H..J. Hotchkyss Cornell University Library 3 1924 031 364 197 olin.anx Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031364197 PRINCIPLES OF EXPERIMENTAL PHYSICS FOR STUDENTS OF SCIENCE AND TECHNOLOGY BY HOMER J. HOTCHKISS Prop ESSOR of Physics in Drbxel Institute FLOYD C. FAIRBANKS Instructor in Physics in Drexel Institute ASSISTED BY JOHN E. HOYT AND FREDERICK LEIGHTON Instructors in Physics in Drexel institute Printed Expressly for Students OF ExPERfMENTAL PHYSICS IN Drexel Institute PUBLISHED BY THE AUTHORS 1913 Copyright, 1913, by THE AUTHORS PREFACE This manual, in common with most laboratory manuals, is written with the purpose of carefully describing for the student experiments which will help him to get a clearer. understanding of the fundamental principles of Physics discussed in the class room, and of teaching him how to apply these principles. But the writers believe that this is' not the most important purpose ; that it is even more important that students in technical courses should understand and use the laboratory methods and principles which are applicable alike to all experimental work whether it be in school practice, in commercial testing, or in re- search. For example, in any experimental work, it is important that one should be able not only to make precise measurements, but also to judge what are proper or most favorable conditions for making measurements, to determine or estimate the relioibility or probable accuracy of his data and derived results, and to interpret and draw concliisions from the data and results. With this latter aim in mind, the course here outlined is in- tended to serve as a preparation for future experimental work in whatever line of engineering or scientific research the student may become engaged. Such preparation relates to methods of observing, recording and graphically representing data; to de- riving and interpreting results ; and to the fundamental principles that serve as tools, guides and safeguards in experimentation. Considerable space has been given to an introduction in which is discussed several general principles relating to experimental work; here also an attempt has been made to formulate some rules relating to the use of doubtful and significant figures in computations, to the end that the reliability of a result obtained through any of the four operations will be approximately ex- pressed by the number of figures retained therein according to these rules. In some of the experiments special prominence is given to the principles and methods underlying experimental work. For in- stance, expression of errors and deviations, proportionality fac- IV PREFACE tors, and the plotting and interpretation of curves are important parts of the first three groups of experiments. Also general methods, such as methods of deriving empirical formulas, of ob- taining constants from plotted curves, of checking results, etc., are made prominent in some experiments. The writers do not consider that a student in elementary physics is capable of doing original experimental work, even to the extent of deducing from his experiments, unaided, the simplest of the fundamental physical laws. However, an attempt is made, by the use of a set of " test questions " upon each experiment, to lead him to think through the principles he has used or verified. These questions are intended to aid the student in writing his re- port upon the experiment as well as to test his knowledge of the principles. In some part of bis written report he discusses each of these questions, or they may form the basis for an oral report. In the course for the Fourth Term the student is given consider- able choice in the method of performing some of the experiments, and is encouraged to devise original methods. 3ome of the experiments have been worked out by the writers, others have been taken from various sources and modified to fit conditions. Practically, the whole manual has been used for several years in the form of mimeograph notes, by day and even- ing students in Drexel Institute. Acknowledgment is gladly made of the work done by Prof. Floyd K. Richtmyer, now of Cornell University, who, when an instructor in the Institute, assisted in making the original draft of these notes. The Authors. Philadelphia, Nov., igi2. TABLE OF CONTENTS PART I — FIRST TERM CHAPTER I. GENERAL INSTRUCTIONS, NOTES, AND PRINCIPLES I. General Instructions SEC. PAGE 1. Supplies 1 2. Pxeparation 1 3. Registering 2 4. Apparatus 2 5. Data 2 SEC. PAGE 6. Credits 2 7. Handing in Reports . . 3 8. Writing Reports ... 3 9. Oral Quiz S II. Notes on Physical Measurements 10. 11. 12. 13. 14. IS. 16. 17. 18. 19. 20. 31. 32. 33. 34. 35. 36. Units Relations of Quantities Proportionality Factors Errors and Mean Values Fractional Deviations . Laws of Deviations . Mistakes Accuracy, Precision, Re- liability Doubtful Figures . Trustworthy Figures . Significant Figures . . 6 6 7 8 9 9 10 10 11 12 12 21. Products and Quotients . 22. Significant Figures in Data 23. Computations; Notation in Powers of Ten . 24. Abridged Multiplication 25. Abridged Division . 26. Cu'rves 27. How to Plot Curves . 28. Interpretation of Curves 29. Estimation of Tenths . 30. Measurements 13 15 IS 16 17 18 18 21 24 25 III. General Principles Law of Causality . Independence of Forces Conservation of Energy Variables Sources of Error . Independent Observations 26 26 26 26 26 26 37. Discarding Data .... 27 38. Using Data 27 39. Measurements and Re- sults . ..'... 27 40. Curves 28 IV. Notes on Units 41. Introduction . 42. Tabulation of Units 28 29 43. Definitions 29 44. Problems 30 CHAPTER II. EXPERIMENTS IN MECHANICS AND HEAT Group I — I. Introductory Measurements EXPERIMENT I — 1. Measurements with Metric and English Scales Errors, Deviations, and Significant Figures I — 2. Measurements with Calipers Study of Micrometers and Verniers 1—3. Relation of Parts of a Right-angled Triangle Measurements of Lengths and Angles PAGE . 33 . 37 . 46 VI CONTENTS Group II — E. Elasticity EXPERIMENT PAGE E — 1. Laws of Flexure of Beams SO Elasticity and Curve Plotting E — 2. Calibration of Springs . S3 Jolly Balance and Curve Plotting Group III — W. Weighing : Mass, Volume, and Density W — 1. Weighing by Method of Vibrations S9 Determination of Mass and Density W— 2. Weighing by Method of Vibrations and Reversals ... 65 Measurement and Calibration of a Capillary Tube W— 3. Measurement of Volume and Density of an Irregular Body 68 Use of the Burette and Pipette Group IV — A. Accelerated Motion A' — 1. Study of the Motion of a Falling Tuning Fork . ... 73 Uniformly Accelerated Motion A — 2 .Study of Acceleration of a Car on a Horizontal Track . . 78 Relations of Force, Mass, and Acceleration A — 3. Study of Uniform Rotation 82 Force and Acceleration toward the Center Group V — F. Composition and Moments of Forces F — 1. Determination of Resultant of Concurrent Forces ... 88 The Parallelogram and Polygon of Forces F — 2. Determination of Various Forces in a Truss 91 Application of the Polygon of Forces P — 3. The Law of Moments of Forces 93 Conditions Essential for Equilibrium F — 4. Determination of Various Forces in a Derrick .... 98 Application of the Law of Moments F — ^5. Equilibrium of Parallel Forces in the Same Plane . . . 101 Resultant of Parallel Forces Group VI — U. Units and Period U — 1. Work against Gravity and Friction 105 Comparison of Systems of Units U— 2. Relation of the Period of a Pendulum to its Length . . . 109 Method of Determination of Period Group VII — M. Machines and Friction M — 1. Study of Fixed and Movable Pulleys 112 Mechanical Advantage and Efficiency M— 2. The Differential Wheel and Axle 114 Mechanical Advantage and Efficiency M-^. The Inclined Plane 117 Mechanical Advantage and Efficiency M — 4. Laws of Friction 118 Determination of Coefficients CONTENTS Vll Group VIII — P. Pressure in Fluids EXPERIMENT PAGE P — 1. Verification of Laws of Fluid Pressure 122 Dependence of Pressure upon Depth and Density P— 2. Study of the Barometer 125 Method of Measuring Elevation P — 3. Verification of Boyle's Law for Gases 130 Pressure and Volume at Constant Temperature Group IX — D. Density of Liquids and Solids D — 1. Determination of the Density of Solids 133 Method by Weighing in Water D — 2. Determination of Density of Liquids and Solids .... 135 Use of the Specific Gravity Bottle D — 3. Density of Liquids 138 Hare's Method by Balanced Columns D— 4. Density of Solids and Liquids .... 142 Density by the Jolly Balance Group X — H. Heat H — 1. Testing the Mercury Thermometer 144 ' Determination of Errors at the Fixed Points H — 2. Linear Expansion for Solids . 148 Determination of the Coefficient H — 3. Determination of Specific Heat 151 Method of Mixtures for Solids and Liquids H— 4. Heat of Fusion 154 Determination, for Ice H — ^5. Heat of Vaporization 156 Determination for Water H — 6. Transfer of Heat by Conduction 159 Conduction in Different Metals H— 7. The Mechanical Equivalent of Heat 161 Determination from Impact of a Falling Body The Tables of Contents, for Parts II, III, IV are placed immediately before the respective parts which begin as follows: Part II begins at page 201. Part III " " " 301. Part IV " " " 401. EXPERIMENTAL PHYSICS PART I— FIRST TERM Elementary Course CHAPTER I GENERAL INSTRUCTIONS, NOTES AND PRINCIPLES I. GENERAL INSTRUCTIONS L Supplies. Each one will need a copy of this manual, a laboratory note book, a supply of report paper, report covers, fasteners, and cross section paper, a good pencil (No. 3 or harder) and a fountain pen. The following supplies are not re- quired, but will be found very useful : a scale graduated in inches and in centimeters, a triangle, dividers with pencil and pen, and a slide rule. Materials should be at hand in the laboratory for writing reports (in the computation and reading rooms) as well as for experimental work. Each pair of partners will be given a locker for their supplies. 2. Preparation. On coming to the laboratory the student should be prepared to state clearly the object of the experiment he is to work upon, the results to be obtained, and the method to be used. He should be able to specify the apparatus needed, and be ready to proceed at once with the experimental work. No one should leave the laboratory earlier than 5 minutes before the end of the full laboratory period, unless excused by the in- structor in charge. Results should be worked out and checked for correctness as far as possible in the laboratory period, so that the work may be submitted to the instructor for inspection. This will often avoid waste of time in the home work due to in- correct or incomplete data. The studenfs diligence and ef- 2 INTRODUCTION ficiency in the laboratory will be important factors in determin- ing his grade. 3. Registering. Register for experiments as directed by the notice posted in the laboratory. Note also any other in- struction posted with this notice. 4. Apparatus. Apparatus is drawn by check from the ap- paratus room, and in general should be returned about five min- utes before the end of the period. All the apparatus required for a given experiment should be drawn out at one time. The number stamped upon each piece of apparatus should be written upon the check after the name of the piece required. Apparatus and reference books from the apparatus room are for use in the laboratory only. Students are held responsible for apparatus un- til returned and found to be in good condition. The check is then placed on the cancelling file. All breakage should be re- ported at once to the instructor in charge. Breakage that oc- curs in doing any experiment that has been assigned to more than one will be charged against all who are working together on that experiment. A double share will be charged against the one directly responsible for the breakage. 5. Data. All observations and data should be recorded and kept in the laboratory note book. Do not take data on loose sheets. Be sure to record the original observations. Distinguish between original observations and derived results. In measur- ing the length of a line, the readings at the ends are the original observations. The length of the line is the derived result ob- tained by subtracting the readings. All supplementary data, such as date of performing, numbers or descriptions specifying the pieces of apparatus used, conditions under which measurements were made, etc., should be kept in the note book. (See also Sec. 22.) 