m3 k8; then will the actual velocity v', acquired by m', be m m2 - m2 o, or V _ V - r2 ~ o; but m2o 0= mn2 X cos n m2o = mn 2 X cos C1m2k2; 156 NATURAL PHILOSOPHY. denoting Cam = C1 m2 = C1 m3 by r, m1 k = m2 k2 by y,, and the velocity of rotation acquired by a point at the unit's distance from C by Yj, then will 2 m3 -- V1 r, and cos Cmc2 k2', Relative velocity which substituted above, give of two elements of a solid body = - y (41) in motion; Moreover, m3 o is the velocity of the ele- Fig. 80. ment m' perpendicular to the direction of C's motion; and A __/' calling this velocity v", and the distance C1 k2, x,, we shall the same in a have direction t = V1 x,. (42). perpendicular to V! 1 X.. (42) the former; Denoting, as before, the weight of the element m' by p', and its force of inertia in the direction CC1 by f', we have 42! --. —- ~ ---- - _ _ - - -- g t gt,), and similar expressions for the inertia of the other elements. Taking the sum of these, and representing the inertia of the entire mass by F, we have, from the principle of parallel forces, F= -.v(p + p + p+ &c.) V1 (r * I" + g "&c); t 1gJI /JU IECIANICS OF SOLIDS. 157 or, denoting the entire mass of the body by MH and the masses of the several elements by mn', mn", in"', &c., this reduces to value for the force F = Al.1 (i' y, + qi" y,, + m"' y,,, + &c.). of inertia of a t t body; 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 n' y,, m" y,,, &c., are taken, passes through the centre of gravity, we have ml'y, + m"y,, + i"' y,,, + &c. = 0; and the above equation reduces to F= If, v; always equal to t 7 the mass into ratio of the increment of velocity to that which is identical with Eq. (39). of the time. 158 NATURAL PHILOSOPHY. The body will We conclude, therefore, 1st, that when a body i acted have a motion of translation; it upon by one or more forces, its centre of gravity will move will also rotate as though the forces were applied directly to it, provided their about the centre of gravity; directions remain unchanged; 2d, that when the line of direction 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 V = (42)'; translation. in which v is the velocity impressed in the very short interval t, from which we may pass to the velocity acquired 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 simple motion of translation. For v being the common velocity of any one element, - X v2, will be the living force of that whose weight is p; -p x v2 the living force of that whose weight is p', &c.; so that the sum of all these living forces, or the total living force, denoted by L, will be v2 x p + p + P + &c.; and representing the entire mass of the system by 1;, as before, Living force in a simple motion of = J 2v translation. If the body have a motion of rotation as well as of MECHANICS OF SOLIDS. 159 translation, then will the living force of mn', in the direc- If the body have tion of the motion of translation be, Eq. (41), rotation; M V =M (V- y )7 = m2 v - 2 v../ m' y, + T/I M7 Y,; and in the direction perpendicular to the motion of translation, Eq. (42),, v =2 = m' V12X 2; and similar expressions for the elements whose masses are ln", im'", &c. Taking the sum of these, denoting the living force, as before, by L, and reducing by the equations m'y, + m"y,, + &c. = 0, y2 + x, 2 r 2 y, 2 + X,,2 = r i2, &c. &c. = &c., mn' + in" + mn"' + &c. = /; we find L = Mv2 + 12(I' r,2 + in" r,, 2 + &c.); or, making m' ri2 in" r,,2 + &c.. i r2 the living force is equal to that due yL = A2 + V2 m... (43). to translation, increased by that due to rotation. ~ 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 greatly simplified, since we are permitted to disregard the shape of bodies, to suppose them concentrated about 160 NATURAL PHILOSOPHY. their centres of gravity, and to reason upon these points as upon the total masses. Generaltheorem ~ 149.-We have seen that in all questions affecting — the qutantif the circumstances of simple motion of translation, we may regard the mass as concentrated about its centre of gravity. 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 performed 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 passing through the centre of gravity, we may still reason upon the motion of this point as though the mass were concentrated 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 Fig. s1. by taking the product of equal to the the weight into the path weight,into the described by the centre of projection of path of the centre of gravity, estimated in a vergravity. tical direction. If, for exam-. ple, the centre of gravity of any body, as a bomb-shell, pass from the position G to 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 multiplying 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 bars, levers, &c., connected with each other after the man- Applies to all ner of ordinary machinery. If the quantity of work kihinds oy; 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. In general, let p, ip', p", &c., be the weights of the several pieces connected together; h, h', i", &c., the vertical 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 z, the vertical space described by the common centre of gravity: then will P' = - h + _p' h' h + p h" + &c... (44). mathematical expression of the rule; To demonstrate this, let m, m', i", &c., be several bodies so connected as to be acted upon by each other's weights. Let P denote the weight of the entire system; Fig. 82. p, p', p", &c., the X weights of the sever- iM al bodies nz, m', m", &c.; Z, the distance ( T of the common cen- A -- tre of gravity from a horizontal plane A B; and HE, f', 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 P Z = pH +v'H' +p"H" + &c.; demonstration of and for a second position of the system, PZ= p, + p' + p"H, + + &e.; 11 162 NATURAL PHILOSOPHY. and subtracting the first from the second, P(Z,-Z) =p (, — ) +_p' (_H'-H') + p" (,"-'-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; HI,'- H', H," - H[" &c., are the corresponding distances h, h', h", through which the centres of gravity of the bodies rn, m', m", &c., have ascended or descended. Moreover, the products P (Z.-Z), p (E, - H), p' (H,' -H-'), &c., are the quantities of work due to the entire weight and to the partial weights. conclusion and 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 wzeights which ascend, diminished by the sum of the quantities of work of the weights which descend. ATIII. EQUILIBRI'UM OF A SYSTEM OF HEAVY BODIES. Equilibrium of ~ 150.-If'the system of heavy bodies be so connected, heavy bodies; 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 -z = 0, and Eq. (44) becomes, ph + p'h''+ p" h" + &c. =.. (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 equieach other;st extraneous effort librium in the system, and the least extraneous effort MECHANICS OF SOLIDS. 163 will impart motion. Such is the condition of equilibrium of a system of bodies acted upon only by their own weights. This equilibrium presents itself under different states according to the positions of the system. If the slightest the position be such that in a slight derangement the effort suffcient to ZD I n give motion. 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 unst ble, 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 equilibrium; 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 system 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 indffferent. indifference. 151. —Take a rod suspended at one end so as to Fig. 83. turn freely about a hori- zontal axis A, and supporting at the other a body which is symmetrical in Illustration of stable reference to a line drawn eqaiibium; from the axis A to the common centre of gravity G. X-. It is obvious that there will i be an equilibrium when the i rod is vertical. It is moreover stable; for in deflecting the system, the centre of gravity will ascend while 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 removed. Indeed, the system will, when abandoned, perform a series of oscillations, whose amplitude about the vertical A G, will diminish continually till it comes to rest. Now suppose the system inverted; if the rod be perfectly vertical, the line of direction of the weight will pass Fig. 84. through the point of support A, illustration of and there is no reason why the unstbleium; system should move one way equilibrium; rather than another. It will there-.............: —.. fore be in equilibrio, but the equilibrium will be unstable; for, however slight the derangement, / the centre of gravity G will descend along the circular path a G', described about A as a centre, and a certain amount of work will be requisite to bring it back to its primitive position. When a cone ABC, resting upon its base B C is inclined to the position A' B' C, its centre of gravity G will ascend and describe an arc G G', and if, in this inclined po- Fig. 85. sition, it be abandoned by the A,,a7 disturbing force, it will return. loth kinds by When the cone is placed up-' illustrated by,:. means of the on its vertex, with its centre of B cone; gravity directly above that point, it will also be in equilibrio as it Fig. 86. 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 a second of unstable equilibrium. An elliptical cylinder placed upon a horizontal plane MECHANICS OF SOLIDS. 16O is in stable or unstable equilibrium, according as the smaller or longer axis of its elliptical base, also by an is perpendicular to the plane. ellipsoid of Fig. 87. revolution; A spherical ball upon a horizontal plane, is an example of equ6iibriunz of indifference. The centre of gravity remaining at the same level however the ball may be displaced, provided it pre- Fig. 88. serve its contact with the plane, the quantity of work necessary to displace it will always be inappreciable, and the ball will, in consequence, have no tendency either Fig. S9. to recede from or return to its primitive equilibrium of position. A perfectly circular cylinder ildiference exemplified by on a horizontal plane is an example of the sphere; the same kind. Some varieties of draw-bridges are but collections of by some varieties pieces in a state of equilibrium of indifference. And to o'1draw-bridges; insure this state, it is only necessary that the common centre of gravity of the bridge and appendages, shall preserve the same level during the motion, in which case, the system will be in equilibrio in all possible positions. Wagons and carriages should, in strictness, require no work to move them on a horizontal plane, except to overcome their inertia, and should, therefore, be so constructed as to preserve their centres of gravity always on the same level. If, during the motion of a wheel, it is seen sometimes to quicken and sometimes to slack- a en its motion, it is because the centre of gravity G is out of heffent of throwing the axis of motion A, and, there- gravity out of the fore, alternately rises and falls axisof a wheel. during the rotation. A wheel whose centre of gravity is out B of the axis of motion, passes 1.66 NATURAL PHILOSOPIIY. 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 B C, through the axis A, at the lowest point 0, and the latter when it crosses the same line at the highest point 0', of its path. The common The common balance consists of a horizontal arm balance; A B, mounted upon a knife-edge D, resting upon the surface 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 subFig. 91. stance to be weighed, and the other the standard weights previously determined. The balance may be stable, A unstable, or indifTerent, I the position of its according as it tends cerntre of gravity; to return to a horizontal position when deflected from it, to overturn, or to retain any position in which it may be placed. Referring the entire system 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 comwhen stable; mon centre of gravity of the moveable part of the appaunstable; ratus from the plane A'B'. If this distance be less than indifferent; FD, the distance of the knife-edge above the plane of reference, the balance will be stable; if greater, the balance 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 on the vertical through the point line FD, passing through the knife-edge. of support. IX. EQUILIBRIUM OF SEVERAL FORCES, VIRTUAL VELOCITIES, AND MOTION OF A SOLID BODY. ~ 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, forces acting and regard these points as the vertices of an invariable triangle a b c; resolve each force into three components whose directions shall pass through the given point of 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 common point at a, b, or c. Each of these groups, having a common point, may be replaced by a single Fig. 92. resultant, and thus, the equilibrium of the giv- - en forces is reduced to that of three forces. i Call the resultant of the given forces may be replaced the group having the by three groups common point a, X; of components; that of the group having the common pointR b, Y; and that of the group having the com- and these by mon point c, Z. These three forces being in equilibrio, threesingle the equilibrium will not be affected by supposing the three forces; 168 NATURAL PHILOSOPHY. lines a b, b c, and c a, to become fixed in succession. The line a b being fixed, the forces X and Y, whose directions intersect it, will be destroyed by its resistance, and if the third force Z, does not an equilibrium act in the plane a b c, it Fig. 92. requires these will cause the system three to act in the same plane; to turn about a b; the same may be shown of the forces X and Y.; The forces, X Y; Z. must, therefore, act in kk the same plane; and in order that they may be in equilibrio, the rethe resultant sultant of either two zero; of them must be equal and directly opposed to the third; that is to say, the resultant of the three must be zero. If the resultant be zero, the quantity of work is zero. The quantity of work of X, Y, or Z, is equal to the algebraic sum of the quantities 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. may 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 R, A, &c. Whence we conclude, that several forces, acting in equlihrio upon the different points of a free ocdy, will be in equilibrio, when the algebraic sum of when the algebraic sum of the quantities of worTk of the forces the quantities of equal to zero. work is zero. iS equal to zero. Now suppose the body to be slightly deranged from its state of rest, and let A A' be A" A' the path described by the ":' —----- point of application A, of MECHANICS OF SOLIDS. 169 the force P, in an indefinitely short time t. Draw A'n perpendicular to A P; A n will be the space described by the point of application in the direction of the Force, and the quantity of work performed by P during the derangement will be P X A n, or Pp, denoting A n by p. The path A A' is called the virtual velocity of the force P; Virtual velocity, A n=p, the projection of the virtual velocity; and the Pr.oetso ofe l virtual velocity product Pp, the virtual moment of the force P. and virtual Denoting by q, r, s, &c., the projections of the virtual moment; velocities of the forces Q, R, S. &o., the quantities of work, or the virtual moments of these forces, will be, respectively, Q q, R r, S s, &c; and if the system be in equilibrio, we have, from the rule just demonstrated, Pp + Q q ~ Rr + Ss + &c.-O.. (46). This equation is but the mathematical expression of the principle of principle, known under the name-virtual velocities, which virtual velocities. consists in this, viz.: whern several forces are in equilibrio, the algebraic sum of their virtual moments is equal to zero. ~ 153. —Any mechanical device that receives the action of a force or power at one point, and transmits Amachine; it to another, is called a machine. Conceive a machine, composed of wheels whose axes are sustained by supports, and which communicate motion to each other, either by teeth, chains, or straps, Fig. 94. on their circumferences. Suppose a force A. c or power to be applied composed of V.-.2~ wheels; so as to turn the first 2I wheel; this wheel will experience a resistance R' from the second; this resistance, in its turn t becomes, for the second 170 NATURAL PHILOSOPHY. wheel, a power which causes it to rotate also; the second Itle process by will experience a resistance from the third wheel, which which the action of awhich the action resistance becomes a of a meter is transmitted; power to give it mo- Fig. 94. tion, and so on to the end. But each wheel experiences a reaction A at the points of sup-.. port which keep it in position, and it is this reaction that becomes the means of transmitting the power to the following wheel; for if these points were unsupported, the wheels would cease to act upon each other and the power first applied could not be transmitted. poits of support Now, replace the supports, by the efforts of reaction replaced by which they exert: each piece or wheel will become a free active forces; 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 WTV, the resistance of the second wheel B by Rs, and the reaction at the point of support of the first wheel, by C1; the projection of the virtual velocity of W, by wl, that of R, by r2, and that of C1 by cl; then will sum of the virtual moments for first'1Wl 1 1 Cl1 R2 = 0; piece; denoting the resistance of the third wheel D by R3, the reaction at the centre of the second wheel by C2; and the MECHANICS OF SOLIDS. 171 projections of their virtual velocities by r3 and c2, respectively, r2r2 + C2C2 + J3 r3 = 0; same for second; and thus we may continue throughout the entire combination 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 R,, the reaction of the support of the last wheel by C,; and the projections of the corresponding virtual velocities by We, re, and c,, respectively, we shall finally have, Re re + Ce ce + lV, We = O. also for the last; But from the nature of the connection, the points of support must not move; their virtual velocities, and therefore the projections, must be zero. Hence, UC cl- 0, C2 c2 = 0,... ce - 0 and the preceding equations become WI w1 ~ P2 A 2 = 0, R r2 T,+ -3 r3 0= O virtual moments.. (47). of points of. 0, support, zero; R, r,e V, We = O, J 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 W1 wl + W e = 0... (48); relation of motive force to the final which shows us that the quantity of work of 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 quantitj of work of those which tend to turn it in the opposite direction. 172 NATURAL PHILOSOPHY. An examination of Eqs. (47), will show that the same remark is applicable to each piece of the combination taken separately, and thus starting from the piece which first receives the action of the force, and proTool; work of ceeding to that which does the work, and which, on this power equto account, is called the tool, we see that the quantity of that of tool; t 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 resistances, and we shall always find the work of the first to be equal to that of the second. If the bodies press against each other in a way to produce a change of figure and friction, new resistances the work of arise which must be taken into account, and the work friction and that of these must be subtracted from that of the forces to employed to change figure; obtain the work of the tool, and hence such resistances are, in general, a hinderance to the final work to be accomplished. If the equilibrium is to be maintained while the work estimated machine is at rest, then must the quantity of work be db a lupoeedno estimated by the aid of a supposed displacement, as in that displacement, to avoid ineltia; case, the influence of inertia will be avoided. If the equilibrium is to exist during a uniform motion the same in of the machine, the quantity of work must be computed unifor motion; from the actual motion of the points of application, for then the inertia will again be excluded. But if the equilibrium is to take place during an when themotion acceleration or retardation of the motion, the inertia of is variable,th tle pieces will no longer be zero, and must be comprework of inertia comes into the hended among the powers and resistances. The conaccount. (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. ~ 154.-Whenever the forces applied to a body accel MECHANICS OF SOLIDS. 13 crate 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 w'k of iner'tia to the work of all 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 difference 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 egual to half of the equal to half the living force gained or lost by the body during the same living force lost or gained; 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 rectilineal path A T illustration —the But by virtue of case of a Fig. 95. projectile; the weight, which would act alone O.r in vacuo, the projectile is continually deflected from this path, X B~-XX and will, in consequence, describe a - curve line A BD; A and we know, ~ 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. is passing from A to B, the work expended by its weight will be P X B C, or P X H, by making B C= H. Denoting by V', The velocity of the the living force at projectile at B, its Fig. 95. two points; living force at this point will be P X g and at A, it was P V2 A D g and its loss of living force, in passing from A to B, loss or gain of P (V - V T722 living force; g 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 equal to double p 2 2w the work of the 2 force; g or V2- 2 =2 2 g... (49). relation of Thus the difference of the squares of the velocities in any velocity to difference of two positions of the projectile, moving in vacuo, is equal level; to the difference of level of the two positions, multiplied by twice the force of gravity. When the projectile arrives at D, then will H = 0; and V72- V'2 = 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; From Eq. (49), it is obvious that while the projectile velocity on is on the ascending branch of the curve, its velocity di- ascending' 0 7 I y di- descending minishes, and while on the descending branch its velocity, branch of curve; on the contrary, increases. The description of the trajectory or curve A B D 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 ""acuo found; of these motions, (~ 105 and 108.) The body may be regarded 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 Fig. 9 computed by the principle of the / parallelogram of Yi,-. ---- ---- velocities. After the body leaves the t'.e point A, it will be X subjected to the 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 A T by a; the space described in the horizontal direction A x by x, and the time required for its description by t, then will space described X = VCOS t... (50). horizontally in the time t; But in the vertical direction, the weight will, during equal times, diminish the component of the initial velocity, 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 A7( NATURAL PHILOSOPI-IY. space in same y for KF. Hence, denoting the vertical space by y, we get time in vertical direction; g... (51). y =YVsin a t -... (51). the true position The true positions of the projectile, which are but points of the projectile of the curve A B X, are given by the intersections of a vertical and horizontal line drawn at distances from A, equal to the spaces y = A y,, and x = A x, 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). Eliminating t from these same equations, and reducing, we find the equation of q the curve y = tan.x - -2 X (52); described-a 2 V. COS2 a parabola; 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 19 where the projectile range; attains the same level, is called the range. The angle angle of x A I = a, is called the angle of projection. projection; To find the range, make y = 0, in Eq. (52), and find the corresponding value of x. Making y = 0, we have 0 = tan ax - g X2, 2 V2 cos awhence x 0= 2 V2 sin a. cos ao x = - AiD= range; and representing by h, the height, due to the velocity V; we have V2 = 2gh..... (53); MECHANICS OF SOLIDS. 177 and denoting the range by R, and recollecting that 2 sin. cos c = sin 2 a, we have, finally, R 2 h.sin 2 4 te value of the range; This value for Fig. 97. the range will be is! a maximum when a = 450, in other 2' words, the greatest range corresponds to an angle ofprojection equal A_ to 450. Since complementary sin 2 c = sin 2 (90~ - a), angles give the same range; it follows that the same range may be attained by two angles D A T and D A T', which are complements of each other. If in Eq. (54), we make = 450, then will greatest range P = 2 h, given by an angle of projection whence equal to 45a; h =R; and this in Eq. (53),. will give value of initial V = Rg ***.. (55). velocity in terms of greatest range; 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 12 = Rg; 12 178 NATURAL PHILOSOPHY. 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 W.g = RW; and the effective quantity of action impressed, denoted by Q, effective quantity of action = R W... (56). impressed upon the projectile; It is from this relation that are obtained the results of the eprouvette, a small mortar constructed to test the eprouvette; relative strength of different samples of gunpowder. For this purpose, a heavy solid ball is projected from it under an angle of 450, with small but equal charges of different kinds of powder, and the relative strength is inferred from the effective quantity of action impressed. For example, suppose equal charges of two different examples of its samples of powder, give R= 1050 feet, and R= 1086 use; feet; these values substituted successively in Eq. (56) give Q= W. 525 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. these results but This supposes the motion to take place in vacuo. If approximations the trajectory be described in the air, the resistance of this in air; e 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. 179 close as we please by using small charges and very dense these ~~~~~~~~~~~~projectiles. m~approximations projectiles, made close by Taking the general case, without limitation as regards giving small the velocity of a body in air, the curve may still be tlocies; described, provided we have a table giving, in pounds or general case in any other unit of weight, the resistances corresponding to which the projectile is different velocities of different calibres. thrown into the Thus, knowing the initial velocity and its two com-_h.; ponents, find from this table, in pounds, the value of the initial resistance, and its horizontal and vertical components 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 atmospheric motive force in the vertical direction. With these forces, atmospheric; )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 obtain the place this a fourth, and so on indefinitely. We thus obtain a of theprojectile series of components, forces acting for a short time with t any instant and the curve. constant intensity in the horizontal and vertical directions; 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 consecutive intervals from the beginning of the motion, will give the distances, A x, and A yl, which determine the points of the curve. The actual space described by the trajectory will be the development of this curve. 180 NATURAL PHILOSOPHY. X. MOTION AND EQUILIBRIUM OF A BODY ABOUT AN AXIS. The work of ~ 155. —The principle demonstrated in ~ 113, of the forces which tu.n aforces which turn work of forces acting upon a body, may be extended to fixed axis 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. Conceive a force R, acting upon the point A of a body free to turn about a fixed axis L M; resolve this force into two others, the one Q, parallel to L M, the other P Fig. 98. in a plane perpendicular to this line, and passing through the point of application A. Doing the same with regard c 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 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 of their componentsin move in that direction, and the path described by the planes point of application of the resultant of the parallel group phependicular to is nothing, and therefore the quantity of work is nothing. Thus, the total quantity of work of the given forces is MECHANICS OF SOLIDS. 181 reduced to that of their components, in planes perpendicular to the axis, and passing through the points of application. ~ 156.-The quantity of work of forces applied to a Theworlkofthe body which can only have a motion of rotation is always, components perpendicular to 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. 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 perpendicular plane containing the component Fi 99 P. Let fall upon PA, the per- L 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 ), is the same as that estimated by 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 proportional to their distances from the axis; hence if kS denote the length of arc described at the unit's distance, and r the distance of the point D from the axis, then will 8 = rSI, and the quantity of work of P becomes the quantity of work of a single P r SI, component; 182 NATURAL PHILOSOPHY. and for forces of which P', P", &c., Fig. 99. are the projections, at distances from L the axis equal to r', r", &c., respectively, we have the quantities of work measured by, the same for other P' r' S1, Pfl r" &c. &c. components; Knowing that the total quantity of effective work of the given forces, Mwhich 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 the effective work T 5 of all the q = (Pr + P' r' + P" r + &c.).. (57). components; 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 arisingy from multiplying the arc described at the unit's distance fromnz the axis, into the momnent, in reference to the sarne line, of the projection of the force oan the perpendicular plane, and conclusion; Eq. (57) shows that the effective quantity of work of several forces, applied to turn a body about an axis, is equlal 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. The sign of the moments of those forces which tend to turn the body in one direction, must be different from signs of the the sign of those which tend to turn it in an opposite moments. 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 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 + P'r' + P"r" + &c.= O.. (58); that is to say, several forces will be in equilibrio about a Forces in fixed axis, when the algebraic sum of the moments, n eqfuieibrioin reference to this axis, of the projections of the forces on per- fxedaxis. pendicular planes, is zero. ~ 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 principleofliving XI~~~~~~~~~ ~~~~forces to a motion 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 inrotation, the rather by their projections on planes perpendicular to the work of the perpendicular axis, it is easy to perceive that in the motion of rotation components, is of a body, the work of the perpendicular components of half the increment of the forces is half of the increment of the living force living force. of the body. The process of estimating the living force 184 NATURAL PHILOSOPHY. of a body having a motion of rotation will now be given. Estimate of the ~ 159. — Consider an element m of a body, situated at a living force of a distance r from an axis of rotation L XL Denote by V the body turning about a fixed velocity which it has at any inaxis; stant, and by p its weight, m its Fig. 100. mass = i. Then will its living force be. 2 r m V2. g If S denote the space described by m during a very short interval of time t, and AS the space described in the same time by a point at the unit's distance from the axis, we shall have S S=r.s, and dividing both members by t, r..... (59); t t 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 the angular velocity; X t in which V1 is the velocity of the point at the unit's distance from the axis-in other words, the angular velocity; relation of and Eq. (59) becomes angular to absolute'velocity; V = r. -VF, MECHANICS OF SOLIDS. 185 and the living force of m becomes i r2 Vj2. The simultaneous living force of m', is m' r' V2, and so on of others; and the total living force of the entire body, denoted by L, becomes value of the L = V,2 ( 2 + t' r'2 + m" r"2 + &c.). (60). living force of a rotating body; In which it is to be remarked, that if the living force changes, the factor V1 will alone vary, while the factor (m r2 + M'r'2 + m" r"2 + &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 I= mr2 + M,'r2 + m" r"2 + &c... (60)' L = V.1.....(60)"; 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 ofthe angular velocity angular velocity, multiplied by its moment of inertia. into the moment Let us suppose that at the end of a certain interval, of inertia; the angular velocity becomes V1', the living force L', will be L = E312 I; and subtracting the preceding equation from this one, we get L L' - ~ = I. (T2 - V2)... (61), increment of 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. of application, estimated in its own direction duming the interval in question; then will equal to twice the quantity of work I * (Vl' - V) = 2 F.. (62). of the motive forces in same time. From this expression it is easy to deduce the nature of the quantity 1. For if we suppose the change in the angular velocity to be unity, 1/2 -V' = 1, and I= 2 F.E; What is meant by whence we conclude, that the moment of inertia of any body, the moment of is twice the quantity of work exerted by its inertia, during a its measure? change in 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 of its distance from the axis, Eq. (60)'. ~ 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 Example we know the angular velocity at illustrative of Fig. 1 1. iustrative of any two given instants of time, and the preceding principle; 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, we may deduce the angular acceleration. Suppose a wheel, for example, mounted upon a horizontal arbor and turned around its axis by a weight P, suspended from a cord wound around the arbor; required MEJCHANICS OF SOLIDS. 187 the angular velocity V1 of the wheel when, moving from wheel turnedby a state of repose, the weight shall have descended through the action of a weight; a vertical height H. Let I denote the moment of inertia of the wheel, then will the living force acquired be I V12, and we shall have, I. V12 = 2P. TH; whence 2 2P. HF Y,~ I and consequently 2 p. IT value of the 1 = ] 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 moter 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 moter. It is a kind of store-house in which to husband wdrk 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 ti~ and V1'. The increment of the living force of the fly-wheel will be equal to double the effective quantity of work of the moter, and we shall have, retaining the notation of ~ 159, I(Vl2 - F1) = 2FE; increment of or living force in any interval;,2 2 _ 2 F. E 188 NATURAL PHILOSOPHY. which gives the difference of the squares of the angular velocities. If the quantity of work developed by the moter 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 2.. squares of the I velocities; 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 be made to to make its motion approximate to uniformity, even approach uniformity; though the motive force be very great. While the motion is accelerated, it is obvious that the work of the moter 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 moter, which now becomes less than the resistance. There are certain machines whose tool cannot perform its work without the fly-wheel. This is strikingly exemplified 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 moter exceeds that of the tool or saw during one semi-oscillation, exemplified in while the reverse takes place during the other; in the saw-milln 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 meter 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 m f Distance from f= t the axis at which the resultant inertia of a and for any other masses n', mn", &c., whose acquired rotating body is velocities in the same time are V', V", exerted; fM TV' t I f4'" b" T — I f --, &c. If, moreover, the masses m, n', n/", &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 = V1; V' = r' V; V" = r V1; which in the above equations give, V E Ip V value of the f = in r. Jel; f' I MY' r f m." r" -; &c. partial forces of t i t i t inertia; But, if this increment of angular velocity V1, has been impressed upon the body by a force F, whose direction is 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,', in", &c., about the axis, and hence, ~ 157, FR - (f r + f' r' + f" r" + &c.) = 0; equilibrium of these with the or motive force equal to their resultant; FR -- -V (Mr' + rm'r'2' + Mt"r'" + &c.) = O; t 190 NATURAL PHILOSOPHY. but the expression within the brackets is the moment of inertia I, and therefore the moment of V the inertia F.FR = —.I. K.. (63); actuaally exerted; t 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 distinction of the time during which it is impressed. Notwithstandbetand what is 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 moment of inertia; of inertia, they must not be confounded with each other. The former is converted into the latter by making - equal to unity. From Eq. (63) we find value for the. R X t 4) angular velocity I.. from which, having given the motive force that impresses 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, how used. 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 rotation. ~ 163. —The moment of inertia of a body with referMeasures of the ence to any axis, we have seen, is measured by the sum of moments of inertia; f the products which arise from multiplying each elementary mass into the square of its distance from the axis. MECHANICS OF SOLIDS. 191 Of all the moments of inertia of the same body, those are thosein reference easiest obtained which refer to axes through the centre of to axes through centre of gravity gravity. It is, therefore, important to be able to find the easiest obtained; moment of inertia with reference to any axis, Fig. 102. by means of that taken with reference to a parallel axis through the centre of gravity. Let GfE be this latter axis, L M any parallel axis, z an elementary mass of the body GKIUi, through which element conceive a plane to be passed perpendicular to the axes, and cutting them at the that in reference to any axis, in points a and b. Join terms of the m with a and b, and let fall from m, the perpendicular moment in reference to a.r e upon a b. Designate m b by r, mN a by r,, a b by eparallel axi D, and a e by d; we shall have through centre of gravity; r2 = r,2 + D2 + 2Dd, and multiplying by the mass m, mr2 = -nr,2 + mDa + 2mDd; and for the masses m', m", mn"', &c.,,,2, 12 in r =' r,' + I' D2 + 2m' Dd', m" r,2 = m" r,'2 + m"aD2 + 2m"Dd", &c., &c., &c. ), which is the distance between the two axes, remaining obviously the same in all. 192 NATURAL PHILOSOPIHY. Adding these equations together, and denoting the moment of inertia in reference to the axis 61 H by I1, and that in reference to ~L X by I, we find the sum of all thepartial I-1 + D2 (m + m' + m" + &c.) + 2 D (md + m'd' moments; + in" d" + &c.); but m - + n' -+ " + &c. Fig. 102. is the entire mass of c the body, and m d + m'd' + m"d" + &c. is, a the sum of the products 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 resulting value; - = r 1 + iJfD2.... (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 conclusion. the centre of gravity, increased by the product of the entire mass of the body into the square of the distancefrom 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 of the bodies are small, in from the axis of rotation, we may take, for the moment compariston cit of inertia, simply the product of the mass into the square their distances from the axis; of the distance of the axis from the centre of gravity. 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 L M; we shall have V2. I = EV2~1 + M.D2 DV2.. (66); valueofthe living force; but V121 is the living force of the body; V12 1 is the living force it would have, if it rotated about a parallel axis through the centre of gravity with the same angular velocity V; M11. D2 V12 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 wouldW~rds; 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 V12 may be neglected in comparison with I. 02 V2, we have simply V21 = M.D... (66)'; valuewhen the linear dimensions of the body are that is to say, the living force of the body is equal to the vely small as product of its mass into the square, )2 V2, of the velocity compared with its distance from of its centre of gravity. the axis. ~ 164.-Thus far the moment of inertia of a body has been expressed in terms of its elementary masses. If the Momentofinertia body be homogeneous and the specific gravity or weight of their in terms of a unit of its volume be denoted by 6, its elementary dimensions and volumes by a, a', a", &c., and masses by m, m', m", &c., density; we shall have 6 a 6 a/ 6 a" inm= —; m'-; m =- e.; g g g and these in the general expression h, of the moment of inertia, give I -= (ar2 + a'r 2 + atr2 + &.); its general value; 13 194 NATURAL PHILOSOPHY. that is to say, to find the moment of inertia of any homorule; geneous body, find the value of a r2 + a' r'2 + a" r"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. noment of inertia ~ 165. —l st. The moment of a straight bar, of inertia of a straight bar in reference to a Fig. 103. perpendicular whose length is a and cross axis through its A middle; section b, in reference to an axis passing through its middle point A, and perpendicular to its length, is given by 1 = b. (12 a), very nearly. 2d. The moment of inertia of a right cylinder Fig. 104. having a circular base, with that of a right respect to an axis through cylinder, is its centre of gravity, and own axis; coinciding with its axis of figure, is given by the equation 71 -Cr4 X - 2 9 in which r is the radius of the base, c the length of the cylinder, and qr 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 rectangutoa,in refer lce ]ar, taken with reference to an axis through its centre of perpendicular gravity and perpendicular to its plane, gives axis through its centre; b = 2 r= 2~r ab(r 2+ 7) x-; MECHANICS OF SOLIDS. 195 in which r is the mean radius, Fig. 105. or that of a circle whose circumference is midway between the inner and outer / surface of the ring, a the j / thickness parallel to the axis, i and b the thickness in the direction of the radius... —.. Fig. 106. 4th. That of a spherical segment taken in reference to that of a spherica segment, in a diameter passing through reference to its its centre of gravity, or mid- versed sine; dle, gives = /rf3 ( r2 - fr + f1-f2) X,'in which f denotes the versed sine of the segment, and r the radius of the sphere; and for the entire sphere, a of a sphere, in = _ Sr5 X-. reference to a g diameter; Fig. 107. 5th. That of a right cone having a circular base, taken that of a cone, in reference to its with reference to the axis of axis; figure gives, = har4X X; 196 NATURAL PHILOSOPHY. Fig. 108. that ofa and that of a truncated right truncated right cone, having circular bases, cone; r _ -r5 =a- a X- 10 r - r' gx in which r and r' are the radii of the greater and smaller bases respectively, and a the altitude. Fig. 109. 6th. The moment of inertia of an ellipsoid is given by that of an ellipsoid; I= 14jr 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 Fig. 11o. parallelopipedon, of which the three contiguous edges are a, b, and c, taken with reference tllhat of a to an axis passing through its rectangular on; centre of gravity G, and parallel to the edge a, -= abc (a2 + b2) X -. The same taken in reference to an axis through the middle of the face a b and parallel to a, the same for a & different axis; I = -- ab c (b2 + 4c2). -. g MECHANICS OF SOLIDS. 197 8th. That of a right prism Fig. 111. *F* n I'. that of aright having a trapezoidal base of t1 of a rit prism with which the greater and less. -----— trapezoidal base; parallel sides are respectively b and b' and distance between them c, the altitude of the 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 + bf' ~b2 + b?' c b + 3 b' a 2 24 b + b' 9th. If the trapezoidal base of the above prism Fig. 112. be replaced by a segment the same when of a parabola, of which the base becomes'C a segment of a c is the length of the parabola. transverse axis, and b that of the chord perpendicular 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 3. 5. b + 16C2 a I 2a b * X= ~abc(3 70 g ~ 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- Fig. 113. on a handle in the shape of a rectangular parallelopipedon which turns that of a common freely about an axis O, at - ip-hammer; right angles to its length. Denote by R, the distance of the centre of gravity of the head B from the axis 0. 198 NATURAL PHILOSOPHY. If the linear dimensions Fig. 113. of the head be small comrnpared with this distance, its moment of inertia will [ l not differ much from - the moment of p inertia of the X R2, head; g and that of its handle is given in reference to an axis through its centre of gravity by the 7th case, or that of the handle 6 with reference to 1 a b c (a2 + b2) _ its centre of g gravity; and denoting by K; the distance of the centre of gravity of the handle from the axis, its moment of inertia, with reference to the axis 0, becomes, Eq. (65), 6 abe 6 - X (b2 + a2) + abcK2, g 12 g or with reference to P 2 + a the axis; g (2 1 since abec =P', the weight of the handle. The total moment of inertia is, therefore, given by the moment of 2 + the entire I = R2 + a2 hammer; g +g 12 The process for finding the moment of inertia of the fly-wheel is much simplified by the fact that all its parts moment ofinertia 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' r 2+ m" r"2+ &c.; and hence, U MECHANICS OF SOLIDS. 199 P -= R2; its value; and denoting the angular velocity of the wheel by V1, its living force will be, ~ 159, P 2.-2 living fqrce of X 2. the fly-wheel; To find the angular velocity of the wheel, count the number'of its revolutions in a given time, multiply this number by 2 %t, and divide the product by the number of seconds in the given time; the quotient will be the angularvelocity angular velocity. Let JV equal 9 feet; the weight p fouientally; experimentally; 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 2000 2000 X 36 = 2250; 32 and for the living force, example. 2250 X 81 = 182,250; 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 mr'? r'2' + in" r" 2 + &C. =JfK HiCentre and radius mnr -I' r~2 - in"?" + &c. = XK2, of gyration; in which H is the entire mass of the body, and K = - /~mr_+ nm'r'2 + m" r"2 + &c. But this is equivalent to concentrating the entire mass into a single point whose distance from the axis is X, without 200 NATURAL PHILOSOPHY. changing the value of the moment of inertica. This point is called the centre of gyration, and the distance E, is called definition; 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 adgyration; o Denoting the principal radius of gyration by K', we may write MK'2 for I1, in Eq. (65), and we have moment of inertia 2 2 in terms of radius I = M'K + MI) (66)" of gyration. XI. CENTRAL FORCES. ~ 168.-Conceive a body, whose weight is P, attached Central forces; to a fixed point C by a rigid bar A C, and suppose it to have any velocity whatever in the direction AT, perpendicular to the bar. If the body were free, it would, in virtue of its Fig. 114. inertia, move in the di- M IT rection A T with a constant velocity. But not a body made to being free, the bar will c revolve about a.. fixed point by keep it at the same dismeans of a bar; tance from C and cause it to describe the circumference of a circle about this point as a centre. There are, then,:,during this constrained 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 ] 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 centripetalforce; 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 inertia; path. 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 A B CD E, substitution of a of a great number of very small sides and having its circle;ygofohe 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 Fig. 115. I - --- 13 c;; 1 to prove there is no loss of velocity D from the reaction of the curve; te _ vli _ the velocity V at the moment of its arriving at B, the beginning of the side B C, it will be animated, while 202 NATURAL PHILOSOPHY. describing this side, with two simultaneous velocities. One of these is the primitive velocity V= B IV in the prolongation of A B; the other B U, in the direction B O, 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 simultaneous vemloities; side B C of the polygon. But we have seen, ~ 106, that when a body receives two simultaneous velocities in different directions, its resultant velocity will be the same as if it Fig. 115. W..' —....; iA possessed them successively, and as though they were communicated one after the other in their respective directheresultant of tions. Thus the resultant velocity B C', with which the which has the side B C is described, coincides in direction with this direction of the side next to be side, otherwise the body would take some other path, described; which is contrary to the hypothesis. B U and WC' are equal and parallel, from the parallelogram of velocities. The radius 0B divides the angle ABC into the two equal angles A B O and O B C; the angle A B O is equal to the angle B TV C', and the angle O B C is equal to the angle B C' l/W; hence the angles B C' W and B W C' are equal, and the side B C' is equal to the side B W,' in other words, the resultant velocity B C', with which the no loss of side B C is described, is equal to the velocity B W which velocity; the body had at the end of the side AB. Whence it MECHANICS OF SOLIDS. 203 results, that the velocity communicated to a naterial point is velocity not not altered in its circular motion; a result easy to foresee, ltered by the 7deflecting cause; since the centripetal force, acting in a direction perpendicular to the direction of the motion, cannot work efficiently; it can neither accelerate nor retard the motion, and, therefore, can neither increase nor diminish the living force of the material point. Now observe, that B U = W C', is the velocity generated 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 1,1 then will the value of F. be given, Eq. (39), by the equation F=..t Draw the radius C 0; the triangles B 0 C and B W C' to find value of are similar, because the angle 0 CB = 0B C is equal centrifugalforce; to the angle B C' W, and the angle 0 B C is equal to the angle B WC'. Hence we have the proportion, BO BO:: BWY WC'; or, denoting the radius B 0 by R, and replacing B W by its equal V, R: BC V W'; whence W:'- BCx V and this, in the value for F, gives BC V F=H. ~ X; t R 204 NATURAL PHILOSPHY. but B C, the element of the space, divided by t, the element of the time, is equal to the velocity V; whence the measure of M1 V2 the centrifugal F = (67). force; _ Such is the expression for the centrifugal force. The numerator is the living force of the body, and the denominator 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 equal to the living force impressed upon the body, divided by living force the radius of the circle described by its centre of gravity. divided by radius 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. 100 M= 32- V= 12; 72= 144; R= 3; 32; illustration; F = 12 X 1 = 150 pounds; 32 x 3 the body, therefore, tends to stretch the bar with an effort of 150 pounds. Denote by V1 the angular velocity; then will V-= 2 R2 and this, in Eq. (67), gives expressed in terms of the F V2R... (68). angular velocity. ~ 169.-Let us next take the Fig. 116. case of a thin layer of matter D A B, rotating about an axis O, A perpendicular to its plane, with Extension to aan angular velocity IV1. Taking (. r body of any dimensions; any one of the elements of the layer whose mass is m, and de- x P0 MECHANICS OF SOLIDS. 205 noting its distance from the axis 0, by r, its living force will be m 2 Vi, and its centrifugal force, m V 2 centrifugal force =tmr 1; of an element; which will act in the direction 0 m of the radius of the circle described by mn about the centre 0. Through the point 0, draw in the plane of the layer, any two rectangular axes, as 0 x and 0 y. 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 q m, and we shall have, denoting Op by x, 0 q by y, the component parallel to the axis Ox by X, and that parallel to Oy by Y; r: x m:: r V2: Xresolved into rectangular components; h:y mr V1 whence, X = m x V1 iF= y -Vl;2 and for any number of small masses m', in", &c., by using the same notation with accents, Xi = ml x' VT-2 components parallel to the i ItfT7i2 axis x, for other X =- qX V17 elements; &c. = &c.; 206 NATURAL PHILOSOPHY. y' = l' y V, components parallel to the yit = ffit axis y; m &c. = &c.; by this process all the centrifugal forces have been 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 X1 and Y1 respectively, X = V1 (m x + m'''+ m" x" + &c.), Y, = y12 (m y + m'y' f "y" + &c.); that is, resultants X, 2 x'i,, parallel to the axes x and y; = -v2 X; in which M' denotes the entire mass of the layer, and x, and y, the co-ordinates 0 P and 0 Q of its centre of gravity G. The resultant of the forces XI and Y1 is the entire centrifugal force of the layer; and this denoted by F1, is, from the principle of the parallelogram of forces, F-=, V 4 l2'y'2 + T,4A',a - V 2' V //+y, y2; and making /X 2 + y = oG = r measure of the centrifugal, force; whence, the centrfzugal 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 0 G. Now suppose any body, as A B C, to turn around the centriflgalforce axis L M. Divide the body into thin layers whose planes of a body of any are perpendicular to the axis. These layers will Fig. 117. give rise to as many centrifugal forces acting at c their centres of gravity, / —-------— o a, G', G ", &c. All ---- these forces are perpen- "dicular to the axis L, / without being parallel to _ each other. Sometimes they have a single re- WI may reduce to a sultant, sometimes they single force, two \._7~~~~~ ~~~forces, or to zero; will reduce to two forces, and sometimes they will A O reduce to nothing, de- - pending upon the form and density of the body, 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 piessure on axis; pressure upon the axis. If the centres of gravity G, G', G", &c., be all on the same straight line parallel to L M 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 = V f2 R (if+' + M" + "~' + &c.); and making the centres of 2f' + l" -I f"/ - &c = X gravity of the *' X1-.M-'~-'~. —.layers on same line parallel to F = R72 P. AL the axis; 208 NATURAL PHILOSOPHY. that is to say, the centrifcalT force of a body, whose secthe centrifugal tions perpendicular to the axis, have their centres of force the same as though the gravity in a straight line parallel to body were the axis, is the same as though the Fig. 118. reduced to centre J~ reduced to centre entire mass were concentrated at the of gravity; common centre of gravity. This simplification is peculiar to the sphere, examples. the cylinder, and surfaces of revolution generally whose axes of figure are parallel to the axis of rotation. Illustration of the ~ 170.-The centrifugal force accounts for a multitude centrifugal force; 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 centrifugal 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 the horse from the centre C. It is to resist this effort that horses, travelling ina 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 hisinclination; G P and GF, to represent his weight and centrifugal force respectively, and construct the rectangle P G FR, the diagonal G R will give the inclination sought. Denoting 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 p V2 F=. g R' and consequently the pressure or his oblique pressure on the TA ~4 ~~grolund; GR = PI + PR2. Finally, in order that the horse may not slip, the surface surface of his BA of his path, must be perpendicular to G R.. path When a horseman rapidly turns a corner, he leans his body towards the centre of the curve which he is ahorseman describing, to bring the resultant of his weight and turning a corner; centrifugal force to pass between his points of support in the stirrups. 