6. Credits. The report on each experiment is read and given a grade. The report may then be returned to the student if (1) mistakes have been made in important parts of the report or (2) certain omissions have been made. In general reports will be given a low mark and not be returned if the student de- GENERAL INSTRUCTIONS 3 liberately omits matter he knows to be a required part of his re- port. He should consult an instructor, before the report is handed in, regarding any points he does not understand, or at least specify in the report what is not understood. If the report is returned to the student for correction, the first grade given may be raised if the corrections are carefully made. If the report is not handed in on time (see Sec. 7) a reduction is made in the grade which depends upon the number of days the report is overdue. If a schedule of the experiments to be performed and the dates upon which they are to be done is posted, a given term's work will be completed if these experiments have all been performed and the reports upon them accepted with a passing grade. If some of these experiments are given a grade below 60 per cent, the student may be required to perform additional experiments. The final grade will depend not only upon the mark given for the reports, but also upon oral and written tests and general diligence in the laboratory. In case the student is making up the work of a previous year, only those experiments already done will be credited upon which he has , received a grade of at least 70 per cent. 7. Handing in Reports. Reports should be written as soon as possible after pverforming the experiment, and should be handed in within the week (vacations not included) following the date of completion of the experimental laboratory work ; un- less, for sufficient reasons, a written extension is given by the in- structor before the time limit is reached. Corrections or addi- tions should be made as soon as possible and the corrected re- ports returned promptly ; not later than one week from the date marked on the cover. As soon as reports are read, those not accepted are placed in the apparatus room to be called for by the student, who will have notice by finding his laboratory number turned into view on the number rack in the apparatus room. If these requirements are ignored by the student, apparatus for further work may be refused him. 8. Writing Reports. Reports should vary in form to suit the different experiments. However, they should follow as 4 INTRODUCTION closely as possible the following outline: (1) Title of experir ment; (2) Object and Method; (3) Apparatus and Procedure; (4) Data, Computations and Curves; (5) Conclusions and Check. If a more explicit outline of what the report should contain is desired, the following is suggested. (a) The report should be headed by the title of the experi- ment given in the manual, and the date on which it was per- formed. (b) A concise statement should be made of the object of the experiment and the general method of accomplishing the desired end. This may include or be followed by a statement of any principles or laws not previously given, together with an ex- planation of how they are related to the experiment. (c) A definite specification of what apparatus is used should be given that it may be identified later if necessary. Briefly de- scribe the general essential features of apparatus not previously described. Sketches and diagrams should be freely used as aids in describing apparatus. (d) Describe the mode of procedure in performing the ex- periment. This should include precautions taken to eliminate sources of error, or efforts to minimize them so that they will be negligible. It is well also to mention what difficulties were met and the mode of dealing with them. (e) The observed data should be recorded; preferably in tabu- lar form. No matter how simple the derivation, no derived re- sult should be entered as an original observation ; but may follow the original observations in the same tabulation if desired. The units should be clearly indicated, and sufficient information should be given to make the report intelligible to any person reasonably familiar with the general subject. (f) The steps taken in the computations required to reach the final results should be clearly indicated. If expressed alge- braically they should be followed by at least one numerical sub- stition and solution. Some check upon the experiment, or test of the reliability of the results should be furnished if it can he obtained. If the results differ from commonly accepted values, or from what an incomplete or incorrect theory may have led one to expect, it is best to seek for the probable cause and ac- GENERAL INSTRUCTIONS 5 count for the deviation as far as possible ; at least mention any discrepancy observed. When data admits of graphical repre- sentation by curves, they should be neatly plotted and inter- preted in accordance with the commonly accepted methods re- lating to curve plotting and interpretation. When any part of a report can be more easily and clearly expressed by a diagram than by words, the use of the diagram is preferable. (g) The drawing of conclusions is an important part of a re- port. These conclusions should be the result of careful study and should relate first to the most important things. Under the heading, " Topics for Conclusions," at the end of some experi- ments, suggestions are given as a guide in drawing conclusions. (h) Two purposes of the Test Questions placed at the end of each experiment are, (1) to suggest topics which should be included in the reports, and (2) to furnish a basis for an oral quiz. Unless otherwise directed, they should all be answered in the report. In general, it is undesirable to answer them in a list, in question and answer form, at the end of the report ; but rather they should be woven into the report where they would naturally occur in the discussion. When this is done, place in the margin the number of the question to which the matter opposite is tp be the answer, together with marks to indicate where the answer begins and ends. Answers should always be given in complete statements so that it will be unnecssary to refer to the question in the manual. (i) The report as a whole should show: (1) Originality in composition and mode of presenting the subject. The student should not copy the language or mode of expression of the manual, but should cultivate individuality in his writing. He should avoid the common fault of using the imperative as though writing directions for performing the experiment. (2) A thorough understanding of the principles and methods involved, and how they apply to the particular case. (3) Ability to rea- son clearly, to interpret results and to draw logical conclusions. To this end an effort should be made to get the main purpose of the experiment clearly in mind, and to follow it through both in performing the experiment and in writing the report. 9. Oral Quiz. Students may be called upon to answer O INTRODUCTION orally questions on any experiment assigned or for which they have registered. The aim in questioning on any experiment is usually to call attention to some point that is liable to be over- looked; or to start a line of thought or reasoning that will help the student to reach the desired end. A real difficulty is most effectually located and removed when the student cooperates by going as far as he can along the line indicated by the questions. In general, the difficulties will be removed by the student's own efforts, which is far more valuable to him than any direct aid from the instructor (or others) that would weaken instead of strengthen his ability to overcome difficulties. The student's an- swers to oral questions will have an important bearing upon his grade. II. NOTES ON PHYSICAL MEASUREMENTS 10. Units. In physical measurements the quantity meas- ured is compared with an arbitrary fixed quantity of the same kind called a unit. The number of units (integral or fractional) in the quantity measured is called the numeric; and both the numeric and the unit are necessary to completely express the magnitude of a physical quantity. For example, to express the distance between two points, a unit of length must be chosen and the corresponding numeric de- termined. If the distance is 30 inches, it is also 2.5 feet. The numeric is 30 if the unit is the inch, and is one-twelfth as large if the unit is twelve times the inch. Note that when the unit is large the numeric for a given quantity is correspondingly small. 11. Relations of Quantities. Many phenomena are as- sociated as cause and effect. Experience has led to the con- clusion that under the same conditions the same cause always produces the same effect. This fundamental fact, sometimes called the law of constancy of nature, is the basis of all experi- mental work. Experiments have been described as " questions put to nature." We get the answer not in words, but in facts which must be interpreted, classified, or summarized in general rules called principles or laws. The laws of nature are usually inductions from observation and experiment; and are simply the PHYSICAL MEASUREMENTS _ 7 Statements of the coexistence or sequence of certain phenomena, or the relations of certain physical quantities. Note the follow- ing quotation from Roger's Inductive Physics, page 6 : " Suc- cess in the study of Physics demands the closest possible atten- tion to the relations of physical quantities as expressed by laws and definitions. The passive attention of the senses is not suf- ficient; the active cooperation of the reasoning faculties is re- quired." 12. Related Variables and Proportionality Factors. The most important problem in experimental work is to determine the relations of certain quantities which are associated as cause and effect, or as independent and dependent variable quantities. For example, if a spiral spring is stretched by a variable force, there will be a corresponding variation in the elongation of the spring. In this case the force is the cause, or independent vari- able which may be controlled by the experimenter, and the elonga- tion is the effect or dependent variable. It is found by experi- ment that, for any given spring, the increase in length is propor- tional to the force, so that the increase in length per unit force is the same for each unit of force added. If this constant increment of length per unit force is denoted by k, the total elongation (£) produced by any given force (F) may be expressed by the equation E = kF, in which k is the proportionality factor, or the factor by which the value of one of the related variables must be multiplied to give the corre- sponding value of the other variable. The equation, E = kF, expresses the fact that the elongation varies as the force, which may be written E oc F; or the elonga- tion is proportional to the force. E a: F is equivalent to E^ : F^ :: E, : ^2 ; or better, E^/F^ = E^/F^, in which the subscripts de- note corresponding values of the variables. Since E is the change of length ik-h) and the force to produce it is the change of load (Lj-^i) ^^^^JiIlA (1) In the general case, if F oc Z it is possible to find a factor K of such value that Y = KX; thus the variation is transformed into 8 INTRODUCTION an equation by inserting the arbitrary constant, or proportionality factor K; and js:=| = >^^ (2) Jl A"2 Xi in which ^fj, y^, and ^itj, y^, are two pairs of observed values of Z and Y. In the variation F oc X, Z may be a power, root, or reciprocal of the variable, as illustrated in the following examples : (1) The areas of circles vary as the squares of their radii. A oc r^; A = Kr"; A = n^^ (3) If A and r are expressed in units of the same system, n is 1, and K = ir; but if A is in square centimeters and r is in inches, n = 6.45, the number of square centimeters per square inch, and K = 6.457r = 20.27. (2) The relation of radius and area may also be expressed by r oc ^/A; r^K'^A; in which K'=l/y/K (3) If one quantity increases as another decreases, so that the increase of one is proportional to the increase of the reciprocal of the other, the quantities are said to be inversely proportional. For example, if the pressure (P) upon a gas increases, the volume ( V) will decrease so that P = K l/F, or P^yP, = Fj/^i- To compress a given volume of air to half of its initial volume requires a pressure tTvice as great as the initial pressure. 13. Errors. Average or Mean Values. In physical meas- urements an error is the difference between an observed value of a physical quantity and its true value. In general, the exact amount of error cannot be known because the true value cannot be determined. Some of the sources of error may be eliminated or made negligible by proper choice of apparatus, method, or conditions; but after all precautions have been taken there are still possibly some unknown sources of error left. If the errors from these sources are constant, it may be very difficult, or even impossible, to detect them. If, however, they give rise to a varia- tion in a series of independent observations of a given quantity, it is probable that some readings will be too large and others PHYSICAL MEASUREMENTS 9 too small, so that the arithmetical mean, or average, will prob- ably be nearer right than a single observation. The theory of probabilities leads to the conclusion that the error is probably decreased in proportion to the square root of the number of ob- servations. For example, the error of the mean of one hun- dred observations may be only one-half as large as that of twenty-five observations. If the judgment is biased by previous observations or by a supposed knowledge of what the result should be, the mean of many observations may be very little if any better than one ; hence, it is important that each reading shall be affected as little as possible by previous ones or by a biased judgment. In order to detect or reduce constant errors the same quantity should be measured by different methods if possible. For further discussion of errors, see Sec. 301, 302, 303. 14. Deviations and Fractional Deviation. If the mean of several observations of a given quantity is subtracted from each observation, the differences are called " deviations." Some are positive and others are negative. The average of the several deviations (without regard to signs) is called the "average deviation." Let this be denoted by d^. The deviations are not the errors, since the mean of the readings is probably not the true value. The average deviation (d^) expressed as a frac- tional part of the mean reading (m) is called the " fractional deviation" (dt). Therefore dt = d^/'m. For example, if the average deviation is .0054 and the mean reading is 2.7150, the fractional deviation is about 5 parts in 2500., or 1 part in 500 approximately. This expressed in per cent, is ^ per cent., and is then called percentage deviation (df= lOOrfj). Exact per- centages are usually not required in computing deviations as in almost all cases approximate values give all the information nec- essary. Hence, the per cent, is usually computed mentally. In computing deviations, do not retain more than two figures, and do not retain in any observation a figure of lower order than the second figure of the deviations. 15. General Lavtrs of Distribution of Deviations. In a series of maxny observations taken with equal care, lO INTRODUCTION (1) + and - deviations of the same magnitude are equally fre- quent. (2) Small deviations are more frequent than large ones. (3) Very large deviations occur very seldom. (4) Deviations larger than four times the average deviation occur not more than once in 1000 times. If observations for the determination of average deviation (rfa) cannot be made, it may be approximated by estimating the maximum deviation that would probably ever occur, and taking one fourth of it for d^. Any observation having a deviation greater than four times the average of the others may be rejected as a mistake. 16. Mistakes. In a series of carefully made observations it is improbable (from the theory of probabilities) that there would be more than one deviation in a thousand that is greater than four times the mean deviation of the other terms of the series. It is, therefore, reasonable to assume that if a deviation is more than four times the average of the other deviations in a series, that it contains a mistake and may be omitted. Mistakes are not errors of observation, as the term is above used, but blunders that misrepresent the observation that would have been obtained if due care had been taken. An evident mistake should be left out ; but do not discard an observation without good rea- son, unless you discard the entire set of observations and repeat. 17. Accuracy, Precision, and Reliability. The term ac- curacy is used in referring to the result of a set of measure- ments to denote its supposed closeness to the truth and always implies a consideration of possible constant errors. The term precision is used when referring merely to the concordance of readings among themselves without reference to constant errors ; hence, the precision of a set of measurements may be very great and yet give a result that is very inaccurate. The accuracy of a result is measured by the ratio of the error to the quantity; or the accuracy is of a high degree when the fractional error is very small. The precision is high when the fractional deviation is small. The unavoidable errors of observation are probably no greater PHYSICAL MEASUREMENTS II for one part of a scale than for another; but the constant limit* within which these errors lie is a larger fractional part of a small length than of a large length. Consequently in meas- urmg length, the extent to which the result is unreliable is directly proportional to the limit of error* and inversely proportional to the length measured. It may be expressed in either fractional or percentage form. Reliability is measured by the smallness of the fraction or per cent. 18. Doubtful Figures. The number of units in some quan- tities is necessarily integral, for instance, the number of per- sons in a room ; while the number of units in other quantities may include a fractional part, the exact value of which can never be determined with certainty. The distance between two fixed points may be measured to a very small fraction of a millimeter if they are not too far apart and a reliable scale is used, yet the absolute distance cannot be known. A scale divided in milli- meters can be read with the unaided eye to one-tenth of a milli- meter with a fair degree of certainty, and by very careful in- spection an expert observer might in some cases reasonably es- timate hundredths of a millimeter, but the estimate would be very uncertain or doubtful in the hundredths place. It might possibly be right, but the chances are much greater that it would be in error by several hundredths. The figure in hundredths place would then be called a " doubtful figure" ; which, in gen- eral, may be defined to be a figure that is probably in error by an amount greater than unity in that place. A doubtful figure should be indicated by some distinguishing mark, such as a dash or dot above the figure. Figures in places to the right of the doubtful figure are meaningless and worse than useless; since they would misrepresent the degree of ac- curacy of the work. If it cannot be decided which of two figures is the doubtful one, the fact may be indicated by marking both. From the definition above the doubtful figure will be liable to differ from the true one by any amount greater than 1 in that place, and less than 10. If two are marked, it would imply that if the first is not doubtful the second may differ by a large *W6 will use the term "limit of error" to express the limit within which the errors in a given set of measurements are probably confined. 12 INTRODUCTION amount (say 5 to 10) ; and if the first is doubtful, the second is worthless. 19. Trustworthy Figures. Figures in places to the left of the doubtful figure may be called trustworthy figures. The last trustworthy figure must not differ from the correct value by more than 1. For example, if a reading is 24.38, the correct value may be 24.40, but cannot be as large as 24.50 without making the 8 worthless and the 3 doubtful. 20. Significant Figures. The term " significant figures " includes both the doubtful and the trustworthy figures. Signif- icant figures may be defined to be any of the ten digits except zeros used to locate the decimal point and digits in places to the right of the doubtful figure. For example, .00470 and 2540000. each have three significant figures. If, in the number 2543760., the 4 is doubtful, the figures to the right of the 4 should be re- placed by ciphers to locate the decimal point. If the first figure to the right of the doubtful figure is more than 5, add one to the doubtful figure when the figures to the right of it aire dropped. Thus, 5257360. should be written 5260000., or 5.26 x 10«. The sum of a trustworthy figure and a doubtful figure is doubtful. A like rule applies to their difference. Therefore, the doubtful figure in a sum or difference is directly under the doubt- ful figure farthest to the left in the numbers added or subtracted. For example, if the mass of a dish is known to be 85.465 grams, and the mass of the dish and its contents is found to be 234.7 grams, by weighing on balances for which the smallest standard mass is one gram, the mass of the contents is 149.2 grams, in- stead of 149.235 grams, which would misrepresent the reliability of the work. The reliability of the result cannot exceed that of the least reliable weighing. The number of significant figures in a mean of ten or more observations may be taken as one more than the number in the single observations. The product of a doubtful figure and a trustworthy one con- tains a doubtful figure and may contain a worthless one if the trustworthy figure is large. In general it is true that ,the sum, difference, product, or quo- PHYSICAL MEASUREMENTS I3 tient of two figures of which one is doubtful will contain a doubt- ful figure. 21. Significant Figures in Products and Quotients. When the final result of multiplications and divisions is obtained the question arises, which figure is probably the doubtful one, or how many figures should be retained as significant? The fol- lowing general law, while it is necessarily more or less arbitrary and may be subject to several exceptions (besides those illus- trated below) in special or complicated cases, will serve as a sim- ple rule covering the great majority of operations. General Law. In the final result (R) of multiplication or division or both, retain as many significant figures as there are in the least reliable ^ factor or number (L) entering the opera- tions, except when the initial figure ^ of one of the two numbers, R and L, is greater than three times the initial figure of the other, in which case the number having the small initial figure will have one significant figure more than the other. The exceptions are illustrated by the following problems : (a) Find the area of a card 8.425 cm. by 12.630 cm. Area = 8.425 x 12.630 = 106.41 sq. cm. (b) Find the cross section of a wire .331 inch in diameter. Sectional area = 3.14 x .167 x .167 = .087 sq. inch. (c) Divide .0001574 by 2160000. 1.574 X 10-* ^ .216 X 10' = 7.3 x 10"" * While any fixed rule must be to some extent arbitrary, there are two facts that have an important bearing upon the question. ( 1 ) A given fractional error in a factor will cause the same frac- tional error in the product; and the product is, in general, no more reliable than the least reliable factor; also (2) the obvious fact that when the product of two or more variables grows to 1 The least reliable number in an operation is the number in which the percentage error is greatest.. 2 By initial figure is meant the first figure of a number, or_ the first figure plus one if the second figure is more than 5 (see illustration b.). * Notation in powers of 10 is explained in Sec. 23. 14 INTRODUCTION the point where the first figure changes from 9 to 10, the num- ber of significant figures is abruptly increased by one. The first would lead one to expect that, in general, the product and factor would have the same number of significant figures; while the second would indicate that there are cases in which the number of significant figures in the product or result is one more or one less than the number in the least reliable factor or number. How these facts lead to the statement of the law is shown as follows : Since a product or quotient cannot be more reliable than the least reliable number entering its determination, we are not justi- fied in keeping in a product (P) or quotient (Q) any more significant figures than will indicate a reliability most nearly in agreement with that of the least reliable number (L). The rela- tive reliability of two numbers may be approximately expressed by the ratio of the fractional changes {fi/f,) due to a change of unity in the doubtful figure of each. For example, ii L = 1.05 and P = 842., a change of unity in the doubtful figure of L will cause a fractional change of 1/106 of L; and a like change in P will cause a fractional change of 1/84 of P- The ratio of these changes is f^/f^ = 1/106 ^ 1/84 = 84./106. = 1. nearly ; but if 2 were the doubtful figure in P, the ratio would be 842./106 = 8. nearly, indicating that the reliability of P would be about eight times that of L instead of once ; therefore 4 should be considered doubtful rather than 2. Consider in like manner the case when L = 1.00 and P = 3.33. Then A/A = 333./10O. = 3^ ; but if F = 33., fyf^ = 100/33 = 3J. In this case we have the boundary where there seems to be no choice between retaining in P the same number of significant figures as in L, or retaining one less. Note that the same would be true for any other numbers of which the initial figures have the ratio ^. If the first figure of P were more than three times the initial figure of L, there would be a choice in favor of two significant figures in P instead of three. Like consideration of other cases leads to the following con- clusion, in which R is the result (either product or quotient) and the term initial figure is used to denote either the first figure of a number, or the first figure plus one when the second figure is PHYSICAL MEASUREMENTS 1 5 more than 5. // the initial figure of one of the two numbers R and L is greater than three times the initial figure of the other, the number having the small initial figure will have one signif- icant figure more than the other. In more than two-thirds of the cases R and L have the same number of significant figures; hence, the cases named above are exceptional and occur only when the first figure of one number is 1 or 2, and the other first figure is more than three times as large. It is possible that cases may occur in which the reliability of the result is less than that of any factor. For example, the stiff- ness of a wire when twisted is proportional to the fourth power of its radius. An error of one per cent, in the measurement of the radius will introduce an error of four per cent, in the fourth power. The possible cumulative effect on the result due to er- rors in several factors may be taken to be the sum of the sepa- rate effects. This is discussed further in Part III. See Sec. 303. 