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 a wagon making upsetting is directly propor- Fig. 120. tu and rapid tional to its weight and the square of its velocity, and inversely proportional to the radius of the curve. This is why the exterior of the roadway is usually elevated - inclination of,,,/ I roadway; in short turnings, and car- / / riages diminish their speed / when approaching them. The sling, the axe, the other examplessabre, &c., exert upon the Fig. 121. thesling, axe, A sabre, &c.; hand, when we give them a circular motion, a traction A A equal to the centrifugal force. The common wheel is usually composed of fellies A, A, &c., A connected with the nave NI V by means of radial arms,,, common X aa14 carriage-wheel; 14 210 NATURAL PIILOSOPHY. &c., and the centrifugal force is constantly acting when actionupon the the wheel is in motion to draw these arms from their fellies of the common wheel places, to enlarge the circumference, and thus to detach the fellies from each other; hence the tire not only protects the wheel firom 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 AA' once in twenty-four hours, and that the cirCentrifugal force cumferences of the parallels at earth's surface; of latitude, that is to say, Fig. 122. the circles perpendicular to the axis, have velocities which diminish from the equator to ------- the poles. To find the law /. of this diminution, let P be the weight of a body on the Q surface of the earth in any parallel of which R' is the radius, its centrifugal force will, Eq. (68), be that of a body Rwhose weight is in which V7 is the angular velocity of the earth. Substituting M for -, we hayve F - Af V12 R'. Denoting the equatorial radius CdE C= P, by R, and the angle CP C' = P CE which is the latitude of the place, by p, we have in the triangle P C C', R' =.Rcos; MECHANICS OF SOLIDS. 211 which substituted for R' above gives F= - M l72 R cos.... (69). Now, the only variable quantity in this expression, law of variation when the same mass is taken from one latitude to another, of centrifugal force; is p; 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- Fig. 123. longation of this ra-, A dius, the distance P, to represent t R' this force, and re-'. solve it into two components P N the centrifugal force resolved and P 1T the one into a vertical nornial, the other and horizontal component; tangent to the sur-.' face of the earth; the first will diminish 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 HPRN is equal to the angle P CE which is the latitude, denoted by p; whence the normal component value of vertical PN = - P x cos p = F. cosg = O Vj2 COs 2(p component; and horizontal P T =PHsin = F. sin q = X V2' R. sin gp cos p; component; 212 NATURAL PEiILOSOPHIY. but, sill n. cos p = a sin 2 Ip; therefore its value; P T = 2 i f2 R sin 2 P whence we conclude, that the diminution of the weights of bodlies 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 consequence of the ceneffect upon tho trifugal force, urged Fig. 123. weights of bodies towards the equator and figure of thee eq,,tor earth; by a force which varies as the sine of t: twice the latitude. At the equator and poles this latter force is zero, and at the latitude of 45~\ it is a maximum, A' and equal to half of the entire centrifugal force at the equator. 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 cause of the suppose the earth to have been at one time in a state of rthesentg.retof fluidity, or even approaching to it, its present figure is readily accounted for by the foregoing considerations. The force of gravity 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 I greatest at the equator than at the poles. poles and least at To find the value of the centrifugal force at thequat equator, make, in Eq. (69), X = 1 and cos: = 1, which is equivalent to supposing a unit of mass on the equator, and we have F = V712 R. centrifugal force at the equator; The angular velocity is equal to the absolute velocity, divided by the equatorial radius of the earth. The absolute 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..................Log. 3.8989993.............. 3.1416...Log. 0.4971507 Feet in one mile..........5280..Log. 3.7226340 Circumference of earth in feet....................Log. 8.1187840 Length of a sid. day in Sol: seconds, 86400 X0.997269, Log. 4.9353259 Absolute velocity in feet.......................Log. 3.1834581 computed; Radius of earth in feet.......L............og. 7.3206032 Angular velocity V1..Log. 5.8628549 Square of angular velocity V.......Log. 1.7257098 Radius of earth in feet.............Log. 7.3206032 f Centrifugal force at equator..0.1112................. 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 arglar make the centrifugal force of a body at the equator equal velocity slffcieti, to destroy to its weight; for by the present'rate of motion we weights at the find equator; f 0.1112 = R, 214 NATURAL PHILOSO PHY. and by the new rate of motion f 32.1937 = V'2 R; f in which 32.1937 is the force of gravity at the equator. Dividing the second by the first, and we find 32.1937 V_ 1'2 0.11.1.2 = 289, nearly; 0.1112 ll whence result; V' = 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 weights of bodies at the various latitudes bodies affected. 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. ~ 172.-If we now suppose the body, instead of Fig. 124.:lotion in a being connected with the argroove, point C by means of a rigid bar, to move about the same point in a circular groove, c. the effects, as regards the centrifugal force, will ob- viously be the same, since the body will be constrained, by the resistance of the when plane of groove, to remain at the same distance from the centre. grove is If the plane of the groove be horizontal, the pressure of horizontal; the body against the side will be constant and equal to the centrifugal force, that is to say, to MECHANICS OF SOLIDS. 215 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, the effect of the tion to the centrifugal force, while the former will. some- body' weight; 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 tofin the thers; others. Take the body at its lowest point m', and denote its velocity, supposed known, by V', and let it be required to find its velocity at any other point in, whose vertical height above i' is H. Denote the velocity at this latter point by V, then will the loss of living force in passing from m' to m be M V'2 - 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 WVI, we have, v(V'2- Vr2)= 22wH; replacing Hf by its equal W and reducing V'2 - V2 2 g1, r~ —-- - value of velocity V V= - H 2gt. at any point; 216 NATURAL PHILOSOPH-IY. Denoting by H', the height due to the velocity VI, we have V'2 = 2y H'; which in the above equation gives same in terms of difference of level of the points; V = v 2 gy(H -H). Thus, the velocity of the body will be diminished Fig. 124. by the action of its weight during its ascent, while, on the contrary, it will be increased during the descent, being always the same at points situated on the same velocity greatest horizontal line. The'veat lowest point; locity will be greatest at the least at the lowest and least at the highhighest. est point. During the descent, the body will acquire living force by absorbing the gain and loss of work of its weight, which living force will again be livin force. destroyed during the ascent because it is opposed to the weight. ~ 173.-When a body, in vir- Fig. 125. tue of the motive forces which act upon it, describes a curve in A" Centrifugal force space, the effect is the same as / of a ody which though it passed over the arcs of /, describes any t curve; the successive osculatory circles /, of which the curve is composed. / I / 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", i, &c., the curve will be given MECHANICS OF SOLIDS. 217 by the series of arcs A A', A' A", A" Ai"', &c., clescribed about these centres, and terminated by these radii. And it will be easy, from the consideration of the centrifugal 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. Let P denote the resultant of the motive forces which to trace the curve act upon the body at any particular point as A; the and centrifugal mass of the body; V its velocity, of which the direction forces: is A T; and r the radius A C; then will, the centrifugal force be measured by fV2 But the body, in describing the curve, does not abandon the small arc AA', 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 resolved 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 IV'r V2 value of the.. (70); noral r component of the motive force; whence _M V2...radius of P (. c rvature; 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 tangential component of the motive force will either increase or diminish the velocity. It will be sufficient to make n + n' 2 218 NATURAL PHILOSOPHY. in which n and n' denote the velocities of the body at the te.minal velocity beginning and ending of the arc. The former of these to be fund;itial must be given, being the initialc velocity; the latter must Fig. 125..be found, and for this purpose we, remark, that as the Amp" are is described in a very short time, say the tenth of a second, l / T the motive force, and therefore A its tangential component, mayI | /| be regarded as constant during ii/ this interval. Denoting the,/ tangential component by q, and the time by t, we have, from i the laws of uniformly varied a motion, Eq. (11), its valh e; n' = n +9% t and valueofmean n + n' + (; n + t (72) velocity; 2 2 iff which, in Eq. (71), gives 2 value of radius; ( 2 -) r,=.. (73). This distance being laid off from the point A, upon the perpendicular to the tangent A 1 will give the centre C. The length of the arc, denoted by s, is found from Eq. (10), or value of are 8 = n t 1 t2. (74). described; 2 =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 T V'2 value of normal P = t X component of motive force; or Mi V'2 in which r' is the radius of the are 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, ni = n' + q terminal velocity on second arc; nV "' = n + +;q 2 2211' which in the equation above gives 21 (' + 2f t) radius of second a (n' + r -; _.; arc; 2P/ 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 5'r ~ n't+~. length of second s = n t $+ 2 2X t2. arc; Finding the value of the motive force at A", its normal and tangential components 1'" 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 fo'other arcs; 220 NATURAL PH1ILOSOPHlY. osculatory arc, and thus the description of the curve may be continued to the end. application to the To apply this general case to a particular example, case of a bomb- take the instance of a bomb thrown into the air. The shell thrown into the air; 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. Let A be the mouth of the piece, of which the axis coincides with the Fig. 126. line A T. This line will T be tangent to the path described 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 co air due to this velocity by ". The value of f may w be taken from a table giving the resistances corresponding to different velocities and calibres. Through A draw A H parallel to the horizon, and denote the angle TA Ht by c; 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, A c = p, normal to the tangent A T; and the other, A n = Ic, in the direction of this line. The angle WA c is equal to the angle TA H = a; and hence, components of the weight of the bomb; = Tsin u; 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 q = k + f = - (Wsin a + f); tangential component; therefore n' -- ni - S- ~ f t, terminal velocity; and W sin + f~ s V=n- t; mean velocity; 2 HM this value and that of p, in Eq. (71), give /sin. a - f~ Mf (n - -Ts ft)2 radius of initial 2 H _____r _are; WV COS a 7 and writing, in Eq. (74), for q its value, we find T -V siln l + f 2length of initial s = n t f-' ilt arc; Through the Fig. 127. point A, draw an incefinite perpendicular to the line A T. and lay off from A the dis / / tance A C, equal / / / to r; with C as a centre, r as radius, con ~ construction; describe the arc / sction; A A' equal to s./ / This will give the / / initial arc. The linear dimension of an arce at the unit's dis- C 222 NATURAL PHILOSOPHY. tance from C, is length of arc at unit's distance fiom the centre;' and denoting the ratio of thle circumference of the circle to its diameter by %, we have 2': -:: 360~0 z r 360~ X s its value in are; 2 r r in which z denotes the number of de- Fig. 128. grees in this are, T or the value of the angle A CA'. But = angle of the this angle is equal I tangents at the tan elntsatthe to that made by the initial points of -t —-.... two consecutive tangents A T and arcs; 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 horizon, or the angle T' A' H', will be angle of projection at the beginning of the- second are; Pursuing the same operation as before, we find -' = Wcos cu' k' = 1Wsincc'; and taking from the tables the resistance f', corresponding AMECHANICS 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 successively by a, a', &c., diminishes in passing from are to arc, it will presently become equal to zero, at the suimmit, and afterward 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 anges of projection; 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. 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 atmosphere and pound shot, projected under an angle of 45~, in vacuo, 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 Fig. 124. centrifugal force may be illustrated experimentally by means of the whirling-tcble. ~.' This consists of a framework upon which are mount- / Whirling-table to ed two vertical axes. Upon illustrate centrifugal force; the top of each axis is fastened a circular block B, B, having a groove cut in the circumference for the reception'of an endless cord C( C, C, which also passes round a wheel 1sV. This wheel is 224 NATUR AL PHILOSOPHIY. provided with a crank and handle 1, 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 veloarrangementof city of rotation. A piece of wood d d, is mounted upon each the parts of the of the circular blocks, by means of screws, to support two polished horizontal metallic bars b, b, along which a small stage S may slide with as little friction as possible. This stage is connected with another 5', which slides freely Fig. 124. on a pair of vertical bars b', b', by means of a piece of flexible catgut passing over the pulleys p, p', in such manner as to lift the stage S' in a vertical, when motion is comnmunicated to S in a horizontal, direction. The stage S' is placed with its centre immediately over the axis of motion. On the piece d d is a graduated linear scale, having its scale and zero in the axis, for the purpose of measuring the distance wveights of the stage S from the centre of motion. A series of weights WT', TW', 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 first; Examnple 1st. Load one of the stages, 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 division 5; make the circumference of the first circular block double that of the second. The angular velocity of the first being K1, that of the second will be 2 V1. When motion is communicated, the centrifugal forces will, Ecq. (68), be, respectively, MECHANICS OF SOLIDS. 225 5 X 8 v1' and 2 X 5 X 4 V2, or 401-12 and 40 V12; 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 1V, load one of the stages 5 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 centrifugal forces will be, respectively,. 6 x 8 Vi= 48 VI, 3 X 7 X 4 V1- = 84 T17, the ratio of which is 48 12 84 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. Such a body of a spherical form, revolving about one when arotating 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 flte shape; perpendicular to the axis, that is, in the plane of the body's equator, have the greatest centrifugal force, while those 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 a b, is an armillary sphere, composed of elastic wires, fitting round the axis by means of a ring, which holds them all together. By Fig. 130. experimental this contrivance it is possible for the b illustration. elastic wires to assume an elliptical figure, having a shorter vertical diameter. Screw this apparatus into the middle of the circular block of the j whirling table, and give to the whole a rotatory motion; the wires, instead a of their original form represented by the dotted lines, will assume, in consequence of the centrifugal force, the figure shown in the dark lines. ~ 175. —When a body Principle of moves with uniform mo- Fig. 131. the areas; 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 C into equal spaces A n', m, m", emin"', &c., these A spaces will be described a body in motion in equal times. If the several points of division be joined unde the action with any point as C, off the line, a series of triangles A Cm', of the central o force; m'Cnm", a"Cm"', &c., will be formed, all having a common vertex and equal bases lying in the same straight line. The areas of these triangles will, therefore, be equal, and 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. 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 rn' 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' n[,, the forces first of the parallelogram constructed upon m' n, and m' z" impulsive; as sides. The line m" n,, being parallel to m' C, the triangles Cm'm " and Cm' n,, will have the same base C m', and equal altitudes; their areas will therefore be equal; hence the triangles C A m' and Cm' m,, will be equal. In like manner, if when the body arrives at m,,, it receive another impulse directed towards C which would cause it to describe m,,n,,, in the time it would have described m,, 0 = m' m,, if undisturbed at m,,, it will describe the diagonal m,, m,,, of the parallelogram constr-ucted upon m,, 0 and n,, n,, as sides; the triangle Cm,, m,,, will be equal to the triangle Cm,, 0 = Cm' m,, = CA m'. 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 thlenincessant; A mn',' m,,, n,, n,,,, &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 centripetalforce. Whence we conclude, that when any body having received areas described a motion, is acted upon by a centripetaZ force, of which the by radius ector proportional to direction is oblique to that of the motion, its radius vector will the time of describe equal areas in equal tifnes. description; 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 2point, the body must be urged towards this fixed point by a centripetal force, for the equality of the triangles 228 NATURAL PHILOSOPHY. conversely, the C m' m" and C m' ino, Fig. 131. areas being equal a,, O and C n,, m,, in equal times, the force must &c., depends upon the teltto the fixed lines ",,: 0,,,.m point; J.k;,7. being respectively parallel to t'-o, C n,, C,- &c., drawn from the positions in- which the'body re-: c ceives the deflecting im- pulses to the centre- C. Denote%. the area by-: A, and the time in 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 A. the times. = a or 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 describe the curve AL B under Fig. 132. Measure of the the action of a centripetal centripetal force; force directed to the centre C; and suppose m and n' to be two positions -of the body very near to': each other. Draw the tangent C- -\ mn Q to the curve at the place .MECIANICS OF SOLIDS. 229 m, and draw m' Q parallel to the radius vector Cm, and Mnt 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 in'. Again, if the conlponlelts of body had been moved from rest at m by the centripetal theoit~l; velocity; force alone, it would have described the path m n - m'Q, in the same. time; the path m n is, therefore, the path due to the' action of the centripetal force. The places en and n' 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 centripetal force can generate in the body at m, in a unit of time, by v, then, Eq. (7), will Am = i v, t2, whence 2 m n. - t2 i but, Eq. (74)', A t-a and substituting this for t, we find 2.a2 X mn value ofthe V, =. acceleration due.a2 to the centripetal force; Multiplying both members by the mass of the moving body, denoted by 2, we have 2:Afa2X mnn iV, = A2 Draw from in', the line m' h perpendicular.to Cm, then, because A is the area of the triangle C m, A', will -A = 3 Cmn X m'h, 230 NATURAL PHILOSOPHY. which in the above equation gives the intensity of 2mn n the centripetal vi = 4 1 a X 2 (74)". force; Cm X 7-h The distance m n is called versedsine; the versed sine of the arc Fig. 132. in m', and m' h the altitude altitude of the of the sector; the first mem- B sector; ber, or lMv,, is the quantity of motion which the centripetal force can generate in m a unit of time, and there- a fore measures its intensity; whence we conclude that, the intensity of the centripetal force by which a body is made to describe a curve, is always value of the equal to four times the nmass of the body into the square of the fintenity 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. XII. MOTIONS OF THE HIEAVENLY BODIES. Phenomena of ~ 177.-The phenomena of the heavenly bodies may the heavenly be divided into three classes: the first, comprehending the bodies; motion of revolution round the sun; the second, the motion 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 ofKepler: called KEPLER'S laws, viz.: 1st. The planets move in plane curves, and their radii 1st law; vectors descrmbe round the centre of the sun, areas proportional to the times of their description. 2d. The orbits of the planets are ell~ipses 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 motion of ~~~~~~~~~~~~~~~~~7 7 ~~~translation. gravity of the planets.' 178.-From the first of these laws, and the principle of areas proportional to the times, explained in ~ 175, 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 first law; body is, therefore, the centre of the system. The consequence of the second law relates to the variation 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 in', be two con- Fig. 133. secutive places of the Q planet moving in an D ellipse of which C A / and CB are the semi- that of the second transverse and semi- 8 A deduced; conjugate axes, and having the sun, towards' which the centripetal force is directed, in the focus S. Draw m' n parallel to the tangent m Q, and produce it till it meets m C, drawn to the centre of the ellipse, in the point v; let fall the perpendicular m' h upon the radius vector Sm; join the body at m with the other focus 232 NATURAL PHILOSOPHY. S'; draw S'N and.CD parallel to the tangent - ig. 33..... Q, and produce m C to the curve at -G. The tangent Q Q' Construction of makes - equal.'. angles, the figure; QinS and Q'mS with the'.line. drawn from'. the. place in to the foci, and. because S'.ZV is parallel to. this tangent, the. triangle in S' N is isosceles, making' m = Nm;. and because CD is parallel to S' N, and CS is equal to CS', the distance NL is equal to L S; hence m L = is equal to the semi-transverse axis CA = A. Denote the semiconjugate axis by B. In the similar triangles m n v and m L C, we have, in mv:: m'L mC; whence, writing A for nL, we have value of the A.. m V versed sine; m C Again, drawing mi F perpendicular to D c we have; from the similar right-angled triangles m L F and m'-h n, -72 -;-2 -2 2 i' n mi r:: m i: m; whence, writing A for m L, we have value of the X F altitude of sector; m' h A2 MECHANICS OF SOLIDS.. 233 -and, dividing the last equation by this, one, we have rmrn my Iv ratio of the __" -- 2Z3A3 X X - - versed sine to the m' h2 m C m n x n F square of alitude of sector; The equation of the ellipse, referred to the conjugate diameters Cm zand CD, gives, because the points n and v will sensibly coincide for consecutive places of the body, -2 =CD C n D X mv X V C; CUn which? substituted for' ~n above, we find mn - x Cm the same, in other h2 - A3XCD2 X E 2 terms; and, because the rectangle of the semi-axes is equivalent to the parallelogram constructed upon.the semi-conjugate diameters CD and Cm, we have -2 — 2 A2 r1 X h2 XB2; moreover, the points m and m' being contiguous, G v will not differ sensibly from 2 Cm. Making these substitutions, the above equation reduces to mn A m'h2 = 2 B2' f its final value; m' 2 2 B2 4ml7a2 and, multiplying both members by /a2, 4 Ha2 X m _ 21 fa2A 1 a X X, Cn -2 B C2 234 NATURAL PHILOSOPHY. The first member we have seen, Eq. (74)", is the intensity of the centripetal force at m. Calling this force F and writing r for the radius vector Cm, we finally have value of the F 2 2a2 A 1 force; F2 - _2 consequence of Every thing being constant in the second member but; r, 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 r2 A2 B2, and we have to find the 1 consequence of F.qr2 A B2 = 2 rf2 iCa2 A3 X the third law; divide both members of this equation by Fat, and there will result r2a A2 B2 22 9,2 2f 1 -- =- X A3 X 2Now, r A B is the area of the entire ellipse; a is the area described by its radius vector in a unit of time; hence periodic time; is the number of units of time in one entire revolution of the planet, called the periodic time. Denote this by T. and substitute it for A B and we get the value of its 2 2 Jf 1 T2 A3. square; F r,2 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 1 A2'M F' r'2 MECHANICS OF SOLIDS. 235 and dividing this equation by the one above T'r2 A3 ratio of' the - x squares of T2 — --.t/2 ~ fF' periodic times; But, by the third law, T,2 A'3 T2 - 13; whence i' Fr IF' rF' or F F' r2 centripetal Ad X -2 =r X ^acceleration; F Now F is the velocity which the centripetal force can generate in one unit of time, or, which is the same thing, it is the measure of the acceleration due to the force which acts upon the planet Al; so, likewise, F is the acceleration due to the centripetal force which acts upon the planet l'; and resolving the above equation into the proportion X F' I 1 F~~ F' 1 1consequence of f -m rA r'2 the third law; 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 NATU RAL PHILOSOPHY. at same distance, planets were brought to the same distance from the sun, th celerntietal each unit of mass would be urged towards that body with equal; 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 Newtonian other with an energy which is directly proportional to their universal omasses and inversely proportional to the squares of their disgravitation; tances from each other. 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 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 parathe orbits might bola, an ellipse, or hyperbola, according as the square of the have been ellipses, body's velocity is equal to, less, or greater than, twice the pahabolas, or attractive force, multiplied by the distance from the:sun. hyperbolas. ~ 179.-Let C mm' be the sector described in the Fig. 134. unit of time: -take the distance C b equal to unity, The angular and describe, with C as a velocity; centre and C b as radius, a the arc b-d = s,, which will B y AK measure the angular velocity. With C as a centre, and Cm' = r as'radius, describe the are 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 Cmn m', 2 Cm X m'h' =' 122s, = a; whence 2a S _ = __; 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 viation radius vector. Supposing the planet to describe the ellipse Fig. 135. A BPD, having the sun at the focus s, the extremities A and P of the transverse axis are called, the former the Aphe- J aphelion; lion, and the latter the Perihelion. The angular perihelion; velocity of the planet is angular velocity greatest at the least at aphelion and perihelion an greatest at perihelion. z~ least at aphelion; Again, denote the angle Cm Q by a, and suppose Fig. 136. the velocity of the body on absolute velocity; the small arc m m' uniform,. which we may do without,a sensible error, the length of m m' will measure the velocity of the planet at m, = since it is described in a unit -of time. Hence mm'sin = m' h = V. sin C; 238 NATURAL PHILOSOPHY. and the area of the triangle or sector Cm m' will be ~ V. sin a X r; whence V. r sin a 2 -a, or value of the 2 a absolute velocity; r. sin a' Draw the tangent m Q to Fig. 136. the curve at the point n, q and from C let fall the perpendicular C Q, then in the right-angled triangle C Q m, will c CQ = r. sinc = p, which substituted above gives the same in V 2 a different terms; p 7 its law of that is to say, the velocity of a planet in its orbit, varies invariation; versely as the length of the perpendicular let fcll from the centre of the sun upon the tangent drawn to the orbit at the body's place. greatest at From this it follows that the velocity of the planet will perihelion and be greatest at perihelion and least at aphelion. 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 directed to the towards the centre of that curve, the force will vary centre of an elliptical orbit; directly as the length of the radius vector, and that the periodic time will be the same for all ellipses, great and small. MECHANICS OF SOLIDS. 239 Let the body, under the action of a force di- Fig. 13l. rected to the centre, B describe the ellipse of which CA and CB are the semi-axes, denoted to fnd the law; respectively by A and. B; and suppose m and c, m' to be two of its consecutive places. Draw the tangent m Q at the 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 Cm in n, and let fall upon the same radius vector the perpendicular m' h. The equation of the ellipse, referred to its conjugate diameters Cm and CD, gives -2 CD2 m'n- - X -n X r m G; Cm whence m' n X Cm2 value of the mn = -- versed sine; D2 X n G 1 Because n' 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'n h and Cm KX are similar, and give the proportion mu n: mo: Cm i K whence n-_ 2 X value of the 7n' h Ad S --; square of sector's C m altitude; 240 N'ATURAL PHILOSOPHY. dividing the last equation by this one, we have ratio of the m n C _4_ versed sine to 2 2 X n G square of sector's altitude; But the rectangle of the semi-axes is equivalent to the: parallelogram described upon the semi-conjugate diameters, hence -2 2 2 CD x mK A2 X B2; moreover, n G is sensibly equal to 2 Cm; making these substitutions above, there will result same in different.: 7 n -- - C. terms; m' ha 2 A2 B2' multiplying both members by 4 iMa2, and dividing by:Cn, we have, Eq. (74)", mn F 22Ma2 4i:Ma X vT- _ = = X Cm CmX2 X h A2B2Cm, in which H is the mass of the body. Finally, writing r for Cm, we find value of the 2 fl a2 centripetal force; F -- A32 X r the law of its that is to say, the centripetal force which will cause a body to variation; describe of that curve, aescrtbe an ellipse when directed to the centre of tha t curve, varies.directly as' the radius vector. to find the Multiply both members of the last equation by periodic time; q2 A 2 B2, and we have F. q2A2 B2 = 2? 2 iHa2 X r. Dividing both members of this equation by Fa2, and we have MECIHANICS OF SOLIDS. 241'r2 A2 B2 r =22 X.r = 2 7r2 X 2 FF _AB taking the square root, and recollecting that - is the periodic time =, we find 2r T = -r / value of the periodic time; The quotient X is the measure of the acceleration due to the centripetal force, which we have just found to vary directly as the radius vector. This makes the radical expression constant; hence T must also be constant. 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 performs their revolutions in the same time. XIII. THE PENDULUIM. ~ 181.-A body M Ql suspended Fig. 188. from a horizontal axis A, about which, Compound it may swing with freedom under the pendulum; action of its own weight, is called, in general, a conmpound pendulum. When the body is reduced to a material heavy point, and the medium of con- / nection with the axis is without T simple weight, it is called a simple pend- penum 242 NATURAL PHILOSOPHY. The simple pendulum is but a Fig. 139. has no real mere conception, and yet the ex- A existence; / pression for its length, which may / easily be found in a manner soon to be explained, is of great practical importance. When the pendulum is at rest, ar/ in such position that its centre of Q gravity G is below and on the vertical line passing through the axis A, it will be in a state of stable equilibrium, ~ 151; but as soon as it is Fig. 140. deflected to one side, as indicated, effect of friction in the figure, and abandoned to and resistance itself, it will swing back and forth about the position of equilibrium, into which it will finally settle in consequence of the resistance of p i the air and friction on the axis.. If these causes of resistance were - removed, the pendulum would con- -Y figure of tinue its motion indefinitely; but pendulum andof this cannot be accomplished in suspension; practice, and hence such figure and mode of suspension are resorted to as to give these impediments the A A least possible influence. B The pendulum is usually mounted upon a knife-edge A as an axis, knife-edge and resting upon a well-polished plate A bob; 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 lenticularshaped mass D, called a bob. One entire swing of the pendulum, by which its centre of MECHANICS OF SOLIDS. 243 gravity is carried from the extreme limit G of its path, on one side of the vertical A L, to G" on the other, is oscillation; called an oscillcation. To find the time of to find time of a a single oscillation, call Fig. 142. single oscillation; the weight of the entire pendulum, W; its mass, AM; its angular velocity at any instant, TV1; its // moment of inertia with /'/ reference to the axis of f /!t&. suspension, I; the dis- i notation; tance of its centre of gravity from the axis, D; the vertical distance P G', through which the centre of gravity must descend from its highest point a 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. V12~ living force; and the quantity of work of the weight will be lWu = Ka, Fwork of the weight; and hence I VT2 = 23 gy. 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 a is passing to G', will be given by Ad = D yj;fall of the centre ~/= L"-'.D,, of gravity; 244 NATURAL PHILOSOPHY. and this, in the above equation, gives I. VI2 = 2 fg Dy,; whence square of the 2.D angular velocity; i2 - I 2 g y,. Denoting by s, the small distance described by the point C during the very short interval t, succeeding the instant at which the angular velocity is IV, we shall have which, in the preceding equation, gives s82 M. D t2= -- 2g y,; whence square of the time required to t2 ___, describe a very -. D 2 g y small arc; Taking A M equal to unity, let CB C" be Fig. 143. the arce described in one oscillation by the / point 3 and MN the / small arc s, described / in the time t, immedi- at ately succeeding the f instant at which the I L angular velocity is 1V.' to fid the arc Draw ME perpendidescribed in the small time; cular to the vertical MECHANICS OF SOLIDS. 245 A B, and N Q perpendicular to ME: then, in the similar triangles A ME and MN Q, we have QN * EM MN: AM; and because A 1f is unity, and MN is s,, Q N one value for i EM thearc; But from the property of the circle EM= 2AB.EB-EB2 = 2 EB- EB2, and if we take the arc OB C" 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 EAL= 2BB; which, in the value of s, above, gives Q IN another value for s, -- /2-EB; B the arc; and this, in the value for t2, gives I 1 Q N2 another value for t2 =k 4M.D- the square ofthe' ]/~. D 4 g y, BE' time; Upon B D as a diameter, describe a semi-circumference D m n B, and through the points M and N, the extremities of the arc s,, draw the horizontal lines Mm and Nn, cutting this semi-circumference in the points m and n. Draw the radius Om, and the vertical n q. From the property of the circle we have 2 BmE=.=BE X ED = BE x PM=.BE X yd; 246 NATURAL PHILOSOPIIY. Fig. 143. whence / / I / I 2/ vertical distance m E E / P from last point; y. I D I / which, substituted for BE in the equation B above, gives I 1'QN2 H..D 4g m 2' and, taking the square root, value of the I QN element of the V) X Etime; The two triangles m O E and r q n are similar, and give qn = QN: E:: nm Om; whence QN _u.r mE Om' and this substituted above in the value of t, gives / I nnm t = 2.M.D x O' proportional to Such is the value of the time required to describe the the projection of arc on the circle elementary arc M1V which we see is proportional to whose diameter the arc m n, or to the projection of MN on the semiis versed sine of a.~ofosillation; circumference described upon D B as a diameter, every other quantity in the second member of the equation being MECHANICS OF SOLIDS. 247 constant; and hence, the time required to describe the whole arc CJMB, which is obviously the sum of all the the time of elementary times of describing the elementary arcs N; makingca &c., must be equal to found; 1 ~ g.'M. its. X'Ore" Om into the sum of all the projections of MilN] &c, on the semi-circumference Dn mB; but this sum is the semicircumference itself; and denoting the time from C to B, or that of a semi-oscillation, by ~ T, we have I D m B 12T -= g1 X; its value; but DinB Om =' = 3.1416, Om the ratio of the circumference to the diameter; whence,,_ ~(75\~ time of a single gT = C *D * - \* oscillation; From this formula we see that the duration is independent 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. oscillations; If the number of oscillations performed in a given interval, say ten or twenty mninutes, be counted, the duration of a single oscillation will be found by dividing the whole time of a single oscillation found interval by this number. from Thus, let 0 denote the time of observation, and N the observation; number of oscillations, then will 248 NATURAL PHILOSOPHY. T - = Ml; N g. and if the same pendulum be made to oscillate at some other location during the same interval 0, 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, the same for a 0 I second place; N' 1. 1D Squaring and dividing the first by the second, we find N'2 g' N2 g ***. (76); forces of gravity that is to say, the intensities of the force of gravity, at number of different places, are to each other as the squares of the oscillations in number of oscillations performed in the same time, by the same time. same pendulum. Hence, if the intensity of gravity at one station. be known, it will be easy to find it at others. Simple ~ 182. —Resuming the general value for 1; Eq. (65), we pendulum; have I= j + D2JMf; which value of I, in Eq. (75), gives 1'= M.i.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 consingle point; nected with the axis by a medium without weight, we have MECHIANICS OF SOLIDS. 249 I= 1,zmr2 = 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 1 for the new value assumed by D, which now becomes the distance from the axis to the single heavy point, we have time of oscillation T of=.. (78); of the simple g pendulun; which is the expression for the time of oscillation of a simple _pendulum of which I 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), g.I M.D or I. + ~af9.. (79); in which case I is called the equivalent ssimple pendulum; equivalentsimple that is to say, the length of a simple pendulum which will pendulum; oscillate in the same time as a compound pendulum whose moment of inertia in reference to the axis of suspension is 1 whose mass is M, and of which the axis of suspension is at a distance from the centre of gravity equal to D. The point situated on a line drawn through the centre centre of of gravity of the pendulum, perpendicular to the axis of oscillation; suspension, and at a distance from that axis equal to i, 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. ~ 183. —4 line drawn through the centre of oscillation, and Axes of parallel to the axis of suspension, is called the axis of oscillasuspension and tion. The axes of suspension and of oscillation are reciprocal. of oscillation are reciprocal; Let D' denote the distance of the axis of oscillation from the centre of gravity; then will I = D + D'. Invert the pendulum and make the axis of oscillation the axis of suspension, take I' for the new equivalent simple pendulum, then will new equivalent IT + @f2 simple Z' D'. pendulum; Hf.D but we have, from the foregoing equation, D' = I — D; and this, in the preceding value for i', gives - I1 + H( - D)2 - -. (I X, Again, from Eq. (79), we have I_ D; l-D== MD; substituting this in the above value for 1', we finally get the simple f1 + ID2 pendulum the- D = same; iD that is to say, when the axis of oscillation is taken as the MECHANICS OF SOLIDS. 251 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 mplefou nd 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 circumstances 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 pendulum will be a minimum, when, in Eqs. (78) and (79), Ah+ D2 X ~D + 2 value of -= — _ _ _ - _ = 1 equivalent simple HiD D pendulum; is the least possible; or replacing A by its value K', deduced from Eq. (66)' by making D = 0, the expression K'a 2+ D2 D must be the least possible. But it may easily be shown, either by trial, or by a simple process of the calculus, that this expression is a minimum when KI' = X, and consequently I = 2 K'; 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. of the equivalent simple pendulum is twice the principal radius of gyration. usual form of the Let A and A' be two acute pendulum; parallel prismatic axes firmly con- Fig. 144. 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 pendulum. Also, let l be a sliding mass capable of being retained in /. any position by the clamp-screw device to change. For any assumed position of \ / the position of l /r the ceptioe of X let the principal radius of gyra- gravity; tion be G C; with G as a centre, G C as radius, describe the circumference CUS'. 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 circumference, being a minimum, or least possible, when on it. By moving the mass M, the centre of gravity, and therefore 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 position of gyratory circumference, keeping the centre of this circumcentre of gravity; ference between the axes, as indicated in the figure. In this position, it is obvious that any motion in the mass 211 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 gyratory circumference. pendulum made The pendulum thus arranged, is made to vibrate about to oscillate durin each axis in succession during equal intervals, say an hour same time; or a day, and the number of oscillations carefully noted; if these numbers be the same, the distance between the axes is the length 1 of the equivalent simple pendulum; MECHANICS OF SOLIDS. 253 if not, then the weight H 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 1; the number performed in the same time by the second's pendulum, whose length we will denote by i', is of course 3600, being the number of seconds in 1 hour, and hence, from Eo. (78), N = 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,,. N/2 3' 60* 7\. (80). its length; Such is, in outline, the beautiful process by which KATER determined the length of the simple second's pendulum at the Tower of London to be 39.13908 inches, or 3.26159 value at London; feet. As the force of gravity at the same place is not supposed 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 Enflseghsted of their system of weights and measures. For this purpose, measures; it was simply necessary to say that the.ith part of the si-mple second's pendulum at, the Tower of London' shall 254 NATURAL PHILOSOPHY. English linear be one English foot, and all linear dimensions at once refoot; sult from the relation they bear to the foot; that the gallon the gallon; shall contain 1223th 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 ounce; one thousand ounces, and all weights are fixed by the relation they bear to the ounce. apparent force It is now easy to find the apparent force of gravity at of gravity at London; that is to say, the force of gravity as affected by London; 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 3.26159 whence g - q2 (3.26159) = (3.1416)2. (3.26159) = 32.1908 feet. From Eq. (78), we also find, by making T one second, g =f 12, and assuming length of the 1 = X + y cos 2 +, simple second's pendulum, a function of the we have latitude; g = x + ycos 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 gravity at these places; and these values and the cor- force ofgravity responding latitudes being substituted successively in fod laces; nt Eq. (81), give a series of equations involving but two unknown quantities, which may easily be found by the method of least squares. In this way it has been ascertained that x2.x = 32.1803 and i2.y = - 0.0821; whence, generally, f g = 32.1803 - 0.0821 cos 2 +.. (81)'; Force of gravity in any latitude; and substituting this value in Eq. (78), and making T = 1, we find f Z = 3.26058 - 0.008318 cos 2 +,. (82). length ofsimple 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 4 = 400 42' 40", the latitude of the CityHall of New York, we shall find ft. in. length at City I = 3.25938 - 39.11256. 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 momentofinertia 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 1, we have, Eq. (79), M. D. I = I; 256 NATURAL PHILOSOPHY. or, since AM=W g7 itsvalue;. D. = I.... (83); g in which W denotes the weight of the body. simple second's Knowing the latitude of the place, the length I' of the pendulum known simple second's pendulum is known from Eq. (82); and from latitude; p counting the number N of oscillations performed by the body in one hour, Eq. (80), gives the body's00) equivalent 1. (3600)2 simple N2 pendulum; 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 Fig. 145. maximum reading of the distance from balance. Denoting by a the centre of gravity distance from the axis C to axis found; distance from the axis d' to the point of support R, and by b the maximum indication of the balance, we have, from the principles of moments, ba= IWD. The distance a may be measured by a scale of equal parts. Substituting the values of WD and 1 in the expression for the moment of inertia, Eq. (83), we get value of the moment of b. a. Z'. (3600)2 (84). inertia; N-,. MECHANICS OF SOLIDS. 2657 If the axis pass through the centre of gravity, as, for the moment of example, in the fly-wheel, take Eq. (79), inertia found passes through 1I1 + HL____ 2 the centre of D- 1if]); gravity; whence 1 = H ).D. -.-2 (85). Mount the body upon a parallel axis A, not pass- Fig. 146. ing through the centre of gravity, and cause it to vibrate for an hour as before; from the number of these vibrations and the length of the simple second's pen- fly-wheel; fly-wheel; dulum, the value of 1 may be found as before; i is known, being the weight IW divided by g; and D may be found by direct measurement, or by the aid of the spring balance, as already indicated; whence I1 becomes known. ~ 184. —When a body, centre of B Q N C receives a motion Fig. 147. percussion; of rotation about an axis B A, which is here supposed - -------- perpendicular to the plane of the paper, each elemen- i tary mass n, 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 inertia exerted by 17 an elementary M. r. mass during an 7t elementary time; '258 NATURAL PHILOSOPHY. notation; in which r is the distance Fig. 147. of m from the axis, and VTj the elementary amount of angular velocity generated ----------- in the very small portion, of time denoted by t. \'. co-ordinate Through the axis A, -- q... planes; draw two planes at right angles to each other, and i let their traces on the paper be A X and A Y. Denoting the co-ordinates Ap and A q of on, referred to these planes, by x and y, respectively, we shall have x cos mAp cos n A - r Resolve the force of inertia, above given, into two components in the direction of these planes. The component parallel to the plane of which the trace is A y, will be component ofthe 1 X inertia parallel to rn r ='t. x - the plane A y; t r and that parallel, to the plane whose trace is A x, will be that parallel to Y V the plane xz; r -* - and for other elementary masses in', m", &c., of which the co-ordinates are x' y', x" y", &c., we shall have the components m' x'. X &c m',. &c., the same for t - other elementary masses; -t7-, &C. m'y m"y &c; Y t t' MECHANICS OF SOLIDS. 259 the resultant of the components parallel to the plane A y, will be TV! resultant of the ( x + + m' x' + M" " + &c.) - i,, components t t parallel to A y, and of the components parallel to the plane A x, __ (I y + 1 " +- &c.) =- j MY; resultant of those t t parallel to A X; in which Ai 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 F1/f/2__ 2 L _2,M.D. resultant of the I [ ~t 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 A G, 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 A x found above. The moment of this force, with reference to the axis, will therefore be its intensity multiplied into some distance as A O = L, on this line, Jl J. ifD. L. its moment; 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 -j multiplied by the ratio ~, therefore t V Ig. D.L. I-, t t 260 NATURAL PHILOSOPHY. Or L =. D.(86); pointatwhich whence we conclude that, the point at which the resultant the resultant inertia of 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 exelrted; 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. This being understood, suppose a force F applied at the point C in a di- Fig. 148. rection perpendicular to the line A 0, and immediately opposed to the direc- / a tion of the motion;. this force would obviously tend tc& bend the line A 0, the point A being retained by the axis, and- the point 0 being urged shock onward by the inertia concentrated by the axis when at it. If the force be suddenly apthe body is plied, the axis: must receive a shock, struck; and to estimate its intensity S, denote by X the distance A C; then,: from the principles of parallel forces already explained, we have L: - X:F S; whence SF.L-X_ F( X). (87);: or, substituting the value of L, Eq. (86), its intensity; =F -)... (88)) 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 the blow applied so as to communicate no S - 07 shock to the axis; a condition that can only be satisfied by making MED 1 —I XX=O; whence I distance from the' — MD- - L = A 0. axis at which it B/y —~ must be applied; There being no shock to the axis, it can oppose no resistance to the motion of rotation, and hence we infer that this latter will be the same as though the body were perfectly free. The point 0 is, on this account, called the centre of percussion, which may be defined, that point of centre of a body retained by a fixed axis, at which it may be struck in a percussion direction perpendicular to the plane of the centre of gravity and axis without communicating any shock to the axis. The centre of percussion may be found Fig. 149. experimentally thus:lay the axis C upon a oentle of support A A, and per- percussion found experimentally; mit the body to fall upon a moveable edge B, resting on a horizontal plane; when this edge 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 distance of the centre of percussion from the axis is equal to Anid, 262 NATURAL PHILOSOPHY. to put a it must be at the centre of oscillation. To move a penpendulum te dulunm without communicating action to its axis, the force motion, the force should be must be applied at the centre of oscillation. applied to centre of oscillation. ~ 185.-Resuming Eq. (87), we see that the shock upon the Fig. 150. axis A will be positive, that is to say, will act in the direction of the impressed force X, as long as X is less than L: when Xis 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- E tion opposite to that of the im- ------- pressed force. Now these shocks The shock may in opposite directions, with a neutral point A, can only be positive, arise from an effort of the particles, which are situated on nothing, or negative; 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 general, begins to move for the instant, but only an instant, centre of about a single point. This point is called the centre of spontaneous rotation; sporntaneous rotation. If the blow be impressed at any point, as 0, the centre of spontaneous rotation will be upon the axis corresponding to the point 0 as a centre of oscillation, and hence its distance from the latter will be given by distance of centre I of spontaneous. (8 rotation from HID axis; 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; = JF; that is to say, no matter where the force F be applied, its the entire shoclk entire effect will be communicated to the centre of gravity, always communicated to which is a confirmation of the result given in ~ 146. centre of gravity; 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 impact pass through centre infinite distance from the point of impact; in other words of gravity, the the body will not rotate, which is another result of ~ 146. Iody will not ~186.-Let Q be a body suspended from an axis A Fig. 151. perpendicular to the plane of the figure. This body being at rest, suppose it to be'\ struck at the point T by another body P, moving in the direction TL at right angles to the surface of contact, Q and in a plane perpendicular to the axis A. Denote/.." by m and w the mass and weight of the impinging L body, and by V its velocity Collision oia before the impact. At the body having a motion of instant of meeting there will translation.., against another be developed a force of com- retained by a pression F, which will act fixedaxis; equally upon each body along the line ITL, but in opposite directions. The pressure upon both bodies, which is nothing when they begin. to touch each other, will aug 264 NATURAL PHILOSOPHY. the action and ment by degrees as they ap- Fig. 151. reaction arible; proach to the state of greatest compression; so that X, although always representing a number of pounds \ weight, is, nevertheless, not \~ a fixed, but a variable quan- s tity. We may disregard for Q \ a moment the body Q, and I 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 measure of the F m v force of reaction; t But the force F also acts upon the body Q, and turns it about the axis A, generating in it, during the same interval 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), moment of action F. A C I. VI equal to moment t of reaction; 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 MECHANICS OF SOLIDS. 265 or, finally, Mp. n. v - I. v,. result for a single instant of time; Denote by v', v", v"', &c., the small degrees of velocity lost by the body P, during the second, third, fourth, &c., intervals of time t, supposed to be always of the same length; and by v,', v,", v,"', &c., the angular velocities acquired by the body Q during the same intervals; we shall have * m'. V-' I= IV,, the same for other instants of p. n v"-= IV,", time; &c. = &c.; by taking the sum of the whole, the sum of the p (v + + +, + + &c.) m = I(v, + v,i + v, + &c.); whole; and denoting by U the whole velocity lost by the body P, and by VF the whole angular velocity gained by the body Q during the entire action, we shall have U = V + v' - v" + v"' + &c., velocity lost; Eli = v, + v, +v, + t v + &c.; angular velocity gained; whence, by substituting above, result for the p.. U= I-.... (90). entire duration 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 becomes imbedded; for, as soon as the body P has taken the 266 NATURAL PHILOSOPHY. ir the bodies be angular velocity of the other about the axis, there will be not elastic, they no effort to regain lost figure, and the two bodies will will ultimately constitute a turn about A as though they constituted but a single single one; one. But the angular velocity of Q about A being V1, that of P will be p V1, and we shall have U= V-pI 7; substituting this value of U in Eq. (90), we find pM( V - pI/) = 1V1; whence angular velocity p. V generated by the 2.. (91); impact. -. P2 + f' 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. Application to ~ 187.-In artillery, the balistic vlc pendulum; the initial velocity of Fig. 152. projectiles is ascertained by means of the balistic A pendulum, which consists of a mass of matter suspended from a horizon- / tal axis in the shape of a knife-edge, after the manner of the compound its coustruction; pendulum. The bob is either made of some unelastic substance, as *R wood, or of metal provided Nrith a large cavity 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; V1 the angular velocity of the balistic pendulum after notation; the blow, I and i 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 muetre ckaof through this centre and perpendicular to the line A 0. oscillation; This condition being satisfied, we have p = r, and Eq. (91) becomes rmV mr2 + I' from which we find ( r+I) Vvalue for the ~V -(P )...(92); velocity of mBn r projectile; 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 Hf 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 m ri2, we have, from the principle of the living force, (I ~+n rt ~2) ~V2 = 2 (AL ~ an) g H; equation of living force; whence (I + A r2) V2 _ 2. (i:l + en) g 268 NATURAL PHILOSOPHY. Denoting by T the time of a single oscillation of the pendulum after it receives the ball, we have, Eq. (75), time of a single oscillation of I ~ nr2 balistic T V ( + ) t). pendulum; D being the distance from the axis to the centre of gravity; whence, I + mr2 DT2 (M + ) g = q2 and this value, substituted in the equation of the living force, gives 2 T T2....) V12 = 2;; whence angular velocity 2 of the pendulum; VI also moment of inertia (MA + m) g. D. T2 of the whole; I+ r 2; and because, Eq. (78), time of oscillation of the equivalent T='K, simple g pendulum; we find length of this T2 g pendulum;'Kr2 Substituting these values of V1, I + m r2 and r in Eq. (92), we find V = /2HffD. il +;:~~~7 1~~~~ NMECIANICS OF SOLIDS. 