22. Significant Figures in Data. All the significant figures obtained in a reading should be recorded and no others. For example, if a scale is read to the nearest 32d of an inch and the reading obtained is 6%2 in., when this is changed to a decimal the result should be 6.156 and not 6.15625, which would indicate a reliability far in excess of that actually obtained. Also, sup- pose the smallest divisions on a scale are millimeters and estimation of tenths of mm. is made. If the reading as near as can be judged is exactly 8 cm. it should be recorded as 8.000 cm. to indicate that in the judgment of the observer the reading was not as much as .1 mm. greater or less than 8 cm. 23. Methods of Computation. Notation in Powers of Ten. The use of logarithms is recommended whenever possible and especially in a series of operations involving multiplication and division, where a considerable saving of labor may be accom- plished. The logarithm table used in any case should have at least as many places in the mantissa as there are significant figures in the least reliable factor or number entering the computation. For example if a number has four significant figures at least a four place table must be used and to avoid interpolation a five l6 INTRODUCTION or six place table should be used for numbers with four signif- icant figures. The number of significant figures in the result may be determined by the rule given in Sec. 21. The smair slide rule (10 in.) is useful in checking products and quotients and should be freely used for that purpose. It may be relied upon for computations involving numbers having three significant figures. However it should not be depended upon at all until the student is thoroughly familiar with the scales used and has mastered the estimation of tenths of divisions. The student should get into the habit of adding or subtract- ing horizontally when the numbers to be used are in adjacent col- umns on the data sheet. Also multiplications and divisions by one figure or by powers of ten should be performed mentally without rewriting the factors. Instead of writing the ciphers used to locate the decimal point it is the common practice to write the significant figures only and to express the other ciphers by multiplying by the proper power of 10. Thus 25400000 could be written 25i.xlO» and .000625 could be written .625 x 10'. In many cases it is desir- able to place the decimal point after the first significant figure. Thus the first number could be written 2.51 x 10' and the second 6.25 X 10"*. Note that the positive power of ten indicates how many places the decimal point must be moved to the right and the negative power of ten indicates how many places the point must be moved to the left. , It is possible and very often desirable to carry this notation through one or several of the four fundamental operations. In addition or subtraction the numbers must be expressed with the same powers of ten. In multiplication the powers of ten are algebraically added and in division, algebraically subtracted. For example, 40xl0«x6x 10-3 = 240x103 2S.6x 10-5 ^8x 102 = 3.2 xlO-f .328 X 10« + 62 X 10* = 32.8 x 10* + 62 x 10* = 94.8 x 10* 24. Abridged Multiplication. This method is useful when figures to the right of the doubtful one are to be discarded. It differs from the common one in three respects, as follows: (1) The left hand figure of the multiplier is used first. (2) PHYSICAL MEASUREMENTS 17 For each successive figure of the multiplier the partial product is set one place to the right. (3) All figures to the right of the first partial product are omitted. Rule. (1) Multiply first by the left hand figure of the multi- plier. (2) Omit the last figure of the multiplicand when using the second figure of the multiplier; the last two figures when us- ing the third figure of the multiplier, and so on; but add to each first product the number carried from the product with the figure omitted. (3) Write the right hand figure of each partial prod- uct directly beneath the last figure of the first partial product. Note that if the figures in the lower line (multiplier) be num- bered from left to right, and those in the upper (multiplicand) from right to left, then the products of like numbered figures will give figures of the same order. These should be placed under each other for the right hand figures of the partial products, after they have been increased by the number carried from the multiplication of the next lower order as in ordinary multi- plication. For example, in the first illustration below : 1x1, 2x4, (3x3) + 1, and (8x2) +2 give 1, 8, and 8 respectively in thousandths place. The 1 after 3 x 3 is carried from 3x4 = 12, of which the 2 is not written because it is of lower order than the doubtful figure of the result. The 2 after 8x2 is carried from 8x3 = 24^ because 24 is nearer 20 than 30. The method retains enough figures in the result in any case; but in some cases the last figure may not be significant. (See No. 2 below.) A study of the following applications will help to make clear the method. 2,456 X 10' 2.341 1.235 12.630 ,8.42S 2456000 .0001674 - 2.341 468 70 18 101.040 5.052 253 63 2456 1474 172 10 2.456 1.474 172 10 2.897 106.41- 411.2 4.112 X lO'' (1) (2) (3) (4) 25. Abridged Long Division. The method differs from or- l8 INTRODUCTION dinary division in one respect, namely : one figure is cut off of the divisor after each subtraction instead of adding a figure to the remainder. Carry from the next lower order as in multiplica- tion. 7.250 -i- 2.853 = ? 10.406^-4.100=? -:°25M_=? 6250000. 2.541+ 2.536 .615 X 10'^ Z.^J 7.250 '^.i0J 10.406 /.;^ X 10*/ 3.84 X lO'^ 5.706 6 200 ^—^^ -^ 3 75 1 544 2 206 9 1 426 2 050 6 118 156 ~3 114 123 3 4 33 _3 ^ (5) (6) (7) 26. Curves. Plotted curves * are very useful in the study of experimental data and greatly assist in picturing to the mind the important relations of the quantities plotted, as soon as the student has learned how to plot and interpret them properly. Such curves are an important part in all physical and engineer- ing work, and it is therefore very important to be able to derive from them the information they may give. After becoming fa- miliar with a few general principles relating to plotting and in- terpretation, the student will be able to devise for himself new ways of graphically expressing data, and new methods of solv- ing graphically experimental problems. 27. How? to Plot Curves. (A) Definitions. Many ex- periments consist in arbitrarily changing one quantity and ob- serving the various values of another quantity related to and de- pendent upon the first. A pair of these related quantities may be expressed graphically by the position of a point with refer- ence to two lines at right angles to each other. The two lines are called Axes, and their intersection the Origin. The hori- * See Sec. 40 for the definition, etc. PHYSICAL MEASUREMENTS IQ zontal distance (x) of a point (p) from a vertical axis is called the abscissa of the point. The vertical distance (y) from the horizontal axis is the ordinate of the point. By use of scales plotted along the axes a point may be so located that its dis- tances from the axes, or the x and y coordinates, represent a pair of "corresponding values of two related variables. It is custo- mary to plot the independent variable horizontally (to the right) as abscissas, and the dependent variable vertically as ordinates, unless there is some special reason for plotting them differently. (B) Scales. The vertical and horizontal scales need not, and generally should not, be the same. They should be chosen (1) so as to be easily plotted and read; (2) so as to have the curve as large as the sheet will permit; (3) so that the curve will not lie too close to either axis; and (4) if some particular part of the curve is of special interest, so as to make that part most useful. Note the number of spaces available on the sheet in a given direction, also note the largest number to be plotted in that di- rection; then, the scale should be so chosen that each space will represent 1, 2, 5, or 10 (perhaps in some cases 4) units, or some decimal multiple of these numbers; especially if the paper is ruled with every tenth line heavy. In plotting very large num- bers or very small fractions, the unit for plotting may be some power of ten or some decimal part of unity. For example, if an ordinate and the corresponding abscissa are 1860000. and .00745 respectively, they may be written 18.6 x 10= and 7.45 x 10'^ ; then the point would be located 18.6 large spaces above the horizontal (or X) axis, and 7.45 large spaces to the right of the vertical (or Y) axis. (C) Location of Points. The scales should be shown by numbering along the axes the lines marking the large spaces ; or it is often sufficient to number every second or fifth line. Write along each axis what quantity is plotted in that direction. The values of coordinates of plotted points should not be written along the axes, and only in exceptional cases should they be written by the points themselves ; since the position of the point 20 INTRODUCTION clearly shows the values of its coordinates when the scales are properly chosen and marked along the axes. Each point should be carefully located (estimating tenths of small divisions) and marked by a very small circle having the point at its center. A large dot or a very small x is sometimes used, but neither is so satisfactory as the circle. In some cases the range of data is within rather narrow limits, so that if all values were plotted from zero the curve would occupy a very small part of the sheet and be too small to be useful. In such cases it is better to plot excesses above certain convenient values that are below the smallest numbers to be plotted each way. Large scales beginning with the chosen values can then be employed and the shape of the curve can be brought out far better. (D) Groups and Titles of Curves. If two or more curves are to be compared, they should be plotted on the same sheet to the same scales. It is frequently desirable to plot two or more curves on the same sheet, using different scales. When this is done it should be made perfectly evident which scale belongs to each curve. This may be done by writing along the curve its title (e.g.. Velocity-time, load-deflection, etc.), always writing the ordinate first ; then opposite each scale should be written the corresponding title. To further distinguish the curves, different colored inks may be used both for the curves and for the scales. (E) Irregular Curves. In some cases one of the variables plotted may not depend directly upon the other quantity, in which case there is no fixed relation between them and the curve may be quite irregular in form. For example, if the out- door air temperature were observed every fifteen minutes for several hours and the points were plotted there might be some marked irregularities in the variations, especially if the time in- cluded a " cold wave." There would probably not be any very abrupt change and each observation would probably be quite reliable and not subject to large errors of observation. In such case a smooth curve should be drawn through all of the points. (F) Regular Curves. In other cases we may be certain that except for unavoidable errors in observation the curve should have a particular form. We then do not attempt to draw a curve PHYSICAL MEASUREMENTS 2.1. through all of the points, but rather draw the one of prpper forrn that most nearly fits the observations. Then, if the ohserve<;i points deviate from such a line more than can be reasonably ac- counted for by unavoidable errors, the cause for such deviation should be sought and accounted for, if possible. For example, the deflection of an elastic beam properly supported, is within certain limits, proportional to the load placed upon it; that is, the deflection is doubled if the load is doubled, and is zero if the load is zero. In this case, the line plotted would be a straight line through the origin (or intersection of axes). It should be so drawn that the deviations of the plotted points on one side of the line will balance those on the other as nearly as possible. If the beam contained defects, or was supported so that other than perpendicular forces acted upon it, or if the measuring ap- paratus were out of proper adjustment, the observed points might deviate from the straight line in such a way as to give a clue to the faulty condition affecting the result. It is often the case that the conditions under which part of the data is obtained are more favorable for reliable results than for other parts. The more reliable data should be given greater weight in determining where to draw the curve. Do not give any more weight to the first and last points than to any of the other points unless there is some special reason for doing so. For example, if it is known that the curve passes through the origin, it should be drawn ex- actly through the intersection of the axes. 28. Interpretation and Uses of Curves. When the co- ordinates of a plotted line have a known relation, it can generally be expressed algebraically. (A) Straight Line Through the Origin. The simplest case is a straight line through the origin. (See Fig. 1.) The ratio of any one ordinate (yj) to its abscissa (x^) is the same as the ratio of any other ordinate (y^) to the correspond- ing abscissa (^2)- Therefore, yi/^^i = y2/'^i = 3. constant = a; then in general yy^ = a, or y = ax (4) This equation holds true for any pair of coordinates and is called 22 INTRODUCTION the equation of a straight line through the origin. The slope of the line is a, since ,^y- = y,jzji (5) X X^ Xi a is the proportionality factor used in the same way that k is used in Sec. 12. // two variables when plotted give a straight line through the origin, the interpretation is that one of the quantities varies di- rectly as the other. (B) Straight Line not Through the Origin. (See Fig. 2.) If a straight line is drawn above the origin and parallel to the line y' = ax through the origin, then any ordinate y of the upper line less the vertical distance (&) between the lines equals the ordinate y' of the lower one, and y-b = y' = ax, from which y = ax + b (6) This is the general equation of a straight line having an inter- cept b on the F-axis. The F-intercept is the value of y when x = Q. The X-intercept (c) is the value of x when y = 0. Problem. Draw a figure that expresses graphically what is given above, and prove from the general equation and similar triangles that, a A - ^^ (7) X X + c c=-bla (8) y^ a(x + c) (9) a-y^^^ (10) // two variables give a straight line not through the origin, the interpretation is, that one of the quantities varies directly as the other plus or minus a constant. (C) Any Straight Line. Referring to Figs. 1 and 3 and equations (5) and (10) it will be seen that the following gen- eral statement will apply to any straight line, whether it does or does not pass through the origin. // two variables give a straight line any change of one vari- able is proportional to the corresponding change of the other variable. PHYSICAL MEASUREMENTS 23 If the ordinates decrease toward the right the slope is nega- tive and is expressed in the equation by -a. A negative slope means that the decrease of the ordinates is proportional to the increase of the abscissas. The slope at any point of a curved line is the slope of the tangent line at that point. Fig. 1 Figf. 2 Fig. 3 (D) Examples of the usefulness of the intercepts and slope of a straight line occur in electrical measurements. (1) If a cell of unknown E.M.F. (£) and internal resistance (r) is connected in series with a known variable resistance (i?), the X-intercept of a curve plotted with reciprocals of cur- rent (1//) as ordinates, and R as abscissas, is the unknown in- ternal resistance r, and the slope of the line is the reciprocal of tihe unknown E. This is shown in the equation of the line, which is, if written in the form corresponding to y ■■ (11) is commonly written R + r (11) o {x + c). Equation (12) (2) If the differences of