269 or, replacing the masses by the weight divided by the force of gravity, H.r D tff +- w simpler value for / =-D X;W'.qt_ velocity of W projectile; 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 Fig. 153. which D is the radius. Let G G'K 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 to find the radical ascent; draw G''R perpendicular to, part of thisvalue K G. Then, because A G = D, and GR- = H, we have, from the proper- G ty of the circle, R G' = (2 ) - ); and if the pendulum be made large, so that the arc G G' shall be very small, which is usually the case, IT may be neglected in comparison with 2 D, and therefore R GU = H. ); value of radical'2R G — ~2Hf. 7; part found; v/2 ID 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 velocity of -WC: + W projectile in Vr = 2*' C-. -. terms of the T tW chord of the arc of vibration; From this equation, we may find the initial velocity V; and for this purpose, it will only be necessary to have the 270 NATURAL PHILOSOPHY. 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 are, it will be sufficient to attach to the lower exvibiration; tremity of the pendulum a pointer, and to fix on a permanent 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 0, and the value of D, found as already described, ~ 184, we have R G'~ - C- =D. sin0 8; whence its value found; C = 2 D. sin- 0; and finally final value of V ='D q- w+ 1* ( velocity. T W in SIMPLE MACHINES. A machine ~ 188.-A machine is any device by which the action defined. 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. ~ 189. —This consists of an assemblage of cords or bars; Funicular the former united by knots, and the latter by joints or machine; hinges. The cords are supposed, for simplification, perfectly flexible, the bars perfectly rigid, and both inextensible, 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 inertia small as compared with may be estimated and taken into the account by'the the powerand 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 A B be solicited by Fig. 154. several forces. Each force may be resolved. caseof a single into two components one in the direction of the cord, the other 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- forcesmustact tensity and immediately opposed. They must also act to to Stretch the cord; stretch the cord, for compression would only bend it, and 272 NATURAL PHILOSOPIY. the action of one force could not be transmitted to the point of application of the other. in the case of a If instead of a cord we suppose a bar, the conditions bar, theforces of equilibrium will be the same, only that the bar being may also act to compress it; 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 action of the the molecular forces as so many springs which, as soon as molecular spl.ings; an effort is made to disturb the particles from their positions 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 the tension the effort by which any two consecutive elements are urged to same throughout, except when approach each other or to separate, in the direction of the vertical; 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 beL equally strong throughout, the rupture ought to take place simultacords never neously at all its points, and yet this is never found to be equally strong 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 bars are weaker as they and ropes, which to all external appearances are the are longer. same in kind, are generally found to be weaker as they are longer. ~ 190.-We have seen that when forces which act upon the extremities of a cord are in equilibrio, the resultant of those acting at one end, must be equal and directly opposed to that of those acting at the other; and MECiHANICS OF SOLIDS. 273 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 foes which act forces which act at one end of a cord is equal to the cord mustbe work of those which act at the other. The work of each equal; resultant must also be equal to that of the tension of the cord at any one of itspoints, as C; and to find the value of this Fig. 155. work, it is only necessary to multiply this tension by the path described by the point * i C in the direction of the tension. Thus the quantity of work of several forces a2plied to one end of a quantity of work cord, is equal to the quantity of work of its tension. In the oapplied to one example of the common device for ringing large bells, end of a cord is equal to that of it is usual to attach to one end A of a rope, which con- the tension; nects with the machinery of the bell, several cords C, upon Fig. 156. each of which a man may pull. It would be difficult to estimate the work performed example of the example of the by each man, because bell-ropes; his effort, as well in A intensity as direction, varies at each instant; but there is a general c aC c tension exerted upon 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 producing useful work. The perpendicular components only produce fatigue, and exhaust uselessly the strength of the men. Fig. 157. effect of And, although the tocomponents perpendicular to tal quantity of work is the main rope of transmitted to the main the bell; rope, yet the disposition of inclined cords is a source of real loss, which is the greater in proportion as the inclination is greater. It r is for this reason that Cl C effect of a hoop. a rigid hoop m n is so c o introduced as to separate the cords, and give the portions to which the efforts are immediately applied parallel directions. ~191.-When several forces act upon cords which meet Equilibrium of in a point and are united by a knot, the tension of any several cords meeting to a one is equal to the resultant of the efforts exerted upon point; 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 liding knot; the force P shall bisect the angle A D B formed by the portions of the cord separated by the bend at D; for MECHANICS OF SOLIDS. 275 the force P must be equal and directly opposed to the resultant of the tensions on D)A and D B; but the whole cord A D B being continuous, these tensions must be equal, Fig. 158. since the tension is the same throughout; if, direction of the force applied to therefore, the distance /.* the knot, must "'~ the knot, must DC be laid off on- - bisect the angle of the two parts P D produced, pro- /ID of the cord; portional to the intensity P, and from C, the lines C m and Cn be drawn parallel to DB and D A respectively, the figure Cm D n will be a rhombus, because D m and D n, which represent the tensions, must be equal. 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 D B, whose Fig. 159. ends are fastened to Xr hooks at A and B...... example in the ~L' —"x.-''~'"-,.,~mode of The weight of the lan- modeof suspending the tern will cause the pul- D common lantern; ley to move till the (p direction of the weight bisects the angle made by the branches of the cord; the pulley will then come to rest and remain in a state of stable equilib- the pulley will rium. The equilibrium will be stable because, being a equilibrium whee 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 Point; length of the entire cord being constant, the point D will, when in motion, describe an ellipse of which A andB are the 276 NATURAL PHILOSOPHY. foci, and as the direction P C, of the weight Fig. 159. of the lantern, bisects C the angle A D B, it will. be perpendicular to the position ofthe tangent to the curve at horizontal D, which must therefore tangent; be horizontal, and no point of the curve can lie below it. If the pulley be removed and the lantern when the pulley be attached by a knot Fig. 160. is replaced by a arbitrarily to some point knot; arbtrarlyto soe point as D, the freedom of motion will be destroyed, tension will not the tension will no lonbe the same throughout; ger be the same through- D out, and the conditions of equilibrium will be those of forces applied P to three cords meeting at a single point. Produce the vertical PD, and lay off the conditions of D C to represent the weight of the lantern. Denote its thillee ihisabme weight by TV; the tension on D A by a, and that on D B as those of three by b; the angle A DB by A, and AD C by I; then, obliq~e forces; o drawing Cn and Cmn, parallel respectively to D A and DB, we have, from the parallelogram of forces, W: a:: sin p: sin(cp-8), IF: b:: sin: sind; whence tension on one W. sin (lp - ) branch; asin (94) sinq)l MECHANICS OF SOLIDS. 277 IV.V sin 6 tension on the SII1 Cp (95). sin cp other; If 6 be less than g - 6, a will be greater than b; that is to say, the tension will be the greater up2on that branch with, branch most inclined has tile which the direction of the weight makes the least ange. greatestinclined hs tension; greatest tension; If the cord AD B be drawn into a straight horizontal line, p will become equal to 180~, the sine of which is zero, and the tensions a and b will become infinite; in other words, there is no force sufficiently great to bring the no forcesufficient to make the cord whole cord to a horizontal position. horizontal. ~192.-Let us now consider a Fig. 161. polygon A B CD, composed of an To find assemblage of conditions of cords obasequilibrium of cords or bars, and / p the funicular actedupon at the Coo polygon; angular points by the forces P, Q, X, S. Moreover, t... let N and N' be two forces drawing 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 conditions 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. If, equilibrium must subsist at each therefore, any one of the forces, as R, be resolved into two angle; components in the directions of the sides D C and B C, adjacent to its point of application, these components will 278 NATURAL PHILOSOP HY. be equal and directly opposed to the tensions of the is independent of sides. The equilibrium is entirely independent of the length of sides; lethe sides, and will subsist when these are reand will subsist when the sides duced to zero, in which case, all the forces and tensions are zelo; 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 conditions of forces shall remain in equilibrio when transferred parallel to equilibrium in words. their primitive directions and applied to a sincgle point. 193. —If all the forces P, Q, R, &c., be weights, and the polygon in ecquilibrio, since the force R will be in the plane of the sides B C and CD, adjacent to Fig. 162. the angle C; the X, force Q equally in the plane of the sides B C and Al Whe the forces A B; the sides \ C/i are parallel, the polygon and AB, B C, and direction of forces CD, will be in p are in same'i plane; the plane of the / parallel forces Q o and R. In the / same way it may / / be shown that the v' / entire polygon M and the forces applied to it are the polygon a in the same plane. If the polygon be a collection of collection of heavy bars; 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 A B; this weight must pass through the centre of gravity of A B. Resolve it into two 3MECHANICS 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 w. In like manner, if w' be the weight of the side B C, weight of sides the components at B and C will be ~ w', and so on for the,esolved into parallel other sides. Thus the angles B and C will be acted upon components; by the weights ~- (w + w') and 1 (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 D D' respectively, the resultant of these tensions must be equal resultant of and directly opposed to that of all the weights. If, there- extlme tensions, equal and fore, the lines AA' and D D' be produced, their inter- opposed to that section 0 will give one point through which the resultant agll the of the weights P, Q, R, S, and that of the polygon, will pass; and this resultant being vertical, if the distance 0 i 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 + w +'w' + w", &c., and two lines H U and M V be drawn value of extreme through i parallel respectively to A A' and D D', the dis- tensions found tances 0 V and 0 U 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 O1 may be drawn vertically through its centre of gravity. ~ 194.-It is often of great practical importance to Method of finding the tensions of know the tensions on the sides of a funicular polygon the sides; the sides; subjected to the action of weights, in order to proportion the dimensions of its several parts. Let A B CD E be a polygon in equilibrio, under the action of the weights P, Q, R, 8, T, including the 280 NAT URAL PHILOSOPHY. weights of the sides, and the extreme forces N Fig. 163. and N', of which the di-, funicular polygon rections are A A' and Ef j in equilibrio _E/', respectively. Dunder the action, es y. of weights; note the tension of the p side AB by t1, that of B C by t2, that of CD by t3, &c. Since the equilibrium subsists about 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 t1; and if A n be taken on the prolongation of A' A to represent N; the parallelogram Ap) n o, constructed on An as a diagonal, will give Ap for the determination of weight P, and p n 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, a b, b c, c d, d e, proportional to the weights P, Q, R, 8, and fT From the point a' draw a'S perpendicular Fig. 164. to A A', and proportional St in length to the tension N;, and join S with the sevgeneral eral points a, b, c, d, and - construction for e; then will a S b S c AS... finding the tensions; diS and e, represent,'N " N" respectively, the ten-.. sions 4, t2, t3, t4, and N'. For the two triangles Apn and a'Sa are similar, because a' S and a' a are respectively perpendicular to A n and Ap; hence the angles Sa' a andp A n are equal; moreover, the sides about these equal angles are proportional by construction and we, therefore, have An = N pn -:: a'S: Sa; MECHANICS OF SOLIDS. 281 and if a' S represent the tension N; a must represent the tension tl. For the same reason, a b being proportional to demonstration; Q, the third side b S, of the triangle a Sb, will be proportional to t2, since the three forces 4, Q, and t2, are in equilibrio about the point B. Finally, since a a' and a' S are perpendicular to the directions A 2p and A n of the forces P and 1N 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 A, and so on. Therefore, when a funicular lines which polygon is in equilibrio under the action of weights, if a represent the tensions, series of distances be taken on a horizontal line propor- perpendicular to tional to these weights, the lines drawn through the points the idea of the funicular 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. ~ 195.-The sides of the polygon may be very short and only subjected to the action of their own weight, Thecatenary; which would be the case with a heavy chain A CB suspended from its extremities. The polygon of equi- Fig. 165. librium then becomes a curve, called the catenary. This curve is em-' its use in the arts; ployed to give form to arches and domes. The use of the catenary for such purposes may c be illustrated by conceiving a series 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 string of balls; other. Such a collection of balls would resemble a string of beads, and if supported at the ends would, under Fig. 165. the action of their own weight, assume the form of the catenary, or A rather funicular sides of the polygon, of which polygon, the the sides would be chords of the the sides would be balls; the chords of the spheres joining the c points of contact. If the whole arrangement be reversed, and the balls, instead of being suspended, be supported upon the ends as fixed points, after the manner inthe stringofballs dicated in A' C'B', the figure will remain unchanged and reversed; 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, points of contact and the spaces extended to tangent planes; between filled up with solid matter, Fig. 166. as wood, stone, or metal, we shall have a perfect system of voussoirs arch-stones or or arch-solids in Voussoirs; equilibrio under the action of their own weight, requiring no aid from friction or any other principle of sup MECHANICS OF SOLIDS. 283 port. The tangent planes or joints of the voussoirs will be position of the normal to the curve. The catenary is also employed in joints; suspension-bridges supported upon two or more parallel also used in chains stretched across a river. In the construction of suspensionbridges. such catenaries it is important to determine the tension at the ends, in order to secure an adequate resistance at those points. ~ 196.-The catenary A CB, suspended from twoGeneral points A and B, is nothing more, as we have seen, than properties of the a heavy polygon in equilibrio, and whose sides are indefinitely small; so that, if upon a horizontal line, a length A'B' be taken proportional to its weight, and this length be divided into a Fig. 16. number of equal parts, B3 L there will exist a certain point S 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 catenary, and that the lengths construction construction to SA', SF', C', &c., of find the tension tfhese lines, are propor- /- -' of the different'these lines, are propor-'c,, -, points of the tional to the tensions of /. catenaly; the same elements. Of all - - "-" -- -c -----—; —-' the tensions, the least is given by the line S C', 6 drawn perpendicular to the horizontal line A' B'. But the element of the catenary to which this tension least tension at corresponds being itself lowest point; horizontal, it will occupy the lowest point of the curve. This length becoming greater 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. as the element is at a grester is Whence it follows, that distance from the the tension is the greatest lo west point; possible at the extremi- Fig. 16I. ties A and B. Two equal B L tensions S F' and S G', appertain to two elements equally distant from the lowest point C: moreover, ------ elements of equal these elements form equal tensions form angles with the vertical equal angles with the vertical the vertical L C passing through this through lowest point; point; hence, these elements, i and N; are situ-," ated on the same horizontal line MN;, and the chord MN; as well as all similar -' / -- --- -- chords, will be divided equally by this vertical catenary line. The catenary is, symmetrical in therefore, a symmetrical reference to this line; curve in reference to a vertical line passing through its lowest point. It follows, also, that when 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 are equal; position of the is upon the perpendicular drawn through the middle point oftensions. of the horizontal line A'B', which is proportional to the weight of the catenary. A and D being, for example, the two points of suspension, and A' D' being the length proportional to the weight of the catenary A CD, 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'. ~ 197.-Two catenaries, (last figure), A CGB and a c b Similar are similar when the points of suspension A and B of the cateiiaries; one, and a and b of the other, are situated upon parallel right lines, and when their lengths A GB and a cb are proportional to the distances AB and a b, 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 proportionally diminished indefinitely, ~ 192. Therefore, when equilibrium A CGB is reduced to the size a c b, the equilibrium will not independent of size; 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 a cb 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. WVe 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 from those of a weight of the smaller catenary, as A'B' represents the similarone; 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 elements of the two curves. These are called homogeneous homogeneous tensions. But because A'B' and a'b' are parallel, we tensions; have the proportion Sf' SF:' a'b': A'B'; whence we conclude that, in two similar catenaries, the ten- tensions of sions of elements similarly situated are to each other as thle elements similarly situated weights of the catenaries. are as the weights of the entire curves. 286 NATURAL PHIILOSOPHY. 0 198. —Let A'B' To construct the be a horizontal line procatenary from its teeg weight, length, portional to the eight and the point of of the catenary, S the G telnsions; point of tensions. Divide the line A' B', and the length of the cate- A/,r ~' B' 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 corresponding weights. Draw the lines SA, 81', 82', 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 SI' 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 required catenary the nearer in proportion as the number of divisions is greater. to draw a tangent The point of tento any point of sions 5 gives the means the catenary. Fig. 169. of drawing a tangent to the catenary at any A --- point. Let E be the given point, and let A' e represent the weight of the portion A E of the a catenary; through e and S draw the indefinite line e G, and from F draw E G perpendicular to e A, E G will - be the tangent line. MECHANICS OF SOLIDS. 287 ~ 199.-The point Fig. 170. Determination of of tensions in the cate- the point of US/ tensions; nary depends upon the intensity and directions of the extreme tensions. For A' B' being the horizontal line proportional to the weight of the entire catenary, if from the extremities A' and B' arcs be described with radii proportional to the extreme tensions, their intersection S will give the point of meeting. The process for find- to find the ing the extreme ten- Fig. 1. extreme tensions ing the e~xtreme ten- from the curve sions must of course A L traced; depend upon the data given. Let us first suppose the catenary A CB to be given and traced out. It is evident from the conditions of equilibrium, that the vertical 0 L drawn through the intersection 0 of the extreme tangents A 0 and B 0, will pass through the centre of gravity of the catenary. If, therefore, a distance 0 G be taken on this line to represent the entire weight of the catenary, and the parallelogram 0 B' GA' be constructed upon the tangents, the sides 0 A' and 0 B' will represent the tensions at A and B respectively. But if only the two Fig. 17to find the the Fig. A and o extreme tensions points A and B ofI B from the points suspension, the weight, D of support, the and entire length of the wlengh of the catenary be given, the curve; process for finding the extreme tensions is as follows, viz.: Take a d a small chain and suspend it against a ver- c 288 NATURAL PHILOSOP/HY. tical plane from two Fig. 172. points a and b, situated figure found by upon a right line paral- _ means of a small lel to A B, and whose chain; distance apart shall be to the distance from A to B, as the length of the smaller chain is d to the length of the longer. The smaller chain being thus suspended, measure by 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. Instead of measuring with a spring balance the tensions at the ends of the catenary, we may proceed as follows: the tensions Draw through the lowfound by fcostruction; est point of suspen- Fig. 173. sion a, a horizontal A: line cutting the oppo- o;.. site branch of the small chain in the point d. --- Upon a horizontal line take the distance a' b' to represent the weight b' d- of the entire chain, and lay off the distance a' d'.C r e proportional to the i — length a c d. The portion a c d of the catenary would be in equilibrio if the point d were fixed and the remainder d b removed; the point of tensions for a c d, and therefore for a c b, will, from what has already been explained, be found MECHANICS OF SOLIDS. 289 somewhere on the perpendicular C' K' drawn to the middle of a' d'; assume it at 0, 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 0 draw a perpendicular to C' K', and lay off upon it from the point 0, the distance 0 g = ae - a b, to the right when a e is greater than a b, 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 Fig. 174. The smallest and catenary the tensions at the different greatest tension of a vertical points are different, and that the small- chain. 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 Q and link C, A 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 Fig. l 75. Direct measure of the point of meeting of the the tension on any point of the tensions, and A' B' a hori- catenary; zontal line representing the weight of the catenary, we have seen that the tension at a' -- - 19 290 NATURAL PHILOSOPHY. D is represented by the length D'S, and that at C, construction; the lowest point, by C', perpendicular to A'B', the lengths A' D' and D' C' representing, respectively, the weights of the portions AD and D C of the the tension at any curve; that is to say, Fig. 176. point is the hypothenuse of a the tension at any right-angled point D, is representtriangle; ed by the hypothenuse of a right-angled triangle, of which one-, side represents the tension at the lowest point Fig. 176. of the curve, and the A other the weight of that portion of the cate- -..... nary included between the lowest point and the X point whose tension is to be found. Hence, the C tension in tension at any point of horizontal direction; the curve, estimated in a horizontal direction, is constant and equal to the entire tension at the lowest point; in vertical 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 effect of these piers, the vertical components will promote their stability tensions on piers. by pressing them down, while the horizontal components will tend to overturn them. ~ 202.-It is comparatively easy to compute the extreme tensions of the catenary when the versed sine of its are is small. Let A CB be a catenary, of which CD, the MECHANICS OF SOLIDS. 291 distance of the lowest point below the horizontal line BA, To find extreme is very small. The curve being in equilibrio, the equi- thevnsed siene Of the curve is Fig. ECUl. small; B ~ u ~ ~D A. ID-a' (V librium of the part B C 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 D C, the curvature must be very small, and the centre of gravity of B C may, without sensible error, be regarded as at the middle point G. The tangents CH and B G', 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 B C. Because the three forces p, 1, and To0 are in equilibrio about the point G', we have p::: Bt:.HG', x p T:: BH: BG'; whence H ElGT' tension at lowest BITB' point; T- B- p' tension at highest B H' point; Observe that BET is the versed sine, which denote by f; and, because B G C may be regarded a right line, H G' is half the semi-space BD, which semi-space denote by i. Then, since the triangle B UG'H is right angled, BG' BIT2+UI[2= 12 292 NATURAL PHILOSOPHY. Substituting these quantities in the above equations, we find horizontal T -_ tension or thrust; 2f' tension at highest T - P 1f + point. 7 + 4 -- 1 4f2 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 Application to show the application of the preceding principles, take the bripgesi- 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 attachsuspending rods; ed vertical suspending rods, which are united at the bottom in pairs by transverse pieces called sleepers; these A receive a set of longijoists; tudinal joists, which, in their turn, support the floor plank. The distances between the suspending pieces in longitudinal direction are supposed equal. These equal portions of the roadway included between two consecutive sleepsections; 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 MECHANICS OF SOLIDS. 293 weight is known, and each pair of determines the cross sec- Fig. 1'79. suspending rods supports the tion of the suspending weight of one rods. The weight of the.. section; suspenders being small P compared with that of the roadway, may be A' I neglected, and thus the weight of the bridge A BA' will be equally distributed. Draw a horizontal tensions on the sides of the right line, and take funicular u v proportional to the a polygon; weight of the bridge; let S be a point such that Su shall be perpendicular to the side UA, and proportional to its tension. Take upon u v, the portions u a, a b, &c., proportional to the weights supported at the angles A, B, &c.; the converging lines a X, b X, &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 Sud, will be similar and give AA": A"B:: ud Sd; whence A" B difference of level A A" -S —d- U. between two consecutive angles; Because of the equality of distances between the suspending rods, A" B will be constant. Moreover, u d and Sd being proportional respectively to the weight of the portion A'D', and the tension to upon the horizontal side, 294 NATURAL PHIILOSOPHY. if we denote by X the weight of a unit of length of the bridge, ratio of weight ofd w half the bridge to d _ the horizontal Sd to tension; which in the preceding gives A"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" = P. A' D'; to and denoting the constant ratio of the weight p to the tension to at the lowest point by k,,alue of the'ifference of level A A" = k. A' D'; If two onsecutive angles; 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 polygon. Denoting by 1 the constant length of a section, and beginning at the lowest angle MI, the horizontal distances will be successively 1, 2 1, 3 1... n 1, for the 1st, 2d, 3d,... nth, angle, to the right and left. Thus the difference of level between the lowest angle KI and the next in order C, is k 1; between C and B, 2 k 1; between B Fig. 180. and A, 3 k l, &c. The heights X difference of level Of the angles C, B, A, &c., of the angles of above the lowest point ], will the polygon above the lowest be respectively k 17, k + 2 k 1, —. ---- __-r angle; 3kgle; Jl I 2 kl 3 k, ik + 2 k I + 3 k I + 4k l, and, in general, MECHANICS OF SOLIDS. 295 if there be n sections between the lowest angle and that under consideration, the height of the latter above the former will be given by the expression rn + 1 height of the nth k l(1 + 2 + 3 + 4.... + n) = k I. n + * angle above the 2 lowest one; In this expression, if we make successively n = 1, n = 2, n = 3, n = 4, &c., we have k1, 3 kl, 6 kc, 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 the locus of the angles is a parabola, for Fig. 181. anglesisa parabola; if y= P= MU de-. —---------- note the height of one f of these angles above the lowest point K, n being the number of its place from the latter, we have 1 n+l y =k /.n 2 (96); and making nl = x= KM, o= /(2+)(x ( 7) 1 = or 3k x2 k equation of the Y = + - X; locus ofthe 21 2' angles; this is the equation of a parabola, of which the vertex is 296' NATURAL PHILOSOPHY. to the right of the point,; and at a distance from it equal to 1 2' place of the vertex of the it is below the horizontal side by the distance locus curve; HK' =y 8' a quantity so small that it may be neglected in Fig. 182. practice. 0B Moreover, from the, property of the parabola, e the squares of the ordinates are to each other to find the point as the abscisses; that is in which the vertical through t Say, the vertex cuts the line of A B M2A: NB supports; and from the similar triangles obtained by joining A and B, A - 02 - 2 AP 2 B_ 2:: PO: q O; whence P O 2 QO".. MA: NB; or P O2 X NB = Q O2 X fA; but P 0= O - KP= — O - MA, QO = QK'- OK = NB - KO; MECHANICS OF SOLIDS. 297 which, substituted above, give (OK- IA)2 X NB = (NB - KOo)2. MA; developing the squares and reducing, we get K02 = MiA x NB. That is to say, the distance K O, at which the vertical line distance of this drawn through the vertex of the curve cuts the chord Point above the vertex; joining any two of its points, is a mean proportional between the heights of these points above the vertex. This property furnishes an easy method of finding the lowest point K on the level k-N. 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 ocuts the vertical YN/V in construction for P'. Upon the distance findin the position of the P' V describe the semi- lowest point; circle VTP', and from the point N draw the tangent NT; with N as a centre and NVT as a radius, describe the arc TT' till it cuts VN in T', and through the point T' draw a horizontal line; this line will cut the cord U V in the point 0, through which draw a vertical line 0 P, 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 HN the equal lengths of the and the abscisses sections; the points of division will correspond to the of the angular points; vertical ordinates k1, 3 k1, 6 k,... n. 2 k 1. This last appertaining to the point U, whose height h is given, 298 NATURAL PHILOSOPHY. we have n.(n + 1)kc= h; 2 whence we have ratio of the 2 h weight ofa k = a section to n (n + 1) (97); horizontal tension found; and hence the lengths of the several suspenders k i, 3 k 1, from which the &C., are known. lengths of the We have seen that suspenders are known; _P_ 2h -o n(n + 1)I and therefore (~ + 1). zp and horizontal n 1). (98); tension found; (98); the tension on the horizontal side is, therefore, also known. The tension on the side next in order to the horizontal side is tension on side t02 + 22, next in order; that of the second in order that on the 2t second in order; /to (2 _ ), that of the third that on third; to2 + (3 p), and so on to that on the nth in order; t02 + p)2 MECHANICS OF SOLIDS. 299 which is the tension on the nth side from the horizontal one. 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 0 tf; 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, dimensionofthe 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. XV. OF BODIES RESTING UPON EACH OTHER, AND UPON INCLINED PLANES. ~ 204. —When two bodies touch and compress each Action and reaction of bodies other, there is immediately a depression or yielding in a forced into direction perpendicular to the surfaces at the point of apparentcontact; contact, which indicates that the reaction of the two bodies takes place in the same direction; that is to say, in the direction of the normal common to both surfaces. Let us suppose one of the two bodies as A to be solicited by forces of which the resultant shall coincide with this nor- Fig. 184. mal, and that the other body A' is fixed; it is plain that the reaction of A the latter body will destroy this resul- Aaction and tant, and that the body A will remain eaction of two A will remain bodies; at rest. But the equilibrium will also subsist if the body A' be replaced by a 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 300 NATURAL PHILOSOPHY. ciprocal action of each other in directions normal to both the principleof surfaces at the common point of contact, extends to the the reaction of two bodies general case of a single body pressing upon two or more extends to bodies at the same time. The reaction of these last are so several. 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- Fig. 185. the bodies having ing upon a level plane A B t ontac point o with a single point of contact m. Since the reaction takes place in the direction of the X B perpendicular to the plane through the point of contact, 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 wll keep a body single point of contact with it, shall be in equilibria, it is at rest against a plane. -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. MECHANICS OF SOLIDS. 301 ~ 206.-But when the body has two points of contact, A Fig. 186. If the body have two points of and B, with the plane, it is not contact the necessary that the resultant of Iesultant leed not pass through the forces shall pass through V either; A4 B either. It will be sufficient if it meet the line A B in any 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- intterset the lind 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- ofcontact; mer, and hence their resultant will have its point of application between A and B; and this resultant being parallel to its components, will be perpendicular to the plane. If the body be laid on a horizontal plane, the equi- when theplaneis librium will subsist whenever the vertical drawn through horizol. the centre of gravity intersects the line joining the points of support somewhere between them. ~ 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 more points; 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, reslltant still its point of application will, from the principles of parallel normal, and within the forces, be within the polygon formed, by joining the points polygon of of contact. If the line of direction of the resultant, pierce contact; 302 NATURAL PHI LOSOPHY. if the resultant the plane in a point m, ex- Fig. 187. pierue the plane terior to the polygon which without the polygon of connects the points of support, contact, the body will tend will overturn; the body will tend to overturn around the edge a b of this polygon nearest to m; if the line of contact be a curve, the Fig. 188. body will overturn about the tangent nearest to nm. The effort by which the body will be urged to overturn is meas-..... ured by the intensity of the resultant of the forces, into b effort by which the shortest distance from its the body is urged line of direction to that about to overturn. which the motion of rotation takes place. ~ 208.- The conditions of equilibrium of a heavy sphere, resting upon a horizontal plane, have already been considered. Let us apExamples; ply the same principles to other examples, and take first Fig. 189. the case of a heavy body resting upon a table having table having but but three feet. If the feet be three feet; tupon a horizontal plane and when the feet are in the same right line, and the in same right vertical line through the -/ y line; centre of gravity be not in the vertical plane passing willoverturn through this line, the table Fig. 190. unless the weightg. 1 unless thweight will overturn towards the pass through this line; side on which the centre of gravity is situated, and with an effort equal to the product of the weight into the distance A g of the projection of the MECEHANICS 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 A g 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 in same right weight pass within the tri- line; angle formed by joining the Fig. 191 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 overturn about the nearest edge a b. In the first case, the equilibrium is stable, because stable Fig. 192. equilibrium; no derangement can take place about the line of either two of the feet without causing the centre of gravity to ascend. And, generally, if the table have any number of feet, there will be stable Fig. 193. equilibrium whenever the line of direction of the weight passes within the polygon in case of aly number of feet formed by joining them. numhe r esultant The effort with which the must pass within table or any other body will thepolygon; resist a cause which tends to Fig. 194. upset it, is measured by the product of its weight into the effort by which shortest distance Ag from the A abody resists a sh s d c cause to overturn line of direction of the weight it; to the line a b about which the motion is to take place; 304 NTATURAL PHILOSOPHY. moment of and this effort will be smaller in proportion as the disstabiliyOdfya tance Ag is less. For this reason, the 9moment of stability heavy body; 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 the same same if the body rest upon to solidsresting a plane face bounded by a Fig. 195. onplanefaces; polygon or curve. The equilibrium 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 Fig. 196. example of the its stability diminishes as the cube and right cubeadrigs h height increases; for, in pro- / prism; portion as the centre of gravity C is more and more elevated, the angle G A B becomes less and less, and the centre of gravity will not have stability to be raised so much above its liminishes as the position of rest when the body centre of gravity is higher; is overturned about the edge aa', as it would if the angle GA B 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- Fig. 197. serve its equilibrium as long inclined prism; as the direction of its weight falls within its base. The difficulty of overturning it will be less in proportion as the 2A MECHANICS OF SOLIDS. 305 distance A g becomes smaller. will overturn When g falls without the base, Fig. 198 when weight falls without the base; the prism will overturn of itself. The Tower of Pisa, Tower of Pisa; though considerably inclined, preserves its equilibrium because the line of direction of its weight passes within its base. A pile of dominos or bricks, in which each one pro- Fig. 199. jects beyond that immediately inclined pile of below it, will preserve its brick; equilibrium till the line of direction of the weight of the entire pile falls without the domino or brick at the bottom, when it will overturn. stability increases We see, therefore, that the natural stability of bodies in- asthe baas I increases and as creases as their bases increase, and the heights of their thecentre 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. 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. forces by other forces than their 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 G g of whose weight falls without the base A B, would, if abandoned to itself, overturn; whereas, if it were acted upon by a force in the direction G.E, of such intensity as to give, with the weight, A O B increase the a resultant which intersects stability; 20 306 NATURAL P HILOSOP HY. the base' at 0, it would be equilibrium supported, and the equilib- Fig. 200. stable; rium would be stable. Reciprocally, the weight TV of the prism is opposed to the /. force GE =F. when the latter acts to turn the solid about the edge A. The measure of o BA this opposing effort is moment of. A g stability; and in this view, we see that the moment of the natural stability will increase as A g increases. In walls destined to support an embankment of earth or 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 conforegoing in the venience will permit from the construction of sustaining walls; vertical line G g of the weight. This is done either by an Fig. 201. exterior slope B A, or by 2 masses of masonry C, called principle of counterforts, attached to the counterforts/; back of the wall. It will be sufficient, in general, for the stability of the wall, if the resultant of its weight W and A g D the pressure against it, intersects the base A D. The Fig. 202. moment of moment of natural stability natural stability; of such structures is always equal to the product of the weight into the distance A g; and therefore the figure of the cross-section of the wall may be varied at pleasure _.- y without injury to the sta MECHANICS OF SOLIDS. 307 bility, provided this product remain the same. Hence the external slope may be suppressed, if the thickness of the externalslope wall be so increased that its augmented weight shall com- and weight; pensate for the diminution in A g. If the ground upon which the wall rests be compressible, Fig. 203. it will not be sufficient that the resultant of the weight and pressure pass within the e base; it must also pass through / when the ground is compressible, its centre of figure; otherwise / o mressle, the resultant there would be more pressure should intersect.4 O.D middle of the on one side of this point than middle of the base; on the other, and the wall would incline in that direc- Fig. 204. tion. If the load of a two- a wheel cart be such that case of a level; cart on a level; the direction of its weight does not intersect the axletree, it will tend to overturn on the side of the weight, and will either exert a presFig. 205. sure upon the horse or an effort to lift him from the ground, according as the on an inclined weight passes in front or in 1 / road ascending passes in front the tendency is rear of the axle-tree. If the to lift the horse; centre of gravity of the load be immediately above the axle-tree on a level road, then, when the cart is as- Fig. 206. cending a slope, the weight will pass behind, and the ten- dency of the load will be to descending; lift the horse; while, on the contrary, when the cart is 308 NATURAL PHILOSOPHY. the tendency is descending a slope, the tendency of the load will be to thorse.spo 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. ~ 209.-Let A B represent Fig. 207. the section of an inclined B Inclinedplane; plane in the direction of its greatest declivity. Although the plane be indefinitely prolonged, it will be sufficiently defined by ratio defined by the relation of the of heighttobase; base A C6 to the height CB, c corresponding to a given length A B. 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 contact; 2d, that the weight shall be perpendicular to the a body on an plane. This last condition cannot be satisfied for any but inclined plane; a horizontal plane, since the weight is always vertical. If the weight be replaced by its two components, one perpendicular 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 may the body to slide, otherwise it will cause it to roll. This slide or roll; last will happen in the case of a spherical ball, since the weight will not meet the plane in the single point of contact m. Let a force P be applied in the direction G S, next figure, conditions of tO prevent the body from moving down the plane. Since equilibrium; 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 it within the polygon of contact, or in the case of the planein which sphere, at the point m. The force P must, therefore, be he foapplce mst applied in a vertical plane which passes through the centre of gravity, and which is, at the same time, perpendicular to the inclined plane. 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 Fig. 208. line G' parallel to the di- s rection of the force P; from Q B intensity of the the point X draw Mi Q paral- force found by lel to G G'; the distance G Q construction; will represent the intensity of the force P, and GM that -' of the resultant, R, of WTV and P. From the principle of the parallelogram of forces, we have intensity of the WI: R: P:: sin Q G f: sin G' G Q: sin G'GJII; force found analytically; but G G' and G M being respectively perpendicular to A C and A B, the angle A is equal to the angle G' G M; and we have sin G' GM = sin BA C= A= B AB; and this substituted in the foregoing proportion gives, after reduction, W: R: P AB.sin Q GM: AB. sin G' G -: BC; from which we find P = WB (99); value of the - AB. *sin Q GM - force; sin U'UGQ value of the =W i Q a. (100). pressure against sin Q G H''the plane; 310 NATURAL PHILOSOPHY. If the power P be ap- Fig. 209. power applied plied parallel to the s plane;l to he plane, the angle Q G MX = 90~; and the angle ) G' G Q becomes the supplement of the angle A B C; whence we have A sin Q GM = sin 90~ = 1; AC sin G' GQ = sinAB = -; which, in the above equations, give BC value of force; P -- VAB' value of the A C pressure against R -= W - the plane; A B' That is to say, when the power is applied parallel to the relation of power, plane, Ist, the power will be to the weight as the height of the weight, and plane is to its length; 2d, the resistance of the plane will be to rlesistance of plane; the weight as the base of the plane is to its length. If the power be applied parallel to the base Fig. 210. of the plane, the angle P Q G M11 becomes equal to power applied the angle A B C, because G parallel to the base; G Q and G 1 are respectively perpendicular to B C and A B; and the angle G'G Q becomes 900~, whence AC sin QG/=- sinABGC A C relation of the B angles; sin sIn (4Ct (} 6_ 1 MECHANICS OF SOLIDS. 311 which, in Eqs. (99) and (100), give B C P-= WT A value of power; A C' = yr AB pressure on A B' plane; That is to say, when the power is appied 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 resistance; be to the weight as the length of the plane is to its base. 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 which the power 45~. When the inclination is 45~, the power and weight weight. will be equal. When the inclination exceeds 450, the power will be greater than the weight. ~ 210.-Let us now consider the notion of a Fig. 211. Motion of a heavy body on the in- f heavy ody on / an inclined dined plane. The body plane; being acted upon by its weight G G' alone, this may be resolved into two components, the one G /M perpendicular, the other A (C G NV parallel to the plane. The first will be totally destroyed by the resistance of the plane, while the second will be effective in giving motion. Denote the weight of the body by WTV the height B C of the plane by h, and its to find the length A B by 1; then, from the similarity of the triangles component of the weight A B C and G G',V will parallel to the plane; W~ G IV h; 312 NATURAL PHILOSOPHY. whence h its value; GN =- 1 T; and because the inclination of the plane is Fig. 211. the same throughout, the 3 ratio 7- will be constant, from the top to the bottom; whence we see that themotion is that the motion of the same arising firom the t action ofe body down the plane, is a' action of a A C constant force; that arising from the action of a constant force. it will be It will, therefore, be univauniformlyd; formly varied, and the circumstances of motion will be given by the laws of constant forces. Substituting Mg for W; we have A GN= - Mg; and making M equal to unity, and denoting by g' the corresponding value of the component G N, we find component of the force of gravity in h direction of the g' g plane; Such is the intensity of the force of gravity in the direction of the inclined plane. This may be varied at pleasure by changing the ratio l; in other words, by altering the inclination of the plane. Now, since the velocities impressed during the first unit of time on the same body, the motion may moved from rest, are proportional to the forces producing be regulated by varying the them, the motion may be made as slow as we please by inclination of the h plane; diminishing -y-. It was in this way that Galileo discovered MECHANICS OF SOLIDS. 313 the laws which regulate the fall of heavy bodies. These in this way being the same as for bodies moving on an inclined plane, Giscovered the it was easy so to regulate the inclination of the plane as to laws of falling bodies; enable him to note and compare the spaces described, times elapsed, and velocities acquired, with each other. If the body be mounted upon wheels, Fig. 212. as in the case of the loaded cart referred to in ~ 209, it will be arged to roll along the when the body inclined plane by an is mounted on measure is r W. D; in which 1V7 denotes example of the example of the the weight of the cart loaded cart; and its load, and D the perpendicular distance m b from the point of contact moment of the m, to the line of direction G b of the weight. 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 manifestedwhen almost entirely from the inequalities in the surfaces of two bodies are pressed together 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. guish two kinds, according as it accompanies a sliding or sliding and rolling motion. The first is denominated sliding, and the rolling friction; rolling on 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 importance in machines, will principally occupy our attention. the measore of The intensity of friction, in any given case, is measured its intensity. 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. Intensity ~ 212.-The friction between two bodies may be measme.reid lby ured directly by means of the spring balance. For this spring balance; purpose, let the surface CD of Fig. 21S. one of the bod- Fig. 213. ies M, be made B perfectly level, so that the oth- / 1 " er body HI', /c / //< X when laid upon it, may press with its entire weight. To some point, as E, of the body l/i', attach a cord with a spring balance in the manner the indication of indicated in the figure, and apply to the latter a force Ek the bmlaine'iwhe of such intensity as to produce in the body i' a uniform uniform is the motion. The motion being uniform, the accelerating and measure; 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 experiments are 1831 1832, and 1833. They were made by the aid of a those of M. Morin;, T contrivance, first suggested by M. Poncellet, which is one of the most beautiful and valuable contributions that 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 expe,'imentsmyy Experiences sur le Frottement." Paris, 1833. The following conclusions have been drawn from these experiments, viz.: The friction of two surfaces which have been for a conclusions r.om considerable time in contact and at rest, is not only differ- thex. experiments; ent in amount, but also in nature from the friction of surfaces 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. The slightest jar or shock, producing the most imper- in machinery, ceptible movement of the surfaces of contact, causes the th friction which accompanies friction of quiescence to pass to that which accompanies motion to be motion. As every machine may be regarded as being considered; subject to slight shocks, producing imperceptible motions in the surfaces of contact, the kind of friction to be employed 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 zniform and definite. These laws friction are uniform and are: definite; Ist. Friction accompanying continuous motion of two firstlaw; 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 selarcrated from one another 316 NATURAL PHILOSOPHY. by an inter2posed stratum of the unguent. The friction in these two cases is not the same in amount under the same influence of pressure, although the law of the independence of extent unguents; of surface obtains in each. When the pressure is increased 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 an apparent apparent exception to the law of the independence of the exception to extent of surface, since a diminution of the surface of consecond law; 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, therefore, to be received with restriction. three conditions There are then three conditions in respect to friction, of spect toaesi under which the surfaces of bodies in contact may be friction; 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. fourth law; 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 remarkable laws obtain, and the accuracy with which the phenomena instance of the uniformity of of motion accord with them, may be inferred from a single these laws; 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 - 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 fic- first law tion be expressed by the following equation, expressed by an equation; F.... (101); whence F= f. P. This constant ratio f is called the coefficient of friction, coefficient of because, when multiplied by the total normal pressure, friction; 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 Wf denote the weight of any body placed upon the inclined plane Fig. 214. its value found AB. Resolve this weight by means of the inclined plane; G a' into two components, 13 one G K perpendicular to the plane, and the other parallel to it. Because the angles G' G M and BA C are equal, the first of these A c components will be' comnponent of the weight perpendicular to W c. cos A, the plane; 313' NATURAL PHILOSOPHY. and the second, Fig. 214. that parallel to W. sin A the plane; in which A denotes the angle BA C. The first of these components determines the total pressure upon the plane, A and the friction due to this pressure, will be the friction on f. Wos A. the plane; 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 velocity, the total friction and this component must be equal and opposed; hence friction and parallel f. W. cos A = W. sin A; component equal whence value of the sin A coefficient of = -- tan A. friction; COSA 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 uniform velocity, the body to which the moving surface belongs being acted upon by its own weight alone. This angleof friction; angle is called the angle of friction or limiting angle of limiting angle of resistance. resistanc; res 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. ~19 TABLE I. EXPERIMENTS ON FRICTION, WITHOUT UNGUENTS. BY M. MORIN. The surfaces of friction were varied from. o3336 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. FRICTION OF FRICTION OF MOTION.4 QUIESCENCE.t SURFACES OF CONTACT. _ a a 95~ 33t 3 90 xfO P Oak upon oak, the direction of the.4 2 3 o. I fibres being parallel to the motion 0.478 2 Oak upon oak, the directions of the) fibres of the moving surface being I perpendicular to those of the quies- 0.324 I7 58 o.54o 28 23 cent surface and to the direction of the motiont - -.J Oak upon oak, the fibres of both sur- faces being perpendicular to the o. 336 i8 35 direction of the motion - - - - Oak upon oak, the fibres of the) moving surface being perpendicular I to the surface of contact, and those 0. 92 Io 52 0.27I i5 Io 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 43 23 end to end.