potential {V) at the terminals of the cell are plotted as ordinates and corresponding currents (/) as abscissas, the curve has a negative slope showing that the decrease of the ordinates is proportional to the increase of the abscissas. The F-intercept is the E of the cell, and the equation of the curve is V = -rI + E (13) 24 INTRODUCTION if written in the form corresponding to y = ax + b. The slope of the lind is -r, from which the resistance of the cell is determined. The common form of this equation (as used in Exp. O — 21, etc.) is V = E-rI (14) Note that in the two cases above the two unknowns, E and r, were obtained indirectly by measuring / and R in the first case and I and V in the other. (E) Physical Interpretation. It may be noted that in general when related physical variables are plotted, a physical interpretation or meaning may be found for each characteristic of the plotted line, such as the slope and intercepts of a straight line, and the curvature, maximum and minimum coordinates, and area beneath a curved line. For instance, the slope of a distance- time curve is the velocity and the slope of a velocity-time curve represents the acceleration. The student should endeavor to interpret in physical terms the characteristics of his curves and find their numerical values. (F) Curved Lines. If the observed values give a curved line when plotted, it is often desirable to plot also such functions of them as will make the curve a straight line ; because that may be easily drawn and shows the relative accuracy or reliability of measurements and results better than any other line. If X and y give a parabola, (y = ax^ + b), plot x^ and y for a straight line. If :ir and y give an hyperbola, (xy = c), plot iy/x and y, or l/y and x. If x and y give a logarithmic curve (y = e^), plot log y and x, etc. 29. Scale Readings and Estimation of Tenths. A scale reading expresses the location of a point, p, on a scale, opposite a point or line, on something else to which the scale is applied. If p falls exactly upon a line of the scale, the reading, in small divisions, is integral ; but if p falls between two lines, a fractional part must be added to the designation of the preceding line. This fraction is usually estimated and expressed in tenths of the small division. For example, if /> is midway between the 37 mm. mark and the 38 mm. mark, the reading is 37.5 mm., or 3.75 cm. If the reading point, p, falls in either half of a small space. PHYSICAL MEASUREMENTS 25 note the ratio of the parts into which it divides the half-space. If their ratio is 2 to 3 or 1 to 4, the number of tenths is one of the following: .2, .3, .7, .8, or .1, .4, .6, .9. If the observer is sure that the reading is greater than a given tenth and less than the next, and if there is no reason for choosing one rather than the other, 5 may be used in the next decimal place of the read- ing. In general it is not necessary to read a millimeter scale to less than .1 mm. It is possible, however, for a careful observer to estimate hundredths of a millimeter, especially when a reading glass is used, but the hundredths would certainly be doubtful. Estimation in tenths of the smallest divisions should also be made when a scale divided in 8ths, 16ths, 32ds, etc., of an inch is used. Suppose the smallest division on the scale is 1/16 of an inch. If estimation of this division is made in tenths the result is in 160ths of an inch. A convenient way to record this reading is to place a decimal point in the numerator of the fraction. Thus, 6 -^ means that the reading was 6 whole inches 16 plus 3/16 inch, plus .3 of a 16th. (See Sec. 22.) 30. Measurements. The length of a line between two points cannot be gotten from a single observation. A reading must be taken at each point and the length is derived from the two original observations by subtracting one reading from the other. In estimating tenths of the smallest divisions on the scale, con- sider the distance between the centers of the division lines, and not the distance between their edges. Take scale readings oppo- site the center of the line limiting the length to be measured. The line of sight, from the eye used to the point observed, should be perpendicular to the edge of the scale to avoid parallax, and the edge of the scale should be as close as possible to the points for which readings are taken. Repeated measurements add very little, if anything, to the ac- curacy or reliability of a measurement unless each observation is independent of those that precede it. The theory of prob- abilities leads to the conclusion that the reliability increases as the square root of the number of independent observations in which the judgment is kept free from bias. Any expedient that 26 INTRODUCTION will aid in obtaining independent observations is of value. It is then assumed that there will be as many observations in ex- cess of the true value as there are below the true value, and the most probable value is the arithmetical mean. Serious defects in the graduation of a scale may be detected by using different parts of the scale to measure a suitable dis- tance between two fine parallel lines. III. GENERAL PRINCIPLES RELATING TO EXPERIMENTAL WORK 31. The Law of Causality. I. Every effect has a sufficient cause. Any given cause, acting under the same conditions, al- ways produces the same effect. Any resultant effect is the sum of the effects of all- the causes acting. 32. Independence and Transmissibility of Forces. II. If several forces act simultaneously upon a body, each force pro- duces its own effect independently of all the others. "When a force acts upon a rigid body its effect is the same whatever point of its line of action is taken as the point of application." (Ferry's Dynamics, p. 7.) 33. Conservation of Energy. III. The amount of energy lost by one system of bodies always equals the amount of energy gained by another. Energy can neither be created nor destroyed. 34. Variables. IV. If possible, always avoid having more than two variables in any one step of an experiment. The third variable is the chief source of uncertainty and error. If the re- lations of several quantities are desired, experiment first on one pair of related variables, keeping all other quantities constant; then take in like manner each other pair of related variables, in which one of the pair is a cause and the other is an effect. 35. Sources of Error. V. Arrange the experiment so that the results will be affected as little as possible by unavoidable sources of error. 36. Independent Observations. VI. The average of many (m) independent observations of the same quantity is probably more reliable than any one observation in the ratio V" : 1 ; but GENERAL PRINCIPLES 27 little or nothing may be gained if the observations are affected by a biased judgment, or by previous readings. Any expedient that will secure independence of observations is helpful. 2>7. Discarding Data. VII. Do not discard parts of a set of data without evidence that will justify such a procedure. It is better to take a complete set of new observations in which an attempt is made to eliminate possible sources of error. An un- expected, or apparently incorrect, result may prove to be right when all of the conditions are properly considered. In many (m) independent readings of the same quantity the deviation of a single reading will, according to the theory of probabilities, not exceed four times the mean deviation more than once in a thousand readings ; hence, the . arbitrary rule, " that we are justified in discarding a reading if its deviation exceeds four times the mean deviation of the others," is sometimes used. The large deviation is probably due to a mistake instead of an unavoidable error in reading. 38. Using Data. VIII. When possible, avoid methods of using data that will cause the result to depend to an undue ex- tent on any one observation or item of data. 39. Measurements and Results. IX. A result cannot be more reliable than the least reliable factor that enters its deter- mination. The degree of reliability may be expressed ap- proximately in terms of significant figures, percentage of error, fractional difference, or probable error. It is useless to expend labor to reduce the percentage of error in one part of an experiment far below that in another part entering the same result as a factor unless it enters more than once as a factor. Quantities to be added or subtracted should be measured to the same number of decimal places. Quantities to be multiplied or divided should be measured to within ap- proximately the same fractional part of themselves without any regard to the decimal point. Factors that are to be squared or cubed should be measured with especial care; for an error of 1 per cent, in the factor will cause an error of 2 per cent, in the square and 3 per cent, in the cube. 28 INTRODUCTIOK '40. Curves. X. A line having the distances of each suc- cessive point from two fixed rectangular axes, proportional to the corresponding values of two related variables, is called a "curve"; whether it be curved or straight. Curves, properly plotted, are very useful for studying data, detecting mistakes, etc. ; and should be used wherever the data is suitable. If the two variables observed give a curved line, try to find a way of plotting that will give a straight line. See Sec. 28 (F). The slope of the straight line and the intercepts on the axes gen- erally have a useful physical interpretation. IV. NOTES ON UNITS 41. Introduction. There are four systems of units in com- mon use. Two of these, called absolute systems, are based upon units of length, mass, and time as fundamental units; and the other two, called gravitational systems, are based apon units of length, force, and time. Note that the fundamental units of the two kinds of systems differ with respect to mass and force. Mass may be defined to be the quantity of matter in a body. Force is that which produces, or tends to produce, motion, or change of motion. Weight is the force due to the earth's at- traction ; it is not mass and in any one system, is not even numer- ically equal to mass. Weight, however, is proportional to mass in any one locality where the weight of unit mass (^) is con- stant. Weight varies from place to place while mass does not. The ratio of the weight (w) of a given mass to the weight (gr) of unit mass at the same place is constant, and represents the number of units .of mass. Therefore m = Wy/g. This relation is true in any one system; but it is not true for w in one system and w/g in another. The relation w = w/g is usually obtained in a manner quite different from that given above. After defining the units of mass (gram and pound) in the absolute metric and the absolute English systems and after the term acceleration is clearly defined and understood, the relations of force, mass, and acceleration, as expressed in Newton's second law of motion, may be considered. This law may be stated as follows: The acceleration (a) of a mass (w) when NOTES ON UNITS 29 acted upon by a force (F) is directly proportional to the force, and inversely proportional to the mass. Therefore a oc F/m and F oc ma; or F = kma, in which k is the arbitrary constant, or proportionality factor, that must be introduced in transforming an expression for variation into an equation. In this equation the units for m and a are already fixed, but the unit for F is not; therefore, a unit for F may be chosen such that k will become unity. That will be the case if the unit force is defined to be, — The force which will impart unit acceleration to unit inass. Then, for any one system in which the unit of force is thus defined, F = ma. This equation is not true if units from different systems are mixed in the same equation. For the special case of gravitational force or weight, w may be used in place of F, and the value of a for a freely falling body is expressed by the symbol g. Then w = mg, or m = w/g; but when m=\, w = g. Therefore, g expresses the weight of unit mass, as well as the acceleration of a freely falling body. 42. Tabulation of Units in Mechanics. TABLE I. FUNDAMENTAL UNITS Systems Length Time Mass 1 Force A. M. A. E. G. M. G. E. Absolute Metric (orC.G.S.) Absolute English (or F.P.S.) Gravitational Metric Gravitational English Centimeter Foot Centimeter Foot Second Second Second Second Gram Pound Wt.oflgm. Wt. of lib. TABLE II. DERIVED UNITS System Mass Farce Work and Energy PoiJjer A. M. A. E. G. M. G. E. g - gramst g- pounds (or slug)t Dyne Poundal Erg (Joule*) Foot-poundal Gram -cm. (Kgm-m*) Foot-pound Erg/sec (Watt*) Ft.-pdl/sec Kgm-m/sec* Ft. -pound/sec (H.P.*) * Denotes large units, multiples of small units, t Use G to indicate this unit. % Use S to indicate the slug. 43. Definitions. For the definitions of the fundamental units, see text books on Physics. In the Introduction, the unit of force in any system has been defined to be the force that will impart unit acceleration to unit 30 INTRODUCTION mass. From experiments with falling bodies we find that the weight of one gram will increase the velocity of a mass of one gram 980 cms. per second (approximately) each second that it acts. Since the force equal to the weight of one gram imparts 980 units of acceleration to a mass of one gram, the same unit force will impart only unit acceleration to a mass 980 times as large, according to Newton's second law, which states that accel- eration is inversely proportional to the mass accelerated. Now the weight of one gram is the G.M. unit of force, which, by definition, will impart unit acceleration to a G.M. unit of mass; therefore, the G.M. unit of mass is 980 (or g) times as large as the A. M. unit, if acceleration is measured in centimeters per second per second. (At Philadelphia g is 980.18 cm/sec^). In like manner it may be shown that the G.E. unit of mass is 32.16 {or g') times as large as the A.E. unit, if acceleration is measured in feet per second per second. Note that 980.2 cms. = 32.16 ft.; also that g is the weight of unit mass; or gr = 980.2 dynes per gram., or 980.2 gm. wt. per G.; and g' = 32.16 poundals per pound, or 32.16 lbs. wt. per slug. The number of mass units in any given body is inversely pro- portional to the size of the units, consequently, to find the num- ber of gravitational units of mass from the number of absolute units, it is necessary to divide the number of absolute units by m' (in grav. units) = m (in abs. units)/5r But the mass m in absolute units is numerically equal to the weight {w) in gravitational units, since the gravitational unit of force is the weight of the corresponding absolute unit of mass. Therefore, m (in abs. units) =w (in grav. units) ; and m' (in grav. units) =w (in grav. units )/5f. This shows that m = w for mi.ved units as indicated ; but wt = w/^g if m, w, and g belong to the same systems. 