- - Elmn upon oak, the direction of the 0 3 22 o 694 34 46 fibres being parallel to the motion o.43 2 Oak upon elm, ditto, - - - - - 0.246 i3 5o o.376 20 37 Elm upon oak, the fibres of the moving surface (the elm) being perpen-I dicular to those of the quiescent o.45o 24 I6 o.570 29 4I surface (the oak) and to the direction of the motion - -.- - -. * The friction in this case varies but very slightly from the mean. t The friction in this case varies considerably from the mean. In all the experiments 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. FRICTION OF FRICTION OF MOTION. QUIESCENCE. SURFACES OF CONTACT. 0 Ash upon oak, the fibres of both sturfaces being parallel to the direction 0.4o 2I~ 49' o.570 290 I' of the motion -... Fir upon oak, the fibres of both sur-) faces being parallel to the direction o.355 i9 33 o. 520 27 2 of the motion - - - - - - - Beech upon oak, ditto - - o.36o 9 48 o.53 27 56 Wild pear-tree upon oak, ditto - o.370 20 i9 o.44o 23 45 Service-tree upon oak, ditto - - - 0.400 21 49 0.570 29 I Wrought iron upon oak, ditto* o.6I9 31 47 0.6I9 3i 47 Ditto, the surfaces being greased and 0.256 4 22 o.649 33 well wetted - - Wrought iron upon elm - o0.252 I4 9 Wrought iron upon cast iron, the) fibres of the iron being parallel to o. I94 Io 59 0.I94 Io 59 the motion - - - - - - - - Wrought iron upon wrought iron, the fibres of both surfaces being par- o. I38 7 52 o.137 7 49 allel to the motion - -. Cast iron upon oak, ditto - - - - o.490 26 7 Ditto, the surfaces being greased and.646 3 52 wetted - - - - Cast iron upon elm - - - - - - o. I95 I 3 Cast iron upon cast iron - - - o.i52 8 39 o.I62 9 I3 Ditto, water being interposed be- o.3I4 7 26 tween the surfaces - - - - - - Cast iron upon brass - - - - - o.I47 8 22 Oak upon cast iron, the fibres of the wood being perpendicular to the 0.372 20 25 direction of the motion - - - - Hornbeam upon cast ironl-fibres par- 2 31 allel to motion - - - - -. Wild pear-tree upon cast iron-fibres 0436 23 34 parallel to the motion - o.436 - - Steel upon cast irh-on - - - - - -0.202 II 26 Steel upon brass - - - - - - - o. 52 8 39 Yellow copper upon cast iron - - - o. I89 Io 49 Ditto oak - o. 6I 7 31 4I O.6I7 31 41 Brass upon cast iron- - - - - - 0.2I7 I92 i5 Brass upon wrought iron, the fibres of ) the iron being parallel to the mo- o. i6i 9 9 tion - - - - - - - - - -. Wrought iron upon brass -.72 9 46 Brass upon brass - - - - - - - 0.20 II 22 * 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 experiments, coincided. The results were remarkably uniform. MECHANICS OF SOLIDS. 321 TABLE I.-continued. FRICTION OF FRICTION OF MIOTION. QUIESCENCE. SURFACES OF CONTACT. -0 O O E C % o a~ v o Black leather (curried) upon oakl - 0.265 14 5i' 0.74 360 3i' Ox hide (such as that used for soles ) and for the stuffing of pistons) upon. 0.52 27 29 o.605 3I I oak, rough - - - - - Ditto ditto ditto, smooth o.335 i8 3 0o.43 23 17 Leather as above, polished and har- 0.296 i6 30 dened by hammering - 3 - - - - Hempen girth, or pulley-band, (sangle de chanvre,) upon oak, the fibres of the wood and the direction of the.52 27 29 o.64 32 38 cord being parallel to the motion - J Hempen matting, woven with small + o. cords, ditto -. 0.32 17 45 -o.50 26 34 Old cordage 1 inch in diameter, dittot 0.52 27 29 0.79 38 19 Calcareous oolitic stone, used in build- ing, of a moderately hard quality, o.64 32 38 0.74 36 3i called stone of Jaumont-upon the I same stone -. J. Hard calcareous stone of 13rouck, of) a light gray color, susceptible of o.38 20o 49 0.70 35 o taking a fine polish, (the muschel- ( kalk,) moving upon the same stone J The soft stone mentioned above, upon o..65. 33. 2. 0.75 36 53 the hard - - -.... The hard stone mentioned above, up- - o.67 33. 5o 0.75 36 53 on the soft - - -. Common brick upon the stone of Jaumont - - - - - - - - - o.65 33 2 o.65 33 2 Oak upon ditto, the fibres of the wood being perpendicular to the surface o.38 20 49 o.63 32 i3 of the stone -... Wrought iron upon ditto, ditto - o.69 34 37 0.49 26 7 Common brick upon the stone of Brouck o. 60 30 58 o0.67 33 5o Oak as before (endwise) upon ditto - o.38 20 49 o.64 32 38 Iron, ditto ditto - - 0.24 i3 3o 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 coefflicienit of the friction of motion was equally apparent in the rough and the smooth skins. f 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 surfaces necessary to, and dependent upon, their motion upon one another. 12 322 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. FRICTION OF FRICTION OF MOTION. QUIESCENCE. SURFACES OF CONTACT. _ i |_''000.03 t ~ Oak upon oak, the fibres being paral- o. o8 6~ Io' o.390 2IO I9t lel to the motion - -. Ditto, the fibres of the moving body being perpendicular to the motion Oak upon elm, fibres parallel - - - o. i36 7 45 Elm upon oak, ditto -.II9 6 48 0.420 22 47 Beech upon oak, ditto - - - - - o. 33o0 8 i6 Elm upon elm, ditto - - - - o. I40 7 59 Wrought iron upon elm, ditto - o. i38 7 52 Ditto upon wrought iron, ditto - 0 o. 77 Io 3 Ditto upon cast iron, ditto - - - - - o. I I8 6 44 Cast iron upon wrought iron, ditto - o. I43 8 9 Wrought iron upon brass, ditto - o. 6o 9 6 Brass upon wrought iron - - - o. I66 9 26 Cast iron upon oak, ditto - - - 0 o07 6 7 o. I 00 5 43 Ditto upon elm, ditto, the unguent o.i25 7 8 being tallow - - - - - - - Ditto, ditto, the unguent being hog's o.37 7 49 lard and black lead -. I377 4 Elm upon cast iron, fibres parallel - o.I35 7 42 o.o98 5 36 Cast iron upon cast iron - - - -o. 44 8 I2 Ditto upon brass - - - - - - - o. I32 7 32 Brass upon cast iron - o. I07 6 7 Ditto upon brass - - - o.I34 7 38.o.i64 9 I9 Copper upon oak - - o. Ioo 5 43 Yellow copper upon cast iron - - o. I5 6 34 Leather (ox hide) well tanned upon 0.229 cast iron, wetted - - - - - - 5 Ditto upon brass, wetted -- 0.244 i3 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 MECHIANICS OF SOLIDS. 823 appears to be the true explanation of the difference between his results and those of Coulomb. He conceives, cause of the that in the experiments of this celebrated engineer, the discr.epaolcy requisite precautions had not been taken to exclude un- res.lts of Alorin and of Coulomb; guents from the surfaces of contact. The slightest unctuosity, 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. Thus, for instance, surfaces of oak having been rubbed example with hard dry soap, and then thoroughly wiped, so as to itlliste; show no traces whatever of the unguent, were found by its presence to have lost Zds of their friction, the coefficient having passed from 0.478 to 0.164. This effect of the unguent upon the friction of the effectof f.ictiou,pon surfaces surfaces may be traced to the fact, that their motion upon uithouct 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. Instead of a new surface of contact being continually presented 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. FRICTION FRICTION OF OF MOTION. QUIESCENCE. SURFACES OF CONTACT. m m UNGUENTS. O a O Oak upon oak, fibres parallel o. I64 o. 440 Dry soap. Ditto ditto - - - - 0.075 o. I64 Tallow. Ditto ditto - - - - 0.067 - - Hogs' lard. Ditto, fibres perpendicular o. o83 o. 254 Tallow. Ditto ditto - - - - 0.072 Hogs' lard. Ditto ditto - - - o. 250 - - Water. Ditto upon elm, fibres parallel o. i36 - Dry soap. Ditto ditto - - - - 0.073 0. I78 Tallow. Ditto ditto - - - - o. o66 - Hogs' lard. Ditto upon cast iron, ditto - o. o8o - - Tallow. Ditto upon wrought iron, ditto o. o98 - - Tallow. Beech upon oak, ditto - - - o. o55 - - Tallow. Elm upon oak, ditto - -. 37 0.411 Dry soap. Ditto ditto - - - - 0.070 0o.42 Tallow. Ditto ditto - - - - o. o60o - Hogs' lard. Ditto upon elm, ditto - - o0.39 0.2I7 Dry soap. Ditto upon cast iron, ditto - o.o66 Tallow. ( Greased, and Wrought iron upon oak, ditto o. 256 o.649 s Saturated with water. Ditto ditto ditto - 0.214 - Dry soap. Ditto ditto ditto - o.o85 o. I08 Tallow. Ditto upon elm, ditto - 0.078 - - Tallow. Ditto ditto ditto - 0.076 - - Hogs' lard. Ditto ditto ditto - o. o55 - - Olive oil. Ditto upon cast iron, ditto - o. Io3 - - Tallow. Ditto ditto ditto - 0.076 - - Hogs' lard. Ditto ditto ditto - o. o66 o.00oo Olive oil. Ditto upon wrought iron, ditto 0.082 - Tallow. Ditto ditto ditto - 0o.o8 Hogs' lard. Ditto ditto ditto - 0.070 0.11 5 Olive oil. Wrought iron upon brass, to.io3 - - Tallow. fibres parallel- - - - Ditto ditto ditto - 0.075 - - Hogs' lard. Ditto ditto ditto - 0.078 - - Olive oil. Cast iron upon oak, ditto - 0. 89 - Dry soap. ( Greased, and Ditto ditto ditto - 0.2I8 o.646 saturated with water. Ditto ditto ditto - 0.078 O.100 Tallow. Ditto ditto ditto - [ 0.075 - - Hogs' lard. Ditto ditto ditto - 0.o75. Ioo00 Olive oil. Ditto upon elm, ditto - o0.077 - - Tallow. MECHANICS OF SOLIDS. 325 TABLE III. -Continued. FRICTION FRICTION OF OF MOTION. QUIESCENCE. SURFACES OF CONTACT. UNGUENTS. o o Cast iron upon elm-fibres. o6.oI - Olive oil. parallel - - Ditto ditto ditto - -.0 -- Hogs' lard and Ditto ditto ditto - 0.09 plumbago. Ditto, ditto upon wrought iron - - 0.o00 Tallow. Cast iron upon cast iron - - o.3i4 - - Water. Ditto ditto - - - - o.97 - - Soap. Ditto ditto - - o. I 0 o. I 000 Tallow. Ditto ditto - - - - 0.070 o. Ioo Hogs' lard. Ditto ditto -- o. o64 -- Olive oil. Ditto ditto - - - - o. 055 - La d and plumbago. Ditto upon brass - o. Io3 - - Tallow. Ditto ditto - - - - 0.075 - Hogs' lard. Ditto ditto - - - - 0.078 - Olive oil. Copper upon oak, fibres parallel o. o69 o. IOO Tallow. Yellow copper upon cast iron 0.072 o. io3 Tallow. Ditto ditto - - - - o. o68 Hogs' lard. Ditto ditto - - - - o. o66 - Olive oil. Brass upon cast iron - - - o. o86 o. Io6 Tallow. Ditto ditto - - 0.077 - Olive oil. Ditto upon wrought iron - o. o8I - - Tallow. Ditto ditto - - - - o. o89 - Lumbago Ditto ditto - - - - 0.072 - - Olive oil. Ditto upon brass - - - - o. o58 - - Olive oil. Steel upon cast iron - - - o. o5 o. Io8 Tallow. Ditto ditto - - - - o. o8 - - Hogs' larld. Ditto ditto - - - - 0.079 - - Olive oil. Ditto upon wrought iron - o. o93 Tallow. Ditto ditto - - - - 0.076 - - Hogs' lard. Ditto upon brass - - - - o. o56 - - Tallow. Ditto ditto - - - - o. o53 Olive oil. Lard and Ditto ditto - - - - o.067 pl'uLbago. Greased, and Tanned ox hide upon cast iron o.365 - - saturated with water. Ditto ditto - o. I59 - Tallow. Ditto ditto - - - - o. i33 0. 122 Olive oil. Ditto upon brass - - - -. 24I - - Tallow. Ditto ditto - - - -. I - 91 - Olive oil. Ditto upon oak - - - - 0.29 0.79 Water. Hempen fibres not twisted, moving upon oak, the fibres of the hemp being placed in (Greased, and a direction perpendicular to o. 332 0.869 saturated with the direction of the motion, water. and those of the oak parallel to it - -J NAT U R AL PHILOSOPHY. TABLE III.-continued. FRICTION FRICTION OF OF MOTION. QUIESCENCE. SURFACES OF CONTACT. E UNGUENTS.,, 3 The same as above, moving} o0.4 - Tallow. upon cast iron - - - - - Ditto ditto - - - - o. 53 - Olive oil. Soft calcareous stone of Jaunmont upon the same, with a layer of mortar, of sand, and 0.74 lime interposed, after firom 10 to 15 minutes' contact A comparison of the results enumerated in the above table leads to the following remarkable conclusion, easily ounlotionS in fixing itself in the memory, that with the unguents hogs' regatd to olive larcd and olive oil interposed in a continuous stratum between oil and lard; them, surfaces of wood on metal, wood on wood, metal on wood, and qnetal 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 40 and 4~ 35'. talliow not so well 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 mnetallic 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 resistance 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 Familiar illustrations of the existence of this force are illastralions of the force of furnished by the pertinacity with which sealing-wax, wa- adhesion; 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. The intensity of this force, arising as it does from the its intensity affinity of the elements of matter for each other, must vary depends upon the ofIthe elements ofextent of the with the number of attracting elements, and therefore with surface of contact; the extent of the surface of contact. This law is best verified, and the actual amount of adhesion between different substances determined, by means measured by the of a delicate spring-balance. For this spring balance; purpose, the surfaces of solids are reduced to polished planes, and pressed together to Fig. 215. 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 area of the surface divided by the effort necessary to produce the separation gives a constant ratio. mode of Thus, let X denote the area of the surfaces opelation; 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 A S = a, or, A - ca. S. The constant a is called the unit or coefficient of adhesion, coefficient of and obviously expresses the value of adhesion on each adhesion; unit of surface, for making S= 1, 328 NATURAL PHILOSOPHY. we have A - a. adhesion between To find the adhesion between solids and liquids, sussolids and pend the solid from the balance, with its polished surface liquids; downward and in a horizontal position; note the weight of the solid Fig. 216 then bring it in contact with the horizontal surface of the fluid and note the indication of the balance when the separation takes place, on drawing the balance up; the difference mode of between this indication and that of amoutinaty the weight will give the adhesion; case; and this divided by the extent of surface, will give, as before, the coefficient a. But in this experiment two opposite conditions must be carefully noted, else the cohesion of the elements 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 eltmentsd for contrary, the solid on leaving the fluid be perfectly dry, those of solids; the elements of the fluid will attract each other more powerfully than they will those of the solid, and a will denote the unit of adhesion of the solid for the liquid. diversity il the It is easy to multiply instances of this diversity in the action ofbodies action of solids and fluids upon each other. A drop of in this respect; 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 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 diversity; 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. Solids which become wet on being immersed in a fluid, effect of covering lose this property if covered with any matter not similarly surfaces with 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 inl the lard or tallow, and the fluid will flow clear of the tumbler. flow of water from a tumbler; 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 poured from of the outer surface; but the contrary will happen if a tin different kinds of vessel be used. 330 NATURAL PHILOSOPHY. The adhesion of solids is appctrently increased by introeffect of ducing a liquid between them. The fluid fills up the exinterposing a fluid between isting inequalities of the surfaces, and thus, by increasing surfaces in the number of points of contact, increases the adhesion by contact; 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 It is difficult to ascertain the precise value of the force find the adhesion beteend the of adhesion between the rubbing surfaces of machinery, rubbing surfaces apart firom that of friction. But this is attended with little of m~chinery; of machinery; practical inconvenience, as long as a machine is in motion. Thle 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 adhethis adhesion sion of unguents to the surfaces of contact, and the oppoinay be disregarded; 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 conand the lilre. siderations rise into importance, and cannot be neglected. Friction ona ~ 214.-Let any body M; rest with one of its faces in plane; contact with the inclined plane A. Denote its weight by W, and suppose it to be solicited by a force F in the direction G Q, making with the inclined plane the angle Q G y', which denote by p. Denote the inclination BA C0 of the plane to the horizon by a. Resolve the weight W= G G' into two components, Gp and GI,', one perpendicular and the other parallel to the plane. The angle G' Gp being equal to the angle BA C, the first of normal these components will be, component of the weight; WF. cos ca MEC)HANICS OF SOLIDS. 331 Fig. 217. and the second, parallel WV. Sill a. component of the weight; In like manner, resolve the force F G Q, into two components G q and G q', the first normal and the second p'allel to the plane. The first of these will be, normal F. sin p; component of the force; anad the second its parallel F_-. COS (. component; The total pressure upon the plane will be WV. cos -.sin; pressure upon the plane; and the friction thence arising f (TV. cos M - F. sin p); corresponding in which f denotes the coefficient of friction. The force which solicits the body in the direction of the plane 332 NATURAL PHILOSOPHY. will be, whole force in direction of the F. cos p - W. sin ct. plane; This will tend to accelerate the body; the friction will tend to retard it. When they are in equilibrio, the body will dither have a uniform motion or be just on the eve of motion; which condition will therefore be expressed by F. cos - WVsin = f(TVcos C - F. sinl ); whence force necessary to hold the body in equilibrio, or to = (f cos r + sin 102 cos + -f sin q~ keep it in Cos p + f. SilL. uniform motion up the plane; iHere 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 G Q making with the plane A B the angle Fig. 218. Q G B equal to p; from Q G lay off the distance b equal to unity, and e draw b c perpendicular to find under to A B; then will what angle to the plane this force may be applied G = cos P, to greatest advantage; be = sin p. Take the distance G e equal to f, and we have ed = fsin p. MECHANICS OF SOLIDS. 333 Make G h equal to e d, and there will result value of the h = 0cos (p + f. sin p, denominator; which is the value of the denominator in Eq. (102). Draw G k perpendicular to G Q, and erect at h a perpendicular to A B, then, because the angle k G h is the complement of B G Q = p, will kh = Gh cot p; or, substituting the value of G h, as given above, kh = f. sin. cot =fcos. Join Ic and b, and it will be obvious that he is the projection of the line kb on AB, and that this projection will be the greatest possible when k b is parallel to A B; that is, when k h and b c are equal; which condition is expressed by the equation, fcosg = sinp, or sin e, = tan p;the value of the f = = tan; tangent of this COS qP angle; that is to say, the power will be alpplied to the greatest ad- conclusion; vantage, when its direction makes with the inclined _plane an anzge 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 fvalue of the force F_ = when plane is cos p + f sin p horizontal; 334 NATURAL PHILOSOPHY. when in Finally, if the body is to be retained in equilibrio on eqeilibrim o the eve of motion up the plane, the condition for this eve of' mnotion down the plane; purpose is given by Eq. (102) as it stands, but if the equilibrium 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 t (sin c - fcos r) for,; F =. (103); cos p - fsin (0) 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 imthe equilibrimn. portance of this will be perceived when we come uo treat of the screw. ~ 215.-The inclined plane is one of the most useful machines employed in the arts, and facilitates the transQuantity of work portation of the heaviest burdens to considerable elevaon the inclined pla ine; 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- Fig. 219. 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- A c king the angle qp equal MECHANICS OF SOLIDS. 335 to zero, that is by F = W(sin a + f cos ci). its value; Multiplying both members by AB, the distance through which F is exerted, we have, F X A B W[A sin a + f. A B cos]; which reduces to F x AB Y W. B C ~ f. TVW. A C..(104). tsquanltity of 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 performed if raised vertically through the same height, in- this vane creased by the quantity of work which the friction due to expressed in 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, f would be zero, and we should have value when the F. A B - W. B C. body is rolled; 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 plane; 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 prlane is equally true of a curved surface, 336 NATURAL PHILOSOPHY. such as that of a all equally true of common road or rail- Fig. 220. illclilled curved L liurfaces; road over an undula- l surfaces; ting piece of ground. 6h B'For, portions of the road, as A b, b b', b' i' &c., may be taken so short as to differ in- i e C sensibly from a plane, in which case we shall have, by denoting the intensities of the forces on these several elementary planes by F', F", F"', &c. F' x Ab = W.bc + f. W.Ac, forces on elementary eF" x bb'Y = W. b'c' + f. W. b c', su rface;.F"' x b'b" = W. b"c" + f W. W.'c", &c., = &c., + &c. 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 total quantity of workon entire Q' = W [bc + b'c' " c" + &c.] + f W [A c + b c' surface; + b' c" + &c.]; and supposing the burden to reach the highest point L, we shall have be + b'c' + b"c" + &c. = LM A c + be' + b' c" + &c. = AM; which, in the above equation, give MECHEANICS OF SOLIDS. 337 Q = WV. L + f. W. A.. (105). quantity of work in the ascent; 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, Q" = - W. LB' + f. W. B' B.. (106); dqsettyintlle 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'+ " = W [ML - LB'] + f W [A BB'], or quantity in the Q -- X B C + -. W. A.. (107). 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 quantityofwork though the load had been transported from A to B along the same as though the path one continuous plane. Nothing is said here of the resist- had been ance of the atmosphere, which, like the friction, would be straight. a cause of opposition to the motion. ~ 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 PA 1,1 resting ig. 221. the tension ofa upon any surface of -- cord arising from which A is the high- - its own weight; — 7 est point, and consider 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 i the question will consist 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 A F. Designate by W, the weight of a unit of length of the cord; then considering the element whose length is MlN, its weight will be weight of a given portion; W. MN. Through the centre of gravity 0 of this element, draw the vertical 0 G to represent this weight, which resolve into two components G Q and Q 0, 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 will determine its tension. Draw MAlN' perpendicular andNN' parallel to the horizon; then will the triangles - Q 0 and iNNV' be similar, both being right-angled this weight triangles, and the angle Q G 0 of the one, equal to the rcomponents; angle HNN' of the other, because the side G Q is perpendicular to i N and 0 G to NN'; hence the proportion, QO: 0:: MN': MN; MECHANICS OF SOLIDS. 339 whence O a X 21 N' component of Q 0 -- weight parallel M_/iN to the cord; Denote the tension by t, which will be equal to Q 0; 0 G represents the weight, equal to TV X AVN; and projecting the points A, M, Nt F, P, upon the vertical by the horizontal lines A a, Mm, Nn, Ff and Pp, we have TMN' equal to m n, and the last equation becomes, W1 X MN x m n value of the t MN W X m, f. 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 M1.I: Now the length A F 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 cord point ismneasured below it. Thus, if F be the end of the cord, the ten- the veltical sion at A will be measured by the weight of a portion of rojection ofball the cord equal to af, provided no motion take place. In like manner, the tension at A, arising from the weight of A P, 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 Fig. 222. a force equal to W(af - ap). If the ends of the cord be upon the an endless cord is same level, or if the cord be endless, it in equilibrio. will be in equilibrio. 4o t NATURAL PIIILOSOPHY. ~ 217.-We shall now take into.consideration the fricF':-iction of a cord tion of a cord when sliding around ally body, say a fixed slidingl around a i lind ica cylindrical beam in a horizontal position. Let the cord support Fig. 223. at one end a weight W; and be subjected to the action of a force F applied at i; - the other end. If the force communi- t o cate motion, it must not only raise the weight W, but must also overcome the friction between the cord and solid. If c:onstruction of the surface were perfectly polished, the friction would be notation; 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, 4, t2, t3, &c., into an indefinite number of very small and equal parts, and draw through the points of division, t4, t2, t,3 &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 t2, t2 t3, &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 b b'; this tension must overcome the weight WT and the friction on the arc t a, comprised between the points of contact. to find the Denote by p the pressure exerted by this element upon tension on a the cylinder, and by f the coefficient of friction, then single lement ofill the cord; will .MECllHANICS OF SOLIDS. 34] 4 = 1V + fr. its value; To find the pressure, we will still disregard the weight of the cord, and remark that the two tangents a b and b t are equal. Moreover, if we construct the rhombus a t4 b, and consider a b as proportional to the weight W; this same side will represent the tension of the cord from a to 4t. The diagonal b mn, will be normal to the chord a 4, and therefore to the surface of the cylinder, and being the to find thel orn.a, resultant of the tensions at a and 4 will be the pressure pres'sre risingt from the tensi(ss; arising from the tension, and consequently equal to -p. The triangles a 0 q and m a b are similar, because they are both isosceles, and the angle 0 of the one is equal to m a b of the other; hence mb: at,': ab: Oa; m b represents the pressure 2p; a t may be taken equal to the arc of which it is chord, which denote by s; a b represents the weight W; and Oa is the radius of the cylinder, which denote by R, and the proportion may be written s:: R; whence S. W value of this = R; normal pressure,; and this, substituted in the value of 4, gives 4= W(= + a ~j. value of the Denoting by t2, the tension along the third tangent telensionlefirst b, and at the third point of division ta, this tension nustelement nearest 8" t, and at the third point of division t2, this tension must the reitance; 342 NATURAL PHILOSOPHY. overcome the tension Fig. 223. t and friction produced by the elemen- 17 io hod tellsion on tary are t2 4, equal in / next element in g /, order; length to a X, or s. / In a word, t2 will be / i circumstanced in re-': spect to t1 as t1 was in regard to'W. Hence ta = /a 1 +; -{;. value; = +1( ) and if t,t t5, t, ~. t be the tensions on the consecutive tangents, and at the points t3, t45,... t, in order around the beam, we shall have t- = t2(1 +X2), values for the suzccessive A iensions ir older = t +, i.rorunld the beanm; t3 = k-(I + ). Multiplying these equations together and dividing out the common factor, we have value of the tension on the tn = v( 1 + -) last element of \ R contact; 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 the power and the resistance; Developing this by the rules for the binomial theorem, we have fTS + +n( -- 1) f2 ithis value = W[l + n -R + l. 2 RB2 developed; + (n 12- ) (n - 2).f3+& It must be remembered that s was taken indefinitely small, and therefore for any definite extent of contac: 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 rF= [1+ + 2. RRs + C.]3 3 ~ under a different R 2. R2 + Y 3 R form; but n s is equal to the entire are enveloped. Denote this by iS and the above becomes F= W [1 + S + f2. 2 + 2. s + &c.]: the quantity within the brackets is the development of the fs function e A; whence f s final relation F TV x e -P (108), between the F= Wx e.... (108), kpower and the resistance; in which e = 2.71825, the base of the Nap. system of logarithms. 344 NATURAL PHILOSOPHY. example to Suppose the cord to be wound around the cylinder illustrate; three times, and f = -, then will S = 3 n.2R = 6 X 3.1416. R = 18.849 R, and F = ] X e X 18s.849 = W x (2.71825)62832; or F = TV. 535.3; that is to say, one man at the end W could resist the combined 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, importance of by a few turns of her hawser around a dock-post, is friiction; 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 perits absolute form no work; nor, indeed, could any animal walk or necessity. even stand with safety, if they were deprived of the aid of this principle. MECHANICS OF SOLIDS. 345 XVII. 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. Such, for example, is the case Fig. 224. description and with the Wedge, which consists A B use; of an acute right triangular prism A B Ca usually employed in the operation of separating and splitting. The acute dihedral angle A Cb, is called the edge; the opposite plane face A b, the back; and the planes definitions; Ac and OGb, which terminate in the edge, the faces. The more common application of the c wedge consists in driving it, by a blow upon its back, into any substance which we wish to split or divide into parts, in such manner that after each common advance it shall be supported against the faces of the application oftthe wedge; opening till the work is accomplished. ~ 219. —The blow by which the wedge is driven forward 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 be perpendicular ther to practice, we will suppose the wedge to be isosceles. to the back; 346 NATURAL PHILOSOPHY. to find the The wedge A CB being inserted in the opening a hb, resultant of the and in contact with its jaws at a and b, we know that the reactions on the faces; resistance of the latter will be perpendicular to the faces of the wedge. Through the points a Fig. 225. and b, draw the lines aq and bp A B normal to the faces A C and B C; from their point of intersection 0, lay off the distances m 0 q and Op equal, respectively, 0 to the resistances at a and b. Denote the first by Q, and the second by P. Completing the constructionand parallelogram Oqm p, Om will notation; represent the resultant of the resistances Q and P. Denote this resultant by R', and the angle A GB, of the wedge, by I, which, in the quadrilateral a Ob will be equal to the supplement of the angle a O b = p 0 q, the angle made by the directions of Q and P. From the parallelogram of forces we have, R'2 = p' + Q2 + 2 P Q cosp Oq = P + Q2 - 2 P Q cos i; or value of the 1 -R p2 + f2 - 2P(Qcos resultant; The resistance Q will produce a friction on the face A C equal to f Q, and the resistance P will produce on the face to find resultant B C, 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 C2p' to represent their intensities, and complete the parallelogram Cq' O'_p'; C O' will repre MECHANICS OF SOLIDS. 347 sent the resultant of the frictions. Denote this by IR", and we have, from the parallelogram of forces, R',-2 f2Q2 +f2p2 + 2f2PQcos0; or value of the R = f /P2 + 2 + 2 P Q cos 8. resultant of the firictions; The wedge being isosceles, the resistances P and Q will be equal, their directions being normal to the faces will intersect on the line CD, which bisects the angle 0=; and their resultant will coincide with this line. In like manner the wedge being the frictions will be equal, and their resultant will coincide isosceles; with the same line. Making Q and P equal, we have, from the above equation, R' = P 2 (1 - cos ), these values lresult; R"f= P I/2(1 + cos d). But I - cos O = 2 sin2 ~ 0, 1 + cos 0 = 2 cos2 I; whence we obtain, by substituting and reducing, R' = 2 P. sin 8, or these; R" = 2f. P. cos0 8; and further, AB sin 28 =2 A c cilrcular functions in terms of elements of the CD- w edge; cos 0 = A' 348 NATURAL PIHILOSOPHY. therefore,' AB R -- P.AC' final value of these resultants; CD R"= 2f.P CD. Denote by F the intensity of the blow on the back of the wedge. If this blow be just sufficient to produce an equilibrium bordering on motion forward, call it F'; the friction will oppose it, and we must have, value of the blow when the wedge A B CD is on the eve of F"= R' + " = P A C + 2f P (109). moving forward; if, on the contrary, the blow be just suficient to prevent the wedge from flying back, call it F";, the friction will aid it, and we must have, value, when on A B C.D the eve of moving =1 _ _ P 2C. P. e ( 10). back;AC The wedge will not move under the action of any force 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, limits within which the blow C D may vary to F' - F = 4 fP () produce no a C motion; which becomes greater and greater, in proportion as CD and A C become more nearly equal; that is to say, in proportion as the wedge becomes more and more acute. The ordinary mode of employing the wedge requires that it shall retain of itself whatever position it may be driven to. This makes it necessary that, Eq. (110), MECHANICS OF SOLIDS. 349 A B 2 CD o A B CD conditions that P' = — P. or P < 2f. P the wedge may A C A C' A C AC' retain theplace to which it is driven; or, omitting the common factors and dividing both members of the equation and inequality by 2 CD, AB r AB C D-J U or CD ' the radius of the wheels. Ca Resolve, by the parallelogram of forces, the pressure upon the larger trunnion 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 f Nandf'N'; and the quantity of work consumed by this friction would be work of friction when wheels (fN + 1 N') RI s,; cannot turn; in which R, denotes the radius of the larger trunnion, and s, 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 friction on the r Tt smaller Nf. and N'f' ~ trunnions; 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 work offriction; (Nf + A'f' ), s,; ~ _'- Rlf' Tj8 MECHANICS OF SOLIDS. 379 a quantity obviously much less than that obtained comparison of above. results; 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 r work of friction, + (N + N'). R s R.S when whenhels PR~~~~~ ~~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 f 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, machine' which will be referred to hereafter. XVIII. STIFFNESS OF CORDAGE. ~ 229.-Let us now con- Fig. 244. IResistance from sider a wheel turning freely stiffness of cordage; about an axle or trunnion, and having in its circumference a groove to receive a cord or rope. A weight W;, being suspended fromE one end of the rope while E 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, 380 NATURAL PHIILOSOPHY. 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. measure of the The measure of the resist- Fig. 244 rigidity of cordage; ance due to the rigidity of 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 remains constant, while the other varies with the weight W; and is directly propor- tional to it; so that, designating the constant part by.; and the ratio of the variable part to the weight W by I, the measure will be given by the expression the value of E + I. W; this measure; 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 vK+ ~. Coulomb also found that on changing the wheel, the stiffstiffness on a ness varied in the inverse ratio of its diameter; so that if wheel whose diameter is E J. W unity; be the measure of the stiffness for a wheel of one foot diameter, then will stiffness on a K + 1. W wheel of any diameter; 2R be the measure when the wheel has a diameter of 2 R. Ai MECHANICS OF SOLIDS. 381 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 tfind theal foot diameter, and subjected to a tension arising from a measure of unit of weight, would, therefore, enable us to find the stiff- Stiffness; ness answering to any other wheel and weight whatever. But as it would be impossible to anticipate all the different 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 necessary 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. d- d2 Do W that * (127). thatfonew 2R white rope; Half worn white rope, dry or moist. 3 K~+I..W S d= 2d. 2 (128). old whiterope; 2.R Tarred rope. S = n. (129). tarred rope; 2R Packthread. S — d. W (130). packthread; 2 R 382 NATURAL PHILOSOPHY. TABLES OF WVEIGHTS NECESSARY TO BEND DIFFERENT ROPES AROUND A WHEEL ONE FOOT IN DIAMETER. No. 1. WHITE ROPES —NEW AND DRY. Stiffess proportional to the square of the diameter. Diameter of rope Natural stiffness, Stiffness for load of in inches. or value of K. 1 lb., orl value of L Squares of the ratios fo new whitbs. lbs. of diameter, or valfor new white ropes, dry; 0.39 0.4024 0.0079877 ues of d2 o0.79 I. 6097 o. o31950I I.57 6.4389 0.1278019 igSquare 3.I5 25.7553 o.5I220Ig Rtiosd. Sq2. No. 2. WHITE ROPES-NEW AND MOISTENED WITH I.00 I.00 WATER. 1.20 I.44 Stifness proportional to square of diameter. I. 30 I. 69 1.40 1.96 Diameter of rope Natural stiffness, Stiffness for load of I. 50 2. 25 in inches. or value of K. 1 lb., or value of I. I.60 2.56 Ibs. lbs. I.70 2.89 for new white o.39 o.8o48 0.0079877.80 3.24 ropes, moist; 0.79 3.2I94 o. o350I I. 90 3.6I I.57 12.8772 0.12780I9 2.00 4.OO 3.15 5I.5III 0.5II20I9 No. 3. WHITE ROPES-HALF WORN AND DRY. Stifness proportional to the square root of the cutbe of the diameter. Diameter of rope Natural stiffness, Stiffness for load of in inches. or value of I. 1 lb., or value of A Square roots of the cubes of the ratios old white ropes, 0.39 0.40243 0.0079877 of diameters, or valdry; 0 0.79 I. 1 o3801 0.0525889 ues of d2-. 1. 57 3.2 844 0.0638794 3.15 9.1o150 0.1806573 Ratiosor Power 7 _j o. 3. or d 2. No. 4. WHIITE ROPES-HALF WORN AND.MOISTENED WITH WATER. I.00 1.000 I.0o ].I54 Stiffness proportional to the square root of the cube I O I.315 of the diameter. I.30 I.482 I.40 i.657 Diamet;erl of rope Natural stiffness, Stiffness for load of I.50 I.837 in inches. or value of K. 1 lb., or value of I.. 6o 2.024 lbs. lbs. 1.70 2.217 old white opes, 0o.39 o.8o48 0.0079877.8o 2.4I5 moisteIned; 1 0.79 2.276I o.o525889 1.90 2.619 I.57 6.4324 0.0638794 2.00 2.828 3.15 I8.2037 o. 806573 MECHANICS OF SOLIDS. 883 No. 5. TARRED ROPES. Sti'ness 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.] Weight of 1 foot in Natural stiffness, or Stiffness for load of No. of yarns. length of rope. value of K. 1 lb., or value of I. ibs. Ibs. - Ibs. for tarred ropes; 6 0.02I o. I534 o. oo8598 I5 0.0497 0.7664 o.oI98796 3o i.OI37 2.5297 0.04II799 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 of Table No. 1; 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 1.18 d.-.79 = 1.5 nearly; and from the table at the side, elements d2 = 2.25. obtained from the tables; From Table No. 1, opposite 0.79, we find K = 1.6097, I = 0.03195; which, together with the weight V = 882 lbs., and ft. 2 R = 1.64, substituted in Eq. (127), give 384 NATURAL PHILOSOPHY. lb. lb. result; 1. 2 6097 + 0.03195 X 882 lbs. _-164 = 40.817, which is the true resistance due to the stiffness of the rope in question. Ecxcample 2d. What is the resistance due to the stiffness of a white rope, half worn and moistened with water, example to having a diameter equal to 1.97 inches, wound about a illustrate Table wheel 0.82 of a foot in diameter, and loaded with a weight No. 4; of 2205 pounds? The tabular diameter in Table No. 4, next below 1.97, is 1.57, and hence 1.97 d = 1.57 = 1.3 nearly; the square root of the cube of which is, by the table at the side, data from the = 1.482. table; In Table No. 4 we find, opposite 1.57, K = 6.4324, I = 0.06387; ft. which values, together with IV= 2205 lbs., and 2 R= 0.82, in Eq. (128), give lbs. lb. 6.4324 + 0.06387 x 2205 lbs. result; = 1.482 x 0.82 266.109, 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 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 illustrate Table No. 5; the rope being about 0.6 of a pound? By referring to Table No 5, we find the tabular number of yarns next below 22 to be 15, and hence 22 n = -= 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 = 4212, and 2 R = 1.3, in Eq. (129), give 0.7664 + 0.019879 X 4212 lbs. S = 1.466 = 95.188. result; 1.3 Example 4th. Required the resistance due to the stiffness of a new white packthread, whose diameter is 0.196 example to inches, when moistened or wet with water, wound about a illstrate 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- 0390 = 0.5 nearly. In Table No. 2 we find, opposite 0.39, lb. K = 0.8048, data from Table No. 2; I = 0.00798; 25 386 NATURAL PHILOSOPHY. which, with i - 275, and 2 R = 0.5, we find, after substituting in Eq. (130), 0.8048 + 0.00798 x 275 tbs. ~result.~ = 0.5 2.999. Work dueto ~ 230. —The resistance just found is expressed in stiffnessof pounds, and is the amount of weight which would be cordage; necessary to bend any given rope around a vertical wheel, so that the Fig. 245. portion A E, between the first point of contact A, and the point E, where the rope is attached to the weight, shall be perfectly straight. The entire process of bending takes place at the bending takes this first or tangential place at the first point A; for, if motion s point of contact; be communicated to the wheel in the direction indicated by the arrowhead, 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 A', the working point of the force Swill have described in its own path describedby direction the distance A D. Denoting the arc described the working 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 have AD = Rs,; and representing the quantity of work of the force S, by MEOCHANICS OF SOLIDS. 387 L, we get L = S. Rs,; replacing S by its value in Eqs. (127) to (130), L R K + I. TV. (131); work of the L = s' 2, stiffness; 3 in which d, represents the quantity d2, dt, n, or d, in Eqs. (127), (128), (129), or (130), according to the nature of the rope. Exatmple. 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 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. WHEEL AND PULLEY. ~ 231.-A plane wheel, Fig. 246. free to turn about its trunnion or axle, support- A ed in a fixed box, may Wheel and be moved in either di- pulley; rection by two forces F and Q, which act in its 388 NATURAL PHILOSOPHY. plane, and tangent to its Fi 246. circumference at A and B. These forces, acting in the A same plane, perpendicular eqnilibrium of to the axle, and tending two fors cting to turn the wheel in oppoupon the circumference of site directions, will be in -F Q a wheel; equilibrio when the elementary 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 describe 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 whenbe equal; ever the forces are in equilibrio, they must also be equal. This supposes the wheel free to turn, without obstruction 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 tackcelnto the 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 V, we shall have relation of work of power, FRs, = s, QR + f N. r s,. (132); resistance, and friction; in whichf is the coefficient of friction at the axle or trunnion, and s, the arc described by a point at the unit's distance from the axis during motion. Dividing by R s,, we find relation of these r forces; Q + 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 arise; of N would but slightly affect the value of ], since only r the small fractional portion R of N is taken. We may, therefore, be content with an approximate value for N. To obtain this, we first to find the omit the consideration of resultant of the power and friction, which will make Fig. 247. resistance; f= O, in the above equa- > tion, which then reduces to F= 9. Denote by g the angle / AJB, which the two X? B forces F and Q make with each other; then, from the parallelogram of forces, will Q N VF= 2' + ~Q2 + 2 F Q. cos; itsvalue; and, because F and Q are, in this case, equal, N = Q 2 + 2 cos; orthis; but 2 + 2 cos q = 4 cos2; whence N = Q X 2 coS 1 p; and finally this; 390 NATURAL PiHILOSOPIIY. but joining A and B by a right line, as also H and C, we have the angle A MCl 0 equal to ~ %, and AD AD cos 2p - sin MCA A —R AC which, substituted above, gives value of resultant A B under a more V' Q convenient form; 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 rule; product by the radius of the wheel. This value of N; substituted in Eqs. (132) and (133), gives A -B quantity of work; FRs, = QRs, + f. rs,Q R (134), value of the F A +f r Q B 1.35); force; _F —\ -p the first of which will give the quantity of work Fig. 24m7 of the power, and the latter the relation of the ~ power F to the resistance 9, necessary to produce inferences; an equilibrium. The first shows that the work of... \ the power is equal to the \ work of the resistance, increased by that consumed A B by friction. We now come to the second case, viz.: that in which' r is not very small Q MECHANICS OF SOLIDS. 391 in comparison with R. And first we remark, that F is case in which always greater than Q, and that the resultant obtained trunnion isnotso small; under the hypothesis of F being equal to Q, is, therefore, too small. Calling Na this latter resultant, we have A B first N = Q.. (136); approximation to the resultant; and this value, substituted in Eq. (133) for N, gives r first F = Q + f N1 = 1.. (137). approximation to the power; Now if N, be too small, it is obvious that F1 will also be too small. But this value of F1 is greater than Q, and if we find the resultant of two forces each equal to F1, or make F, A B second N = F1 -N2... (138); approximation to R resultant; it is obvious that N2 will be too great, and so of the value second F = Q +. * f* N2 =. appr.oximation to the power; Thus tne 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, + F2 mean of the 2 approximations; or r. N, X N2 value of the F2 = +. f 2; power; 392 NATURAL PHILOSOPHY. and eliminating NV and N2, by means of Eqs. (136), (137), and (138), we find final value of R A ] AB); power; -a+ Q J 2 &R) and multiplying each member by Rs,, quantify of = FRsRs, +fr s,Q [i + r AB work; R f R conclusion. The first will determine the condition of the equilibrium, and the second the quantity of work. ~ 232.-The pulley is a small wheel having a groove Pulley; 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; Q the resistance Q. The pulley may turn either upon trunnions 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 same as that of a simple wheel, and to the discussion principlethe of ~ 231 we have but to add the consideration of the ae asthat of the wheel; 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 quantity of work of the latter must be equal to the work of the resistance Q, increased by that of the friction and stiffness 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 are described at the unit's distance from the axis by s,, we must have C.s Qis, + d, - K 4 IQ R's + f N. r. s,; quantity of work.ns2 = 1ts, + d, R f r. s, of power; in which d, denotes either d2, d 2, 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 FK +T.I.Q+ r value of the F Q + d, + R.Q + power; Make K+ IQ Q +td, 2ec 2R and the above becomes 394 NATURAL PHILOSOPHY. different form of r same;=, +f and replacing N by its value resultant of all Fig. 249. the forces except / 2 + Q,2 + 2 F Q, cos friction; in which g denotes the angle A MB, made by the branches of the rope not in contact with the pulley, and A B we get r FX= Q, +f +Q +2Q cos. Transposing Q,, squaring and solving the equation with reference to F. and we have 1 + (f Co )s G9 the most general Q, (141). value for the 11 power; (w) 1of ) Taking!-the upper of the double sign, because the motion takes place in the direction of; replacing Q, by its value, and calling the angle A CB, enveloped by the rope, 0, in'which case, cos p = - cos, we finally obtain K+IQ 1 -(f ) Cos 0 the same in QF R (142.) known terms; 1(f 2 (1-s 0) - ( R.|1 +. f'_ Z [ _ /....O MECHANICS OF SOLIDS. 395 Fig. 250. When the two branches of the rope are parallel, then will 0 = 180~; cos B = - 1; and the equation becomes, K IQ value when the F = -Q c1 2 Rt /).. (143). branches of the = (8 2R f r rope are parallel; If the rope be perfectly flexible, and the friction be zero, then will Ki= 0, 1= 0, f= 0, and F= Q; that is, the power wilt 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 R s,, and there will result r l _ R )( co general value for Q R s, J d, (K I Q)s, (144). the quantity of -'FRs= 214_ work of the + (-cosO) [2-( f )2(1+cos)] 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 neglected; desirable, however, in any case, to take it into account, it is easily done. For this purpose, find, by the parallelogram of forces, the resultant of the weight of the but it may boe 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. Required the quantity of Fig. 251. example for work necessary to illustration; raise 500 pounds of coal, through a ver- ( tical elevation of 50 feet, by means of a rope passing over a fixed pulley, in such a position that the conditions of the power F shall be approposition; plied in a horizontal direction; the pulley, which is of lignum-vitse, is 1.25 feet in diameter; the radius of its eye is 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. tabular elements; Here 8 = 900, and cos 0 = 0; in Table IV. ~ 225, f = 0.11; Table No. 3, ~ 221, d, = d= ( ) = (1.2)y nearly = 1.315; K 1.13801; I = 0.0525889; R= 0.625; r = 0.05; Rs, = 0.625 X s, = 50; whence numerical values 50 r of thedata; S = 0625 = 80 feet; Q = 500 lbs.; and f r - 0.0084. These data in Eq. (144) give quantity of work; FR s, (500 X 50 + 1.315 1.13801-0.0559 X 500 80) (1 2 0.0084 v2-(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; value without friction and stiffness; whence 26250.17 - 25000= 1250.17 is the loss due to stiffness of cordage and friction, which would be sufficient loss due to to raise 1250.17 pounds through 1 foot of altitude, or frtifness and 1250.17 125 7 = 25 pounds through the given height of 50 feet; 50 a result well calculated to impress us with the necessity of including these resistances in all estimates of work. 26250.17 26250.17 lbs. numerical value s, - 525 nearly. of the power. _R s. 50 ~ 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 perforated near the ends for the reception of the axle, and Fig. 252. whose shank terminates in a A hook to which the resist- Q ance W is attached. The pulley is, in this case, said to be moveable. Denote the notation; resistance to be overcome and put in motion, by W; the tension of the rope between the fixed hook and tangential point H by Q; let the other notation be the same as in the case of the fixed pulley. The quantity of work of F must be equal to that of the 398 NATURAL PHILOSOPHY. tension Q, increased by the work dlue to the stiffness of the rope and friction; that is, quantityofwork; FRS, = Q Rs, + d + QR, 1 + rf s,.. (145). 2 2 Dividing both members by R s,, valueofthe FK + IQ r power; Q + 2R R The pulley being supposed either on the verge Fig. 252. of rotary motion in the di- A rection of 1F, or rotatingQ uniformly, it is obvious that W will be equal and directly opposed to the result- ( 0 ant of F and Q; and that H e Q will be equal and. directly opposed to the resultant of to findthe tension F and W. This latter reof rope atthe sultant being found by the parallelogram of forces, Eq. / (31), and in its value that of F, in last equation, substituted for F, the force Q will become known in terms of W, Fig. 253. the friction, and stiffness of Q cordage; and this value of Q, being substituted in Eq. A 1B (145), will give the work in terms which are known. The method here indithe same cated is perfectly rigorous, found byio; but is somewhat long, and approximation; may be avoided by resort- w MECHIANICS 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 -'_ R. W. approximate - AB''value for tension; which, substituted for Q in Eq. (145), gives W'' Rs, A B FRs, = K +I.W.AB + d, 2.. (146); quantity of work; + r.f W.s, dividing by R s,, K+f.W.R F Y AB r valueofthe A Jr, 2 -~R + ~R. * ( ) ~power; If we suppose the stiffness of the rope and friction zero, there will result, 7) power, when F = W stiffness and A B friction are zero; or 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; Exavmple. Let the pulley be of cast iron and turn example; upon a wrought-iron axle, greased with tallow; the di 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 A B be equal to the diameter of the pulley. By reference to the proper tables, we find data from the 2 f = 0.07[ d, = dL= (4~ - (1.8)2 nearly = 3.24; tables; \0.79 K = 1.6097; I = 0.0319501; and from the given data, R = 0.65; r = 0.0225; A B 1.3; Rs, = 40; data of the example; 40 example; S - 5 61.538 nearly; and WTV = 3462; 0.65 which, substituted in Eq. (146), give 3462 X ()2 X 61.538 0.65 inumerical result; FRs, = 1.6097 + 0.031950 X 3462 X 1.5 = 71279.35; -+ 3.24 X X 61.538 + 0.07 X 0.0225 X 3462 X 61.538 same with with neither friction nor stiffness of cordage, the quantity neither stiffness of work would be simply nor friction; Rs, = 3462 X(065) 61.538 = 69239.5; 1.30 work ofstiffness 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 llufle is a Themuffle; collection of pulleys in two Fig. 254. separate blocks or frames. A One of these blocks is attached to a fixed point A, by which all of its pulleys become fixed, while the other block is attached to the resistance Q, and its pulleys thereby made moveable. A definition and rope is attached at one end / description; 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 is made to act upon each pulley of the combination. The power F is applied to the other end of the rope, arrangement of and the pulleys are so pro- the rope; portioned that the parts of the rope between them, when application of the stretched, are parallel. Now suppose the power F to pwerandve f 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 t1; the radius of this pulley by R1; that of its eye by r1; the coefficient of friction on the axle by/; the constant and coefficient of the stiffness notation; of cordage by K and 1 as before; then, denoting the tension of the rope between the last point of contact with the first moveable, and first point of contact with the first fixed pulley, by t2, the quantity of work of the tension t2 will, Eq. (145), be 26 402 NATURAL P HILOSOPHY. work of the + K t tension on first t2R1s, = R s + d, 2 Rs, +f(t+ t2) rs,; ascending 2 R branch; dividing by s, moment of this + it tension; t2R1 = t1 + d, R +f(t1 ~ t2)r.* (148). Again, denoting the tension of that part of the rope which passes from the first fixed to the second moveable pulley by t3; the radius of the first fixed pulley by R2, and that of its eye by r2, we shall, in like manner, have moment of tension on second K+ It descending 23 R = t2 R2 + f -( + (t2 t3)r2 A. (149). branch; And denoting the tensions, in order, by t4 and t5, this last being equal to F. we shall have moment of tension on second K ~ Ift3 ascending t4R3 = t3 A d(t3 +, r3 (150)7 branch; same on third - It descending tR t 4 = J4 (t4 + F) r4. (151); branch 2 4 so that we finally arrive at the force F, through the tensions 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 thcomponenesis of tance, t t2 + t =. (152); 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 approximation which is found, in practice, to be sufficiently accurate. If the friction and stiffness be supposed zero, for the method of moment, Eqs. (148) to (151) become approximation,; t2 R1 = t R1, t3 R2 = t2 R2, t4 R3 = t3 R3, friction and stiffness zero; from which it is apparent, dividing out the radii R1, R2, the tensions R3, &c., that t2= t1, = t2 4 t, F — t4; and hence, Eq. become equal; (152) becomes 4t = Q; resistance equal whence to tension on one l)ranch multiplied __ Q by the number of t~ pulleys; 4' the denominator 4 being the whole number of pulleys, moveable and fixed. Had there been n pulleys, then would Q general value for tl — n.-' the tension; With this approximate value of t1 we resort to Eqs. (148) to (151),, and find the values of t2, t3, t4, &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 404 NATURAL PHILOSOPHY. large. If we now suppose the true tensions to be proportional 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.: to find the true from the QI Q: approximate Ql tension; or, which is the same thing, multiply each of the tensions found by the constant ratio the product will be the true tensions, very nearly. The value of t4 thus found, substituted in Eq. (151), will give that of F. example to -Example. Let the radii Aq, R2, R3, and R4, be respectively iliustrate; 0.26, 0.39, 0.52, 0.65 feet; the radii r1 ='2= r3 = r4 of the 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, XK = 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 2400 lbs. value of first - 600. tension; 4 Dividing Eq. (148) by RI, it becomes, since d, = 1, t2 +K 1 f (t + t ). Substituting the value of R1, and the above value of tX, and regartding in the last term t2 as equal to ti, which we may do, because of the small coefficient -, f we find AMECHANICS OF SOLIDS. 405 600 1.6097 + 0.0319501 x 600 t2 - 2 X (0.26) = 628.39. value of econd tension; + 6 X 0.09 x (600 + 600) Again, dividing Eq. (149) by R2, and substituting this value of t2 and that of R12, we find lbs. approximate t3 = 673.59. value of third tension; Dividing Eq. (150) by R3, and substituting this value of t3, as well as that of R,, there will result lbs. approximate 44 -'709.82; value of fourth tension; whence 600 ] ++ 628.39 26= 21180; resultant of these Q1 2 + t 3 + t4- 2611 tensions; + 109.82 j and ratio of the Q 2400 0919 approximate to - 2- - = 0.919; the true 2611.80 resultant; which will give for the true values of t, = 0.919 x 600 = 551.400 t2= 0.919 x 628.39 = 577.490 t3 = 0.919 x 673.59 = 619.029 iruetension; t4 = 0.919 x 709.82 = 652.324 2400.243 406 NATURAL PHILOSOPHY'. The above value for t4 = 652.324, in Eq. (151), will give, after dividing by R4 and substituting its numerical value, 652.324 1.6097 + 0.03195 x 652.324 F= 2 x 0.65 0.06 + oX6-x 0.09 X (652.324 + F); and making in the last factor F = t4 = 652.324, we find numerical value lbs. lbs. lbs. lbs. oftlhepower; F = 652.324 + 17.270 + 10.831 = 680.425. Thus, without friction or stiffness of cordage, the intensity of F would be 600 lbs.; with both of these causes of 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 work absorbed proportional to its intensity, we see that to overcome fricby trictio, and stiffness of tion and stiffness of rope, in the example before us, the cordage. moter must expend nearly a seventh more work than if these sources of resistance did not exist. Wheel and axle; ~ 235. — Wzeel and Axle is a name given to a machine, 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 description; wheel; the resistance Q is applied to another rope, wound dof powel and in the opposite direction about the arbor, and also acts in resistance; a plane perpendicular to the axis of motion. The power is generally applied in the plane of the wheel, otherwise, 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 AMECHANICS OF SOLIDS. 407 nent would press the shoulder of the arbor against the effectofan face of the box, and increase the effect of friction by oblique application increasing its "lever arm." It may happen, however, that of the power; the particular object to be accomplished will sometimes make it inconvenient to process where satisfy this condition of power does not keeping. tie action of Fig. 255. act inplane of the power in the plane t. of the wheel, in which event, it will be easy to / find the pressure arising from the parallel com- Do - ponent of the power or resistance, and to compute the friction by the rules already given. A Supposing the power henthe power 7 acts in the plane and resistance to act in of the wheel; planes at right angles to the axis, we remark, that the plane of the wheel in which the power acts, and the plane perpendicular to the axis, through the direction of the resistance, will cut from the arbor equal circles. Through the point L, 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 the two equal and parallel forces Q and Q1 will be equal to their sum, will pass through ], 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 408 NATURAL PHILOSOPHY. forces which librium, if the machine be at rest, or its uniform motion, maintain the if at work, must therefore be maintained by the power 1, equilibrium or t b b motion uniform; the force Q2, the friction, and the stiffness of cordage. To this end, the resultant of X, Q2, and stiffness of cordage must intersect the axis. At the point of intersection, conceive this resultant to be replaced by its primitive components, and there will then act upon the axis the forces X, Q2, Q + Q1, and the resistance due to stiffness of cordage. Each of these forces being resolved into two pressure upon parallel components acting on the trunnions A and B, the trunnions; there will result two groups of forces, one applied to each trunnion. Denote the resultant of the group acting on friction on the the trunnion A by M, that of the group acting on the workunnions andts trunnion B by M', then will the frictions be respectively filf and f' M'; and, employing the usual notation, the quantities of work will be fMr s, and f' C1' r' s,, the radii of the trunnions, and their friction being unequal. 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, K2 f' QR' s,+frs,+f'M'r's,; but if the trunnions and boxes are supposed of the same size and material, quantityofwork; FRs, = Q2 R' s, +, + Q R', + (+')rs,. The quantity M2+ M', being the sum of the pressures friction of upon the trunnions, the last term shows that the friction is ftrunnionsame the same as though the resultant of all the forces were effect wherever applied; applied to a single trunnion in any arbitrary position, and, therefore, at the centre of the wheel. But this would reduce all the forces to the same plane, in which case Q would take the place of Q2, and Q9 and Q2 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, - R's, + N r s, (153); quantity ofworms; 21R and, dividing by Rs,, I 2+ IQ' fN. (15). value of the F dK Ij ~'f p (154). power; Now, Nbeing the resultant of all the forces of the system except friction, it is the resultant of I, Q, and d, 2+ Q. or, since Q and d, 2K Q act in the same direction, it is the resultant of F and Q + d, R Q To find V, we will pursue the method explained in ~ 231. Make K+ - IQ Pfind the resultant Q + d, 2 = Q (155); of all the forces but friction; then will F R' r = Q + r. (156). If we neglect the consideration of friction for a moment, by and find the resultant N1 of F and Q1, or of approximation; R' Q1 and Q1, we shall have, denoting the inclination of the power to the resistance by p, first N=/ Q2+ Q'-+2Q' 2 cos -Q, = + (-+2 cos.. (157); approximation ~ A Yi\ R\R for resultant; 410 NATURAL PHILOSOPHY. and this for XN in Eq. (156), gives first R approximation F = Q1 - +N1 = 1.. (158). for power; the first Now the value of N1 was too small for V; because we approximation sufficient; omitted the term fN -, in the value for F; and, hence, F1 is too small for F; but the deficiency is less and less, in proportion as the fraction f is smaller and smaller. In ordinary practice there will be but little difference between the true value of F and that given by Eq. (158). when it is not, In cases wherein r is considerable in comparison with a second approximation R, a further approximation will be necessary; and to acmust boe made; R' complish this, we remark, that F1 is greater than Q1 i, and Qi therefore less than F1,; and that if this latter be combined with F1, to obtain a second resultant N2, this last Fig. 256. will be too large, and when substituted in geometrical Eq. (156), for N, will indication,, A' give a value F2 for /"' X, which will also be too large. The mean of the two values of Fi and F2 will be the practical value of F. The value of N is given by the equation, second approximation N2 = F 1 2 cosp (159); for resultant; and second approximation to a' r value of the F2 = 1-R- + fNT2 -; power; R f MECHANICS OF SOLIDS. 411 whence F, + F2 QR''rVIV+ N2 (O mean of the F= Q1 +.. (160). two 2 W Z R 2 approximations; To find the quantity of work, multiply both members by Rs, replace Q1 by its value, and we have Q +iK+ IQ Rs~f, N+N2 FRs,=QR'Is,+d, +QR' s, +fr sl2.+ (161). quantityofwork; EIxample. 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 Fig. 257. diameter of the wheel is 4 feet; that of example to the axle, 1 foot; illustrate; that of the trunnion, which is of wrought iron, working in castiron boxes and lubricated with hogs' lard, 1.5 inches; that of the rope, which is white, half-worn, and dry, 1.5 inches; and the power acts in a horizontal direction. Here R = 2 feet; R' = 0.5 feet; r = 0.125 feet; dataofthe J3 0 7 15 X3 a question and f 0.07; di, =d= _ 0.79) ( 1.9)2 = 2.619; tables; K = 1.13801; I = 0.0525889; Q = 4000 lbs.; 80 R's, = 80 feet; s, 80 - 160 feet; and p = 90~, or cos p = 0. These data, substituted in Eq. (155), give 412 NATURAL PHILOSOPH Y. Ibs. lbs. lbs. 1.13801 + 0.0525889 X 4000 Q1 = 4000 + 2.619 1 = 4553.89; and this, in Eq. (157), making cos - = 0, and substituting 0.5 for -, its value - = 0.25 feet, we find ~~R 2 value of first N1 = 4553.89 Vi + (0.25)2 = 4694.04. resultant; This and the values of Q,, -,, and W, in Eq. (158), give F1 = 4553.89 X 0.25 + 0.07 X 4694.04 X 0.0625 = 1159.008; which, substituted with the values of R and cos P = 0, in Eq. (159), gives valueofsecond N2 = 1159.008 1 + (4)2 = 4778.68; resultant; hence, N = N, + N2 4694.04 + 4778.68 = 4736bs. N = =24736.36; 2 2 which, with the values already found for Q1, in Eq. (160), gives lbs. 0.125 lbs. vale of power; F = 4553.89 X 0.25 + 0.07 X 4736.36 = 1159.19. 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 lbs. ft. FRs, = 4000 X 80 + 44311.20 4- 663.07 = 364974.27. quantity ofwork; 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 friction and tion and stiffness of cordage, to raise more than a quarter stiffness of. of a ton through the given height. cordage. If, in Eq. (154), we make f = 0, and disregard the stiffness of cordage, we find RI F= Q ~. (162); 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. ~ 236. —Wheels are often so combined in machinery as Combination of to transmit the motion impressed upon some one of them, wheels; according to certain conditions, determined by the object motion to be accomplished. This is usually done by one or other transmitted by endless bands, of the following means, viz.: 1st. By endless ropes, bands, ropes, and or chains, passing around cylindrical rollers, called drums, chains; mounted upon arbors; 2d. By the natural contact of these by natural drums; 3d. By projections called teeth or leaves, accord- contact; and by teeth. 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. 414 NATURAL P HIILO S OPHY. ~ 237. —When two Fig. 258. Resistance due to wheels are connected d c stiffness of bands with each other by and ropes; with each other by means of an. endless AB ) o 1 band or rope d c b e, passing around the drums A and B,:r mounted upon the arbors of the wheels, a sufficient force F applied to one of them 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 the bands and iS sufficient to prevent the former from sliding over the drum; latter, and thus a resistance Q, applied to the second wheel, may be overcome. The motion of the drum B is obviously due to the difference of the tensions in the two branches d c and e b; and applying the power as indicated in the figure, the tension of d c must be greater than that of e b. Denoting the first of these by T and the latter by t, motion due to the force which moves the drum B will have an intensity difference of equal to T - t; and the quantity of its work must be tension; 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 R2; that of the wheel to which Q is applied by R"; that of its trunnion by r2; the arc described by the point at the unit's distance from the axis of motion by s2, &c., then will work of difference of (- t) R2s2 QR" s2 + fN2 r2s2. (163). tension; The action of the force F produces the difference of tension T- t, and its work must, therefore, be equal to that of T - t augmented by the work of friction on the trunnions of the arbor of the wheel to which F is applied. Denote the radius of this wheel by R', that of its drum by RI, that of its trunnion by r1, the are described at the MECHANICS OF SOLIDS. 415 unit's distance by s8, and we have F.R' s = (T- t) RI s1 + fN. r1s. (164). workofthe power; Adding these equations together, we get FR' s, + (T- t) R2 s2 = (T- t) R, s1+ Q R"s2 + fN, r 2+ fNr,, s; but because all parts of the band have the same velocity, the circumferences of the drums must move at the same rate; hence circumferences of X' 2 RI i?.the drum have the same velocity; which will reduce the above equation to F-R' s =Q RI' R 2 + fN2 82s2 + f, r,1 s, (165). workof the power; Whence we see that the work of F is equal to the work 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, N2 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. The difference T - t only exists while the system is in motion; when at rest, and the force does not act, this difference 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 7= 1T, + 11t and t T - H. (166); tensions; 416 NATURAL PtHILOSOPHY. from which T and t may be found when T, and II are known tension at rest The tension T1 is entirely arbitrary. It should be as arbitrary; small as possible, to produce the requisite friction between the band and the drums to avoid sliding during the motion, for if greater than this, it will only increase the pressure and, therefore, the friction on the trunnions, unnecessarily. In general, it will be sufficient if this friction shouldbe just be great enough to prevent sliding under the effect of Q, sufficient to pevent sliding; at the surface of the drum of the wheel to which Q is applied. But this effect, neglecting friction on the trunnions and stiffness of cordage, is Q -. 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 must be at least equal to this force to prevent sliding. The branches d.c and eb of the band are solicited respectively by the two forces T1 ~+ E, and T1 - H; and these substituted in Eq. (108), the first for F and the second for T;V we find, fs relation of the T f T H) e"2 two tensions; 1 I- = 1 e subtracting Ti - H from both members of this equation, and we have fs + f-(T - (T - H) (T - H) eR2 - (Ti - H); the first member reduces to 2 H; that is to say, to the difference 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 difference of (167 tensions; (167), Q.R t MECHANICS OF SOLIDS. 417 Rit" /s same in terms of Q = (T -- H) - 1). (168); the friction, &c.; 2 tension at rest from which two equations we may compute H and T., and therefore, Eq. (166), T and t; and, finally, the resultants N2 and N, by the rules for the composition of forces. 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 2 = 2 feet; R" = 0.5 feet; Q = 1000 lbs.; f - 0.265, (see Table I, ~ 212;) s = P 2, = 3.1416 R2; R"i 0.5 z~8. Q - - 1000 x 2 = 250; lbs. R2 2 t tbs.. half difference of H 2 = ~ Q 125; tensions; QA 250 Afs (2.7182818)0.265 X 31416 - 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 value found by = 0.4342942 X 0.83251 the aid of logarithms; = 0.361554 nearly; the natural number of which is 2.2991, whence 250 250 lbs. T -H H- t = 192.44. value; 2.2991 - 1. 1.2991 27 418 NATURAL PHILOSOPHY. Adding 2 H-= 250 lbs., we have Ibs. greater tension; T1 + 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 T2= T+ t= 634.88 lbs., which being combined with Q = to find the 1000, according to the principles of the composition of esultant; 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 valueofpower; F = Q Q +fN 8 1 - - Rs but velocity of the circumferences R2s2 = 4s1; equal; whence 8! -'~2 S1 R2 and by substituting above, final value of R p I r2Rr power when the F Q/* +" ~f N2. - -' -+ fN 1V i motion begins in * 2 2 R P its direction; 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 without friction; F- Q P 1 R'. R MECIANICS 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 withouteelriction;xle the band about the axle of the wheel to which the power Fig. 259. F is applied, and the wheel of that to whose axle Q is applied, then will R1 be the radius of the first axle, and FZ R, that of the see- Q ond wheel, and the preceding equation gives us this rule, viz.: 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 to the resistance. radii of the axles to that of the radii of the wheels, in the case of an equilibrium or uniform motion. ~ 238.-In the preceding discussion, no mention is made Rigidity of of the resistance arising fromi the stiffness of cordage. bands my be neglected; 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 endless rope, the op- Fig. 260. position to motion takes place at the ~~~~~~~~~~~points where the 6 < 0 ( ( wrigidity of ropes; points where the rope bends in passing on to the drums, and not at all at the points where a it leaves the latter 420 NATURAL PHILOSOPHY. and becomes straight. Thus at the point a, the resistance is value of the resistance at one K +- I. Q point; 2 R" at the point b it is at another; 2 RIa and at the point d it is K'~+ IT at another; 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. rigidity of chains; If the connection be made by an endless chain, each link, as it turns in the next one in order, may be regarded Fig. 261. ~7[/ e b X.ich link a as a trunnion revolving in its box; and each, as it comes box;nion inits to be applied to the drum, revolves about the next one through an angle E'HE, equal to D CD', 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. Denoting by r the radius of the inner circular arc in notation, &c.; which the end of each link is shaped, s2 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 tensions on the two branches of the chain,; then will the work of friction among the links at the points b and frespectively, (figure before the last,) be work of friction f Tr s2, and ft r s; among the links, at one set of points; and denoting by s1 the arc described by a point at the distance of unity from the axis of the drum A, the work of friction at the points e and d will be, respectively, f Tr sl, and ft r s,; the same for another set; and the whole amount of this kind of work will be f r (T + t) (s2 + s1). whole work of this friction; 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 velocity of S2-2 s2 = 1; circumference of drums equal; in which R,2 and R, are respectively the radii of the drums B and A. From this relation we find R2 $1 - s2; which, substituted above, gives value of the work frs (7:/' J+ t) + 12) *4* (169) of friction amons / the links; 422 NATURAL PHILOSOPHY. Exampn2e. Let T and t have the values of the last example, (that of the strap,) and suppose r = 0.03, the chain gxample; of wrought iron, for which we find in the table of ~ 225, (assuming thatf 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 data; the expression (169) for a single revolution of the drum B, in which case s2= 2 X 3.1416 = 6.2832, become quantity of work lbs. lbs. of friction. 0.07 X 0.03 X 6.2832 (442.44 + 192.44) (1 + 4) = 41.88; 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; a force being impressed upon one Fig. 262. wheel, it cannot move without communicating motion:1 The teeth are usually curved, and so shaped as to have a common normal D1 D2, at conditions of their point of conconstruction of the teeth; tact n, where the action of one and 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 Fig. 263. conditions of a * 1 preserving a this normal does — preservng a constant normal not change, but re-, 4,..,at the point of %/-'-._', ~ -..contact mains stationary, ntact; and the point of z contact is always \ - on it. Wewillnot ".. stop to explain the constructions by which this is accomplished; it will be sufficient for our present purpose to be assured of its practicability, and that we may proceed on the supposition that it has been executed in the case under consideration. From the centres C1 and C2 of the wheels, let fall upon the normal D1 D2, the perpendiculars C D1 and C2 D2. The points D1 and D2 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 C1, 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 C? D2; or, because of the similar triangles C1 B D, and C2.B D2, inversely as the distances C1 B and C2B. The circles described about C1 and C2 as centres, with radii C B and C2 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 Q the resistance acting at the distance R from the axis of the wheel whose centre is C2. The effect of this resistance acting at D2, in the direction of the normal D, D,, will, from the principles of the wheel and axle, be Q1, given by the relation R resistance at the Q1 = Q - 2.. (169)'; distance of Ct2 Z)2 common normal; 424 NATURAL PHILOSOPHY. and this Q1 will be the pressure at the point m. Its friction will be value of the friction on the f Q1, teeth; acting in the direction ql q2, tangent to both teeth at their point of contact. The elementary quantity of work of this friction will be equal to its intensity, multiplied into Fig. 263. the elementary dis-.. tance by which the / rubbing points now I — at n, separate in. the direction of this / to obtain the tangent; which dis- Z quantity of work tance is obviously of this friction; equal to that by which the points ql and q2, the extremities of the perpendiculars let fall from C1 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 sI, and that described at the same distance from C2 by s2, the paths described by q1 and 2 will be, respectively, C, q1 X s,% and C, q X s2; and because the points q1 and q2 must move in the same direction when the tangent q2 q, passes between the centres C, and C2, the elementary path of friction will be equal to the difference of these paths, and its elementary quantity of work will equal the value of this f Q1 [12 q2 X S2 - Cl q X si] Designating the radii of the primitive circles whose centres are C, and C2 by R1 and R2, respectively, we have, because of the equal velocities of the circumferences of these circles, MECHANICS OF SOLIDS. 425 R1 s = R2 8s2; relation of the whence paths at unit's distance from the two centres; s2 -- z Sr Moreover, drawing through the point B the line z2 %z, parallel to the tangent q2 ql, and denoting the angle m B C1, which is the complement of the angle C, B zl, by p, and the distance m B by h, we find lever arms of the friction; 01q1 = RPcosp - h; these values of s2, C q2, and C6 ql, substituted in the expression for the elementary work of friction, give f Q1 h s,( R + ) work of friction; Denote by w 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; R1 s8, and hence oo RI s, = f Ql h sl (R + R2) or tangential force ___RI R_ at the - R(170). circumference of primitive circle; Represent the angle B C1 m by 0. In practice, the angle m B QC does not differ much from 90~, and we may take h = R1tan; 426 NATURAL PHILOSOP HY. and because 0 is generally very small, the tangent may be replaced by the arc, and h R1I; which, substituted above, gives another form for tangential force R+ (7 at primitive X -- Q1 +R X' * ) circumference; The value of 0 varies from a maximum to zero on one side of the line of the centres C1 C2, 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 O 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 81, and value of the angular distance the second by 82, lay of point of off the distance A E contact; =,1; erect at A the Fig. 264. perpendicular A G = 01; draw GE: then will the ordinates of this line which are parallel to A G repre- sent the different val- A B B ues of 0, and the area of the triangle EA G will be the sum of all the values of 0 between 01 and zero. Again, make EB = 02; erect at B the perpendicular B G"'= 2; draw G"'E: the area of the triangle EB G"' will be the sum of all values of 0 between zero and 02. Make altitude of a mean 12+ A 2 triangle; BO - 1 + - 01+ #,' MECHANICS OF SOLIDS. 427 complete the rectangle B 0', and draw A O; then will the construction; triangle A B 0 be equivalent to the sum of the triangles A E G and ]EB G "', and therefore equivalent to the sum of all values of 8 between 01 and 02, the mean of which is obviously the middle ordinate. xy- = B 0 + 2 _ (01+ 02)2 - 2 0= 2 _ 81 + 0 02 mean value of 2 (01 + 02) 2 (81 + 02) 2 01 + 02 angular distance of point of contact; Neglecting the last term as insignificant, 81 + 02 XY = Multiplying by ]R, we find that R1 (08 + 02) 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 substituting in Eq. (171), we find R,_ + 2 a a a\ tangentialforce 2R 2 2 = at f 2 2R2 + - 2 RI -at pimitive BRI2~~~~ 2 A ~ ~ ~ ~ I~~~~ circumference; Denote the number of teeth on the wheel whose centre is C1 by n,, and the number on the wheel whose centre is C2 by n2; then, because the teeth and intervals between them must be the same on each circumference, in order to work freely, distance from the 2 r R1 _ 2 ~ R2. place of first to a =- 2 - vthat of last point of contact; which, substituted above, gives ~- Qf + 1= =f.~ Q1 + n. Replacing Q by its value give n in E. (169)', and recollectReplacing Q, by its value given in Eq. (169)', and recollect 428 NATURAL PHILOSOPHY. ing that, within the limits supposed, C2I D2 becomes R2, we finally have final value of tangential force R n2 +n 7 which is = i. Q. 1 (172). equivalent to nj n2 friction; To find the quantity of work, multiply both members of this equation by R2 s2, which will give its quantity of R2S2 = frs QR 2 n, work; = fS yi - - yLU3). n2 nl example; Example. Required the work consumed in each revolution 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. data; Here, R = 0.8; Q = 1000 lbs.; s2= 2 x 3.1416; - = 3.1416; f= 0.152; n2 = 64; n1 = 192; and, therefore, abs. 64 - 192 work; wR s2 - 0.152 X 3.1416 X 6.2832 X 1000 X 0.8 4- 19 — 50; 64 X 192 that is to say, the quantity of work consumed in one revoresult. lution by friction on the teeth, in the case supposed, is sufficient to raise 50 pounds through a vertical distance of one foot. XX. THE SCREW. 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 ~ 240.-To form a clear idea of the figure of the screw Screw with and its mode of action, conceive a right cylinder a k, with square fillet; circular base, and a rectangle a b c m 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 rectangle to rotate uniformly about Fig. 265. the axis, and the rectangle itself to move also uniformly in the direction of that line; and let this twofold motion of rotation and of translation be so regulated, that in mode of one entire revolution of the plane, 1 generating; the rectangle- shall progress in the i — direction of the axis over a distance greater than the side a b, which is in the surface of the cylinder. 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 distances d' in', d" rI", &c., of the several points of the helix from this plane, are respectively proportional to the circu- properties of a lar arcs mnd', md", &c., into which the portions mm', m m", helix; &c. of the helix, between the origin and these points, are projected. The solid cylinder about which the fillet is wound, is called the newel of the screw; the distance min n m', 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 PHILOSOPHIY. 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 of the different axis of the screw, increases, therefore, from the newel to helices; the exterior surface of the fillet, the same helix preserving its inclination unchanged throughout. The screw is received into a hole in Fig. 266. a solid piece B of metal or wood, called the nut; a nut or burr. The surface of the hole Q through the nut is furnished with a wind- -- fillet of the nut; ing fillet of the same 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. From this arrangement it is obvious that when the nut relative motion of is stationary, and a rotary motion is communicated to the screw and nut, screw 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. In the first case, one entire revolution of the screw will carry it longitudinally through a distance equal to the helical interval, 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 1OF SOLIDS. 431 The resistance Q is applied either to the head of the screw, or to the nut, depending upon which is the moveable element; in either case it acts in the direction D C of the axis. The power F is applied at the extremity of a bar application of the G H connected with the screw or nut, and acts in a plane resistance and power; at right angles to the axis of the screw. Denote the perpendicular 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 revolution, supposing it to retain the same distance from the axis, be work of the F X 2 R -d; power in one revolution; and the quantity of work of the resistance will be work of the -l~ x h. resistance; The power F and resistance 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 F1, 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 Fi. helix; The resolution must, therefore, be made with reference to a normal at a helix midway between the newel and outer surface. This helix, like all others, is situated upon the surface of a cylinder of which the axis coincides with that of the screw. Denote the radius of this cylinder C mTv by r 432 NATURAL PHILOSOPHY. construction; Conceive a tangent plane to this cylinder at any point, as mnIV, and two cutting-planes normal to the axis, and at a Fig. 267. projection of intermediate Ic V B helix; distance from each other equal to a helical interval, and equally distant from mIv. If we now develop the portion of the cylindrical surface, included between the cuttingplanes, on the tangent plane, the surface of the cylinder Fig. 268. will become a rectan- X gle whose base A E development of is equal to 2 q r, and the intermediate whose altitude B is B, helix; whose atitude B s equal to h; and the It helix will become the diagonal A B. De- A __ note the length of the resolution of the helix A B by l. Then powerscndo draw the normal mIv L, and resolve Q and F1 as before resistance into components; stated. Since Q = in'I K is perpendicular to A E, and L mtv perpendicular to A B, the angles L m'v K and EA B are equal; also, since F1 = ImIv is perpendicular to BE: the angles Im'v L and A B E are equal, and the triangles A B E, ImIv 0, and L m'vE, being right angled, are similar, and give the proportions 1' 2r r" Q. L'S, h. nv mvO; MECHANICS OF SOLIDS. 433 whence L I 2, r rQ normal L bmw = component of resistance; m = h. F normal m77 0 h= component of power; and the total pressure, which is equal to the sum of miv 0 and mIV L, becomes 2 ir r Q h Es total normal I 1 pressure; and the friction 72f rrQ + hF1) friction; and since in one revolution the path described by this friction is the diagonal A B = 4, its quantity of work will be its quantity of f(2c r Q + h Fl); work in one revolution; and because the work of the power F must equal the work of the resistance Q, increased by that of the friction, we have work of power equal that of 2 m R. F Q h + f(2 % r Q + F, h). resistance increased by work of friction; But the effect of F and F1 being the same, their quantities of work must be equal, and hence 2,1RF = 2~vrrF1; whence F = FR 28 434 NATURAL PHILOSOPHY. which substituted in the preceding general equation, we get workofpower; 2'RRF = Qh + f(2 r r Q + F R 1); and finding the value of ], value of the hr~f rr2fA power; = Q 2rRr -fRh Multiplying both members by 2 r R; then adding and subtracting Qh, in the second member of this equation, we find workofpower; 2vrR.F= Qh +f Q -h (15), in which the work absorbed by friction is given by the last term; that is to say, by work absorbed by h2 ~ 4,72 r2 friction; 2tr -fh If we neglect the consideration of friction, or make f = 0, we find, from Eq. (174), simply relation of power h and resistance Fi Q X without friction; 2 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 the power. From which it is obvious that the power of 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 f 2 + 4 r2r2 2 - r-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 intermediate consistently with sufficient strength. helix should be Let b 0 be the radius of small; the interior helix, or that of the newel, and a 0 that Fig. 269. of the exterior helix; it is usual to make the projection a b, of the fillet, equal to the thickness a d, measured in,, D. the direction of the axis; and for facility of execution,.' e.... proportion of the different parts of the dimensions of the chan- the screw; nel are made equal to those C 0 of the fillet, that is to say, a b is made equal to a d; in which case, the helical interval aa' will be equal to 2 a d = 2 a b, when there is but a single fillet. Should there be two fillets, which are often employed to increase the helical interval without changing the size of the newel, and therefore of r, then.ule; will the helical interval be 4 a b. Considerations affecting the union of sufficient strength with least friction, have suggested this general rule in regard to the projection of the fillet, viz.: make the projection a b equal to one third of the radius 0 b of the newel, or ab = ~ Ohb. This will give radius of the Ob = 3ab; newel; 436 NATURAL PHILOSOPHY. and Ob + lab = r = 3ab + — ab - ab; and because h = 2 a b, radius of intermediate r 7h helix; which substituted for r, in the expression for the friction, gives work of friction; Xf Qh 7 4 22 and making ~ = to which it is very nearly equal, the expression reduces to its finalvalue; f x Qh 122 11 -fTo apply this to a particular example, let the screw be example; 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 X Qh; which, substituted in Eq. (175), gives result. 2qR. F= Qh + 1.152 Qh = 2.152 Qh; whence we see, that friction occasions a loss of work greater than the whole work performed by the resistance. Endless screw; ~ 241. —The endless screw is employed to transmit a very slow motion, and, at the same time, to overcome considerable resistance. It is a short screw, with square fillet, MECtHANICS 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 description; of a crank. The fillet passes between teeth on the circumference of a wheel Fig. 270. of which the axis is perpendicular to that of the screw. The resistance Q is applied to the circumference of the arbor of the wheel. The rubbing faces of the teeth, instead of being parallel to the axis of A the wheel, are slightly in- surface of teeth inclined to axis clined to that line, so as of inclined; of motion; to make them parallel to the surface of the fillet when the latter is brought in contact with the teeth. 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 ~~~~~~~~~7 ~~name; 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- Fig. 271. pendicular to that of the, wheel, will cut from the rubbing surfaces of the a fillet and teeth a profile; and if we confine our- section by a plane through the axis selves to what takes place of the screw in this plane during the perpendicular to motion, we shall find that w theel; the circumstances will be 438 NATURAL PHILOSOPHY. the same as those of two circumstancesof wheels acting upon one Fig. 271. action same as -A those of two another through the interwheels with vention of teeth; for, as teeth; the screw turns about its axis to bring different;j parts of the fillet in this cutting plane, the section a b will move in the clirec- B tion from A to B, driving the section b e of the tooth before it. Let Q, be the force applied at b in the direction A B, which is tangent to the circumference whose centre is on the axis of the wheel, and whose radius is Ce = R,, and which will sustain the resistance Q in equilibrio: then denoting by N the resultant of Q, and Q, by r the radius of the arbor, and by r, that of the trunnion, will quantityof work; Q. R,s, = Qrs, ~ fir,s,; in which s, is the arc described at the unit's distance from the axis of the wheel. Dividing by R, s,, value of the + Q., - _ (176). power; RI A Find, by the process explained in ~ 234, 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 produces a friction upon the teeth of which the value is n + n' t11\ friction; f*Q +' -= f Q'~ ( + wherein n denotes the number of teeth on the wheel whose centre is C, and n' the number on the other. But the cir MECEALNICS OF SOLIDS. 439 cumference of this latter wheel being a right line, is infinite as well as the number of its teeth; hence reciprocal of the number of teeth n' 2 on section of screw; and the foregoing becomes 1 value of the f.. 7 friction; which must be added to Q, to obtain the force necessary to turn the wheel and to obtain the total pressure on the fillet of the screw. This sum, which is Ir firf &I &,( total pressure on Q. + f -' Q. Q 9 r f-, ~the fillet; being substituted for Q in Eq. (175), will give 2 r F = Q( 1 f + f - h + f Q, ( + f) 2j 4 n n 2,,Yr -fh or 2. R F= Q, 1 +f X h+ h. 2 + 4 - f A (177). qu`antity of work RF Q1-(1 ~f-) [h,f-j -' h' of power; In the discussion of the screw, no reference has been made to the friction on the pivots and collars by which frictionon pivots the screw is kept in position. It will always be easy to nd collars neglected. find this, in any particular case, by the rules for finding the friction upon pivots, sockets, and shoulders or rings, explained in ~ 223. 4:40 NATURAL PH:ILOSOPHY. XXI. THE LEVER. The lever; ~ 242.-The Lever is a Fig. 272. solid bar A B, of any form, supported by a fixed point 0, about which it may freely fulcrum; turn, called the fulcrum. Sometimes it is supported upon trunnions, and frequently upon a knife-edge. 0 ( levers divided Levers have been divided into'different orders; into three different classes, F called orders. Q first order; In levers of the first order, the power F and resistance Q are applied on opposite sides of the ful- E crum 0; in levers of the second; second order, the resistance (3) Q is applied to some point Q between the fulcrum 0 and point of application of the third; power F; and in the third order of levers, the power F is applied between the fulcrum 0 and point of application of the resistance Q. examples of The common shears furdifferent orders of levers. nish an example of a pair of levers of the first order; the nut-crackers of the see- 0 MECHANICS OF SOLIDS. 441 ond; and fire-tongs of the third. In all orders, the conditions of equilibrium are the same. ~ 243.-When the lever is supported upon a point, the Equilibrium of equilibrium requires that the resultant of the power and the lever in which the resistance shall pass through this point in order to be fulcrum isa destroyed by its reaction; to have a resultant, the power point; and resistance must lie in the same plane, and as the resultant will also be in this plane, the power, resistance, and ful- Fig. 273. crum, must be in the same. If the resultant pass through the power, resistance, /,', and fulcrum in fulcrum, its moment same plane; taken in reference b.,,i thereto must be zero,... which requires that 0 the moment of the power shall be equal to that of the resistance. That is, when a lever A B 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 resistance; this latter point Om and On, perpendicular respectively to the direction of the power and resistance, then will F x On = Q X On. If the lever turn upon trunnions, then will the moment of when lever is the power F, be equal to that of the resistance increased supported on by the moment of the friction on the trunnion. -Designating the radius of the latter by r, then will 442 NATURAL PHILOSOPHY. momentofpower F X Om = Q X On + fN. r; equal to that of resistance, plus that of friction; in which N is the resultant of F and Q. Multiplying both members by.s,, we have wok of the F X Om X s, = Q X On X s, + fNr. s,; power. that is to say, the quantity of worke of the power F, must be egual to that of the resistance Q, increased by the quantity of work of the friction. Use and ~ 244. —The lever is not intended to produce a conadvantages of the tinuous rotation, but is usually employed to move a heavy lever;, burden or great resistance through a short distance during each separate effort of the power. 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 Fig. 274. increases the term f Yr s,, or the quan- o tity of work absorbed i by friction. If the lever be laid upon a simple knife-edge, r becomes zero, and the foregoing equation becomes relation of power to resistance on an edge for a F X 0Om X S, = Q X On X s8, fulcrum; making the quantity of work of the power equal to that of the resistance. The advantage of this machine, the usuallythe 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, loss the power to resistance; a simple change in the point of support or fulcrum, which 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 0 is placed so that the distance On is one thousandth part of O0n, then will F- Q 1000 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 pulp of, a diminution of of work of the resistance must equal that of the power, power increases the path described by the point of application of the latter its path; must increase in the same proportion. 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 illustrate this; 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 66666 a distance of 0.2 ft., he will require 0.2 =333333 seconds nearly, = 92.6 hours nearly, = 9.26 days, supposing the man to labor 10 hours a day: in fact a man left to his individual efforts would never accomplish such a task. This example shows us that the lever is only useful for practical use of momentary efforts, and when the burden, being considera- the lever. ble, is to be moved through a very small distance. 444 NATURAL PHILOSOPH tY. XXII. ATWOOD' S M ACHINE. Atwood's ~ 245.-We shall terminate this branch of our subject machine; with a discussion of an instrument whose object is an experimental verification of the laws of constantforces. This instrument is the invention of Atwood, an English philosoobjects of the pher, and bears his name. Before proceeding to describe machine; 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 A D1) be two inclined planes having a common altitude A E; H' Fig. 275. and H', two wheels f of different diamethe general case ters mounted upon of which this the same arbor to machine is a ho/aro particular which they are example; firmly attached, and of which the W axis is supported upon trunnions parallel to the com- B 1mon intersection of the two planes; 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 bewhile the other ing parallel to the inclined planes. Now if the weight descends; W be made sufficiently heavy, it will overcome all opposition to motion and slide down the plane A B, while MECHANICS OF SOLIDS. 445 the weight W' must Fig. 276. from its connection A move up the plane to investigate the /o,'"~~rzqmr ~ ~circumstances of AD. It is required motion; to find the circum-. stances of motion.' Denote the angle which the planes —' —-' A B and AD make respectively with the vertical A E, by p and p'; the radius of the wheel H by R, that of H' by B', and that of the trunnion by r. The pressure of W upon the plane A B we have seen, is Wsin q; components of that of W' on the plane A D is the weights normal to the planes; T'. sin p'; and the friction on the planes A B and A D will be, respectively, fW. sin p, and f W' sin a' friction due to * /'si ~.these pressures; The stiffness of the cord c', which alone opposes the motion since the cord c unwinds, is, ~ 229, d K +I. (Q) stiffnessof cord 2 R' which winds; in which d, represents d2, dc, n, or d in Eqs. (127) to (130), inclusive, according to the cord or rope used, and (Q) the tension of the cord c'. This latter is equal to the component of W' parallel to the plane A D = W'cos', increased by the friction due to its normal component 446 NATURAL PHILOSOPHY. = - W' sin a'; that is to say, tension of the cord that winds; (Q = W COS {p' W' = Wp (cos p' + sln ); which, substituted in the expression above, for the stiffness of the cord c', gives its stiffness d K + I. IV' (cos p' + f. sin p') its stiffness; dr o 2l 2 RI' At the instant motion be- Fig. 276. gins, let the centres of gray- A ity of W and' be at G' and G,, respectively, and in any subsequent instant at'.. G" and G,,; denote the dis- -- tance' GG" by x, and G, G,, by x', then will x and x' be D the paths described by the centres of gravity parallel to the planes in the interval; and length of paths in direction of X COS p(, and x' cos p, - weights; will be the corresponding distances in the direction of the weights. The quantity of work performed by VW will be quantity of work; Wx cos p:, and that performed by W' in the same time, quantity of work; TV'- xW cos (p, and the total quantity of work of both will be total quantity; TVX COS (2 - WV X" Cos p'. MECHANICS' OF SOLIDS. 447 The quantity of work absorbed by friction on the plane AB is work absorbed by f. W. x sin ), friction on one plane; and that absorbed by friction on the plane A D is 4. W' x sin', that on the other; and the total quantity absorbed by friction will be, supposing the unit of friction the same on both planes, f (Wx sin qp + W' x' sin q'). work absorbed by all the frictions; The quantity of work absorbed by the stiffness of the cord c' will be dK + I. W' (cos 9' + f sin 9q') I work absobed by 2 R' stiffness of cord; The work consumed by friction on the trunnions will be work absorbed by f N. r. s,, friction on trunnions; 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 components of Wcos p -f Wvsin % and W' cos' + W sin, the pressnre on trunnions; for their values, and p + q' for their inclination to each other. s, is the arc described at the unit's distance from the axis. The work absorbed by the inertia of the wheels and arbor, or, which is the same thing, half the living force of 448 NATURAL PHILOSOPHY. the wheels and arbor will, ~ 159, Eq. (60)", be work absorbed 2 by the inertia of V1 I g V1 _ wheels and 2 arbor; 2 in which VI 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 TV'; the living force of the first will be living force of W 2 one body; and that of the second,?W' V'2 that of the other; and the quantity of action in the two bodies, will.be quantity of action W V2 + W' I'2 in the two bodies; 2g 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 W V2 q- Wl t7rl 2g work of the weights equalto + f (Ix sin 0 + WI x' sin t') the living forces of moving parts Wx Cos - W' cos K + I T' (c + f and the work of K (cos fsin') friction and 2 R' stiffness; V2 I +/ fNrs, + 2.(178). The variables in this equation, for the same inclination of the planes, are V; V', VI, x, x', and s,; but these, by the MECHANICS OF SOLIDS. 449 nature of the system, are connected by the following relations, viz.: xR' V' V'".R-.R' ~* V' = R * I (179), xO:: 1 Rx: R.X. x' - (180), relation between the angular velocities and 1: R: s, x. s.. (181), spaces; R - ~R: V:: 1 VJ. V1= R. (182). These values of V1, V', x', and s,, being substituted in Eq. (178), will give R12 WTV2 + WIV'2_ 2g R R' + f (Wx sin 0 + W' - -sin qt) Wx cos - W V x-cosW= B equation (178) in + d K + I WI (cos t' + f sin') differen + di x. V2 1 +f * Nr R -+ Rz; and solving this equation with respect to VI, B'.r W. cos ~ - W' ~ R ~ cos ~' B' — f( W. sin - W ~ sin') value for the 1Tr2 R/ gs _~ j2 B square of the B"W+W R * -. R' K, I.Wl.(cos It+fsinl') velocity; -f i N.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. varies as the square of its velocity. HIence the motion is, motion uniformly ~ 68, Eq. (8), uniformly varied; and the coefficient of x is varied; twice the velocity which the force producing this motion is capable of generating in a unit of time. Making W. cos 4-W' - ~'' cos B' value of the -f ( W sin ~ - W' - sin i') velocity, g generated ARt2 i in a unit of time; W+ - - W/l R-d r' K+'I.W.(cos'.+fsini') B 2.RI -fN. Br the foregoing equation may be written square of the Vi = x. 2A.. (183). velocity; Since the motion is uniformly varied, if T denote the time of describing the space x, then will Eq. (7) become space described; X = 2A T2..... (184); writing A for V1, and x for s: substituting this for x above, we find V2 = A2 T2. or value of velocity; V = A T..... (185). Eqs. (183), (184), and (185), give the laws of unilaws ofuniformly 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 Any device that will make the time in which the Fig. 27i. motion takes place com- \ II paratively great, while the velocity acquired shall be small, will enable us to verify these laws from ob- n servation. For this pur- - pose, A must be small. By ~ reference to Eq. (182), we C find that A may be dimin- ~ _ ished at pleasure by increasing the angle p, or decreasing ~'; this will increase the effect of friction, which opposes, while it will diminish the component of W, which aids the motion. Or A may be diminished, by diminishing the angles p and p', the difference between the weights Wand W' and that between R and R'. OwingI f 7n to the uncertainty of friction it is better to accomplish the object by the latter method, and this Atwood has done. His machine consists essentially of a fixed pulley R, over which passes a cord having attached to each extremity a basin s, for the reception of weights; a vertical graduated scale r of equal parts, say inches, to measure spaces; and a pendulum clock h which beats 452 NATURAL PHILOSOPHY. seconds, to mark the time. Fig. 277. The basins are short cylin\ t i ders of brass, having a wire e coincident with the axis and projecting some three or four inches beyond the w, n A- upper bases; the cord is at0C tached to the ends of these o'G - wires. The weights are either circular plates m, or bars n, -~~I wll Do~i~7' II II 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 cordf to enter, that the weights may be dropped upon the basins. a'ta I II The scale piece r is provided with three sliding stages, two of which a and el{ aI a re rings, and the third c is plane. The rings, whose diameters are less than the length of the bar-weights,!.I Wc 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, ~~.~I\,HCIPIK.SC. _Y is used to support the basin MECHANICS OF SOLIDS. 453 bearing the greater load opposite the zero point of the device for scale. The arbor is also connected by means of a second adjusting the basin to the zero arm with the escapement-wheel of the clock. This stage ofscale; 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, qp and Ip' are both zero, and therefore sin p = 0; sin' = 0; cos c = 1; cos p' = l:reduction of general equation to the case of Atwood's mo.reover R is equal to R', and hence machine; R' 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 - W1 corresponding A = g I value of the TW + TV' + g general coefficient; The circumference of the wheel has the same velocity as the points of the cord, and therefore the same as the basins. Designate by X", the mass which if concentrated in the circumference of the wheel would have a moment 454 NATURAL PHILOSOPHY. of inertia equal to that of the wheel, then moment of inetila of the wheel; f" R2 = I; whence and this, substituted above, gives velocity - - 1 1W generated in A = g - g TW I- W' -- W" unit of time; TV + X + gA 4 + W' TV in which W" denotes the weight of the mass If". This value of A, substituted in Eqs. (184) and (185), gives,TV- TVspace; = W + T W 2 g.. (186), t- TV' velocity; V = g 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 determinatio, of the same number of circular weights in each basin; then the weight T"; 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 Vf+ 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 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 ~ coincidence of ing substituted for T in Eq. (186), will enable us to find W", clock beat with removal of bar since all the other quantities in that equation are known. is obtaineof; If the removal of the bar and the beat of the clock be not coincident, the ring stage must be shifted, and the experiment repeated till the coincidence is obtained. Example. Let each basin weigh 11 units, and suppose 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 TV - W' = 1, data; WV W' = 51; making g = 32 feet = 384 inches;g =g-192 inches. Substituting these values in Eq. (186), we find 1. 192 T2 corresponding value of space; whence 192 TV" - 192 T - 51. value of W"; 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 192 numerical value -TV 27 X (3)2 -- 51 = 64 -- 51 = 13; fmeical W that is to say, the moment of inertia of the wheel will have 456 NATURAL PIIILOSOPHY. conclusion; the same effect to resist motion as the inertia of thirteen units of weight placed in the basins. Substituting this value for WT" in Eqs. (186) and (187), they become space for the WJ - I' particular X T. (188), machine; + W' + 13 2 velocity in the = _T- TV' rT (189 same; W TV+ WT' + 13 and, loading the machine as before, x = - x 192 X T2 = 3T2, experimental verification; V=.a x 384 X T = 6. Making T equal to times; 1, 2, 3, 4, &c. seconds; the corresponding spaces will be spaces; 3, 12, 27, 48, &c. inches; and the corresponding velocities, 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 verification; 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 MECHANICS OF SOLIDS. 457 excess of weight in either basin, and the motion will be- motion uniform come uniform, there being no reason why it should be after the bar is 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 proportional to the times, which is the characteristic of uniform 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 from the ring which takes up the descending bar, equal to that through which the latter has descended, it will be found that the all the laws scale will just reach this stage at the instant the motion tlhicfll gheay is destroyed by the action of the ascending bar. All the bodies may be laws which regulate the fall of heavy bodies may be machiehi verified by means of Atwood's instrument. 458 NATURAL PHILOSOPHY. XXIII. IMPACT OF BODIES. 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 together, are exerted along the same right line, perpendicular to the surfaces of both, at their common point of contact. When the motions of the centres of gravity of the two direct impact; bodies are parallel to this normal before collision, the impact is said to be direct. When this normal passes through the centres of gravity of two bodies which come into collision, and the modirect and central tions of these centres are Fig. 278. impact; 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- Fig. 2l9. mon normal, and the normal does not pass through the j centre of gravity of the othdirect and er, the impact is said to be eccentric impact; direct and eccentric. Fig. 280. When the path described by the centre of gravity of one of the bodies, makes an obliqueimpact; angle with this normal, the impact is said to be oblique. MECHANICS OF SOLIDS, 459 Bodies resist, by their inertia, all effort to change circumstances of either the qcuantity or the direction of their motion. figueduingtho collision; 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. In the impact of bodies, three periods must therefore three periods of be distinguished, viz.: 1st., that occupied by the process the impact; of compression; 2d., that during which the greatest compression 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 restitution and of of distortion; the latter denoting the reciprocal action ex- distortion; 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 defined; The ratio of the force of distortion to the force of restitution, is the measure of a body's elasticity. This ratio is sometimes called the coefficient of elasticity. When these coefficientof two forces are equal, the ratio is unity, and the body is 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 PH ILOSOPHY. Directimpact of to be direct. The forces of distortion and of restitution, two bodies; 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 Fig. 281. which F may be renotation; garded as constant, by v; and the small. - velocity gained by B, in the same time, by the action of the same force, by v'; also denote the mass of A by M, and that of B by i'; then will F, which may be called indifferently the action of one body or the reaction of the other, be measured by Myv, or 11' v'; and, because of the equality of action and reaction, equality of action if V = 21 17'. and reaction; That is to say, the quantity of motion lost or gained by each of the bodies, during each indefinitely small portion of time,,will be equal to each 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 f T fI = 7 motion equal; V.A,. But the force F acts 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 forces producing il1 TT = TT Elf VWXthese act in HI=- MY V,, opposite directions; or iI, + H',, = 0. That is to say, the aZgebraic sum, or the whole quantity of quantity of motion lost and gained will be zero; and in every stage of the motion of the system constant. impact the quantity of motion in the entire system will, therefore, be the same as before the impact began. ~ 248. —At the instant the bodies have ceased to ap- To find the proach each other, they will have attained their greatest ctommoenelocit compression, and, considering their condition before the greatest retrocession begins under the action of the molecular m n; 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 1 and I' as before. The whole quantity of motion before the impact will be M V + M' V', and that at the instant of greatest compression will be (X + I') U. But these, by the last article, must be equal, or (i + 3') U= V +3 M' V'; whence Taise tw + bo iest (mg 90i ). the value of this U= +M' if. ( 90) velocity That is to say, when two bodies moving in the same 462 NATURAL PHIL O SOPHY. direction have a direct impact, the common velocity, at the expressed in instant of greatest compression, is equal to the sum of the quanwords; tities of mnotion 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 X 7 - jf opposite U (191). directions. i = 1+' * ~ 249.