44. Problems. The relations of force, mass, and accelera- tion in each of the four systems are given below and illustrated by. problems. m ■■ NOTES ON UNITS 3 1 Absolute Systems /^^•^•^//'" dynes) = m (in grams) X a (in cms/sec. 2) |(A.E.) f (m poundals) = m (in pounds) X a (in feet/sec. 2) Gravitational |<«-^-) ^<- ^— -•> = ^S^^^^^) ^ '^ <'" -'Z-^) '"""^ [(G.E.) .(in PO-nds wt.,=^l^n^2^D^> x . (in feet/sec. , Prob. 1. Given: , - is kg rei cms/sec.z „ , / . ,> -111 lbs. " -\ 2 feet/sec.2 ^ =? (required) A.M. F=mXa = 5000 X 61 = 305000 dynes = .305 megadynes A. E. f = m X « = 11 X 2 = 22 poundals ^ ,, ^ W 5000 G.M. i^ = — X a = -^g^ X 61 = 311.2 grams wt. W 11 G.E. f =-7- Xa= ^23g x2=.684poundswt. Prob. 2. Given: r (A. M.) 196000 dynes \ ( A. M. ) 500 grams P _ I (A.E.) (.44 X 32.16) poundals _ J (A.E.) 1.1 pounds )(G.M.)200gm. wt. "'-|(G.M.) .510 G. UG.E.) .44pd. wt. I (G.E.) .0342 slug a is required . ,, , F 196000 ,„„ (A.JVl.) a = — = ^„. — = 392 cm. per sec. per sec. m 500 ,, _, F .44x32.16 ,„.. , ^ (A. tj.) a= — = — = 12.86 feet per sec. per sec. m 1.1 tr> T.A\ P 200 200 „„ <«-M-> '^ = ;w^ = 500^ = rsro = ^^^ ^^- per sec. per sec. F 44 44 ^^- ^-^ ^ = :^ = ni32l6 =^342= ^^-^^ *"' P*' '"=• P*' '"• Prob. 3. Given: (■(A.M.) 49000 dynes f(A.M.) 61 cm. per sec. per sec. j._\ (A.E.) 1.1 X 32.16 poundals J (A.E.) 2 ft. per sec. per sec. 1 (G.M.) 500 grams wt. 1 (G.M.) 61 cm. per sec. per sec. UG.E.) 1.1 pd. wt. Ug.E.) 2 ft. per sec. per sec. Required, m = ? m = 8033 grams = 17.69 pounds = 8.20 G. = .55 slug Note: For Absolute System use m in grams or pounds for mass. For Gravitational System use m/g or m/g' for mass. CHAPTER II EXPERIMENTS IN MECHANICS AND HEAT GROUP I— I. INTRODUCTORY MEASUREMENTS experiment 1-1 Measurements with Metric and English Scales Errors, Deviations and Significant Figures The objects of the experiment are: (1) To give practice in reading scales, especially in estimation of tenths of small divi- sions. (2) To compare the units of length and area in the metric and English systems. (3) To study the methods of ob- taining and using data, and the meaning of certain terms relat- ing to measurements and reliability of data. The experiment includes two parts, A and B. (Read "Remarks" preceding test questions before beginning the experiment.) Procedure. Part A. Place a 12-inch scale and a 30-cm. scale with theit^ edges in contact. Adjust them so that the centi- meter scale reads from left to right, and the 12-inch mark on the inch scale is opposite some part of the first or second centi- meter but does not coincide with any line on the centimeter scale. Take readings opposite the 12-inch mark, and also opposite the 6-inch mark. Tabulate these as shown in the table below, then subtract the readings to find the number of centimeters in 6 inches, and place the result in the next column as indicated. Proceed in like manner with readings opposite the 11-inch and 5-inch marks, 10-inch and 4-inch marks, etc., as shown in Table HI. Record readings immediately, and do not change them. For exceptions, see Sec. 15 and 37, Chap. I. Data and Results. When the fifth column is completed, find the mean value, and place it below the column. Subtract the mean from each value in the column and place the differences 33 34 MECHANICS in the next column. Prefix the proper sign before each differ- ence. These differences in the sixth column are called devia- tions from the mean. The average of these, without regard to signs, is called the average deviation. TABLE in Marks on Readings on Marks on Readings on Centimeters Deviations inch scale cm scale inch scale cm scale in 6 inches from mean 12 0.83 cm. 6 16.04 15.21 11 3.37 S 18.S9 15.22 10 5.89 4 21.13 15.24 9 etc. 3 etc. etc. 8 2 7 1 Inch scale No Cm. Scale No Mean length of 6 inches in centimeters. = Average deviation from the mean. z= average deviation Fractional deviation of lengths = = Approx. mean length Centimeters per inch from table (column 5) = Correct number of cm. per inch =:: Difference between the correct value and the observed = Percentage difference = per cent, approx. ; or about 1 part in Procedure. Part B. Measure the dimensions of a rect- angular plate or card, as indicated by the dotted lines shown in Fig. 4, in both centimeters and inches. Check the results by reducing the square inches to square cen- timeters, which may then be compared with the area obtained with the centimeter scale. Use an inch scale divided in sixteenths. Estimate tenths of sixteenths and record as Fig. 4. shown in -the table. Data and Results. Tabulate data and results as indicated in Table IV. 1 '.'■■" f* ;S 6 .7 1 1 1 1 INTRODUCTORY MEASUREMENTS 35 TABLE IV Readings Cm. Scale R — L Cms. Readings In. Scale R — L Line Left (L) Right (R) L R Inches ?1 a: 3.73 14.13 10.40 3^f 5.2 ' 16 *W 4.37 14.78 10.41 ^'-i 8.6 ^ 16 ^fe' 4 12.32 19.60 7.28 .^ ^ 16 '¥ ^5 ^ 6 7 li>t1i"t Plate A, Scale No. 10, Divisions 10 per cm. and 16 per inch. Mean Length cm., or in. Cm. per in. = . . Mean Width . . . ^. . . . cm., or in. Cm. per in. = . . Area = 7.250 X 10.406 = 75.44 sq. cm. piy. Area = 2.853 X 4.100 = 11.70 sq. in. = 75.46 sq. jrf. Fractional Difference in Area is less than 2 parts in /OOO; or iSBS^SHti Difference in Area is about .03 per cent, of Area. Average Percentage of Error in Cms. per inchj= Remarks. Note carefully Sec. 29 and 30, and be able to an- swer the first three test questions before beginning to take read- ings. (See Sec. 2.) Place the title of the experiment and the date at the top of the data page in the note book. Rule Table III (with pencil in the data book) ready to enter the observed data. (See Sec. 5.) In measuring the plate avoid placing a marked division of the scale even with one edge of the plate. Use different parts of your scale. (See Sec. 30.) The procedure for Part A should not occupy more than one- half hour. As soon as all of the data required in this part are observed and recorded, proceed to get the observed data for Part B, which should be obtained in less than one hour, even when the observations are very carefully made. As soon as the observed data is complete and recorded, present it to the Instructor in charge for inspection. 36 MECHANICS Make out Tables III and IV on separate sheets of report paper, heading each with the title of the experiment, the name, and the date when the data were observed. Do not use observa- tions or computations made by another. The tables include the outlines following the data. These should be worked * out in the note book before making out the report sheets. Bind the re- port sheets in a report cover, properly filled out, and hand in. (See Sec. 7.) Test Questions. Part A. 1. State the objects of the experiment. 2. Describe a method that may be used in estimating tenths. 3. State the precautions that should be observed in taking readings, re- lating to line of sight, position of scale, width of lines, and independent observations. 4. State the important facts relating to errors, deviations, mistakes, accuracy and precision. (See- Sec. 13-17.) 5. If the deviation in measuring a line S inches long is .1 per cent., what per cent of deviation will probably occur in measuring a line 2 inches long, also one 10 inches long, with the same scale? Explain. 6. (a) Which of the general principles expressed in Sec. 34-39 are in- volved in this experiment? Explain. (b) Discuss the sources of error, and methods of reducing their ef- fects. (Note that each reading is used only once, and all of the inch marks except the zero are used to get six measurements of a six inch length, which measurements may differ regularly or irregularly.) 7. What means have you of checking your results? Part B. 8. Define doubtful and trustworthy figures, and significant figures. (See Sec. 17-20.) 9. State the principles and rules relating to the number of significant figures in products and quotients. (See Sec. 20-21, and 39.) 10. (a) Explain fully how you made measurements with your inch scale to fractions smaller than the smallest division on your scale. :(b) How large an error of reading might occur even with careful reading? (See Sec. 29-30.) 14 3 .i(c) Reduce the reading 7 j^ inches to the decimal form and mark the doubtful figure. 11. (a) Mark the doubtful figure in your results for area, and explain how you determine it. (b) Note the difference between the determinations of area, and the errors in the number of centimeters per inch. Do they indicate that the difference in area is probably due to errors in scales, or to other sources of error, or both? Explain. ♦Abridged multiplication is required in this experiment. (See Sec. 24.^ introductory measurements 37 experiment 1-2 Measurements with Calipers Study of Micrometers and Verniers To measure small lengths and keep the fractional error rea- sonably small requires some other means than the common scale and unaided eye. For example, to measure the diameter of a wire within one-tenth per cent, requires special apparatus. Many forms of apparatus for precise measurements use either the micrometer, or the vernier or both. Each is a device for reading from the measuring instrument fractional parts of the small divisions of the main scale very much smaller than those that can be estimated with the eye alone. The smallest fraction to which the device reads directly is called its least count. How to determine and use the least count is the key to success in the study and use of micrometers and verniers. The object of the experiment is to enable the student to read, understand and use correctly any form of micrometer or vernier, however complicated or unusual in form. The experiment is divided into three parts, A, B, C. Part A relates to micrometer calipers, Part B, to vernier calipers, and Part C, to verniers for graduated circles. Part A. Micrometer Calipers. The micrometer screw con- sists essentially of an accurately cut screw of definite pitch turn- ing in a fixed nut, or part of the apparatus, and having a gradu- ated head, or barrel, on which small fractions of a revolution may be accurately read. The pitch of the screw is the distance it moves endwise during one complete revolution; or the axial advance per turn. The least count is the distance it moves end- wise when turned only the small fraction of one turn correspond- ing to one small division on the graduated rotating head or bar- rel ; or the axial advance per nth part of a turn, if there are n divisions on the head. The micrometer caliper consists of a metal frame so formed that the object to be measured may be placed between two par- allel faced arms, or jaws, one of which is the end of the mi- crometer screw, and the other a fixed stop. The graduated 38 MECHANICS screw head is a hollow cylinder which covers and rotates around the fixed stem in which the screw turns. The number of com- plete turns of the screw, from contact with the stop, is indicated by a scale lengthwise on the fixed stem. Parts of a single turn are indicated by divisions on the beveled edge of the rotating cylin- der. The number of these divisions isl usually such that one divi- sion will indicate a movement of the screw through a convenient ANVIL RATCHET Fig. 5. fraction of the whole unit (inch or cm.) For example, to make a iriicrometer which will have a " least count " of .001 inch, a con- venient method would be to make the pitch of the screw %o inch; then the circumference of the cylinder is divided into 25 parts. One full turn will advance the screw %o of an inch and %5 of a turn (i.e., through one cylinder division) advances the screw %5 of %o or .001 of an inch. It is customary to estimate tenths of the small divisions in mak- ing readings. Some instruments have a 10-part vernier on the stem on which, tenths of the small divisions are read. (The vernier is explained in Part B.) Some calipers have a friction head and ratchet stop to aid in setting them with the same pressure each time on the object measured. In making a setting turn the head slowly until the ratchet clicks once or twice. If there is no ratchet stop, hold the head lightly, in the fingers so that they will slip on the head as soon as the jaws make good contact. Carefully avoid setting the jaws too tight. The pressure should be the same for the zero reading, with the jaws closed, and the setting on the object. Be sure that the faces, of the jaws are, clean. Pro'cedurfe. For practice in the use of the micrometer 'caliper, measure the diameter of two or more wires and deter- INTRODUCTORY MEASUREMENTS 39 mine their gauge numbers. Take at least six measurements of the diameter of each wire in the set given you in the apparatus, each time measuring a different part of the wire. Be careful to avoid places where the wires are bent. Take measurements with both French and English calipers, as checks on each other. Con- sult the table of wire gauges and note the diameters and gauge numbers most nearly like the mean of your results ; also compute the cross-section as indicated in Table VII. Data and Results. Tabulate the data and results as in- dicated in Tables V, VI, and VII. TABLE V Metric micrometer, No. . . English micrometer. No. . . Least count Least count Wire Number Readings Met. Micrometer Readings Eng. Micrometer Diameter Zero On wire Zero On wire Cm. In. Brass f 21 -.0006 etc. .1650 cm. etc. + .0002 etc. .0657 in. etc. .1656 etc. .0655 etc. TABLE VI Wire Number Mean Diameter Gauge Tabular Diameter Cm. In. System Number Cm. In. Brass 21 .1657 .0656 Birm. 16 .1651 .0650 TABLE VII Wire Brass Number 21 Mean Diameter in Mils 65.6 Cross Section in Circular Mils Cross Section in Sq. -Millimeters Remarks, Part A. A " mil " is .001 inch. It is a unit of length in common use in expressing the diameter of wires. A " circular mil " is the area of cross-section of a cylindrical wire 40 MECHANICS having a diameter of one mil. The number of circular mils in the cross-section of a wire is the square of the diameter in mils. Test Questions. Part A. 1. How may the pitch of the screw be verified? Define pitch. 