-The velocity lost by the body A, up to the instant of greatest compression, is obviously equal to V- U Velocity gained 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 i(v - U), or by force of ( distortion; 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 force of f 7 or HI'.e restitution or',, and if e denote the coefficient of elasticity, then, from the definition coefficient of j[ V elasticity; X H~( - U_)-e MECHANICS OF SOLIDS. 463 f/V,, T coefficient of Jr'll ( - V') eV' elasticity; whence V, = e(V- U). (192), velocities lost and gained; V,, = e( - V')... (193). 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 exprlessed in words; 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 iV - U +e (TV - U); loss of velocity of V - U 1- ev ( -- ~); the impinging body, and the entire gain of B will be U V + e (U - V'). gain ofthe other; 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 = IV- V+ U- e(V- U) = (1 + e) U- eV; and as B must, after the impact, have its primitive velocity increased by its gain, v' =V'+ U -V'+ e(U -') = (1 + e) U- eV'; and substituting for U its value in Eq. (190), we have Hlf' V + if I VI velocity of the = (1 + e) _ + -- eV. (194), impinging body 1 after the impact; 464 NATURAL PHILOSOPHY. velocity of the A 1F+ H 2' other after the' (+ e) e V. (195). impact; i2f + -e (' Thus, the velocity of either body after impact, is equal to the coefficient of elasticity increased by unity, multiplied into the in words; 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) -' eV. (196), when the bodies meet. v' = (1 + e) * H r + MI V97). ~ 250.-If the body B be at rest when the body A impinges against it, then will V' be zero, and v = (1 + e)_y+ M,- eV. (198), When one of the bodies is at rest; v' = (1 + e) M +.. (199). From the last equation we find coefficient of v' ( + 1') -1 elasticity; e -. (200) and when the masses of the bodies are equal, or 31= M', its value when 2 v the masses are e -- 1.... (201); equal; which suggests a very easy method of finding the coefficient of elasticity of any solid body. For this purpose, MECHANICS OF SOLIDS. 465 turn a pair of spherical balls of the same weight from the experimental body whose coefficient of elasticity is to be found; suspend determination of 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 circular graduated arcs whose centres of curvature are at the points of suspension. Fig. 282. The body A being drawn back to any given degree / upon its scale and abandon- s ed, will descend and impinge i description of against the body B with a. instrument, and velocity due to a height modeuit equal to the versed sine of the arc which it describes A before the impact; the body B will ascend on the opposite arc 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 = I//2 g, velocity of impinging body and that of the V = 2gh'; body struck; which, substituted in the value of e, gives e=2~-. (202). coefficient of eh = elasticity; Exacmple. Two ivory balls of equal weights, and therefore of equal masses, were made to collide in the manner 30 466 NATURAL PHILOSOPHY. above described. One descended through an arc of 20 exampleof two degrees, and the other ascended through an arc of 18 ivory balls degrees and 30 minutes; required the value of eo By tables of natural sines and cosines, we find nat. cos 20~ = 0.9396926; versed sin 20~ = 1 - 0.9396926 = 0.0603074; and denoting the radius of the circular scale by R, w;ve have height of fall of the colliding h = 0.0603074 R. body; Again, nat. cos 18~ 30' = 0.9483236; versed sin 180 30' = 1 - 0.9483236 = 0.0516764; height due to the velocity of the h= 0.0516764 R; body struck; andcl numerical value e=2 051676 1 1 0.05167 of the coefficient; 0.060307.R- 1 = 2 0.0603074- 1 = 0.85138; 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. iExcample. Two ivory balls, whose masses are repreexample of the sented by 6 and 4, move in the same direction with bcollson of ivo velocities of 10 and 7 feet a second respectively. What is the velocity of each after impact? The conditions of the question require that the larger mass 6 shall overtake the smaller mass 4, because the former has the greater velocity. Hence MECH:ANICS OF SOLIDS. 467 7 = 6; V = 10; e = 0.85. given data; I= 4; V'= 7; These data, in Eqs. (194) and (195), give 60 + 28 A v = 1.85 - 0.85 X 10 7.78, 10 v-elocities after impact; v' = 1.85 60 + 28 0.85 X 7 = 10.33. 10 Exacnrmle. Let the same balls move ill 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 60- 28.ft v = 1.85 0 0.85 10 - 2.58, 60- 28 v'= 1.85 1 + 0.85 X 7 11.87. l0 ~ 251.-Now suppose the bodies A and B to move, Obliqueimpact; the first with a velocity V in the direction from E towards F, and the second with a velocity V' in the direction Fig 253. from C towards D; and let the collision take place at H/. Through the point;l draw jv description and the common normal Ntatio and resolve each of the velocities V and V' into two components, one in the direction of the normal and the other in the direction of the tangent plane at H. For this purpose designate the angle F ON by p, and D O, N 468 NATURAL PHILOSOPHY. 22 by p'; the components in the direction of the normal, will be normal'VcosOS, and V' cos Q'; velocities; and those parallel to the tangent plane, will be tangentidalV sin pq, and V' sin p'. velocities; 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 0 and 0', respectively, we shall have, from Eqs. (194) and (195), V Cos0 JrVcos 0 = M''- cos' os 23 components of V Cos 0 = (1 + e) 11+ e 1Vcos... (203), velocity in direction of the normal after A + M V 0 impact; V' cos 0' = (1 +- e) V cs- os e V' cos q. (204). 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 tangential V sin 0 - V sin pg.. (205), components of velocity after the impact; V; sin' = V' sin p'... (206). Squaring Eqs. (203) and (205), adding, extracting the MECHANICS OF SOLIDS. 469 square root, and reducing by the relation, cos2 + sin20 - 1, we find -'Jilcs Ve + M'1 V'. Cos 01 ]2 + Vvelocity of the v= [ +(1 s es cos 2 + V sin2 I.. (207); impinging body ill+ q-lf Mt after the impact; and treating Eqs. (204) and (206) in the same way, o + 21~~ VI C~~os t-e o ~ ~TT'2+ velocity of the v'r=Wl; q +i, U _ e V cos 1 1 V'2 sin ip.. (208). body struck after the impact; Again, dividing Eq. (205) by Eq. (203), we have V sin ( direction of the tanO = -.. (209); fi.rst body's ( e+ e) Vcos p1 +' V' cos- - e osp motion and, dividing Eq. (206) by (204), tan'= V sin.. (210). that of the + e) Vos p + M' V' cos' second (1+e) + -e V'cos p' 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 rest; will V' be zero, M' may be written for Ill + 1', and Ht 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 velocity of the impinging body. v = V v2 e2 cos2 c + sir2 p; after impact; 470 NATURAL PHILOSOPHY. and to Eq. (209), direction of the impinging body's tan q motion after ta = - (211) impact; e The tangent of 6 being negative, shows that the angle NHIAf which the direction of A's motion makes with the normal NNB' after the Fig. 284. impact, is greater than 90.graphical degrees; in other words, ltstlrationof that the body A is driven result back or reflected from B. This explains why it is that a canion-ball, stone, or other body thrown obliquely against the surface of the earth, will rebound several times before it comes to rest. If the bodies be non-elastic, or, which is the same thing, if e be zero, the tangent of 6 becomes infinite;:that is to body will not say, the body A will move along the tangent plane, or if rebound when 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 tand = - tan q.... (212); which means that the angle NHF = -EHN' becomes equal ill perfectly to I-KHN'. The angle -EHN' is called the angle of incielastic bodies the angle of deuce, the angle KHANT', commonly, the angle of reflection. incidence equal oincidence equ Whence we see, that when a perfectly elastic body is reflection; thrown against a smooth, hard, and fixed plane, the angle of incidence will be equal to the angle of reflection. If the angles g and p' be zero, then will cos p = 1, cos 1' = 1, sin qp= 0, and sin q' = 0, and Eqs. (207) and (208) MECHANICS OF SOLIDS. 471 become VV ~ 2W' V1e' v= (I ~, e) +e J case of direct impact; J21V + 21' V' v= (1 + e) U~ + - 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 If1V + 211I''V = ~ 11 (213), bodies non-elastic; -if. V+ 2' V' v = (214); and in the second, e = 1, consequently 21 V + 111f' V' v =2 - -. (215), bodies perfectly elastic. M Y + if''I v 2 u+ 2' _ V'F.. (216). ~ 252. —-The equations which have just been deduced, are sufficient to make known the circumstances of motion Oblique and of the centres of gravity of the colliding bodies, for we eccentric impact; have seen, ~ 146, that whenever a body is acted upon in a 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, 472 NATURAL PHILOSOPHY. that when the direction of Fig. 285. in the eccentric the force does not pass P impact the bodies through the centre of gravwill rotate; ity, which is the case in the eccentric impact, the / / body will also have a ro-, tary motion. Employing the same notation as before, and sub- " tracting Eq. (203) from the identical equation, Vcospg= VcosP 9 C we find loss of velocity of one body in (' (cosp - V' Cos') direction of VCOS - V COS 0 = (1 + e) HI( Vco - normal; the first member is the loss of velocity of the body A in the direction of the normal, during the impact; and multiplying both members by the mass of A = IM we have, for the quantity of motion lost in the direction of the normal, motion lost in MK( s ( I'cos c -' cos') h /; (~00cos ( - v cos /) = (l + e) that direction; _/iF'iF i' If the force of which either member of this equation measures the intensity, and of which the direction coincides with the normal, does not pass through the centre of gravity, it will give rise to rotary motion. From the centre of gravity G', of the body B, let fall the perpendicular G'C' construction; 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 I1; then, because the angular velocity MECHANICS OF SOLIDS. 473 i. equal to the moment of the impressed force divided by the moment of inertia, iMmfl' Vcos V'cos. 7 angular velocity s, = (1 + e) b + M (217). of one of the (l + HI b bodies; Also let fall from the centre of gravity G of the body A, the perpendicular G C upon 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, (1~+) M i'X Vcosp- V cosp' (218) angular velocity Al + J''1' ~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. 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 beect regarded; for the bodies acting upon each other in the direction of the normal with a pressure of which the measure is (1 + e) (vCos (g - V' Cos'); this pressure will give rise to friction, whose intensity is measured by f(1 + e) (Vcos p - V' cos ~'); measure ofthe and this acting in the direction of the tangential commonents of the velocities will accelerate the one and retard 474 NATURAL PHILOSOPHY. 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 tallgential force to overcome if Uj. friction; Now if the tangential velocities be equal, it is obvious that the bodies will move together in the direction of the tangent, H~ U, will be zero, the friction will not be called into action, and the bodies will not rotate from friction. If the tangential velocities differ by a quantity that will make A U; equal to the friction, then will the whole of the latter be exerted to produce ro- Fig. 285. tation. If the tangential velocities be such as to limits within give to Af U; any value,vhilh friction *." aywhch frction between these limits, a /. to produce part only of friction will be /. A rotation, 7 exerted, and this part alone B / will determine the rotation. If the difference of the tan- gential velocities be such as to make AL U, greater than the friction, the bodies will slide along each other and rotate at the same c time; the latter motion being due to the entire friction, and the former to the excess of i U, over the value of this force. Denote by n, the ratio of the friction to AL U,, then will quantity of tangential motion AL U; = nf (1 + e) ~ iE+ A' (- V' cos p'). lost; Let fall from the centres of gravity of the two bodies the perpendiculars G T and G' T', upon the tangent T T'; -IECHA 0NICS OF SOLIDS. 475 denote the length of the first by b, and that of the second by a,. Then will the angular velocity of the body B, produced by friction, be HELMMi~T' Vcos p - V' cos pI' angular velocity of ~(ai + e) b M''to friction; and that of the body A, _fd 1 e Vcos p - F' cos Q' angular velocity.f (1 + e) cb, a +' o of the other, due I II+ 1' to friction; whence the whole angular velocities of the two bodies will become s,=(18~M e IVcos - cos (b + fb,), ) (b whole angular velocity of the iwr', V cos - Yv cos ~0 bodies; sea = (1 + ne) *(a +- -fct'). If the balls be spherical and homogeneous, the normal will always pass through the centre of gravity, b and a will reduce to zero, and the rotation will be due to friction alone. If the impact be direct, then p and qp' will be zero, particular cases there will be no tangential components of the velocities of figueH U,, and consequently n will reduce to zero, and the rotation will be due to the eccentricity of the impact. PART SECOND. MECHANICS OF FLUIDS. INTRODUCTORY REMARKS. Condition of all ~ 253.-We have seen, ~ 12, that the physical condition bodies depends of every body depends upon the relation subsisting among upon the molecular forces; its molecular forces. When the attractions prevail greatly over the repulsions, the particles are held firmly together, a solid; and the body is called a solicd. In proportion as the difference 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 action of its own weight, and the resistance of the sides of a vessel into which it is placed, readily take the figure of the a liquid; 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 application of some extraneous force or to be confined in a closed vessel to keep them together; the body is then called a agas or vapor; g7aS or vcapor, 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 MECHANICS OF FLUIDS. 477 that which determines a gas or vapor, bodies are found in solids, liquids, all possible conditions —solids run imperceptibly into intd each slrtun liquids, and liquids into gases. Hence all classification of bodies founded on their physical properties alone, must, of necessity, be arbitrary. ~ 254.-Any body whose elementary particles admit of Deflnitions, &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 different 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. Finally, a body may viscous fluids; approach so closely both a solid and liquid, as to make it difficult to assign it a place among either class of these bodies, as paste, putty, and the like. paste; putty. ~ 255. —Fluids are divided in mechanics into two Classification of classes, viz.: compressible and incompressible. The term in- nuids; compressible cannot, in strictness of propriety, be applied compressi ble and to any body in nature, all being more or less compressible; ncompressible: 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 incompressible; gases and vapors as compressible. compressible ~ 256.-There are many fluids that readily pass from the compressible to the incompressible class, when subjected 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. confined in closed vessels, as in the instance of steam in vapors boilers. Vapors are generally invisible, and must not be distinguished itramiss and ~confounded with the mists and clouds which are often seen clouds; 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 gases temperature to bring them to a licluid form. All such fluids distin.shed; are called gctses. The most familiar instance of this class from vapors; of bodies is the atmosphere which surrounds us on every side and in which we live. It envelops the entire earth, atmosphere; 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, acnd 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 combustion, and, with the various forms of carbon, is one of mechanical use the principal agents employed in the development of meof oxygen; Cchanical power. proof of the The existence of air, gases, and vapors, is proved by a existence of vapors and gases; multitude of facts. Contained in a flexible and impermeable 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 atmospheric districts, are but air in motion. Air opposes, by its inertia, resistance; 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. ~ 257.-Many bodies take, successively, the solid, liquid, iMECIHANICS 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; ftsion,; from a liquid to a state of vapor, vcaqorization or vaporization; volatilization; that by which a vapor returns to a liquid, condensation; condensation; and a liquid to a solid, solicdification or congela- solidification. tion. Some bodies appear to take but two of these states, while others constantly present themselves only undcer 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. ~ 258.-The subject of the mechanics of fluids, is usual- Mode of ly divided, as before remarked, into hycdrostatics and 7Iycdro- ~onsieri" the subj ect; dynciznics, the former treating of the equilibrium of fluids, hydrostatics; and the latter of their motions; and not unfrequently the hydrodynamics; compressible fluids are discussed under a separate head called 2neurmntics. 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 perspicuous by disregarding them. And in the discussions which are-to follow, the fluid will be considered as without 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 modifications can be known only from experiments. 480 NATURAL PHILOSOPHY. II. MIVECHANICAL PRINCIPLES OF FLUIDS. Level surface; ~ 258.-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 vwhen at rest; equilibrio, its free surface is always normal to the resultants of the forces which solicit each of its surface particles. For if the resultant OF of the forces which act upon any one of these Fig. 286. particles 0 were oblique to. nlolrmnal to the the surface A B. this resultwehuiathltofth ant might be resolved into Iforces which act upon the surface two components, one 0 F', -- — B particles; normal, and the other 0 F" tangent to the surface; the 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 inconsistent with the equilibrium. This free surface which every levelsurface fluid in equilibrio presents in a direction normal to the defined;; 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 its upper or free surface normal to the direction of the levelsurfaceof force of gravity. If the earth did not rotate about anheavyfluids; 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 centrifugal force X C, combined with the weight JI G of each element, giving rise to a re- Fig. 287. sultant HN slightly oblique P to the direction of the weight, figure of the level every free surface is in strict- surface of heavy ness a portion of the surface fluids; of a spheroid of revolution, flattened at the poles and protuberant at the equator. 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 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 caseofthe 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 operation 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 A BD C 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 E i, F; H, G communicating freely with each'oth- Fig. 288. er; that is to say, the A B surfaces at E, F; and G, of the communicating fluid would be upon the same level.,B Whence we conclude, A homogeneous that a heavy fluid, as R- F G Is heavy fluid in water or merry vessels water or mercury, communicating poured into several vesfreely will stand in all at the 6ets which communisame level; cate freely with each 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- Fig. 289. tal tube connecting freely with the vessels B and y, and having > a stop-cock D interposed, so that the con- B nection may be inter-, experimental rupted or established illustration; at pleasure. Fill A / with water, the stopcock being closed. When the water in A is at rest, open the 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. To the operation of this principle we are indebted for this principle the transfer of water from remote locations to artificial determinesthe transfer of water reservoirs for the supply of cities and towns. Springs to artificial reservoirs from also owe their existence to it. The greater part of the remote points; solid crust of the earth consists of various strata ranged one above another; many of these are of a loose and porous 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 Fig. 290. the latter, where their fur- i ther progress is for a time i cause of springs; 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- Fig. 291. mote districts. Here, if a channel be provided for the water by boring through the hard crust which confines it, it will spout forth or overflow, in its effort to gain and is the cause the level of its source of the discharge in the distant mountain. from Artwelsian. This constitutes an Artesian well, a name derived 484 NATURAL PHILOSOPHY. from the French province Artois, where, according to account, they were first employed. Principle of ~ 261.-From the principle of fluid level, it is easy to equal pressures; pass to that of equal pressure. Suppose a vessel, A B D CF E, in which the Fig. 292. branches EF and BD C i. cA have a free communica- L; tion with the part A B; then if water, mercury, a _ Q wine, or any other fluid, be poured in either at _E, A, or C, and the whole be a heavy fluid; suffered to come to rest, several vessels the surface at 1K of the f communicating; fluid in the part A B, at L in the branch E7, and () at M in. the branch BD a, I will be upon the same level. Through the point N; taken at pleasure below the surface of the fluid, conceive a horizontal plane to be passed. It is obvious that the weight of the fluid contained in the vessel below P N Q can contribute nothing to the support of the columns L P, I O, and f Q, since this weight acts downward; and the equilibrium would obtain if the fluid severalcolumns contained in the part of the vessel below PNQ were of unequal without weight. This fluid may therefore be regarded as weights supportingeach solely a means of communication between the columns other; L P, I0, and MHQ, in such manner that it will transmit the pressure resulting from the weight of the columns L P and H Q to support the weight of I0, and reciprocally. If now, instead of the columns L P, I 0, and H Q 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 if a pressure equal to that arising from the column M Q weight of be applied to the surface Q, while the columns LP and col""mn'offluid replaced by IO remain, the equilibrio will still subsist, and this, pressurespo whatever be the directions and sinuosities at D, F, &c. pistons; The weight IW of the column. Q X is measured by b. h. d. g; in which b is the area of the base at Q, h the height Q H, d the density of the fluid, and g the force of gravity. The weight W' of the column I 0 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'I~~~~~~,.~~~~ratio of the'.h d.g - [~ (219) weights of W b.h.d.g b **' columns of equal altitudes; 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 I 0 were transmitted by the fluid down the vessel NB, up the sinuous vessel BD Q to Q, and there diminished in the ratio of the base NO to that at Q. In like manner, the pressure from the column i Q must be transmitted by the fluid down the tube Q D H, up the vessel BN to the base N 0, and there increased in the proportion of the base at Q to that at N. 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 the areas of the respectively applied. If the areas b and b' of the pistons pistofis; become equal, the forces will be equal, and this, whatever be the actual dimensions of the pistons. Whence we conclude, that the force impressed upon a fluid, is transmitted by it equally in all directions; and that every surface exyposed to pressure transmitted the fluid will receive a pressure which is directly proportional eqally in all to its extent. Moreover, this pressure will be perpendicular directions; to the surface, for if it were oblique, it might be replaced 486 NATURAL PHILOSOPHY. pressure always by its two components, one normal, the other parallel to normal to the the surface; the former would be destroyed by the resistsurface; 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 W = W. (220); transmitted; b whence we have this rule for finding the amount of presIst. rule; sure transmitted to any surface, viz.: Miultiply the intensity of the pressing force into the ratio obtained by dividing the area to which the pressure is transmitted, by that to which the force pressure is directly acpplied. Making b = 1, W will be the pressure transmitted when the upon the unit of surface, and Eq. (220) becomes pressure is applied to a unit b (221 of surface; --.. whence we have this second rule for finding the pressure transmitted to any given surface, viz.: Multiply the intensity 2d. rule; of the force appolied to the unit of surface by the area of the surface to which the pressure is transmitted. 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, e tube fh, say half an inch in diameillustration by the ter, open at the top. The vessel anatomical is filled with water, and closed siphon; 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, J Ki with its superincumbent weights TV', will be raised by the pressure MIECHANICS OF FLUIDS. 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 erical in. in. PR2 X 50 3.1416 X (0.25)2 X 50 C.ft. 1728 1728 -.00568. Now one cubic foot of water weighs sixty-two and a half pounds, whence the weight of the pressing column or Wbecomes, in pounds, lbs. lb. 1W = 62.5 X 0.00568 = 0.355. weight of the; pressing column; The area of a section of the glass tube is in. b = rR2 = 3.1416 X (0.25)2 = 0.196; area of a section of the tube; or, in square feet, 0.196 b = 144 = 0.00136, nearly. Let the diameter of the vessel A be one foot then will diameter of the larger vessel; ft. b' = 3.1416 x (0.50)2 = 0.7854; and these values of W, b, and b', substituted in Eq. (220), give 0.7854 lbs. W = 0.355 = 204.8, nearly; weight sustained; 0.00136 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 If the bladder were removed, and the vessel extended hewaeremwould upward to the line e d, on a level with the fluid in the rise inthe larger tube, the water would rise in it to that height, when it vessel; would come to rest. The volume of the added water, in cubic feet would be 50 X 0.7854 = 3.272; 12 and allowing 62~ pounds to each cubic foot, the weight of distilled water at 600 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 Muultiplication of may be multiplied at pleasure by this principle of equal power by the transmission of pressure. It will be sufficient for this principle of equal transmission of purpose, to provide a strong pressure; vessel for the reception of a fluid, and to connect with it Fig. 294. a pair of pistons whose surfaces bear to each other any F desired ratio; the power F - I 1 being applied to the smaller piston b will be transmitted to the larger b' and made to r hold in equilibrio or overcome almost any given re- B C sistance R applied to the latter. But we are not, there MECHANICS OF FLUIDS. 489 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 however; 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; RP ~ 8s' = p __ 8l22 work of the bs' _. (222). 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 F upon 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 s b, and, therefore, from what has just been remarked, volumes of the 8Ib' = sb; fluid equal; whence 85 b' b which, substituted above, gives work of power s' = Fs. (223). 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. If the friction of the pistons against the sides of their friction and respective chambers and the viscosity of the fluid be taken viscosity; into the account, the work of these must be added to the 490 NATURAL PHIILOSOI IY. term Rs', which would make the effective quantity of the hydraulic work, measured by Rs', actually less than the work of the machine enables a feeble power to power. What then is gained? The answer is the same perforlm what as before, viz.: the machine gives to a feeble power the it could not without it; 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 t6 raise to the summit of a wall a ton of bricks, by taking a few bricks at a time, wnereas an effort to elevate 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 accomof all machines* plished by the application of a diminished power, the space through which the latter is exerted must be proportionally increased. HIad 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 this principle principle of the equal transmission of pressure; for, the prove that of volume of the fluid remaining the same, we should have equal pressure; 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 R F' whence pressures are i: R: b: proportional to the surfaces. that is to say, the pressures are directly proportional to the areas of the pistons to which they are applied, when MECIHANICS 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 tranismission of pressure is exhibited by the Hydraulic or Brcazah's Press. Bydraulic press; The main features of this machine are the following: A large and small metallic cylinder A and a, are Fig. 295. made to commu- m nicate freely with each other by a duct-pipe r. Water stands in both L ] of the cylinders, description, and and each is provi- mode of applying the power; ded with a strong _ _ _, thepower; piston. The piston S of the larger cylinder carries a strong head-plate 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 = (200) - 160000; and suppose the distance c d to be equal to 50 inches, and 492 NATURAL PHILOSOPHY. c b to be one inch, and let a man throw his weight, say data; 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. 1 50:: 150 F; whence lbs. power applied at F = 150 50 = 7500. smaller piston; Substituting these values for F and - in Eq. (222), and omitting the common factor s', we find value of bs. lbs. resistance; R = 7500 X 160000 = 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 path of the power R. s' at smaller piston; S = 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 1200000000 ft. of path of the =S 160000; resistance; 7500 and becaise the power applied at d must pass over 50 times this distance, we find ft. 160000 X 50 = 8000000, MECHANICS OF FLUIDS. 4-93 or 8000000 miles. path of the 5280 - 1515, 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, usesofthe and guns; and two were recently employed with success hydraulic press. to raise, through a vertical height of more than one hundred feet, the great iron viaduct-tube, weighing upward of eighteen hundred tons, over the Menai Straits. I V. PIRESSURE OF HEAVY FLUIDS. ~ 264.-Let us now examine the pressure which aPressure of heavy heavy fluid exerts on the base of a vessel in which it is fluids; contained. For this purpose, let A B D C be a vessel containing Fig. 296. a heavy fluid, as wa- A o._ o'.o ter, in equilibrio. The:;Zupper surface A B of ------ the fluid will be hori- zontal. Conceive a horizontal plane G H to be passed, and suppose the fluid below this plane, or that contained in the fluid below the portion G CD E, to be devoid of weight; then it is the horizontal stratum devoid obvious, from our previous principles, that the weight of of weight; any slender vertical column, as E,1; will exert a pressure 494 NATURAL PHILOSOPHY. at 1; which is distributed equally in all directions through the fluid G CD H, and that this pressure acts equally upward to oppose the descent of the other each elementary columns which stand Fig. 296. column scoltinmn a vertically over the A 0 0' B sustaining all the | others; plane G H; the column EI alone keeps, therefore, in equilibrio all the other - u columns of the mass A G H B; consequently, the mass G CD 1, being still supposed without weight, there will result no pressure upon the base CD, except that which arises from the weight of a single filament E1, 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 b the base; W 7 in which W is the weight of the column E;, 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 1 will be given by weight of the pressingfilament; 1 -- h. b. D.g in which D denotes the density of the fluid, and g the force of gravity. Substituting this above for EV we find pressure upon h.b'.D... (224. W' = h b'. D g. (224). the base; If now the plane G H 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 h will become the vertical height ElI' of the surface of the fluid above the base. But the product b' h is obviously the volume C O 0' D weight of the of the fluid contained in a right cylinder or prism having column which measures this for its base, the base of the vessel; D. b'. h is the mass of this pressure; cylinder or prism, and D. b'. I. 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 de2lth of this base below the surface of the fluid. 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, iepeet of 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 pressing the base and altitude of the fluid are the same. The right cyl- Fig. 297. inder, inverted and erect truncated cones, illustration; having equal inferior bases B, B, B, and B Bi the same altitude h, will, when filled, con- Fig. 298. tain very different volumes of fluid, yet the bases will all experience the same right cylinder, amount of pressure truncated cone, both erect and from the weight of i tnerete d inverted; the fluid, if it be the - - f. same in kind, or of the same density. The experimental verification of this _ 496 NATURAL PHILOSOPHY. apparent paradox is easy. A D CGB is a glass tube, of which the ends are open and bent upward; the end experimental B is furnished with verification of this fact; a brass ferrule upon Fig. 298. which a screw is cut for the reception of a mate-screw If around the bottom of the vessels F, F', and F", also open at both description of the ends. On the end A r Biy apparatus for the iS a sliding ring of _ Ho,.. metal or wood. At E is a short wire that I may be moved up and down, and is held in any desired position by friction. 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 details of the the mercury will press upon the latter by its weight and experiment; 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 A D CB, the mercury will again be found at the ring. In all these experiments, 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 deductions; 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. ~ 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 against inclined under any angle whatever to the horizon, and thence to surfaces; the pressure on a curved surface. Let A cB C be a vessel with plane or curved sides, and filled with a heavy fluid; Fig. 299. suppose GH and G' H' to A l? be two horizontal planes in- G /1r definitely near each other. J The layer of fluid between these planes may be considered 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 extent of this elementary surface by b', and the depth EI by h', the measure of this pressure will'be pressure upon an D. g. b'. h; elementary inclined surface; 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.; similar pressures; and the pressure upon the entire surface will obviously be the sum of these; or, if the total pressure be denoted by 32 498 NATURAL PHILOSOPHY. P, then will total pressure upon the entire P = Dg(b' h' + b"h" + b"' h"' + &c.). surface; 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 value of this pressure in P = D.g.b.h.... (225); weight; 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 expressed in surface pressed, and for its altitude the depth of the centre of words; gravity of this surface below the upper level of the fluid. Example Ist. Required the pressure against the inner example fist; 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 Fig. 300. 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 4 a2 X 2 a + a2 X a of gravity below h 5 a. the surface; a Again, surface pressed; b - 5 a2; MECHANICS OF FLUIDS. 499 and these, substituted in Eq. (225), give P= D.g.b.h = g. 3a3 valueofthe pressure; Now Dg X 13 = Dg, is the weight of a cubic foot of water - 62.5 lbs. whence lbs. P = 62.5 X 3 a3. inpounds; Make a = 7 feet, then will lbs. P = 62.5 X 3 X (7)3 = 27562.5. its numerical 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. Exarmple 2d. Let the vessel be a sphere filled with mercury, and let Fig. 301. its radius be R. Its centre of gravity is at the centre, and therefore below the upper surface at the dis- example second; tance R. The surface of the sphere being equal to that of four of its great circles, we have b = 4 R2; surface pressed; whence volume whose b. h = 4,f R3; weight is equal to the pressure; and, Eq. (225), P 4. D. g. g R2. whole pressure; The quantity D g x 13= Dg, is the weight of a cubic foot 500 NATURAL PHILOSOPHY. of mercury = 843.75 lbs., and therefore, substituting the value of - = 3.1416, lbs. plessure ins; = 4 X 3.1416 X 843.75. R3. Now suppose the radius of the sphere to be two feet, then will R3= 8, and its numerical value; P = 4 X 3.1416 x 843.75 X 8 = 84822.4. The volume of the sphere is A~r4 R3; and the weight of the contained mercury will therefore be - ~ R 3 g D = W. Dividing the whole pressure by this, we find ratio of weight of p pressing fluid to 3; pressure; W whence the outward pressure is three times the weight of the filuid. example third; Example 3d. Let the vessel be a cylinder, of which the radius r of the base is 2, and altitude l, 6 feet. Then will b. h= f rl(r + 1) = 3.1416 X 2 X 6 X 8; which, substituted in Eq. (225), value of P - 301.5936 X Dg, pressure; and weight ofI W= 3.1416 X 22 X 6 X Dg = 75.398 X Dg; pressing fluid; whence, ratio of weight to P 301.5936 D g _ pressure. W 75.3984. D. 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 EI/F be any plane, and MN Fig. 302. the intersection of this plane produced with the upper........ surface of the fluid which / r presses against it. Denote the area of any elementary, portion n of the plane _EIF. t geometrical by b'; and let m be the pro- representation and notation; jection of its place upon the x upper surface of the fluid; draw mil perpendicular to MN,; and join n with M by the right line n M; the latter will also be perpendicular to YV,; and the angle n Mm will measure the inclination of the plane EIF to the surface of the fluid. Denote this angle by p, the distance m n by h', and Mn by r'; then will distance of an elementary hi = rf sin Ap. pressed surface below the fluid surface; The pressure of the fluid upon the element n will, Eq. (225), be D. g. b'. h = g b' r' sin 9; pressure upon this element; and its moment, in reference to the line MN as an axis, D g b' r'2 sin p; its moment; and for any other elements of which b", b"', &c., denote the 502 NATUIRAL PHILOSOPHY. areas, we have, in like manner, Dg b" r"2 sin p, moments of the elementary pressures D g b'" r"'2 sin g, &c., &c. Denoting by h the depth of the centre of gravity of the area EIF below the surface of the fluid, and by r the distance of that point from the line M1N; we shall have depth of centre of gravity of the rsin whole area pressed; and, for the total pressure upon EIF, whole pressure; P = D.g.b.h = Dgbrsin p, 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 moment of the entire pressure; But the moment of the entire pressure must be equal to the sum of the moments of the partial pressures, and hence Dgbrx sin g = Dg sin p (b' r' + b" r2 + b"' r"' + &c.); whence distance of the point of total b' r'2 + 6b r"2 + b"' r"112 + &c. pressure from the X (226). axis; b r MIECHlANICS OF FLUIDS. 503 The numerator of the second member, is the moment interpretation of of inertia of the plane ElF; the denominator is thethelast equatin; product of the area of the plane itself by the distance of its centre of gravity from the axis, and as a similar expression would result if the pressures were referred to any Fig. 302. other line in the plane EF - as an axis, it follows from - -...... 184, Eq. (86), that the result- -v // ant pressure passes through / the centre of percussion of the surface pressed. This cr point is called the centre of /Cetre of pressure; pressure. It is that point in pressure; the surface to which, if a single force be al2:lied in a direction defined; contrary and equal to the total coincident with pressure exerted upon it, the surface will remain in equi- centre of ibrio. percussionu. ~ 267. —The principles which have now been explained, Application of are of high practical importance. It is not only interest- the preceding principles; 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 applicable; 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. 504 NATURAL PHILOSOPHY. -thickness of the Let A B CD be a section or sustaining wall of areservoir; profile of the wall of a reservoir, 1J1N the upper surface of Fig. 303. the water, and E E' the bottom. A B Denote the length of the wall by 1, the depth NE of the water - against its face, supposed verti- _-_ aG cal, by d; then will the surface l pressed be measured by I d; 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 D.g. 1. d2 the face; 2 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 supposethewall edge passing through C. First, let us suppose it slides. may slide; Denote the depth of the face A D by d', the mean thick-:ness m n by t; then will the weight of the wall be weight of the. d D'.. ld. t; wall; and, denoting the coefficient of friction between the wall and earth byf the whole friction will be friction on the f D' g.. d'. t, ground; 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 D g 1 d stability; 2 = fD'gd't; MECHANICS OF FLUIDS, 505 from which we find t D d2 value of mean D 2fd' thickness; The density of water is usually taken as unity, and on ordinary earth, the value of f for masonry, does not vary much from I, whence 3 d2 value of the t thickness in 29_D' d'' ordinary cases; The thickness is the only unknown quantity, since d and d' must result from the capacity of the reservoir. 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 -A~~~~~~~~~ ~~~the front line of pressure when both are taken in reference to that line. itsbase; Let G be the centre of gravity of the profile B CD, and denote the distance C O of its projection upon the base of the wall from C, by r. Then, from the assumed figure of the profile, we shall have ratio of lever arm -=, or r - n of the wall to its t thickness; in which n is known; and the moment of the weight of the wall will be moment of the weight of wall; The centre of pressure 0', 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 to - d; and adding the distance ED denoted by a, the moment of the pressure, in reference to C, will be D g l da) moment of the 2 7" fluid pressure; 506 NATURAL PHILOSOPHY. and, to insure stability, we must have condition of Dg d2 stability; D' g 1 d' t2 n (3 d + a); whence thickness of the _ / 1 d2 (d + 3 a) wall; v\6 n D' dt If the water come to the bottom of the wall, and the reservoir be full, then will a = O, d = d', and t =d 1 D thickness of Next, let A B C be a secwater-pipes, boilers,&c.; tion of a cylindrical waterpipe or boiler perpendicular Fig. 304. to the axis, the inner surface of which is subjected to a pressure ofp pounds on each superficial unit. Denote by R the radius of the interior circle, and by 1 the length of the pipe or boiler parallel to c surface pressed; the axis; then will the surface pressed be measured by 2 R 1, and the whole pressure, by whole pressure; 2' R P12. 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 r will denote the path described by the whole pressure, and its quantity of work will be 2 qf R 1 P r. quantity of work; The interior circumference before the application of the pressure was 2 n R, and afterward, 2 er (R + r); the difference of which, or path of the 2 7 (R + r) - 2K R = 2 rr, resisting molecular action; 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 1; and its thickness by t. The intensity of the force which a section parallel to the axis is capable of resisting will be B I t, and its quantity of work the quantity of Bi t X 2 r. workof this force; But by virtue of the principle of the transmission of work, this must be equal to the work of the pressure, and we have 2_BBltr = 2 sfRIpr; condition of stability; whence t Rp B lthickness. 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 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. inch in diameter - - in wire, I-10th of an inch 36 to 43 Telford in bars, Russian (mean) 27 Lame English (mean) 25 hammered ON. Brunel rolled in sheets, and cut lengthwise - - - -.i ditto, cut crosswise i8 - in chains, oval links 6in. 2i Brown clear, iron 1, in. dia.e f dith to, Brunt on's, withan 6 5 to low stay across linkee - Cast iron, quality No. a - - 6 to 743 Iodgkinson 38 to 4r Hodgkinso in - rs, Rutoian (men) 27 L 37to48 3 - 6 to 9~ - - 5i to 65 Steel, cast - - - - - - 44 Mitis cast and tilted - - - 6) 25 Rnie blistered and hammered 3 ruel shear ets,- and- - - - - 57 rawl h - - - - 5 itis Damascus - - - - - 3 i ditto, once refined - 36 ditto, twic e refined - 44 lopper, ironast - - - - - Renie 52 Rennie hammered - - - - 25 lo sheet iron, quality No. 1 - 6 to gkingston 38 to 4 odgkinson wire -- 2- to - 37 to4 Platinum wre --- - - 1 - 7 Guyt on Silver, cast - - - - - Miti wire -.- - - - 7 Gold, cast and - - 6 Rennie wire -and hammered Brass, yellow (fine) - 8 Rennie 7 Gun m etal (hard) - - 44- Tin, cast - - - - - - 2 * The ca esrongest quality of ast iro is a Scotch iron known as the Devon Hot Blas No. 3: its tenacgst qality ofs cast iron, is per square inSctch 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. MECHANICS OF FLUIDS. 509 TABLE-continued. SUBSTANCES EXPERIMENTED ON.. |, TE-n Z.r ). Z. Tin wire - 3 Rennie Lead, cast - - - - - - 4-5ths - 31 Rennie milled sheet - - - - I Tredgold wire - - - I.I Guyton Stone, slate (Welsh) - - 5.7 Marble (white) - - 4 -.4 Givry - - - -- I Portland -. I.6 Craigleith freestone -. 2.4 Bramley Fall sandstone - 2. 7 Cornish granite - - 2.8 - Peterhead ditto - - - 3.7 Limestone (compact blk) 4 Purbeck - - - - - 4 Aberdeen granite -- 5 Brick, pale red - - -. 3 -.56 red - - - - - -.8 Hammersmith (pavior's) I ditto (burnt)-.4 Chalk - - - - - - -.22 _ Plaster of Paris - - - - o3 Glass, plate - 4 Bone (ox) - - - - - - 2. 2 Hemp fibres glued together 4I Strips of paper glued together 13 Wood, Box, spec. gravity.862 9 Barlow Ash-.6 8 - Teak - - - -.9 7 Beech - - - -.7 5 Oak - - - -.92 5 -- 1.7 Ditto - - - -.77 4 - Fir - - - -..6 5 Pear - - -.646 41 - Mahogany - -.637 31 Elm - 6.57 Pine, American - - - 6 _.73 Deal, white - - - -.86 - In the result just obtained for the value of t, no attention 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 cylindrical portion of the surface. 510 NATURAL PHILOSOPHY. V. EQUILIBRIIUM 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 subEquilibrium of jected to the action of a homogeneous heavy fluid. But floating bodies; 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- Fig. 305. librio, it is obvious that this state a body wholly will in no respect be altered by supimmersed ina pOSill any portion B to become fluid; posing any Bt to becomeB A 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 1! H1[lllt this solid is urged downward by its weight, which passes through its 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 first result; 1st. The pressures upon the surface of a body entirely immersed in a fluid, have a single resultant, and that this resultant 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., fourth result; 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 continue to ascend, till the weight of the fluid displaced by the part immersed, is equal to that of the entire body. In the first case, the density of the body will be increased, 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 float, according ished; and as the density of the original body was the as itsdensityis same as that of the fluid, we see that when the density of ghan 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 beautifully illustrated by what Fig. 306. is usually called the " cylinder and bucket" experiment.ody will lose the body will lose Place a hollow cylinder a, a portion of its in one of the scales of a weight equal to that of the balance; suspend to this displaced fluid; scale a second cylinder b, of solid metal, exactly fitting the former, and in the opposite scale put a weight c, that 512 NAT.URAL PHILOSOPHY. shall restore the equilibrium Fig. 806. of the balance. Now immerse the cylinder b in a cylinder and vessel lW of water, the scale ebuckeit of the weight c will deexperiment; 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 the fluid to the vessel, in the same way that the weight of a weight of the person in bed is transmitted through the latter to the bedmmnsemd olidhe stead, and thence to the floor. This is proved, experimentvessel; 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 experiment and a weight c; suspend the solid cylinder b by means of a thread from a Fig. 263. detached ring R, and depress it till it is wholly immersed experimental into the water of the tumproof; bler; the scale A will fall; I fill the cylinder a with water of the same temperature and density as that in the tumbler; the equilibrium will A. be restored. This important principle, which determines the circumstances under which a body will rest upon a fluid, is frequently employed to ascertain this principle the weights of large floating masses, such as ships, boats, used to fina edgtofshipsd and the like, which are entirely beyond the capacity of our weight of ships, re.; ordinary weighing machines. For this purpose the volume, in cubic feet, of the immersed part is computed from MECHANICS OF FLUIDS. 513 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, cargo; 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. The upward action by which an immersed body apparently 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 a fluid; 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, and indifferent or indifferent. If stable, the body will not overturn when equilibrium; careened; if unstable, it will; if indifferent, the body will retain any position in which it may be placed. Let l Q N represent a section of any body, as a boat at Fig. 308. rest upon the water, G of which the upper d.. surface is AB, calledI pl of the 2plane of floatation.... floatation; When this plane is i produced through the boat, it will divide her into two partial volumes, the lower of 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. this whatever be her position, whether careened or erect. Whence it follows, that if a series of planes il' N', Mi" N", &c., be passed, making the volumes l' Q N', 1" QN", series of cutting &c., respectively equal to MQ N; these planes will, each planes; 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 many tangent planes to a curved surface a b c, which may be conceived as invariably connected with the boat. Now'the effect, as regards the careening motion, will be the same as though this surface were the boundary of a physical axis which is made to roll back and forth oscillations of the on the plane of float- Fig. 308. boat; ation, regarded as a a physical surface, after X.' the manner of the, pendulum axis on A.. l its supporting plane, during an oscillation. When the boat has position of the a position of equiline of rest ing ofesthe librium, the line of during the equilibrium; 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 O and a' 0, 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 il" N". When one of these normals coincides with metacentre; the line of rest, the point 0 is called the metacentre, being the point of intersection of the line of rest, with an adjacent MECHANICS OF FLUIDS. 515 line of surpjPort. 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 compelled 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 equilibriuqn 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 equilibrium determined by reversed, and incifferent when these centres coirncide, for then a the relative slight derangement will cause no motion in the centre of gravity. positions of the 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 shipwithout cargoes are provided with ballast of stone, sand, ballast; or other heavy matter, to diminish the chances of upsetting. Fig. 309. The buoyant effort'of water is used to buoyant effort great advantage in used to raise CD phi\ 81l at\ A = sunken masses; raising heavy sunken masses. For this purpose it is usual to connect two or more 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-of employing this principle. 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 516 NATURAL PH ILOSOPH T. force empty and water-tight barrels between hu deck and hull. Levelstrata in ~ 269. —We have just seen that when a body is imheterogeneous 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 mixture of the surface of the heavier fluid, where, being freed from the different fluids having no buoyant action of the latter, they will arrange themselves, affinity for each under the efforts of their own weight, into a stratum of other; 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 will form level 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 dense lowest; the least dense at the top. This is confirmed by daily observation, 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. The same conclusion follows from the consideration, thesameresults that these fluids when mixed constitute a heavy system, from etthe which, we have seen, can only come to a state of stable properties of the centreofgravity; 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 tact, counteract the buoyant action of the heavier fluid, theywill not and the lighter will be held in a state of mixture. In- btain when the fluids have an stance wine and water, water and alcohol, brandy and affinityforeach water, and the like. other. VI. SPECIFIC GRAVITY. ~ 270.-The specific gravity of a body, is the weight specific gravity of so much of the body, as would be contained under a defined; 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, therefore, 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 =. measure for the W;C7 = g. D. V; weight of a body; and denoting the like elements of the other body by W,s 518 NATURAL PHILOSOPHY. D,, and,7;, we have weight of a TA D, V second body; w = 9.,.. * Dividing the first by the second, ratio of the _ gDV _ DV weights; 1I g D, V, D, 1 and making the volumes equal, same when the V D volnumes are -. (227). eqoual; W,,. 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; = D... (228); 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 gravities and densities are referred to the same substance same numbers as a standard, the numbers which express the one will for same standard. also express the other. ~ 271. —Bodies present themselves under every variety of condition-gaseous, liquid, and solid; and in every kind Choice of a of shape and of all sizes. The determination of their specific standard; 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 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 volumes of the body and fluid; and lf the latter be taken as the standard, and the loss of weight occupies the denominator, 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 specific gravities Sometimes the gases and vapors are referred to atmo- anddensity; spheric air, but the specific gravity of the latter being known as referred to water, it is very easy, as we shall gasessometimes referred to presently see, to pass from the numbers which relate to raetospheic one standard to those that refer to the other. ~ 272.-But water, like all other substances, changes its Varying density density with its temperature, and, in consequence, is not of water; 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. Let D denote the density of any solid, and S its specific reduction to a gravity, as determined at a standard temperature corre- standard temperature; sponding to which the density of the water is D,. Then, Eq. (227), _D specific gravity at'~ ) -- ~.~ one temperature; Again, if 5' 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 D i) same at another S = s D e. temperature; Dividing the first of these equations by the second, we 520 NATURAL PHILOSOPHY. have ratio of these.DI specific gravities; St D whence S= S' ~ d.. (229); and if the density D, be taken as unity, specific gravity reduced to a S = S ~ D,,.... (230). standard; 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 determnined at any otlzer temperature, multiplied by the expressed in density of the water corresponding to this temperature, the words; density at the standard temperatulre 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 expansion for each degree of Fahr. thermometer is accurately density of water known from experiment; give it the form of a slender at different temperatures; cylinder, that it may readily conform to the temperature of the water when immersed. Let the length of the cylinder at the temperature of 32~ Fah. be denoted by l, and the radius of its base by m I; its volume at this temperature will be, volume of a r m2l2 X Z = r23 slender cylinder; Let n l be the amount of expansion in length for each degree of the thermometer above 32~. Then, for a temperature denoted by t, will the whole expansion in length be its expansion; n I X (t - 320), MECHANICS OF FLUIDS. 521 and the entire length of the cylinder will become I1~ + n I ~t -320) - + -n~ ( 820) ~ its increased z + ~n z(t - L32) = z I1 + n (t - 32)]J; length; which, substituted for I in the first expression, will give the volume for the temperature t equal to'aim2l3 [1 + n (t - 32)]3. Its increased'K' z~ 3[ 1 +.?Z( - s2o)]8. volume; The cylinder is now weighed in vacuo and in the water, at dif- Fig. 310. ferent temperatures, experimental determination of varying from 32~ the density of upward, through any water at different temperatures; desirable range, say to one hundred degrees. The temperature at each process being substituted above, gives the volume 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 volume, gives the weight of the unit of volume of the water at the temperature t. It was found by Stampfer, 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 3875; sity of water at this temperature as unity, and dividing the weight of the unit of volume at each of the other temperatures by the weight of the unit of volume at this, o 38.75, the following table will result 522 NATURAL PH I L O S O P HY. TABLE OF THE DENSITIES AND VOLUMES OF WATER AT DIFFERENT DEGREES OF HEAT, (ACCORDING TO STAMPFER,) FOR EVERY 21 DEGREES OF FAHRENHEIT'S SCALE. (Jahrbuch des Polytechnischen Institutes in Wein, Bd. 16. S. 70.) ~t -D/ Diff.Tr Diff. Temperature. Density. Volume. 32. oo 0.999887 I. ooo I3 34.25 0o.999950 63 I.000050 63 36.50 o.999988 38..oooo0000 38 38.75 I.000000 12 I.000000 12 4I.oo00. 999988 I 2 1.000012 12 43.25 o0.999952 35 1.000047 35 45.50 o.999894 58 I. oooIo6 59 47.75 o.9998 3 8..00ooo87 8I 50.oo 0.999711 Io2 I. 000289 102 52.25 0.999587 124 I. ooo43 124 54.50 0.999442 I45 I. ooo558 I45 56.75 0.999278 164 1. o000723 I65 59.00o o.999095 183 I. 000ogo906 83 61.25 o.998893 202 I.oo io8 202 63.50 o.998673 220. 00oo329 221 65.75 o.998435 238 I.oo567 238 68.oo 0.998180 255 I.001822 255 70.25 0.997909 27I 1.002095 273 72.50 0.997622 287 I.002384 289 74.75 0.997320 302 I. 002687 303 77.00 0.997003 317. oo3005 3i8 79.25 o.996673 330. oo3338 333 8. 50 0.996329 344 I.oo3685 347 83.75 0.995971 358. oo4045 360 86.oo o.99560oi 370 100. oo44i8 373 88.25 0.995219 382. 0048o4 386 0.50 o. 994825 394. 005202 398 92.75 o.994420 405 I. 0056I2 4Io 95.00 0.994004 4i6 I.0 6032 420 97.25 o.993579 425. 006462 43o 99.50 o.993145 434. 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 590, 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 S = -' X 0.999095. MECHANICS OF FLUIDS. 523 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- volumes of the 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 corresponding to the given temperature. ~ 273. —Before proceeding to the practical methods of Instruments used finding the specific gravity of bodies, and to the variations to find the specific gravity of 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 Nicholsonz's Hydrometer. The first is similar in principle and form to the common Fig. 311. balance. It is provided with numerous weights, extending through a wide range, from a small fraction hydrostatic of a grain to several balance; ounces. Attached to the under surface of one of the basins is a small hook, from which may be suspended any body by means of a thin platinum wire, horse-hair, or anlly other delicate thread that modeof attaching will neither absorb nor yield to the chemical action of the the body; fluid in which it may be desirable to immerse it. Nicholson's lycdrometer consists of a hollow metallic ball Nicholson's A, through the centre of which passes a metallic wire, hydrometer; prolonged in both directions beyond the surface, and supporting at either end a basin B and B'. The concavities 524 NATURAL PIHILOSOPH Y. of these basins are turned in the Fig. 312. same direction, and the basin B' is B made so heavy that when the in- c description, and strument is placed in water the conditions the stem CC' shall be vertical, and a instrument must satisfy. weight of 500 grains being placed ) in the basin B, the whole instrument 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 provided with weights similar to those of the Hydrostatic Balance. Process for ~ 274.-(1). If thle body be solid, insoluble in water, and finding specific Will sink in that fluid, attach it, by means of a hair, to the gravity of a solid heavier than hook of the basin of the hydrostatic balance; counterpoise watelbythe 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 temperature 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 corresponding 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 = ),, x. (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 lighter than the like, attach to it some body, as a metal, whose weight wain the air and oss of weight in twater are previously in the air and loss of weight in the water are previously MECHANICS OF FLUIDS. 525 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- described; mersed in the water. Fromn 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. grs. A piece of wax and copper in air = 438 = W + V', example; Lost on immersion in water - - = 95.8 = w + w' thecaseof wax; Copper in air - = 388 = W', Loss of copper in water - - = 44.2 = w'. Then W + W' - W' = 438 - 388 - 50 = W; w + w' - w' = 95.8 - 44.2 = 51.6 = w. Temperature of water 43.25, Di,, = 0.999952, hfV 50 -pecif gravity S = D,, X 0.999952 56 - 0.968. pecificgaity wII 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 the slbnldaid 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 526 NATURAL PHILOSOPHIY. and find the apparent specific gravity with the water thus saturate the saturated. Multiply this apparent specific gravity by the boyiadthpoe h density of the saturated fluid, and the product will be the as before; specific gravity referred to the standard. This is a common method of finding the specific gravity of gunpowder, the water being saturated with nitre. (4). If the body be a fiqtuid, select some solid that when the body is will resist its chemical action, as a massive piece of glass a liquid; 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, prorule; 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. Example. grs. Loss of glass in water at 410, 150 = w', example; e" " sulphuric acid, 277.5 = w, specific gravity 277.5 X 0.999988 = 1.85. of sulphuric acid; 1]50 (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 exa 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 process; of water; divide the weight of the gas by that of the water, and multiply by the tabular density of the water 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 corrections are made, will be discussed when we come to treat moisture; of the properties of elastic fluids. To find the specific gravity of a solid by means of Nicholson's Hydrometer, place the instrument in water, mode ofusing and add weights to the upper basin till it sinks to the mark Nicholson, 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 instrument, 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 mark as before; add these weights to the weight of the instrument, the sum will be the weight of an equal volume 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 specific gravity. The scale ~ 275.-Besides the hydrometer of Nicholson, which areometer; 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 Scade-Areormeter. It consists, generally, of a glass vial-shaped vessel A, terminating at one end in a long Fig. 313. slender neck C, to receive the scale, description; 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 application 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 thisinstr.ument that of the body; for when the body is at rest its weight and that of the displaced fluid must be equal. Denoting 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 conditions of equilibrium; g VD g' D' in which g denotes the force of gravity, the first member the weight of the instrument, and the second that of the MTECHANICS OF FLUIDS. 529 displaced fluid. Dividing both members by D' V, and omitting the common factor g, we have IZ) V ratio of densities equal to that of -D V the volumes; 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 1V', or that of the displaced fluid; and in proportion as D' increases in respect to D, will V' diminish in respect to TI that is, the solid will rise higher and higher out of the fluid in proportion as the density of the latter is increased, 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 the scale; 1; the instrument is then placed in some heavy solution of salt, whose specific gravity is accurately lknown 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 process being repeated for rectified alcohol, will give another point towards the opposite extreme of the scale, which may be completed by graduation. To use this instrument, it will be sufficient to immerse it in a fluid and take the number on the scale which coincides with the surface. To bring into view the circumstances which determine use; the sensibility both of the Scale-Areometer and Nicholson's Hydrometer, let s denote the specific gravity of the fluid, sensibility of the c the volume of the vial, 1 the length of the immersed instrument; portion of the narrow neck, r its semi-diameter, and w the total weight of the instrument. Then will r r2, denote the area of a section of the neck, and n r2j, the volume of fluid displaced by the immersed part of the neck. The weight, 34 530 NATURAL PHILOSQ?;P HY. therefore, of the whole fluid displaced by the vial and neck will be weight of fluid C + Sr2l displaced s but this must be equal to the weight of the instrument, whence condition of the equilibrium; W - (c + f r2 1), from which we deduce w specific gravity; S - + C + r2 1 length of neck W (231). immersed;'r 2 S Now, immersing the instrument in a second fluid whose specific gravity is s', the neck will sink through a distance 1', and from the last equation we have length immersed W - SC for second fluid; r r2 S' subtracting this equation from that above and reducing, we find difference of'W /S S specific gravity; 1 - The difference I- 1' 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 instrument more sensible, therefore, in proportion as w is made inference; greater and r less. Whence we conclude that the Areometer is the more valuable in proportion as the vial portion is made larger and the neck smaller. sensibility of If the specific gravity of the fluid remain the same, Nicholon's which is the case with Nicholson's Hydrometer, and it hydrometer; becomes a question to know the effect of a small weight MECIHANICS OF FLUIDS. 531 added to the instrument, denote this weight by w', then will Eq. (231) become w + wI - sc ~ r2; subtracting from this Eq. (231), we find r o' _- t=_ w q r2s 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 385~ F.=1.000.] Name of the Body. Specific Gravity. I. SOLID BODIES. (1) METALS. Antimony (of the laboratory) 4.2 - 4.7 Brass -- 7.6 - 8.8 Bronze for cannon, according to Lieut. Matzka 8.4I4 - 8.974 Ditto, mean 8.758 Copper, molten - - - 7.788 - 8.726 Ditto, hammered -8.878 - 8.9 Ditto, wire-drawn - 8.78 Gold, molten - I9. 238 - I9. 253 Ditto, hammered - 19.36I - I9.6 Iron, wrought - 7. 207 - 7.788 Ditto, cast, a mean 7. 25I Ditto, gray -- 7. 2 Ditto, white - - 7.5 Ditto for cannon, a mean - - 7.21 7.30 Lead, pure molten - - -.3303 Ditto, flattened - I I. 388 Platinum, native - - - I6.o - I8.94 Ditto, molten - 20. 855 Ditto, hammered and wire-drawn 2I.25 Quicksilver, at 320 Fahr. - - I3.568 - I3.598 Silver, pure molten. 10. 474 Ditto, hammered - io.5i - 10.622 Steel, cast - -7.9I9 Ditto, wrought - 7.840 Ditto, much hardened - 7.818 Ditto, slightly. - 7.833 Tin, chemically pure - 7. 29I Ditto, hammered - 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 Zinc, molten 6.86i - 7.215 Ditto, rolled - - - 7. I 9I (2) BUILDING STONES. Alabaster - - 2.7 - 3. o Basalt - - 2.8 - 3. Dolerite - -- 2.72 - 2.93 Gneiss - -- 2.5 - 2.9 Granite - - 2.5 - 2.66 Hornblende --- - 2.9 -3. I Limestone, various kinds 2.64 - 2.72 Phonolite - - 2.51 - 2.69 Porphyry -- 2. 4 - 2. 6 Quartz -- 2.56 - 2.75 Sandstone, various kinds, a mean - 2.2 - 2.5 Stones for building I.66 - 2.62 Syenite - 2.5 - 3. Trachyte - -.4 - 2.6 Brick I1.4I - 1.86 (3) WooDs. Fresh-felled. Dry. Alder - o. 8571 o. 5ooi Ash - o0.9036 o.644o Aspen.0.7654 0.4302 Birch -0.9012 0. 6274 Box - -- 0.9822 0.5907 Elm - o0.9476 0.5474 Fir -- o0.8941 o.555o Hornbeam -.- -0.9452 0.7695 Horse-chestnut - - -o.86i4 0.5749 Larch -.- o.9206 0.4735 Lime - o. 8 70 o. 4390 Maple -. o.9036 o0.6592 Oak- I.0494 o. 6777 Ditto, another specimen - - - I. 0754 0.7075 Pine, Pinus Abies Picea - - - o. 8699 0.47 6 Ditto, Pinus Sylvestris 0.9121 0.5502 Poplar (Italian).0.7634 o. 393I Willow 0.7 55 0.5289 Ditto, white - o.9859 0.4873 (4) VARIOUS SOLID BODIES. Charcoal, of cork o. I Ditto, soft wood o 0.28 - 0.44 Ditto, oak I. 573 Coal-. - 1.232 - I.5IO Coke - 1.865 Earth, common - i- 1.48 rough sand - I. 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 TABLE-continued. Name of the Body. Specific Gravity. Flint, dark - - - - - 2.542 Ditto, white - - - 2.741 Gunpowder, loosely filled in coarse powder - - - - o. 886 musket ditto - - - 0.992 Ditto, slightly shaken down musket-powder -- I. 069 Ditto, solid 2.248 - 2.563 Ice - - - - - - o. 916 - 0. 9268 Lime, unslacked - - -. 842 Resin, common - - - I. 089 Rock-salt - -- 2.257 Saltpetre, melted - - 2.745 Ditto, crystallized - - - - I. 900 Slate-pencil - - - - - -.8 - 2.24 Sulphur - - 1.92 - 1.99 Tallow - -0.942 Turpentine.. o. 99I Wax, white o. 969 Ditto, yellow -- o0.965 Ditto, shoemaker's - -o.897 II. LIQUIDS. Acid, acetic -- I.0o63 Ditto, muriatic - I. 2 11 Ditto, nitric, concentrated - - - -.52 -.522 Ditto, sulphuric, English.- - - -. 845 Ditto, concentrated (Nordh.) - - - I. 86o Alcohol, free from water - - - - 0.792 Ditto, common - o0.824 - 0.79 Ammoniac, liquid - - o.875 Aquafortis, double - - -. 3oo Ditto, single - I. 200 Beer -. - I. o023 - I.o34 Ether, acetic - - - - - o. 866 Ditto, muriatic - - - o. 845 - 0.874 Ditto, nitric - - - o. 886 Ditto, sulphuric o- 0.75Oil, linseed - - 0.928 - 0.953 Ditto, olive 0.915 Ditto, turpentine -0.792 - 0.89I Ditto, whale - o. 923 Quicksilver - I3. 568 - i3.598 Water, distilled- - -. oo000 Ditto, rain - - I..oo3 Ditto, sea - - I. 0265 -.0o28 Wine - - 0.992 - i.o38 Barometer III. GASES. Water = 1. 30 In. Temp. 38i0 F. Temp. = 340. Atmospheric air =- 7-0 -- - - o.ooI30 1.0000 Carbonic acid gas -. o.o098 1.5240 Carbonic oxide gas - 0.001 26 o0.9569 Carbureted hydrogen, a maximum o -.ooI27 0.9784,Ditto, from coals - o.- -ooo39 o.3ooo o.ooo00085 0.5596 534 NATURAL PHILOSOPHY. TABLE-Continued. Name of the Body. Specific Gravity. Barometer Water =- 1. 30 In. Temp. 38~0 F. Temp.= 340. Chlorine - o. 00321 2.4700 Hydriodic gas - o. 0o577 4. 443o Hydrogen - 0 o. oo0895 o. o688 Hydrosulphuric acid gas o.ooi55 I.I9I2 Muriatic acid gas - - - o. 0062 I. 2474 Nitrogen - - o. ooI 27 0. 9760 Oxygen - - o.ooI43 I.1026 Phosphureted hydrogen gas o. ooI3 o. 8700 Steam at 212~ Fahr. - - - o.ooo82 0.6235 Sulphurous acid gas - 0 o.00292 2.2470 use of a table of The knowledge of the specific gravities or densities of specificgravities; different substances is of great importance, not only for 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 weight of any body; W gD V' in which g is the force of gravity, D the density, V the volume, and IrT the weight, of which the unit of measure is the weight of a unit of volume of water at its maximum density. Making D and V equal to unity, this equation becomes WJ = g; 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 weight of a cubic foot of distilled lbs. water at 62.5 Dbs. maximum = 62.556; density; 0.99914 IMECIANI'CS OF FLUIDS. 535 whence, if the volume be expressed in cubic feet, volume in cubic feet; lbs. weight of a body W = 62.556 x D V... (232), in pounds, volume being in cubic feet; in which W is expressed in pounds; and if the unit of volume be a cubic inch, weight in 62.556 pounds, V -- D V = 0.036201 D),.. (233). vole in cubic ~~~~~~1728~~~ ~ ~inches; Also:WV. volume in cubic V-~= lbs....(234), feet; 62.556. D = W. volume in cubic ls = lb,.... (235). 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. weight of 50 TW = 0.036201 x 0.555 X 50 = 1.00457. cubic inches of fu; Exazmple 2d. How many cubic inches are there in a examplesecond; 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 volume of a 120.036201 45.6. pound cannon0.036201 x 7.251 bal=5 636 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. Gases and vapors differ mostly from liquids, in the contract and readiness with which they yield a portion of their volume toxpandecording and contract into smaller spaces when subjected to an 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 conditions of to be at rest, unless these forces are opposed by the reaction rest; of inclosing surfaces, as the sides of vessels, or the application 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 no marked aggregation. Between solids and liquids, a gradation is variety of observable, and in the degree of fluidity of the latter, a fluidity; strongly marked variety obtains-as in tar, oil, water, ether, and the like; but between compressible and incomusually pressible fluids, no similar connecting links are found. transparent; 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. 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 production of phenomena, makes it not less interesting than important to determine the laws of its equilibrium and motion. (1) The compressibility and elasticity are easily shown omnmpremsibility by inclosing the air in a bag of some impervious substance, sd elasticity shown; as india-rubber, and pressing it with the hand; the hand will experience a resistance, while the volume of the confined air will diminish: on removing the hand, the bag will be distended by the elasticity of the air, and restored to its former dimensions. Air-pillows and Fig. 314. india-rubber hag; cushions, in common use, are familiar illustrations. (2) A is a two-necked bottle containing some liquid, as water, B an inflated bladder, or india-rubber india-rubber bag bag, attached by the neck to one of connected with a the mouths. A glass tube ca b, open two-necked 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. (3) Hero's Ball.-A hollow globe Fig. 315. a, from which the external air can e be excluded by turning a cock b, contains a tube that reaches nearly Hero'sball; to the bottom, and fits in the neck by a screw. Fill the vessel about half full of water, screw in the tube c d, 538 NATURAL PHILOSOPHY. breathe through c, and close the Fig. 315. stop-cock b; the breath will ascend e through the water, mingle with the air in the space a, take from it a portion of its volume and thus in- L 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 princiDprincipleofthe ple depend the operations of the air-chamber in fireie-engine~; engines and similar machines. (4) Hero's Fountain.-In this apparatus, also, the compression of air and consecuent 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- Fig. 316. 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 a the vessel a, and extends nearly to its bottom, as in Hero's Ball. Upon the top of this vessel is a basin n o, from the bottom of which a pipe ef, open at both ends, passes clear through, nearly to the bottom of description; the vessel g. The tube c d, being unscrewed, is removed, and after pouring water into the vessel a till its surface comes nearly to the upper end of the tube t, the pipe c d is replaced, and the stop-cock b closed. mode of action; Water is now poured into the basin n o; this will descend through the tube ef 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 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 rest; the increasing elasticity of the air becomes equal to the pressure arising from a head of water equal to the difference 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 c d. (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 compressed, the figure will descend, and rise again when the compression Fig. 317. 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- description; 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 bladder 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, motion of the the air will be compressed, the increased elasticity thus pro- figure; 54A0 NATURAL PHILOSOPIEY. 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 reexplanation of moving the finger, the air above the water as well as that the motion. 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. VIII. THE AIR-PUMP. Air-pump, or ~ 277.-Seeing that the air expands and tends to diffuse air-syringe; itself in all directions when the surrounding pressure is lessened, it may be rarefied and brought to almost any degree of tenuity. This is accomplished by an instrument called the Air-Pwtnp or Exchausting 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 the receiver; 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 rn n imbedded in a metallic table, on which it stands. 2d. A Bcarrel 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 comthe barrel and munication ofg. An air-tight piston P is made to move piston; backl and forth in the barrel by means of the handle a. MECHANICS OF FLUIDS. 541 Fig. 318. 7t 7 hgraphical 3d. A Stop-cock h, by means of which the communication 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 perforations 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 Fig. 319. direction of the axis from the smaller end, and as it m =,descriptionof approaches the first, i- stop-cock; dines sideways, and runs out at right angles to it, as indicated in the second figure. In this position 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 wvill rush out, in the direction indicated by the arrow-head, through the comimunication of g, into the vacant space within the barrel. The air which now occupies both the 542 NATURAL PHILOSOPHY. barrel and receiver is less dense than when it occupied the receiver alone. Turn the cock a quarter round, the commode of munication between the receiver and barrel is cut off, and operation; 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 TO find the degree of exhaustion after any number of of exhaustion; 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 receiver 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 ratio of the densities; r + b: r 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 - - r + p density; 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 - -pJ + b'r F jr:: D + +.' t; /ratio of densities; in which x' is the density after the second backward motion of the piston, or after the second double stroke; and we find xi D / D + p second kr +.p + b' diminished; and if n denote the number of double strokes of the piston, and xn the corresponding density of the remaining air, then will the nth = D - + " ) diminished r - p +- b 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 compared with the receiver and pipe, and after a few double the air can never strokes, the rarefaction will be sufficient for all practical be wyholly purposes. Suppose, for example, the receiver to contain the receiver; 19 units of volume, the pipe 1, and the barrel 10; then will r + _p 20 2. r+p+b 30' and suppose 4 double strokes of the piston; then will llustration, n= 4, and r + p 2 4 16 density after 4th c+g)- ( 3-)4 - 0.197, nearly; double stroke; )~~~~~~~~~~~a.~lr~ 544 NATURAL PHILOSOPHY. that is, after 4 double strokes, the density of the remaining rarefaction by air will be but about two tenths of the original density. best pumps; p With the best machines, the air may be rarefied from four to six hundred times. The degree of rarefaction is indicated in a very simple gauges; manner by what are called gczges. 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. objects, and The mode in which the gauge acts, will construction; be better understood when we come to discuss the barometer; it will be suffi- Fig. 320. cient here simply to indicate its construction. 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 terminating in this end is filled with mercury. A scale of equal parts is placed between the branches, having its zero at a point scale of the midway from the top to the bottom, the pogaugeitiond 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 airpump. Repeated attempts have been made to bring the airpump to still higher degrees of perfection since the time first inventor; 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 pump was, that the atmospheric air could not be entirely the most serious ejected from the barrel, but remained between the piston defect of the older pumps; 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 communication re-establish'ed with the receiver, this portion of air forces its way in and diminishes the degree of rarefaction Fig. 321. section of one of the most approved pumps; already attained. If the air in the receiver is so far rarefied, 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 sooner, in proportion as the space below the piston is the defect abondueto 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 bardescription of the rel of the pump; within it the piston works perfectly airimproved 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; toothed, is elevated and depressed by means of a cog-wheel use of oil; that is turned by the handle ilM. 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 shape of the air-tight. To diminish to the utmost the space between isto-rod;dof 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 indefinitely small, the oozing of the oil down the barrel contributing still further to lessen it. The exchange-cock Ei exchange-cock; has the double bore already described, and is turned by a short lever, to which motion is communicated by the rod c d. The communication G(H is carried to the two plates comnmunication; I and Ki on one or both of which receivers may be placed; the two cocks N and 0 below these plates, serve to cut off cut-offcock the rarefied air within the receivers when it is desired to leave them for any length of time. The cock 0 is also an exchange-cock, so as to admit the external air into the cockto readmit receivers. the air; Pumps thus constructed have advantages over such as advantages of work with valves, in that they last longer, exhaust better, this kind of pmnp. 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, showing 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 contained in a vessel, and the whole be placed under the 5~48 NATURAL PHILOSOPHY. receiver of an air-pump; when the Fig. 322. air within the receiver is rarefied, showing also the that which was contained in the expansion ofai; 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 one flask to another. Let there be a fluid in the flask A. The' neck of this flask contains a glass tube fitted air-tight into it, and reachthird, illustrating ing almost to the bottom; the tube being bent twice the spme at right angles, the other end principle; passes freely through the neck of a Fig. 323. second bottle B. Place this appa- a c _ ratus under the receiver of an airpump, and exhaust; the fluid will l mount up from the bottle A and m 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 fourth. 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 abandoned 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- Fig. 324. Atmospheric sembling the arms of a windmill, resistance; with this difference only, that the illustrated; 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 contrivance a rapid rotary motion is com MECHANICS OF FLUIDS. 549 municated to them. In order that this may act under a receiver, a rod must be made to pass through an air- descriptionand tight leather stuffing-box e; at the end of the rod is a use of the instrument. 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. ~ 280.-The atmosphere is the ordinary medium through Effects of rarefaction on which sound is transmitted to the ear. In proportion as sound; the air becomes more rarefied, the transmission of sound through it becomes more feeble. 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 instrument by which this may considerably varied: a is the bell, Fig. 325. be illuchthitrated; 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 A upper wooden plate c, and into a lower plate d, by which the whole r apparatus is fastened down to the I_ plate of the pump; lastly, h is the lever by which the clapper is agitated. 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. 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 5I0 NATURAL PHILOSOPHY. ensue unless air be admitted. Varm- Fig. 326. place a bird in blooded animals, as birds, die if the receiver ofa rarefaction be carried to a small pump and exhaust; degree; cold-blooded animals, on the contrary, endure a high degree of rarefaction. Many birds ascend to considerable heights in the atmosphere, and it may be hence inferred 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, combustion. place it beneath the receiver and exhaust qSlickly, after it has been replenished with fresh air; the flame will expire much sooner than before. To the same cause it is owing that in vacuo no light is produced by striking a flint and steel together. 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 has weight; 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 ascertain its weight when filled with air. Exhaust the air, Fig. 321. by means of the air-pump, and the flask will be founcd _ experiment to lighter than before; readmit showthis; the air, it will regain its for- 7 A mer weight. Force into the flask an additional quantity of air, by means of the airpump, used as a condenser, and the weight will be found to be increased. Since the atmosphere has weight, it must exert a pressure upon all bodies in it. To illustrate the truth the air exets 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- Fig. 328. 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 suspended nearly 30 inches experimt e this; above the level of the mercury in the basin. If we consider the circumstances attending this experiment, it will be seen that the tube containing the mercury forms with the basin a system of communicating tubes, as in ~ 259. 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 withdrawing the from the surface of the mercury in the basin, and, by the atmosphere; 552 NATURAL PHILOSOPHY. 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 atmosphere, and the mercury in the tube will again rise to Fig. 329. an instrument its former height. This is exhibit te facts well illustrated by the folof this lowing device. R is a reexperiment; ceiver closed air-tight at the top by means of a metallic b m plate; a is a tube filled with mercury after the manner just described, and terminating 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 description and tube a, it terminates at one end in a vial-shaped vessel, use; 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 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 towards 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. From this we see, that the atmospheric air presses on inference from the mercury, and indeed upon the surfaces of all bodies this experient; exposed to it, with a force sufficient to maintain the quiclk 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 sujface 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, Iorricellian tubes, and the vacant space above the mercury Torricellian in the tube is called, the Torricellian vacuTn, to distinguish tubes; Torricellian 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 of one square inch, the atmospheric pressure up the tube atmospheric will be exerted upon this extent of surface, and will sup- pressure at the 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 atmos1pheric pressure of fifteen pounds to each square inch. The body of a man of ordinary stature has a surface of pressure upon the surface of a about 2000 square inches; whence, the whole pressure to man; 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 entiohere; extent of its surface in square inches, and to multiply this by fifteen; the product will be the weight in pounds. 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 atmospheric mercury being 13.6 referred to water as a standard. This pressure; 554 NATURAL PHILOSOPHY. has been verified by Hanson and Sturm, who actually performed the experiment at Leipzig. The atmospheric pressure is exhibited in a most striMagdeburg king way by means of the Mllcagdeburg hernisplheres. These hemispheres; 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 hemispheres together, connect them with description and the communication-pipe of the air- Fig. 330. g; pump, exhaust the air, and turn the stop-cock, and clisconnect from the pump. It will be found that great force will be necessary to pull the _:__ hemispheres asunder. If the diame- V 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 examples of 24\ Guericke's - 3.1416 X = 452.39, hemispheres; 2 and the force, estimated in pounds to overcome the pressure, would be 15 X 452.39 = 6785.85. Fig. 331. In the experiment referred to above, there were successively from 14 to 30 horses harnessed to the hemispheres, without effecting the separation. B the forcing of The pressure of the atmosphere fluid through will force fluids through such solid pores of solids; bodies as are porous. Let R be a long receiver, provided with a tube MECHANICS OF FLUIDS. 555 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 exhibit 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 receiver, and to curve over at the top to prevent the mercury from falling into the communication-pipe of the pump. The atmosphere presses not only atmospheric downward, but upward, and later- pressure is exerted in every ally in all directions. This is shown Fig. 332. direction; by the following experiment: The two hemispheres A and B, are con- c nected by a tube in such manner that one of them mlay turn about a joint C, while the other is stationary. Place the hemisphere A upon the plate of the air-pump, and upon B lay a plane plate of glass or metal fitting it air-tight. Exhaust 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 si'phons 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- Fig 333. exemplification verted positions upon the ceilings of this in the of rooms, &c. The feet being flat gsn adhesion of and flexible, are brought close a- and ceilings. gainst the wall or ceiling so as to exclude the air, the centre of the 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. 556 NATURAL PHILOSOPHY. X. MARIOTTE S LAW. Mariotte's law; ~ 282.-We have seen that the atmosphere readily contracts into a smaller volume when pressed exter- Fig. 884. nally, that it as readily regains its former dimensions when the pressure is removed, and that it is, therefore, both compressible and elastic. Let us now consider the connecting the law which connects the pressure, pressure, den density, and elasticity. For this purand elasticity; pose, procure a siphon-shaped tube AB D, open at A, the end of the longer branch, and hermetically sealed at the end D of the shorter branch. Place between the branchthe instrument es, and parallel to them, a scale of carlompressing equal parts, say inches, having its the air; zero on the line o o. Pour in, at the open end A, as o much quicksilver as will fill the horizontal part of the tube, and bring its upper surface to the zero 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 the shorter branch, and cause the air above it to be compressed 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 expertionimenofiment;he experiment; 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 primitive volume, the difference of level of the mercury in the two branches is just 30 inches, thus making the compressing force double what it was before; that when it is" compressed into one third of its original volume, the difference 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 volumnes occupied by it are in- volumesare to. 7 7inversely versely proportiona to the pressures. vernal to proportional to This law holds equally when the the pressures; air, instead of being compressed, is per- Fig. 335. mitted to expand. Let a b be a glass tube, about 33 inches long, one end a, being fitted with an air-tight cock, and instrument for the entire length of the tube being expanding the graduated in inches. Open the cock a, air 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 20 and without the tube. Now close the cock a, and so confine a portion of air 6 description and at its ordinary density within the tube' mode of using; above the surface of the mercury. Elevate the tube any distance whatever, taking care that its open end shall be below the surface; the air will expand, and fill a larger portion of the 558 NATURAL PHILOSOPHY. tube, though a column of mercury will still stand at an elevation above the outer level, so that the weight of this column, with the elastic force of the inclosecl air, counterweight of the balances the natural pressure of the atmosphere. The presolumpended of sure therefore which the included air sustains, is equal to mercury plus the weight of a column of mercury 30 inches high, minus elastic force of confined air, that of the column supported in the tube. Let the equal to spice full of air above the mercury in the closed tube atmospheric pressure; 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 becomes 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. experiments By experiments made at Paris, it has been found that made at Paris; this law obtains when air is condensed 27 times, and rarefied 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 produce a density unity, p the pressure corresponding to a density D, then, according to this law, will, expression of = PD.(236). Mariotte's law P (26). 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. 059 XI. LAW OF THE PRESSURE, DENSITY, AND TEMPERATURE. ~ 283.-It is a universal law of nature that heat ex- Law connecting pC ads all bodies, and is ever active in producing changes the pressure, of density. We have now to consider the law of this temperature; chainge in air. It has been ascertained, experimentally, that air, subjected to any constant pressure, will expand 0.00208th of rate of the air's its volume at 320 Fahr., for each degree of the same scale expansion; above this temperature; so that if V1 be the volume of the air at 320, and V its volume at any other temperature t, then will V = V [1 + (t - 320) 0.00208)].. (237). If D, be the density at 320, 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 value for any 1 V~1 [1 + (t 320) 0.00208] D D,;,tempelatuen pressure; whence density at any 1 P (t - 320). 0.00208 under a constant pressure; If p, denote the pressure necessary to restore this air to the density D,, we shall have from Mariotte's law D, 1 + (t - 320) 0.00208 560 NATURAL PHILOSOPHY. whence pressure to produce at a given = [1 + (t- 32) 0.00208].. (239). temperature a density at 320 under a given Again, let the pressure p be produced by the weight of plessure; a column of mercury, having a base unity, and an altitude h,,, taken at a given latitude, say that of 450, in order that the force of gravity may be constant. Denoting the density of the mercury by D,,, its weight will be weight of a column of ]), h1i g; mercury; in which g' denotes the force of gravity at the latitude of 45o. Substituting this for p, in Eq. (236), we have D,, h,, g' = P D; whence pressure to D A, produce a unit of P - D' density; and substituting the value of D, given in Eq. (238), this becomes same in different p _ D,, h,, g form; D From Eq. (236), we have D-Jpi and substituting the value for P above, we get density at 320 under a constant ) p D pressure; D,,i h,, g' [1 + (t - 32~) 0.00208]' MECHANICS OF FLUIDS. 561 Denote by A the height of the column of mercury at 32~, necessary to produce upon a unit of surface the pressure p, then will weight of a column of p = D,9A, h gy; mercury equal to the constant pressure; which, substituted for p above, gives, after striking out the common factors, D, h density at 320 -D under a constant i,, [1 + (t - 32o) 0.00208]' pressure; 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, density answering to a D, given 30 1 + (t - 32~) 0.00208 (240)'; emperature and barometric column; 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. FExamnple. 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 illsstrate the use of this formula; 0.0013, whence 0.0013 26 D X 0.0011. 30 1 + (420 - 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. It will be recollected that, when referred to the same to obtain the standard, the numbers which express the specific gravities tabular spcific, of bodies also express their densities, and that the specific &c.; 86 562 NATURAL PHILOSOPHY. 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 of any body; standard substance. The gases and vapors are incessantly changing their densities, on account of the varying pressures and temperatures to which they are subjected. Tabulated densities must, therefore, correspond to a standard standard of temperature and of pressure. Thirty-two temperature,nd degrees Fahrenheit's scale is adopted for the former; and pressure for density ofgases, the weight of a column of mercury, at the same tempera&c. j ture, having an altitude equal to thirty inches, and resting upon a base whose area is a superficial unit, is taken for the latter. By a very simple transformation of Eq. (240)', we find tabular value for D 30 X [1 + (t - 320) 0.00208] x D. density; h 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- Fig. 336. tities, by the following process: Provide a glass vessel A, whose mouth may be closed by a stopcock B, air-tight, and of which the bottom terminates in a long a vesselforfinding narrow tube 0, closed at the end. the weights and Let the capacity of this vessel be volumes of gases, carefully ascertained by filling it with water, and pouring this water afterward into a graduated vessel;, o also let the tubular portion C be graduated and numbered by tenths, hundredths, &c., so that the num- 0 bers shall increase towards the smaller end, and express that portion of the entire capacity MECHANICS OF FLUIDS. 563 of the vessel, regarded as unity, which is comprised between its mouth B and these numbers. This being understood, denote the weight of this vesgel 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,; the buoyant effort of the atmosphere, under the same pressure and temperature, by e; and the weight required to counterpoise the vessel filled with air by W1, then will weight of vessel Connect with the air-pump, and exhaust as far as convenient; 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 weight of vessel W2 = WI + Wa, - le; filled with riarefied air; in which 1W,, denotes the weight of the rarefied air remaining in the vessel. Subtracting this from the equation above, we find ~~Wl~ - W2 = Wa - W-a~,; ~ weight of the extracted air; which is obviously the 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 W,,, may be 564 NATURAL PHILOSOPHY. found from the proportion V W - W. 1 W.; whence weight of the vessel filled with Wt -W air under the -- _ a. (b). barometric V pressure; Next fill the vessel with water, and weigh again; denote the counterpoising weight by W3, and the weight of the contained water by WO, and we shall have W3 = WV + Wo -e; and subtracting Eq. (a), we find W3 - WI =w- Wa; adding Eq. (b), we find weight of the WI - V vessel full of - Wa- + X water; V and dividing Eq. (b) by this one., we get ratio of the weights of equal W1 V2 Wa volumes of water (W3 - TV) V + V - W2ii and gas; 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 W, - V2, we finally get density of the -- X Vd. air; W3- WV +1 W1 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 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 processfor water, the volume of gas exhausted by the pump; fill finding the density or specific with water, and weigh a third time; then divide the dif- gravity of a gas. ference between the last and first weights by the difference 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 substituted in the value for ),, this latter may be found and tabulated. XII. BAROMETER. ~ 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 atmosphere at 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 mountain, 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 air, and its atmosphere. These, together with its ever-varying tem- change of. perature, keep the density and elastic force of the air in a density; state of almost incessant change. These changes are indi 566 NATURAL PHILOSOPHY. barometer; cated by the Barometer, an instrument employed to measure the intensity of atmospheric pressure, and frequently called weather-glass; a weather-glass, because of certain agreements found to exist 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 description of containing mercury. A scale of equal parts is cut upon a the barometer 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 cistern, 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 consequent elastic force of the air be increased, the column of mercury will rise till it attain a corresponding increase of weight; Fig. 337. column of if, on the contrary, the density of the air mercury in equilibrio with diminish, the column will fall till its di-.71 atmospheric minished weight is sufficient to restore 30 pressure; the equilibrium. 29 In the Common Barometer, the tube and its cistern are partly inclosed in a common metallic case, upon which the scale is mountain barometer; 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 Sipfhon Barometer consists 1= 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 bTorricellian tube; the shorter is open, and on the surface of the quicksilver MECHANICS OF FLUIDS. 567 the pressure of the atmosphere is exert- Fig. 338. ed. The difference between the levels moveable or sliding scale; in the longer and shorter legs is the sliding scale; barometric height. The most convenient 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. Different contrivances have been adopted to render the minute variations in the atmospheric pressure, and consequently in the height of the barometer, o different devices more readily perceptible by enlarging fer appreciating the divisions on the scale, all of which barometric devices tend to hinder the exact meas- column; 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-Barometer. The essential properties of a good barometer are: width of tube; purity of the mercury; accurate graduation of the scale; and a good vernier. 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 atmospheric pressure will sustain the same weight-in other words, the same quantity of mercury; but the same quantity of mercury will occupy different volumes, according efects of to its temperature, and the same atmospheric pressure will, temperature; hence, sustain a longer column when the temperature is high than it will when the temperature is low. The indications 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 that mercury expands gli-th part of its volume for each mercury; 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 observation; b the altitude which the barometer would have at the standard temperature, and b' the observed altitude, then will, barometric column reduced T-T1] to standard b=b' [i~ = b' [1~ (T- T+) 0.00010011 (241); to standIard 9990 temperature; when T' becomes 1 b' will be equal to b. attached A thermometer is usually attached to the barometer thermometer; 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 example for 930~. What would have been the column under the same illustration; pressure, had the temperature of the mercury been 32~? Here we have in. b' = 29.81,',= 93.00, data; 0 T = 32.00, T — T'= - 61.00; and in. in. reduced column. b = 29.81 [1 - 61 X 0.0001001] = 29.63. Barometer used ~ 285.-The barometer may be used not only to measto measure the elasticity of ure the pressure of the external air, but also to determine confined gases the density and elasticity of pent-up gases and vapors, &c.; and furnishes the mnost; direct means of ascertainoing MECHANICS OF FLUIDS. 569 the degree of rarefaction in the receiver of an air-pump. When thus employed, it is called the bcarometer-gauge. barometergauge; 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 its use and temperature, will give the pressure in Fig. 339. application; pounds on each square inch. In the case of the steam in the boiler of an 2 -6 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 1 steam flows and presses upon the mercury; the barometer tube be, open at top, reaches nearly to the bottom of the 1t 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 - scale of inches of atmospheres. and another of atmospheres; If a very high pressure were exerted, one of several atmospheres, for example, Fig. 340. an apparatus thus constructed would require a tube of great length, in which 4~: _ case fariotte's manometer is considered 72 - preferable. The tube being filled with R14 air and the upper end closed, the surface of the mercury in both branches Mariotte's manometer; 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 570 NATURAL PHILOSOP HY. it up the branch b c, and the scale from b to c marks the force of compression Fig. 340. its mode of in atmospheres. The greater width of action. tube is given at a, in order that the.4. level of the mercury at this point may not be materially affected by its ascent up the branch b c, the point a being the / zero of the scale. ~ 286.-Another very important use of the barometer, is to find the differ-'I ence of level between two places on the earth's surface, as the foot and top of a Levelling by hill or mlountain. means of the Since the altitude of the barometer depends on the barometer; 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 plapon the atmosphere, from its utmost limits, presses with its entire height of the barometer; 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. 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 effects of the mean altitude of the barometric column will vary irregularity of the at different places. This furnishes one of the best and earth's si'face; most expeditious means of getting a profile of an extended 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 altitudes of two barometric columns, and the difference of level of the places where they exist, conceive the atmosphere 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 the barometric columns and may without error be regarded as uniform, the density differenceoflevel of the places; varying from one stratum to another. Then, commencing at any elevated position 0, above the level of the sea, denote Fig. 341. 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 point, will be due to this pressure; call this density D, then, from Mariotte's law, Eq. (236), will p = P D; elastic pressure; in which P is the pressure necessary to produce, on a unit of surface, a unit of density. From this equation, we have _0 P density corresponding; The weight of so much of this stratum as stands upon a unit of surface will be gh 1 mweight of a small g h=p*p; column on unit of surface; in which 7A 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, 572 NATURAL PHILOSOPHY. augmented by the weight of this stratum found above; and, denoting this pressure by p', we shall have pressure on unit g h g+h of surface of 1 + p + P second stratum; Denoting by D' the density of the second stratum B, we have again by Mariotte's law P' = PD', or D' P' =TD and for the weight of this stratum upon a unit of surface, weight of second g hD' 2 g h stratum; P and substituting the value of p', found above, same under ghD' = p (I + another form; P P 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 pi", will plessure upon P g=p (1 l (1g+ ( (1J g )\2 unit of surface of P" =- P (-) + P k(1 qI - - -- third stratum; and in the same way will the pressure p"', upon the fourth stratum, be given by the equation same for fourth, + g h stratum; 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 / g h6 \n pressure upon Pn = -p O + p ); unit of surface RI'P~~ ~~of nth stratum; in which p, denotes the pressure at the lowest station. Developing the second member of this equation by the binomial formula, and dividing by p, we have pnu gh n(n-1) g 7 n2 nn -1) (-2) g3h3 ratio of the upper 1 +n+. P --- 2 -3 3 + &. andlower -P P I. 2 P2 1. 2 o 3 P3 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 _n._ n._. g h n g2 g2 82 n g3 h8 p Jr - 1 + nh +r g + 3 +- At &C. same reduced; ~~p P 1.2P2 1. 2. 3P But the second member is equal to nugh e P in which e = 2.7182818, the base of the Naperian system of logarithms. Whence,.-Pn — g same under ____ = e * different form; But n being the number of strata, and h the common height of each, z h will be equal to the difference of level between the first and last points. Calling this z, and taking the Naperian logarithm of both members, we find, after substituting z, log n - z Naperian p p; logarithm; 574 NATURAL PHILOSOPHY. and passing to the common logarithms common H. Log -n g z logarithm; -o - in which M denotes the reciprocal of the modulus of the common system; whence we have difference of f_ Logn level; g 1) Denote by bt the height of the barometric column at the lower station, where the pressure is p,, and by b that at the upper station where the pressure is p, then will ratio of pressures in terms of pn __ n~ barometric columns: and reducing the barometric column b to the temperature of b' taken as the standard, we have, Eq. (241), same reduced to a standard temperature T; 0, Ib [I + (T - T') 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)', value of force of ft. gravity; g = 32.1803 [1 - 0.002551 cos 2 4], or g = g' (1 - 0.002551 cos 2 {); ft. in which g' = 32.1803, the force of gravity in the latitude of 45~. Substituting these values of P, g, and the value of P given by Eq. (240), in the value for D above, and we find value for difference of JMD,, h,, 1 + (t - 32) 0.00208 T 1,n 1 level; D, 1 — 0.002551 cous X 1 q- (T- T') 0.0001001] MECHANICS OF FLUIDS. 575 In this it will be remembered that t denotes the temperature of the air; but this may not be, indeed scarcely ever is, the same at both stations, and thence arises a dif- difficulty arising ficulty in applying the formula. But if we represent, for from difference of a moment, the entire factor of the second member, into the two stations; which the factor involving t is multiplied, by X, then we may write difference of level Z = [1 + (t - 320) 0.00208] X. for constant temperature; If the temperature of the lower statwion be denoted by t,, and this temperature be the same throughout to the upper station, then will temperature z, = [1 + (t, - 32~) 0.00208] X throughout the i-~~~~~~~ ~~~same as at lower station; And if the actual temperature of the uzpper station be denoted by t', and this be supposed to extend to the lower station, then would tempelrature z' = 1 + (t' - 32o) 0.00208]. sae as upper station; 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 valueof nominator of the value X, z, would be too small. Again, difference of level, the true by supposing the temperature the same as that at the one; 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 z' too great. Taking a mean between a, and z' as the true value, we find 576 NATURAL PHILOSOPHY. tlue value for_; t difference of Z - = + (t + t' - 640) 0.00208] X. level; 2 Replacing X by its value, value for difference of -- M D,,, 1 + 2 (t, - t' - 64) 0.00208 XLog. [ 1 D1 leveI; D, 1 — 0.002551 cos 2 b10 The factor z)",, we have seen, is constant, ancd it only remains to determine its value. For this purpose, 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 z; 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,1 h,, = 60345.51 English feet; which will finally give for z, final value for ft. 1 - (t,'J- 64~) 0.00208 b, 1 difference of z 60345.51. (02 Lo. 1 6034.level; 5. 