2. Define least count, and explain how it is found. Explain exactly why the micrometer reads as it does. 3. Explain how you made the zero reading and how it is used in your computations. 4. Specify some of the sources of error or unreliability. 5. State the number of significant figures in the result obtamed for the mean diameter, and explain how it is determined. 6. How do the areas of circles vary with their diameters ? Show how to find the area of cross-section of your wires in circular mils, and also in square millimeters. Which is the easier? Why? How many signif- icant figures in the cross-section? jExplain. 7. Explain how a micrometer could be constructed so that its least count would be 1/512 inch. Topics for Conclusions. 1. How you would proceed to determine how a micrometer reads. 2. Some sources of error or unreliability. 3. What can be done to make measurements with the micrometer cali- per more reliable, and what is the probable extent of the unavoidable error. Part B. Vernier Calipers. The vernier is a device for reading fractional parts of the small divisions of a scale. It consists of a scale arranged to slide along the edge of the scale used to make the measurements. The zero line of the sliding vernier scale marks the main scale reading to be taken. Figs. 6 and 7 represent in diagram the essential parts of a vernier caliper with the main scale divided into units and tenths and a ten-part vernier scale. In Fig. 6 the jaws are closed, and for convenience the zeros of the two scales coincide. This, how- ever, is not essential since anyl other reading, for the zero reading (with the jaws closed) could be used if subtracted from the reading with the jaws open and set on an object to be measured, as in Fig. 7. The vernier divisions shown are shorter than the small main scale divisions by one-tenth of a scale division; therefore, if the jaws are separated .1 of a scale division, coincidence of lines on the two scales will shift from the zero to vernier line 1 ; and if the jaws are separated .2 of a scale division, coincidence shifts to vernier line 2. In like manner, coincidence shifts one vernier division toward the right for each additional tenth of a scale division that the jaws are opened, until the end of the vernier INTRODUCTORY MEASUREMENTS 41 scale is reached, when the jaws are opened one whole scale di- vision ; but at this point the zero of the vernier is also coincident with the line at the beginning of the next division on the scale. If the jaws are separated from scale line 1 to line 2, coincidence will again shift over the vernier scale from to 10; and in like manner this will be repeated as the vernier zero moves over any division on the scale. If the vernier zero stops at any point between the main scale lines, as in Fig. 7, the number of tenths S tjB IIIMIIIIIIMIII|IIIIIIIJII]IIIIMIII|IIII|IIII|IIII|IIII|IIIIIIII|IIII 01234S678 Fig.\ 7 5 1 3 DllJ' M III! lllllllltlllllllllliLlUJ iiii Jill nil 1111 nil nn m\\\\\ 1 ' 1 ' 1 ' 9 12 9' 9 i 8 Fig. 7, 6 B ■ L i | i|i li|i |if im ffii' i ' i iHii|tii|i;/i' i ' i 'li'iV i i // ,^ / y — 60 — z -3 / ^' / y / / y r ,->^ / /• / z -55 r.-*^ ^p*^ f"- /" z -2 ^ ^ s^- 4 x; ^ 6' ^ ^ r ^ z Ul -I -I ,3 ^ r / /■' 45 ^ /" ..^ / -^ / X I 2 LOA D IN Pou NDS 3. 4 i . LOA i D IN Kilo 1. GRAM 2 1 4 1. i 1 B Fig. 12. Part B. The essential parts of Jolly's Balance are a long, delicate spiral spring at the end of which is attached a pan for weights, etc., and some means for measuring, fairly accurately, the elongation of the spring. In one form of balance a pointer is attached to the spring and moves in front of a mirror, upon which a scale is etched. In another form, the spring is sup- ported from the top of a rod, which may be raised or lowered to any desired position by means of a milled head at the base of the standard. A small, short rod with a mark upon it to be used as an indicator is attached to the spring. This rod swings in a glass tube around which is a fine scratch. When readings are to be taken, the mark on the little rod is brought into coin- cidence with the scratch upon the tube by raising or lowering S6 MECHANICS the rod to which the spring is fastened. By placing the eye so that the scratch on the front of the tube appears to be in line with that on the back, parallax is avoided. The distance the rod has been raised or lowered may be read from a scale engraved upon the rod, and thus the elongation of the spring is determined. The calibration of the balance consists in placing various known weights in the pan and observing the elongation corre- sponding to each weight. From this data a curve is plotted called a calibration curve of the balance. The weight of an unknown body may then be determined by observing the elonga- tion of the spring which it produces and reading off the weight from the curve. Procedure. First observe the zero readings of the spring, that is, the reading when the pan is empty. Place a mass of ^ gram on the pan and again take the reading. Repeat in the same way, adding -J gram more each time, until the spring has been elongated not more than 40 cm., observing the zero reading between' each of the other readings. The steps may be larger than i gram if the spring is stiff. The data may be tabulated as shown below: TABLE XIV Jolly's Balance No. Weights (W) Readings with- out Weights Readings with Weights Total Elon- gation (£) Increase for each Added Weight W Plot a calibration curve using weights as abscissas and total elongations as ordinates. Weigh a five-cent piece using the Jolly balance you have cali- brated. Remarks. If the spring is wound so that the turns press against each other when not loaded, the load must increase to a definite amount before elongation will begin. The ratio of elongation to increment of load will not be constant until all the turns are separated enough to remove this lateral pressure against ELASTICITY 57 adjacent turns. If part of the spring is wound close and part open, the slope of the calibration curves will change as load in- creases. If the calibration curve is a straight line the reciprocal of the slope gives the increment of load per centimeter elongation; this is called the constant (k) of the spring. If the length between the pointers when the coils are separated is /„ and the corresponding load is £„, then any greater load L = ^ i^-h) +^0- This is the equation of a straight line in which L and (/ - /„) are variables. In taking each reading on the Jolly balance carefully avoid the error due to parallax. Make the setting and take the read- ing as accurately as you can. The small rod may be made to swing in the center of the tube by adjusting the leveling screws in the base of the standard. Be careful not to stretch the spring beyond the point mentioned above. It may be ruined if elongated beyond its elastic limit. Test Questions. Part A. 1. What is meant by the calibration of a spring? Is the calibration curve necessarily a straight line? Ex- plain. 2. Is it possible to use the spring for measurement of force without a calibration curve? Explain. 3. If a pull of 1.2 kg. stretches a spring to a length of 40 cms., and the constant of the spring is ISO gms. per cm., what is the pull when the length is 52.4 cms. ? 4. If the data limits are 0-3S000. for ordinates, and 100.-170. for abscissas, what scales would you use on section paper 15x20 cms.? What scales if the limits were 60.30-60.42 and 20.8-95.2? 5. Why is it that both of the length-load curves in lbs. do not have the same slope? What is the significance of the slope of the length- load curve and the reciprocal of the slope? What is meant by the con- stant of the spring? 6. What is the meaning of the X- and F-intercepts in your length- load curves assuming they continue straight lines? 7. Why does the Ibs.-kg. curve pass through the origin? Test Questions. Part B. 8. Carefully describe the Jolly balance, used and the method of using it. 9. Explain your curve and show how the balance and curve are used. What is the significance of the slope of the curve? 10. Estimate the percentage error which may occur in reading on the Jolly balance. 11. How is Hooke's Law verified in this experiment? What is the significance of the last column of Table XIV? 12. What quantities included in the above results are total stresses? Total strains? (See E-1.) 13. Does the elasticity of the spring seem to be perfect? Give the 58 MECHANICS reason for your answer. How could you tell when you were near the elastic limit? 14. How many grams would produce an elongation of 1 cm. in the spring as used? How is this determined from your curve? 15. Try to explain why the spiral is tapered. Topics for Conclusions. 1. General method of calibrating a spring. 2. Use of a calibrated spring for weighing. 3. Advantages of a calibration curve. MASS AND VOLUME 59 GROUP III — W. MEASUREMENT OF MASS, VOLUME, AND DENSITY. experiment w-1 Weighing by Method of Vibrations Determination of Mass and Density The measurement of mass is one of the most important opera- tions in the laboratory and consequently must receive the careful attention of each student. Of the various methods in use the one given below is a good one to illustrate the principles involved. Before studying it, however, the student should study carefully the definitions of mass and weight in the text book, and learn how these two terms differ in meaning. The apparatus used for the determination of mass is the bal- ance. Essentially the chemical balance is constructed as fol- lows: Rigidly fixed in an upright position is a central post in the top of which is set an accurately ground, hardened steel, or sometimes agate plate. Bearing upon this plate, and usually of the same material, is a knife-edge placed at the center of a horizontal truss, called the " beam " of the balance. Near the ends of the beam, and as accurately as possible at equal distances from the central knife-edge, hang two pans of equal masses from knife-edges and plate bearings similar to those at the center. At- tached to the center of the beam is a long pointer, the end of which moves over a graduated scale fixed at the base of the post. This will indicate even a very slight deviation of the beam from its position of equilibrium. By means of a lever or milled head at the base of the balance, the knife-edges may be lifted from the plates. A good balance should have a beam as light as possible and at the same time very rigid. The knife-edges supporting the pans should be at the same distance from the central knife-edge. The three knife-edges should be very nearly in the same plane and parallel, and the center of mass of the beam should be only a short distance below the point of suspension. (See Fig. 101.) The beam of the balance, then, is a lever having equal arms and practically frictionless bearings. The determination of mass 6o MECHANICS depends upon the fact that at a given place the force with which the earth attracts a body (i.e., its weight) is directly proportional to the mass of the body. Therefore, if equal masses are placed in the pans of an equal arm balance the beam will remain in equilibrium, since the two forces will be equal and hence their moments equal. The method of measuring the mass of a body is to place it in one pan of the balance and, in the other, set known standard masses or " weights " until the beam is brought to its equilibrium position and the pointer indicates the same reading it had when the pans are empty. This operation would be very simple were it not for the fact that it is very difficult to select standard masses that are nearly enough equal to the unknown mass to bring the beam exactly to its position of equilibrium ; besides, it would be extremely dif- ficult to make and use the exceedingly small masses which would be necessary. Also, when using a sensitive balance considerable time would be wasted waiting for the pointer to come to rest. Consequently, a method is employed known as the " method of vibrations " which largely eliminates these difficulties. This method consists in reading the turning points of the pointer as it swings before the scale and from these readings, by compu- tation, finding the position at which the pointer would come to irest. This position is called the " equilibrium position " when the pans are loaded, and the " zero point " when the pans are aiot loaded. The determination of the zero point should precede or follow each operation of weighing. The zero point is determined as follows : With both pans of the balance empty, set the beam oscillating by lightly fanning or blowing upon one of the pans. Close the balance case and, as soon as the pointer swings not more than five to seven divisions on either side from the middle division (10), begin to take scale readings of the successive positions of the pointer at its turning points. Evidently, if there were absolutely no friction in the bearings or air resistance, the pointer would swing each time the same distance to either side from the zero point. In this case the MASS AND VOLUME 6i point of equilibrium could be found by taking the mean of any two readings on opposite sides of the zero point. Since there always must be some