1 — 0.002551 cos 2 X L 1 + (T- T') 0.000100 level; 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 foritsuse; 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 savedby 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, =, I - 0.002551 cos 2 { mode of 1 a computing one; 1 + (IT - T) 0.0001 Then will z= ABE Log. b, z = A B [Log. C + Log. ba - Log. b]; abbreviated formula; and taking the logarithms of both members, Log. z = Log. A + Log. B + Log. [Log. C- +Log. b - Log. b]..(242). its logarithm; Making t, + t' to vary from 40~ to 1620, 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 the head A, opposite the values t, + t', as an argument. Causing the latitude.+ to vary from 0~ to 900, the variations in logarithms of the corresponding values of B are entered latitude; in a column headed B, opposite the values of hr. The value of T - T' being made, in like manner, to vary from - 300 to + 300, the logarithms of the corresponding values of C are entered under the head of C, and opposite the values of T - T'. In this way a table is easily constructed. That subjoined, was computed by variationsin Samuel Howlet, Esq., from the formula of Mr. Francis temperatureof mercury; Baily, which is very nearly the same as that just described, there being but a trifling difference in the coefficients. 37 578 N ATURAL PHILOSOPHY. TABLE FOER FINDING ALTITUDES Detached Thermometer. t,+ t' A t,+t' A t,+ t' A t,+t' A 40 4.7689067 75 4.7859208 IIO 4.8022936 I45 4.8180714 4I.7694021 76.7863973 III.8027525 I46.8I85140 42.769897I1 77.7868733 112.8032I09 147.8i89559 43.770391 78.7873487 ii3.8036687 I48.8I93975 44.77o885I 79.7878236 114.804126I I49.8I98387 45.77I3785 80.7882979 115.8045830 50o.8202794 46.77I87II 8I.78877I9 116.805o395 I51.8207196 47.7723633 82.789245I I17.8054953 I52.82II594 48.7728548 83.7897I80 II8.8059509 153.8215988 49.7733457 84.790I903 II9.8064058 I54.8220377 50.7738363 85.7906621 120.8068604 i55.822476I 5i.774326I 86.791I335 I2I.8073144 I56.822914I 52.7748T53 87.79I6042 122.8077680 I57.8233517 53.7753042 88.7920745 I23.808221I I58.8237888 54.7757925 89.792544I I24.8086737 i59.8242256 55.7762802 90.7930I35 I25.8o09I258 60.8246618 56.7767674 91.7934822 126.8o95776 I6I.8250976 57.7772540 92.793950o4 I27.8I00287 I62.825533I 58.7777400 93.7944I82 I28.8I04795 163 825968o 59.7782256 94.7948854 I29.8109298 i64.8264024 60.7787Io5 95.795352I I30.8II3796 I65.8268365 6i.7791949 96.7958184 I3I.81I8290 166.8272701 62.7796788 97.796284I I32.8I22778 167.8277034 63.780I622 98.7967493 I33.8127263 168.828I362 64.7806450 99.7972I4I i34.8i31742 I69.8285685 65.7811272 100.7976784 I35.8I36216 170.829ooo5 66.78I6090 IOI.7981421 i36.8140688 17I.8294319 67.7820902 102.7986054 137.8145i53 172.8298629 68.7825709 Io3.7990681 i38.8I496I4 173.8302937 69.7830511 Io4.7995303 I39.8I54070 174.8307238 70.78353o6 Io5.7999921 140.8I58523 I75.83ii536 71.7840098 io6.8004533 141.8162970 176.8315830 72.7844883 1o07.8009142 142.81674I3 177.8320II9 73.7849664 Io8.80I3744 I43.817I852 178.8324404 74 4.7854438 Iog09 4.8oi8343 44 4.8I76285 I79 4.8328686 MECHANICS OF FLUIDS. 579 WITH THE BAROMETER. Latitude. Attached Thermometer. B T —T' C C 00 o.ooii689 + _ 3.00ooII624 00 0.0000000 0.0000000 6.0oo1433 I.0000434 9.9999566 9.ooIIrI7 2.0000869.999913i 12.00o0o679 3.oooI3o3.9998697 i5.00II024 4.0001737.9998262 I8.0009459 5.000217I.9997828 21.0008689 6.0002605.9997393 24.0007825 7.0003039.9996959 27.000ooo6874 8.00ooo3473.9996524 3.00ooo5848 9.003907.9996090 33.000ooo4758 Io.000434i.9995655 36.00036i5 II.0004775.9995220 39.0002433 12.0005208.9994785 42.0001223 i3.0005642.9994350 45.0000000 I4.0006076.99939I6 48 9.9998775 i5.ooo65io.999348I 49.9998372 i6.0006943.9993046 50.9997967 17.0007377.99926II 5I.9997566 i8.0007810.9992176 52.9997I67 I9.0008244.999174I 53.9996772 20.0008677.999I305 54.9996381 21.o0009II.9990870 55.9995995 22.0009544.9990435 56.99956i3 23.0009977.9990000 57.9995237 24.00oo04I.9989564 58.9994866 25.ooio844.9989129 59.9994502 26.ooII277.9988694 60.9994I44 27.00II7IO.9988258 63.9993ii5 28.0012143.9987823 66.999216i 29.0012576.9987387 69.9991293 30.ooI3009.9986952 75.9989852 31 0.001oo3442 9.99865I6 81.9988854 90 9.9988300 580 NATURAL PHILOSOPHY. Taking Eq. (242) in connection with this table, we have this rule for finding the altitude of one station above another, viz.: — rule for Take the logarithm of the barometric reading at the lower computing station, to which add the number in the column headed C oppodifference of level with a site the observed value of T- T', and subtract from this sum barometer; 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.4w; 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. Detached thermometer, t' = 70.4; t, = 77.6. example first; Attached T' = 70.4; T = 77.6. Barometric column, b = 23.66; bn = 30.05. What was the difference of level? Here 0O O 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~ ~ - - = 4.8193975 Add B for 21~ - = 0.0008689 height of 6885.1 — s. - 3.8379103; Guanaxuato; whence the mountain is 6885.1 feet high. MECHANICS OF FLUIDS. 581 It will be remembered that the final Eq. (242) was de- barometric duced on the supposition that each stratum of air pressed formula true onl3 when there is no 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. Moreover, 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- sthldtationsde 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 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. Exanmple. 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; t, = 56 and 61. Attached T' = 57.5; T = 56,5 and 63. observed data; in. in. in. Barometric column, b = 28.94; bn = 29.62 and. 29.63. First, to reduce 29.63 inches at 630, to what it would have been at 56.5. For this purpose, Eq. (241) gives in. b(I + T-T' X 0.0001) = 29.63 (1- 6.5 X 0.0001)= 29.611. reduction, 582 NATURAL PHILOSOPHY. Then 29.62 -1- 29.611 i reducedcolumn; b2 = 29.6105, 0 0 temperature at 56 + 61 lowerstation; 2 58.5, o 0, + t' = 58.5 + 57- - = 115.5, O 0 0 T- T= 56.5 -57.5 - = -1. in. To Log. 29.6105 = 1.4714458 Add C for - 1 = 9.9999566 in. 1.4714024 computation; Sub. Log. of 28.94 = 1.4614985 Log. of - - - - 0.0099039 = - 3.9958062 Add A for 115.5 -= 4.8048112 0 Add B for 41.4 = 0.0001465 ft. heightof thehill. 632.07 2.8007639; whence the height of the hill is 632.07 English feet. XIII. PUMPS. ~ 287.-Any machine employed for raising water from one level to a higher, in which the agency of atmospheric Pumps; pressure is employed, is called a Pumnp. There are various MIECHANICS OF FLUIDS. 583 kinds of pumps; the more common are the sucking, forcing, different kinds. and lifting pumps. ~ 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 perforated in the direction of the bore of the Fig. 342. barrel, and the orifice is covered by a valve F called the piston- piston-valve; valve, which opens upward; a similar valve E, called the sleeping-valve, at the B sleeping-valve; bottom of the barrel, covers the upper end of the sucking-pipe. Above the highest point ever occupied X by the piston, a discharge pipe P is in- L X discharge-pipe; serted into the barrel;. -.. the piston is worked by means of a lever E; or other contrivance, attached to the piston-rod G. The distance AA', 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 lowest 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, thereaiston; fore, be forced open, and air will pass from the pipe to the 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 eqnilibriun; 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 diuingthe this excess of elasticity will force open the valve F, and descent of the 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 the result of a 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 piston; 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 limitof fluid which the atmospheric pressure may support, in the play; vacuo, at the place. From a little reflection upon what has been said of the MECHANICS OF FLUIDS. 585 operations of this pump, it will appear that the rise of factupon which water, during each ascent of the piston after the first, depends the rise depends upon the expulsion of air through the pistonvalve 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 compression, the water must cease to rise, and the working of the piston can have no other effect than alternately to com)ress and dilate the same air between it and the surface of the water. To ascertain, therefore, to find the the relation which the play of the Fig. lay tion ofther piston should bear to the other / dimensions; dimensions, in order to make the pump effective, suppose the water to have reached a stationary level -A X, at some one ascent of the piston ------------- to its highest point A', and that, in its subsequent descent, the pistonvalve will not open, but the air below it will be compressed only to the same density with the external hypothesis in air, when the piston reaches its respect to rise of lowest point A. The piston may be -- water; worked up and down indefinitely, within these limits for the play, without 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 confined air, between the level X and A will be when the piston is at its lowest point; B x (b - x - a); 586 NATURAL PHILOSOPHY. the volume of this same air, when the piston is raised to A', provided the water does not move, will be volume of same air expanded B (b - x). when piston is at hihet point; 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, weight of the is measured by column of water BT.. which the first will support; 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 weight supported B I', g. by the second;' 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'gD: BhgD; whence ratio of the hb = x -A- a heights; b - x 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, omitting the common factors, B, D, and g, condition of x + h' = h x -- h equilibrium; b - x or, clearing the fraction and solving the equation in reference to x, we find altitude of point 7 /2 of stopping; X = 2 -4a.. (243) MECHANICS OF FLUIDS. 587 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 stoppage; the contrary, 4 a h 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.: 4ah > b2, or b2 condition of ac > 4 hA incessant flow; 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 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 found by the 13-, this being the specific gravity of mercury referred to water as a standard, and the product will give the value of h in feet..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. in. 28 Barometer, 1-2.333 feet. data; ft. h = 2.333 X 13.5 = 31.5 feet. bPlay =169 f. 4 X resulting limit for Play =a>~ x 31>5 -- 1.341+; 4h 4 X 31.5 play; that is, the play of the piston must be greater than one and one third of a foot. 588 NATURAL PHILOSOPHY. The quantity of work performed by the moter during the delivery of water Fig. 344. quantity of work through the discharge-pipe P, is easily of the moter in t the sucking- computed. Suppose the piston to have pump; allny position, as M, and to be moving upward, the water being at the level L in the reservoir, and at P in the pump. The pressure upon the upper surface of the piston will be equal __M to the entire atmospheric pressure de- -_ noted by A, increased by the weight of the column of water AP', 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 A + B c' g D, of piston; in which g and D are the force of gravity and density of the water, respectively. Again, the pressure upon the under surface of the piston is equal to the atmospheric pressure 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 prcssure on the under surface of A -B c g, piston; and the difference of these pressures will be A + Bc'g D - (A - B g ) = Bg D (c + c'); but, employing the notation of the sucking-pump just described, pressure to be overcome by the c + c' b; power; 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 p, we shall have the resistance which the force F, applied to the piston-rod, must overcome to produce any useful effect; that is, value of the F = B b g D + p. motive force; Denote the play of the piston by 2p, 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; the quantity of work will, by omitting the effort necessary to depress the piston, be PFnp = nup [B b. gD + p]; quantity of work; 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, quantity requisite Fnp = up [62.5. B b +.. (244); to deliver a gven feet; in which p must be expressed in pounds, and may be determined either by experiment in each particular pump, or computed by the rules already given. 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 irreglar in its exerted at all, not more than is necessary to overcome the friction. ~ 289. —What is usually called the lfiting-purnp, does Lifting-pump; not differ much from the sucking-pump just described, 590 NATURAL PHILOSOPHY. except that the barrel and Fig. 345. positions of the sleeping-valve E are placed barrel and pipe, and reversed in this at thebottom ofthe ppe, and pomp; some distance below the surface of the water L L in the H reservoir; the piston may or R may not be below this same surface when at the lowest point of its play. The piston and sleeping valves open L, upward. Supposing the piston, at its lowest point, it I = mode of action; will, when raised, lift the column of water above it, and the pressure of the external air, together with the head of fluid in the reservoir above the level of the sleeping-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 the result of times, a column of water will be lifted to the mouth of the several strokes of the piston; discharge-pipe P, after which every elevation of the piston will deliver a volume of the fluid equal to that of a cylinder 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 by the same rule water in the reservoir and the discharge-pipe. Hence the as for suckingpump; quantity of work is estimated by the same rule, Eq. (244). 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, p will be zero, and that equation reduces to work for one Fp = 62.5 Bp X b; elevation of piston; 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, measure of the useful effect. 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- Forcing-pump; cing-pump is a fur- Fig. 346. ther modification of the simple sucking- _ pump. The barrel B and sleeping-valve E are placed upon the top of the sucking-pipe M. The piston F is without perforation and valve, description; and the water, after being forced into the barrel by the atmo- B spheric pressure without, as in the sucking- action of the piston and pump, is driven by leepingvale; the depression of the piston through a lat- X eral pipe -H into an -- -I_ air-vessel IV, at the air-vessel; bottom of which is 592 NATURAL PHILOSOPHY. second a second sleeping- Fig. 346. sleeping-valve; leeping-valve; valve E', opening, like the first, upward.'p Through the top of the air-vessel a disdischarge-pipe; charge-pipe Kpasses, air-tight, nearly to the bottom. The water when forced into the, air-vessel by the descent of the piston, rises above the lower ___ -=_ end of this pipe, con- B fines and compresses the air, and this, re- - __ acting by its elasticiaction of the ty, forces the water air-vessel, second up the pipe, while the valve; up the pipe, wile te valve E' is closed -- by its own weight and the pressure from above, as soon as the 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 in forcing-pump; 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 these pressures, which is obviously the weight of this resistance to be column of water. Denote the area of the piston by B, its overcome by the power; 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; its measure; and denoting the indefinitely small space described by the piston from this position by s, the elementary quantity of work will be wBy. s. elementary w B y. s. quantity of work; In like manner, denoting by y', y", y"', &c., the different heights of the piston, and by s', s", s"', &c., the corresponding elementary spaces described by it, the elementary quantities of work of the power will be same for different w By' s', w B y" s", w B y"' s" &.; positions of piston; and the whole quantity of work during the entire ascent, will be w [By s + B y' s' + By"s" + By"' s"' + &c.]; workduri ascent but Bs is the volume of a horizontal stratum of the fluid in the barrel, and Bs X y 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 equivalent Bp y, = By s + By's' + By" s" + &c.; expression for the same; and w. BFp. y, 594 NATURAL PHILOSOPHY. will measure the quantity of work of the moter 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 descent; 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, its measure; w B p, for the quantity of work of the moter during the descent of the piston; and hence the quantity of work during an entire double stroke will be the sum of these, or werk luring one W Bp (y, + z,). double stroke; 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; wBp b; and calling the number of double strokes n, and the whole quantity of work Q, we finally have work for any number of double Q = nW B b.... (245). strokes; If we make z, = y,, or b = 2 y,, which will give y, = - motion made the quantity of work during the ascent will be equal to regula-pmp; in 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 degree regular. In the lifting and sucking pumps, the moter 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- imregular in the lifting and 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. 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 Fig. 34P. its discussion more naturally o appertains to the motion of 7 fluids, its analogy with the, description; pumps, renders a description of it here proper. The siphon having its parallel branches vertical and plunged into two liquids whose modeofusing; upper surfaces are at L M and L' M', the fluid will stand 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 atmospheric pressure without, and when the two columns 596 NATURAL PHILOSOPHY. conditions of the unite and the vent is stopped, the liquid will flow from flow; the reservoir A to A', as long as the level L' 1&I' is below L M, and the end of the shorter branch of the siphon is below the surface of the liquid in the reservoir A. The cause of this apparent paradox will be manifest from the following consideration, viz.: The atmospheric pressures upon the surfaces L 11 and L' M', tend to force the liquid up the two branches of the tube. When the explanation; 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 exerted. The atmospheric pressures are very nearly the Fig. 347. same for a diflference of level A --—.; —--- 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 consequence of the greater density of the liquid. The atmomotion due to spheric pressure opposed to the longer column will therethe excess of fore be more diminished than that opposed to the shorter, pressure up the 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 0 of the siphon; by h' the difference of level between 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 L M; be pressure up the shorter branch; A - a h D g; MECHANICS OF FLUIDS. 597 and that which tends to force the fluid up the branch immersed in the other reservoir, be A-a h'Dg; pressure up the.A -- a h'.D 9',,' longer branch; and subtracting the second from the first, we find pressure which a D g (h' - h), determines the flow; 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 1 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 measured by quantity of actiop, in passing a a D g (h' - h) 1. siphon full from the upper to lower reservoir; The mass moved will be all the fluid in the siphon which is measured by a 1 D; and if we denote the velocity by V, we shall have, for the living force of the moving mass, a I D. V7; living force; and because the quantity of action is equal to half the living force, we find aDg(h'- h) l= 2Dl whence (hi ~~ ~ - h7A~ ~velocity of the v = /2 g (h' - h); flow; 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 598 NATURAL PHILOSOPHY. in the two reservoirs. When the fluid in the reservoirs flow will cease comes to the same level, the flow will cease, since, in that -when the water in the reservoirs case, h' - h - 0. comes to same The siphon may be employed to great advantage to level; drain canals, ponds, marshes, and the like. For this purpose, it may be made flexible by constructing it of leather, practical well saturated with grease, like the common hose, and furthe sappliphon; 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; Fig. 348. when, the ends being plug- 0 ged up, it is placed across the ridge or bank over which mode of using it the water is to be conveyed; purposesg the plugs are then removed, the flow will take place, and thus the atmosphere will be 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, ver whichthe over which the water is to be conveyed, should never water may be raised. exceed the height to which water may be supported in vacuo by the atmospheric pressure at the place. 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 A B D C to be any vessel containing a heavy fluid whose upper level is AMECHANICS OF FLUIDS. 599 A B. If a small opening a b be Fig. 349. flow of liquids made in the vertical side of the from vessels,.z 1 B through vessel, the pressure from within will apertures; urge the fluid out, and this pressure being greater as we descend to a greater distance from the upper surface A B, 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 of fluid discharged in a unit of time, as a second, is called the expense. The liquid on leaving the vessel forms aexpense; continuous stream called the vein or jet, which takes the vein or jet; form of the curve described by a body thrown perpendicularly 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. inshape, a At every point of this parabola, the weight of the fluid parabola; tends to alter its velocity, but at the orifice, the velocity is determined solely by what takes place within the vessel. If the orifice be in the horizontal bottom, as at a' b', the jet will be vertical, and the liquid will flow downward; if, as at d, the orifice be in a horizontal face pressed vertically upward, the jet will also be vertical, Fig. 350. direction of the and the liquid will ascend on leav- vein determined by the face of the ing the vessel. In general, when -e sil. the sides of the vessel are thin, the direction of the vein will be perpendicular to the surface through which the orifice is made. ~ 293. —The interior surface of every vessel containing Motion through a heavy fluid is subjected, as we have seen, to a pressure rifices; therefrom, which depends upon the extent of surface and 600 NATURAL PHILOSOPHY. the distance of its centre of gravity Fig. below the upper level of the fluid. to find the At the moment an orifice a b is A B flowing freelyuid made, the fluid at its mouth is urged through an by this pressure to leave the vessel, orifice in a thin plate; the neighboring particles crowd towards the opening, describing paths which converge towards and lead through it. This movement is soon propagated in some modified degree 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. 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 equal volumes volume must be equal to that disflow through the charged through the orifice. Dedifferent sections in same time; note by A the area of the section Fig. 352. NB of the interior of the vessel, at' the upper surface of the fluid; by a v {-1 —----------- B' the area of the orifice i 0; by s I per stratum NB descends in any - - indefinitely small portion of time; Q and by S the distance 0 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 A s; and that through the orifice, by a 8; and because these must be equal, we have equal volumes; As = aS; whence MECHANICS OF FLUIDS. 601 s a ratio of spaces = —-~~~ --- -~-. and areas of A/'~ -/]~~~~' ~sections; But because the distances s and S are described in the same time, theywill 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 8 V ratio of spaces V -;and velocities; which, substituted above, gives V a ratio of velocities V~~C~~~ A-~~ R~ and areas; 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 V=T a (246) velocity through v=. -.. (246). any section; Again, since the flow is permanent, it is obvious that the living force of the fluid mass N' B' iM Q must always be the same. Denote this by L, and let w represent the weight of the fluid mass in _BB' N', equal to that in M 1' 0' 0; then will the living force of the mass NBHJQ be +W v2 living force of g the interior fluid; and that of the mass B' N' Q M' O' 0 be w 2 that of a portion L + -; within and that g at the jet; and subtracting the first from the second, we find for the difference of living force of the same mass NB/l Q, and 602 - NATURAL PHILOSOPHY. B' N' Q H M' O' O, moving with the velocities v and V respectively, the expression difference of living force of W ( V2 the same V). nass; The quantity of work performed by the weight of this same mass Fig. 352. in the interval between its occupying the space NBMQ, and B'N'Q f' O' 0, is, as we have seen, equal to this weight multiplied by the vertical distance V.. through which its centre of gravity may have descended in the interval. Let G' be the centre of gravity of the whole mass when in the position NBHfQ, and G" when it occupies the space B' N' Q M' 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 work of the weight of the W(h" - h') Wh t- h'; entire fluid; and calling the distance of the centre of gravity of the mass ZfM'0' 0 below the upper surface, h"'; that of the centre of gravity of the mass N' B' iHQ below the same surface, 1; and the weight of each of these equal masses, W'; we have, from the principles of the centre of gravity, work at the beginning of any Wht" - Wl' + wh"', short interval; that at the end; WTh' = vW'Z + w-s; in which ~ s denotes the distance of the centre of gravity of the mass NBB' N' below the surface NB; whence wolrk duling the work dringthe1W]h" - W']' = w (h"' — s); interval; 2 MECHANICS OF FLUIDS. 603 but h"'- is the vertical distance between the centres of gravity of the masses NBB' N' and MM' 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 Hi 0 to ~11' 0', becomes w h. the same; 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 elementary work p As; from external pressure above; in which p denotes the pressure exerted upon the unit of surface. If, moreover, the fluid at the orifice be also subjected 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 elementary work I'J A s; fiom external pressure below; and taking the difference, we have, for the effective work of these pressures, effective work of (p' - p) A s. external pressures; Now A s D g = w, from which W volume of the Ags -';,,stratum; and, substituting this above, we have (p1'-p) A s = ( - p) -; effective work; whence the whole quantity of work due to the weight of 604 NATURAL PHILOSOPHY. the fluid and the pressures at the upper surface and the orifice, becomes total effective W work; Vw h + (p -- 2) D; and because the difference of the living force at the beginning and end of any interval, is equal to twice the quantity of action in this interval, we have quantity of work W 2 _2 equal half gain (V _ v2) = 2 w (h of living force; g.D g or, dividing out the common factor, multiplying by g, and substituting for v its value, given in Eq. (246), we have -e.~2; whence velocity of egress gh p through the = (247). orifice; 1 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 same when the pressures at top / 2 gh and orifice are V 2 (248); the same; V 2.. and if the area of the orifice be very small as compared with that of the upper surface of the fluid, the fraction A will be so small, that it may, without sensible error, be omitted; in which case, the fluid at the surface will be at ME'CHAEANICS OF FLUIDS. 6035 comparative rest while it flows through the orifice, and velocity of egress V = f2 gTh; 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. 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 a heavy body in the distance of the orifice below the fluid level. The falling through distance h is called, in the case of discharging fluids, the the depth of orifice; 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 aF~~~~~~ ~~expense in a,VT; Y:f"volume; and in weight, A v D g. in weight; 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 quantity in Q = a Vt.... (249), volume in a given time; quantity in = a VD g t.... (250); weight in a given time; in which Dg is the weight of the unit of volume. 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 A, we find, from Eq. (247), velocity of = 30 32 + 40 = 31.6 feet; egress; and for one second, quantity in volume in one Q 0.02 X 31.6 = 0.632 cubic feet, second; quantity in lbs. pounds in one Q = 62.5 X 0.632 = 39.5 pounds. second. 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 and vapors; same time, through any interior horizontal section of the 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 though different sections and true for liquids, but are not so in the cases of gases and density vary; vapors. When fluids of this latter class are confined and subjected 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 equal quantities, through the orifice a b; but the den- Fig. 353. by weight, will sities at these places being different, flow through the different sections the volumes of these equal weights in the same time; will also be different. In these par- B ticulars, the circumstances attending A, B thevolumes of -A! * >,,these will be the motion of gases and vapors dif- different fer from those of liquids. To find the velocity of egress ata the orifice, we remark, that the fluid is subjected, as in the case of liquids, to the action, 1st, of its the forces which act: weight, own weight; 2d, to that of the a b pressure from opposing pressures at the piston and the piston, and orifice; and 3d, to the additional repelsion; action arising from the repulsions of the particles for each other, this latter producing expansion whenever the pressure from without will permit it. The quantity of work work of the upon the stratum issuing through the orifice, due to the weight; weight of the fluid mass, is, as we have seen, measured by w h; in which w denotes the weight of the stratum, and h the height of the fluid above the orifice. To find the to flndthework work due to the pressures, denote the pressure upon a prsutore; unit of surface at the piston by p; that on the same 608 NATURAL PHILOSOPHY. extent of surface at the orifice by Fig. 353. p'; the area of the piston by A; that of the orifice by a; the disnotation; trance between any two consecutive positions, as A B and A' B', of the A piston by s; the distance between the two corresponding positions a b and a' b' of the stratum at the orifice by S. Then, because the weights of the volumes A B B' A' and a b b' a' of the fluid are equal, we have a bi the equal AsDg = aSD'g... (251); weights; in which D and D' denote the densities of the gas at the piston and orifice, respectively, and g the force of gravity. Whence volumes As $D inversely as - D densities; aS ) But by Mariotte's law the densities are directly proportional to the pressures, hence densities directly -D _ p as pressures; D which substituted above, gives relation of A s p' volumes and - (2. 52). pressures; a. p Clearing the fraction and transposing, we find loss or gain of work due to p.A s - p'a S = 0. pressure; But p A is the pressure on the Whole extent of the piston, and p A s 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 Sg is the quantity of work of this pressure; and as these quantities of work are produced in the same time, we see that the loss or gain of this is zero; work, due to these pressures, is zero. The quantity of work due to the molecular actions, arises in consequence of the to find quantity expansion which takes place when the gas passes from the of wolk due to molecular action; 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 proportional to the primitive volume expanded during the change of pressure; if the it is directly primitive volume to be expanded be doubled, tripled, proportionial to the primitive or quadrupled, &c., the quantity of the work will be volumeto be doubled, tripled, quadrupled, &c. Hence, taking a cubic expanded; foot of the gas under the pressure p, and denoting the quantity of work due to the expansion, corresponding 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 BB'A' to a bb' a', be measured by work of the A E.s expansion from A, 8..E.. the volume at the piston to that at But since w denotes the weight of the gas in the volume o; A BB'A', we have w = A s g D; weight of the stratum; whence As- w = its volume; and wA. s. E _ E. work due to Ag 8. expansion; whence the whole quantity of action or work due to the weight and expansion of the fluid will be work due to pf /- x pansweight and w II + W W + expansion of the gD L' stratum; at 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 living forces, living force equal W 2 to twice quantity - ( V2) = 2 w -+ of action; gg or _ 2E V2 _ v2- = 2gh +. (253). From Eq. (252) we have relation of s p a elementary; paths; A and the spaces s and S. being described in the same time, they are to each other as the velocities v and V; hence same as ratio of V p'a velocities; V p A or velocity at = * piston; pA' which substituted in Eq. (253) for v, we find V2(1 - p2a2) = 2g + 2E Making,2 a2 l ~ = K 2 p2 A2 A32 the above gives value for the 1 velocity of V=- 2gh+.. (254). egress; MECHANICS OF FLUIDS. 611 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 dxpatsioe of a or cubic foot; and suppose this unit of volume to be con- umit of volume, tained in a tube, of which the area of the internal cross- pressm e to 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 p' 1: 1; whence = _ p 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! - 1. = P_ = P _ p2. expansion during p p the change; Dividing this path into two equal parts, and adding one of them to unity, the original length, we have 1 p -_ 2+P P' p length when the 2 + 2p' 2p' expansion is half "..p' 2'' completed; 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, P +p 1: 2 P p, whence 22p' corresponding pi-Pf + p" pressure; 612 NATURAL PHILOSOPHY. If we now observe that the consecutive pressures are the three 2 consecutive pA + _ and p pressures; + and that the constant space passed over, during the interval which separates the instants in which these pressures are exerted, is spaces described by the pressure - while its value is 2 changing from 21p the first to last; Fig. 354. the computation of the total work becomes easy by the rule given in ~ 46. For this purpose, take -A B AC= CB= -P and erect the perpendiculars A[ =p determination of CfM' 2pp the work; p pjO BM"= p'; join the points H;, A', and M"; the area A B Af" Mf 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 its value; A C (A XL~ + 4 C'M + B'"); and, substituting the values above, we have MIECHANICS OF FLUIDS. 613 1 P _p 4. _2pP- P-E same in other 2* p P? + +I ), terms; which, substituted for E, in Eq. (254), gives 1 12+ 8_pp' value of velocity V= 1 i + 3 D pa ( + +p p + p )'' (255). in terms of _p 231 23 +3-231 pressures: When the orifice is small, as compared with the area of the piston, the fraction,2 p2 a p2 A2 may be neglected, and K will become equal to unity. Moreover, the term 2 g h, in the case of gases, is scarcely ever appreciable in practice; making these suppositions, Eq. (255) becomes 1 - P (p + + p').. (256). velocityincase 3D p j p+p of small orifices: The pressures 2 and p' are usually ascertained by means 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 dete.minethe pressures; terms of the indications of these instruments. For this 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 = g h D,, and p' = gh'D,,; which, substituted above, give velocity in terms.D/,,7 h- h' 8 h h' of the indications V=' 3 A'. -(h+ I+ h'). (257); of the h + h manometer; 614 NATURAL PHILOSOPHY. in which V1 will be expressed in feet, g being equal to 32 feet very nearly, and D,, equal to 13.5 nearly. The expense e, in volume, will be given by the equation expense; e aV..... (258); and the quantity Q in volume, discharged in a given time t,, expressed in seconds, will be known from quantity discharged in Q = a Vt, (259); volume; 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 po; the density D', which the fluid assumes on leaving the orifice, is determined by the pressure p', and is connected with D, according to Mariotte's law, by the relation density on t p hi leaving the - D = =D -. orifice; hHence, the expense Q', in weight, will be given by quantity in weilit Q' = D' g a V = D g a V - (260); discharged in h unit of time; and the quantity Q" in weight, discharged in the time t,, quantity, in A' weight, in time " D g t, (261); in which a must be express- Fig 355. ed in square feet, as above. The density D is computed by Eq. (240)'. example; Examkple. The open gauge, connected with a gasometer, i_ containing heavy carbureted hydrogen, shows a difference of level in the mercury of 8_ L 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 orifice 0.02 of a square foot of area in 20 minutes = 1200 seconds. Here, h - h' = 8 inches = 0.666 feet, h' = 28 " - 2.333 "; data; whence h - 36 " = 3.000 D,,= 13.5 g = 32 t = 52~; and from Eq. (240)', after substituting the value of h and t, above, and that of D, in the table, page 533, for heavy carbureted hydrogen, we find 0.00127 36 D X -= 0.001465; density; 30 1 + (52 - 32) 0.00208 and these values, in Eq. (257), give / 13.5 0.666 8 X 3 X 2.333 ft' 3230.00 X 02.- X (3 + + 2.333) 668.02. velocity; Substituting this and the numericalvalues of a and t, in Eq. (259), we find quantity in Q = 0.02 x 668.02 x 1200 = 16032.00 cubic feet. volume; The quantity D g, in Eq. (261), is the weight of a cubic 616 NATURAL PHILOSOPHIY. foot of the gas, whose density in this case is 0.001465; and as a cubic foot of water weighs 62.5 pounds, the value of Dg becomes 62.5 X 0.001465 = 0.0916, nearly; whence quantity in lbs. 28 s lbs weight. "= 0.0916 X 0.02 X 668.02 x X6 X 1200 = 1142.4. 36 ~ 295.-A stream flowing through an orifice is called Veinal a vein. In estimating the quantity of fluid discharged contraction; 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 viscosity, and does not adhere to the sides of the vessel and theoretical orifice; 4th, that the particles of the fluid reach the suppositions; 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 disresults of charge must, therefore, differ from the actual. Experience experience; teaches that the former always exceeds the latter. If we take water, for example, which is far the most important 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 causeswhichtend thus flowing away from those without, the vein will to contract the eto contract the increase in length and diminish in thickness, till, at vein; 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 MECHANICS OF FLUIDS. 617 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 veirnal contraction. The quantity of fluid discharged yeinal must depend upon the degree of veinal contraction, and contraction; the velocity of the particles at the section of greatest 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. 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 contraction; 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 depends upon; 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 may be discharge 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 aperture be circular, square, or oblong, which embrace all coefficient of cases of practice, provided that in comparing rectangular discharge; 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- dependsupon; 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 circumstances, such as the head of fluid, and the place of 618 NATURAL PHILOSOPHY. the orifice, in respect to the sides Fig. 856. and bottom of the vessel, vary, then will the coefficient also vary. When the flow takes place through - discharge thin plates, or through orifices in thin plates; whose lips are bevelled externally, the coefficient corresponding to given heads and orifices, may be found in the following table, provided the orifices be remote from the lateral faces of the vessel. This table is deduced from the expericoefficient ments of Captain Lesbros, of the French engineers, and deduced from agrees with the previous experiments of Bossut, Micheexperiments; lotti, and others. TABLE. COEFFICIENT VALUES, FOR THE DISCHARGE OF FLUIDS THROUGH THIN PLATES, THE ORIFICES BEING REMOTE FROM THE LATERAL FACES OF THE VESSEL. FValues of the coefficients for orifices whose smallest dimensions or Head of fluid diameters areabove the centre of the orifice, in feet. ft. ft. ft. ft. ft. ft. o.66 o.33 o. 6 0.98 0.07 o.o3 o.o5 0.700 0.07 o.627 o.66o o.696 0.13 o0.6i8 o.632 0.657 o.685 0.20 o.592 0.620 o.640 o.656 o0.677 table of o 0.26 o.602 1 o.625 o.638 o.655 o.672 coefficients; o.33 o0.593 o.608 o.63 0.637 o.655 o.667 o.66 0o.596 o.6I3 o.63I o.634 o.654 o.655 i I.00 o.60I o.617! o.63o o.632 o.644 o.650 1.64 o.602 0.6I7 o.628 o.630 o.640 o.644 3.28 o.605 0.615 1 o.626 o.628 o.633 o.632 5.oo o.603 0.612 0.620 o.620 0.62I o.618 6.65 0.602 0.6i o.65 0.6i5 o.6Io o.6Io 32.75 o.6oo o.600 1 o.6oo o6oo o.6oo oi.6oo coefficients for In the instance of gas, the generating head is always greater than 6.65 ft., and the gas; and for coefficient 0.6, or 0.61, is taken in all cases. orifices not in the 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 table; orifices between those of the table, we take coefficients whose values are a mean between the latter, corresponding to the given head. MECHANICS OF FLUIDS. 619 As the orifice approaches one Fig. 35'7. of the lateral faces of the reservoir, the contraction on that side becomes I Il i less and less, and will'ultimately be- I l l \ II j lj I j 1when orifice is ultimately be- near one lateral come nothing, and the coefficient face; will be greater than those of the \\\\\S;| table. If the orifice be near two of these faces, the contraction becomes nothing on two sides, and the coefficient will be still greater. Fig. 358. Under these circumstances, we have the following rules: Denote by C the tabular, and by C' the true coefficient corresponding to a near two lateral given aperture and head, then, if faces; the contraction be nothing on one side, will CG~~~~~~' = 1.03 C; ~coefficient in the first case; if nothing on two sides, C' = 1.0,6 C; coefficient in the G' = 1.06 O;Gi Csecond; if nothing on three sides, coefficient for no C' = 1.12 C; contraction on three sides. 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. ~ 296.-When the orifice is rectangular, and has no Discharge upper limit, or is open at the top, it is called a sluice-way. through sluice-waysg It is usually a cut made in the edge of a reservoir, through 620 NATURAL PHILOSOPHY. which the fluid may Fig. 359. flow when it rises _A B above a certain level. __-_ estimate of the The expense is estiexpense through mlated in this wise. a sluice-way; Denote by I the length. of the horizontal side of the sluice-way; by h the head or distance B;, 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 sluicenotation; way; and by V the mean velocity: then, supposing the sluice-way filled to the upper level of the fluid in the reservoir, will h 2 H V2 = 2gh = 2g x'H = (2g H); whence value of mean velocity; -V 0.707 2 g H; and the theoretical expense will be theoretical V x 1 x H = 0.707 2~ 2 H x I x H. expense; 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 lE, will be given by the equation, practical E = 0.57 X 0.707 X I X HX /2gH 0.403 1. H. 2gH... (262). expense; The experiments of Dubuat, Bidone, Eytelwein, and Lesbros, show that the coefficient 0.403 should be re MECHANICS OF FLUIDS. 621 duced about 0.39 when H becomes equal to or greater than 0.66 of a foot, and increased to 0.415 when H becomes less than 0.07 of a foot; but that it remains variation in the sensibly the same, whatever be the total contraction or toefficient; position of the sluice-way in regard to the vertical sides of the reservoir, provided H 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.: The foregoing conclusions suppose that the fluid is discharge discharged through orifices in thin plates, and that, du- throughthick b plates; 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 plactes 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 coefficients; to 0.66 of a foot, under heads which give a coefficient 0.619 in the case of thin plates. The cause of this increase is obvious. It is within the observation of every one, 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 considerable pertinacity. This affinity becoming effective between the inner surface of the -tube and those particles 622 NATURAL PHILOSOPHY. effects of of the fluid which enter the orifice near its edge, the latter molecular action. will not only be drawn aside from their converging directions, but will take with them, by the force of viscosity, the other particles, with which they are in sensible contact. 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. XVII. DISCIHARGE OF FLUIDS THROUGH PIPES. Discharge of We have considered the discharge of fluids through 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 less than through equal, the discharge will be less as the pipe is longer. The orifices; 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 parallel filaments, but must be incessantly deflected from their onward course into partial eddies formed by the small ircauses which regularities of surface. Moreover, as the pipes increase in obstruct the motion; the length, will the surface exposed to fluid pressure increase, and as the extent of surface, all other things being equal, 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 Fig. 360. compute the amount of living force Loss of living force arising from resulting from the shock of fluids, -- fothe ising acom flowing with different velocities. fluids; For this purpose, let the fluid in the pipe.LK flow with the velocity V, and denote by M the mass which flows into the vessel B C in a unit. L of time; also let the velocity of the l SjI fluid in the vessel BC be', and its mass M'; then will the corresponding living force be a 2 t t2 living force M V + Ml' VI; before the impact; and supposing the fluid to be water, which we have regarded 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 V - A. V + M'.' common velocity AM - AM' after the impact; and the corresponding living force, ~(M+ MI) /=M. V l M' VTr/ 2 (M + MI (M 2V +- M' V')2 corresponding MX- +M 2 u > + -Mt' living force; and the loss of living force in a unit of time, denoted by L, 624 NAT U RAL P I:ILOSOPHY. /.~rTT1,r/TTI/~f2C4%LrY/ MT (V-T Vt)2 fo.ore L = M -2+ MlI' V'2 _ (-f V +VI VI)2 force -.LVM + A-' i + Mv' and, dividing by M', L Z (V- V')2 (263); samlne: Afl or when the mass M' is very great as compared to A, seame when a small mass flows into alare L (V-')2... (264). mass. ~ 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 Loss of living may be flowing. Let A B CD be contractionof a vessel containing a heavy fluid, Fig. 301. cross-section ofa of which A B is the upper level, \\ and issuing through an opening a b B'' in the bottom C D; 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 a b, a and by a' that of the contraction 0 LV at a' b'. The fluid, in passing hypothesis; through the contraction a' b', imnpinges 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 notation; region of eddies; and let m represent the coefficient of the expense at a b, 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 1, that through the section A"B" will be A" V", 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 the iffent sections of a b, A" B", and a' b', in the same time, we have m a V = A" V", ma V = m'a' V'; whence ma V - A,, velocities; n a V m a and the velocity with which the fluid through a' b' impinges against that below the diaphragm, will be' -- VF - m a - relative velocity \m'a' A" of the impact; Denoting by w the weight of fluid that passes a' b' in any small portion of time, its loss of living force will be W (I _ 7)2 = W. m 2a2( 1 - C) V2; g g and denoting the factor m a by E the quantity of work lost will be W K2 V2. work lost; 2g The work of the weight, during the same time, will be w h, and the quantity of work remaining will be 40 626 NATURAL PHILOSOPHY. the work wh - K2 V2; remaining; 2 g but this must be equal to half of the living force, hence which is equal to W half the living V2 WA= w - K2 force; g 2g whence we find velocity of egress 2 - h (265); through a b. In = /2' and from which we see that the velocity will be less than that due to the height C0, equal to h. ~ 299. —Let us apply this to the discharge of a fluid Loss of living through a short pipe, inserted into the orifice in the side force in short of a vessel. The fluid havpipes; ing contracted to its minimum dimensions at n, again Fig. 362. dilates, and fills the tube at a' b'. Let V be the mean hypothesis and velocity at a' b', where the notation;,,_ ~ _ narea 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 through a section at a b, where the velocity is V', and cross-section a', will be m a' V', in which m is the coefficient corresponding to' the area a'; and, as these must be equal, we have a = mna'V'; whence velocity at the a - entrance of pipe; t n a MECHANICS OF FLUIDS. 627 and the loss of living force, w~O X (' - X (m a' 2) loss of living w-(V'- V)2=a force; The quantity of work of the weight, in the same time, is w X h, and this, diminished by half the loss above, must be equal to half the actual living force; and, therefore, 2g V =h _w * ( a )2 or making,-1 = K, we find m a 2 g h velocity of egress I + K2' from the pipe; When the tube is cylindrical a = a', and 1- 1; when the contraction is complete in n, and the head varies from 3 to 7 feet, it is found that m is equal to valueof m; 0.62 very nearly; whence K -= - 1 = 0.613 very nearly, 0.62 and 10 value of the = 0.85; constant; whence final value for V = 0.85 f 2 g h. velocity of egress; Experiments give the coefficient 0.82, but, in com 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 little less. work of the weight w h. ~ 300.-When the velocity of a fluid is considerable, Flow of fluids and the length of the pipe through which it flows is great, through pipes of; 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 Fig. 363. bore throughout, con-,-_ i. 7 necting two reser- voirs A CD B and A' C' D' B', partly ease stated; filled with fluid, the former to the level \\\\ A B, and the latter to the level A'B'. De- K c D note by H the differ- X notation; ence of level between AB and A' B'; by a the area of a cross-section of the bore of the pipe; by C the contour of this section; by L the length of the pipe; and by V the constant velocity of the fluid flowing 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 frictioninpipes; 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 L X C X V2 proportional to g a this function; and that this loss is a certain fraction n of this function, or is equal to w LxCx V the loss of work g a from friction; 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 by that due to a diminution at the orifice a b, measured by the expression work lost firom W W. V"-2.w2 diminution at thVe 2 g rfn1 22g' < ) K entrance of the pipe; in which 1 _ = v2; m and, because of the principle of fluid level, H is the only distance through which w can act to produce work, we have W2 v~ w=w. VB w E2 _ W.. (V 2 2=wH- (266;-.[_n_''.(266); 2g 2g g a whence 2gH. (267) velocity of /1 + K + 2n ~L C egress; from which the velocity may be found. The expense, denoted by Q, will be given by Q = a V.... (268). expense; 630 NATURAL PHILOSOPHY. Taking the value of m equal to 0.60, (see table,) we find vacle ofthe 1 + K2 = 1.4444. constant; Experiment shows that, for water, value of the coefficient n, for n= 0.0035; water; and for air or gas, and for gas; n = 0.00324; modification in and it is important to remark that, when the question te formula fors; relates to the discharge of gas, we must make D - h - h'(8hih =S2 31D h h+ h + A + h 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 C = I D, and a = 4, so that C 4rD 4 a = rDa 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 8rn, velocity in case forwater,... 47.94 c e (269). of water; for r, = L 51.57. icase of air; for air.. = 49.83 D- (270); L 55.72. D ( 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, forygsee lo 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- Fig. 364. er than a b, or of less section __ than a, V being the velocity when the within the pipe, the expense aperture of final egress is smaller may still be deduced from a than section of slight modification of the value of the velocity, as given pipe; 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'; condition of permanent flow; whence _V' aV m' a" and the living force of the fluid as it issues through a'b', will be ~W 12 W co X'aF2 living force of the ~ V- -W- X 2X V2; discharging g g m2' a'2 luid; which, being placed equal to the second member of Eq. (266), will give V=\. V (271). its velocity; Ma2 +K +2nL.m72 at2 a 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 slightly, or if the pipe term- Fig. 365. coefficient n'; inates at a' b' in a conical - === tube, then will the value of -= - __= m' vary from 0.82 to 0.96. Exacimple. Let the height example; 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, ft. ft. ft. data; D = 0.5; H = 70; L = 1200; which, in Eq. (269), give velocity; V - 47.94.- X70 = 8.102. 1200 + 51.57 X 0.5 The value of a, in Eq. (268), will be given by area of the D2 0.25 section of pipe; a = - = 3.1416 X = 0.196; 4 4 which, in Eq. (268), gives expense. Q = a V = 0.196 X 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. END OF MECHANICS. A. S. BARNES & COTMPAN-Y'S P'UBLICATIOhNS. H'AII BEEl S El UCAT[ION AL COU SE'. NATURBAL SC IEN C ES The 3Alessrs. Chambers have employed the first professors in Scotlandl in the lpreparation of these works. They are now offered to the schools of the United States, ninder the Americanl revision ol D. iAt. MPlcs, s i. D'., LL.D., late Slpe7rinteadeint of Pzlblic Schools ibto the city and COuLty of.New YO?'k.!. 0'I;RiM SBERS TREASURY 0 F KNIOWLEDQE. I1, C _ARfiS ELEM'ENrTS OF DLi'AWIN 8,, t PERSPAEiTIVE, [ 1. CAM aBths`ERS' ELEMENF"IhS OF RA'pTURAL PHI3LOSOPCHY IV.':;E3D t& BAINS-S CkHE M i"lST"RY AND EL EC'TRBICITY, V.' H^MILTONi S VEGTETABLE AIqD ANItMAl PHYSIOLOGYt', VI, OH;A~iB%,ERS ELEM'ENTS OF ZOOLOGYD VI. A-'-S EL. M""iTS OF GiEOL OGY, It is wlII tllnown l that the original publishers of these worlrks (the Mlessls. Chnmbers of Edhm'ili')h). je hble to commaondd the best talellt in tie prepm'ation of their books, n(ut lltht it is theil practice t, deall faittlfully wvith the public. I his series; wvill not disappoint t.he reasonable expectations thus excitedi. They are elementary works prepired by ot iliors in every way crapable of doingl justice to their respective luldlltelldrtrii."F,!lt who hlae evidently bestowed Iponl them the necessa-ly time itnd labor t) atd.t Llen tc their plrpose. \e recrmineills them to teachers and parellts with confidnce. If not intrioducedl as class-books in the school, they may be oUset to excellent adv;mta-S e in geierial exercises, aid ocl sional class exe — cise, for wiich every teacher ool'ht to provide hitmself switlh an aimple store of mnteriai.ls. riThe! volumes mly be had septilately;'iad the one filst l;luned, in t.le hn.irds of a teahller of the yo'ngelr clhsses, miihlt fuliiiish an inexllustihble fllrlt ofL anmiS'oN:ellit irltl inlSt1lction. To-'etlher, tLhey woittld coustittllea rich treasule to a familyiiv of titoeiigent chlilcllen, and ismpart. thirst t'r Irnowledioe.o "-eFec? ostt Chrm/'io. O Of a11 the ioinumiero)ls works of this class ihat have been published, thlee r'e none that have acquired a;l mnole thsorougchly deserved and hig'h repulltationl tlh an'this series.'he Cmllbelr s of Euiibshurg'h, weiL kisowi s tlte calefll alll initelligt'ilt p1)tlsiheirs of a asti numiber of works of much importalsice in the educrtionnil woorld, are. the falthlers of tis series of books, a'nd the Alleliccl editor has exercised ant unusual dcegree of judglllent ill their preparatio'l for tile use of schools as well as private fiamilies in this coteuntcly."-P/ii'ItadcZe/pia Bsi/rllet,. "The titles furniiish a key to the contents, and it is only necessary flor us to say that the milterial or each volume is admirab lly worlked lip, plresenfing w itMh sllfliccielt fulness aind with much lear'nees of' imethod the seve ral sulbjects whicht ale trbeited." -Cinlcit77)nat Ga7zelte.