friction, how- ever, the swings of the pointer gradually decrease in length, so that the positions of three succes- sive turning points might be as shown in Fig. 13. The readings of these three points, in order, are 5.2, 14.4, 5.6. Obviously the mean of any two of these readings is not the equilibrium position. In order to find it we must take the average of the mean of the two readings on one side, and the reading on the other side. Thus, in the case shown in Fig. 13 the equilibrium position is found as follows: Left Right Left Mean Left 5.2 14.4 5.6 5.4 mud ^^ npi nnp 15 to Fig. 13. Average Zero Point i. (5.4 +14.4) '^9.9 Five, or any odd number of successive swings may be taken, instead of three, to obtain the point of equilibrium when a greater degree of accuracy is required. In this case the mean of the readings on each side is taken and the average of these two results is the equilibrium position. Or, several sets of three readings each may be made and the average of all the results taken as the point of equilibrium. It is customary always to begin and end the readings on the left. Estimate to tenths of the scale divisions in reading the turn- ing points of the pointer. Some practice may be necessary to enable you to do this satisfactorily, but after a few trials this should not be difficult. In determining the zero point and the equilibrium position make at least two sets of three readings each. The zero point is seldom exactly at the middle division of the scale, but if it is within about one division of this point, the balance is not seriously out of adjustment. Now if we wish to weigh an object, we place it in, say, the 62 MECHANICS left hand pan of the balance. We add standard masses to the right hand pan until the pointer is within two or three divisions of the zero point. Then by the method of vibrations we find the equilibrium position. Now a small known mass is added to or taken from the pan containing the standard masses, sufficient to carry the pointer past the zero point, and the equilibrium! position again determined. Since we know now (1) how many divisions the small added weight has moved the pointer, and (2) how far it is necessary to move the pointer to bring it to the zero point, we can easily determine by direct proportion what mass would have to be added to the right hand pan to move the pointer exactly to the zero point. For example, suppose that with 28.34 g. on the right hand pan the equilibrium position was found to be 12.4; and with 28.35 g. (.01 g. more) the equilibrium position was 8.7. The zero point is, say, 10.6. It is evident then that 28.34 g. is too small, while 28.35 g. is too large. But how much should we add to 28.34 g. to bring the pointer to the exact zero,— that is, to 10.6? This is found as follows: It is seen that .01 g. pro- duces a deviation of the pointer from 12.4 to 8.7, or 3.7 divisions. The deviation required is only from 12.4 to 10.6 (zero point), or 1.8 divisions. Now, if .01 g. produced a deviation of 3.7 divisions it would require 1.8/3.7 of .01 g., or .0049 g. to bring the pointer to zero. Therefore, the true mass of the object is 28.34+ .0049 or 28.3449 grams. The result may also be obtained by finding in like manner the amount to be subtracted from 28.35 grams. The operation of weighing may then be summed up as follows : 1. Find by the method of vibrations, the zero point with both pans empty. 2. Then put the object to be weighed (the metal plate from experiment I — 2) in the left pan and determine the equilibrium positions (for weights just a little too large, and again for weights just a little too small). From this data compute the mass (X) of the object. 3. Repeat with the object in the right hand pan. Take the mean of these two values as the true mass of the object. From the mass and the volume obtained in experiment 1-2, MASS AND VOLUME 63 compute its density. Compare with a table of densities. Tabu- late data as in the illustration (Table XV). Be sure to indicate which pan the plate is in each time. TABLE XV. (illustration) DATA FOR OBTAINING THE MASS OF A SMALL CYLINDER Copper Cylinder No. 236. Bal. No. 5. Oct. 17 Determina- Left Pan Right Pan Swings of Pointer 1 Mean of Rt. and av- erage Lefts Mean Mass X of tion of Left Right Left the Body Zero Point Equilibrium Position Equilibrium Position 4.32g 4.33g Cylin. Cylin. 5.1 7.3 6.9 4.8 8.1 6.2 15.0 12.6 11.4 14.0 14.3 16.7 5.5 7.4 7.2 5.2 8.4 6.4 5.3 7.4 7.1 5.0 8.3 6.3 10.2 10.0 9.3 9.5 11.3 11.5 10.1 9.4 11.4 ■x= g. 4.32g.+.0035g. 4.33g-.0065g. Mass of Cylinder = 4.3235 grams. (Check, 4.32 + .0035 = 4.33 - .0065) Volume of Brass Plate (From Exp. 1-2) Density of Brass Plate (M/K) Tabular density of Brass Percentage Error Remarks. It is not strictly necessary that the added .01 gram should carry the pointer past the zero point found. If it does not, the computation will be just the same except that both weights will be either too large or too small. However, in all cases the equilibrium position should be within two or three divisions from the zero point. The pointer always swings from the zero point toward the pan containing the smaller mass. (Why?) Observe the following directions for using the balance: Before any readings are taken see that the pans are clean. Remove any dust with a camel's-hair brush. Whenever you put weights on the pan, or take them off, the knife-edges should be raised from the plates. Do not change the weights when the beam is released. The weights should never be handled with the fingers. Al- ways use the pincers or forceps provided with the set. Add the weights systematically beginning with the larger ones. When the weights are taken from the pan they should always be set in their places in the box. 64 MECHANICS Readings should be taken only when the balance case is closed. Observe especial care when releasing the beam not to allow the knife-edges to come in contact with the plates suddenly for there is danger that the knife-edges will be dulled, thus destroy- ing the accuracy of the balance. A mistake is frequently made in summing up the weights in the pan. To guard against this mistake it is a good plan to get into the habit of summing the weights at least three times ; first add them as they lie upon the pan ; then sum up the vacant places in the box ; and finally sum up the weights as they are removed from the pan. The arms of the balance may not be, and probably are not, exactly the same length. Therefore it is necessary to weigh the body in each pan. If fF is the weight found in one pan and W the weight in the other, it can be proved that the true weight is V WW. But when, as in this experiment, W and W are nearly equal, the average of the two values is sufficiently accurate. Test Questions. 1. Explain the process of weighing by the method of vibrations. Ejcplain in detail how to compute the small mass necessary to swing the pointer from each equilibrium position to the zero point. 2. Why is it necessary to use the method of vibrations in order to ob- tain an accurate result? 3. Why is it necessary to take an odd number of readings of the turn- ing points of the pointer? What is the object of weighing the body in each pan? 4. Why is it desirable to have the balance case closed while taking readings ? 5. Do you consider your determination of the mass of the plate by the balance to be a precise measurement? Explain why. How many signif- icant figrures in your mass of the plate, that is, which is your first doubtful figure? How many significant figures in your value of the density? Explain. If the plate were being weighed only for the purpose of finding its density, would it be necessary to use the method of vibrations ? Why? 6. Explain! the principal precautions necessary in using the balance. State some conditions essential for a good balance. 7. Check your results by noting whether those obtained by subtracting the fraction of a centigram from the larger mass agree with those found by adding to the small mass; also whether the data and results are con- sistent with the fact that the pointer always swings from the zero point toward the pan holding the lighter mass. Topics for Conclusions. 1. Advantages of weighing by the method of vibrations. 2. Relative precision of the chemical balance. 3. Some conditions wliich affect the reliability of weighings on a balance. MASS AND VOLUME 65 EXPERIMENT W-2 Weighing by the Method of Vibrations and Reversals Measurement and Calibration of a Capillary Tube The method of weighing described below is a short cut of the method given in Exp. W — 1. Some advantages of this method are that the " no load zero point " is not required ; the error due to the inequality of the balance arms is eliminated; the time re- quired for the experimental work and computations is shorter; and the accuracy is greater with less chance for mistakes. The steps in this method are, (1) Place the mass X to be determined in the left hand pan of the balance, and enough standard masses {M) in the right hand pan to keep the pointer on the scale when vibrating over a moderate range. (2) Set the pointer swinging over a suitable range and close the balance case. When the vibrations have become regular observe the scale readings for three successive excursion limits of the pointer ; that is, two readings on one side and one on the other (negative readings will be avoided if the middle line on the scale is called 10). (3) Interchange X and M and proceed as in (2). (4) Open the door and drop a 2 eg. mass on one pan (prefer- ably the lighter) without arresting the beam. Then close the door and take the readings as before. Table XVI shows the method of recording and gives a sample set of data. TABLE XVI Right Pan Pointer Readings Sum Diff. X- M Left Pan Left Right Left X M M = 23.55 g. X 2 eg. added 6.3 7.8 3.1 11.2 14.3 9.0 6.4 8.0 3.2 17.55 22.20 12.15 -4.65 10.05 = -.46 eg. or - .5 eg. Mass of = X = 23.55 - .005 = 23.545 grams. If a 2 eg. mass is too large for the last step, a 1 eg. mass may be used; but the fraction will then be the part of a half centi- 66 MECHANICS gram and must be divided by two to give (X-M) in centigrams. If the balance is sufficiently sensitive, a 2 mg. mass may be used. Procedure. Weigh a capillary tube on the chemical balance by the method of vibrations and reversals described above. Then have the instructor place some mercury in the bore. Lay the tube on a strip of mirror, place a cm. rule beside it and measure the length of the thread of mercury. The mirror is used to avoid the error of parallax. By placing the eye so that the end of the mercury thread and its image in the mirror appear to be in line, the eye will be in a vertical line above the end of the mercury. Thus its position may be accurately read on the scale. Now place the tube upon the balance pan and weigh it with the contained mercury. In handling the tube to put it on the pan great care must be exercised to keep it horizontal, otherwise some of the mercury may run out. From the data now obtained find the weight of mercury con- tained in the tube. Using the density of mercury (13.6 g. per cc.) find the volume of the mercury in the tube, the mean area of cross-section, and the mean diameter of the bore. The diameter of the bore of the tube is probably not the same throughout its length, and may vary considerably. If the diam- eter is determined at many points along the tube, and the varia- tions in the bore determined, the tube is said to be calibrated. Using this data, a curve may be drawn showing the diameter at each of the points measured. This is sometimes called a calibra- tion curve, and is very useful as a correction curve when, as in the case of a thermometer, accurate results depend upon the uni- formity of the bore. Have the instructor remove all the mercury from the tube ex- cepting about a centimeter. Using this short thread, calibrate the tube for variations in the bore, beginning at one end and pro- ceeding to the other. In each position measure very carefully the length of the thread and the distance of one end from the end of the tube. From your data draw a correction curve, using distances from one end of the tube as abscissas and lengths of the thread as ordinates. In drawing in this curve join the suc- cessive points by straight lines. The result will be a broken line curve. MASS AND VOLUME 6/ Arrange all your data and results in neat tabular form. Remarks. In the method of weighing described above, the " Sum " column is obtained by adding the mean of the two left readings, to the right. The " Difference " column is composed of the differences between the first and second, and the second and third numbers of the Sum column. The first difference is twice the deflection of the pointer produced by the excess of one mass over the other, and the second difference is the deflection produced by the 2 eg. mass added to one pan. Therefore, the first difference divided by the second gives the difference between X and M in centigrams. In this case the difference is subtracted from M because the sums of the pointer readings show that M is larger than X. Note that when the loads on the pans are un- equal the deflection of the pointer is toward the lighter load. If the sums of the pointer readings were each divided by two the results would be the respective equilibrium positions of the pointer, viz., 8.77, 11.10, and 6.07 in the illustration given above. The differences between these are 2.33 and 5.03 ; that is, the pointer is deflected 2.33 divisions by changing the excess mass (X - M) from one pan to the other, and is deflected 5.03 divisions by adding 2 eg. to one side. Now adding two eg. to one side is equivalent to taking one eg. from the other pan and placing it on the one. (Verify this experimentally.) Therefore the transfer of one centigram would produce a deflection of 5.03 divisions; and the deflection of 2.33 divisions must be produced by the trans- fer of 2.33/5.03 of one eg., or .46 eg. Of course, the fraction of a centigram obtained from the dif- ferences between the sums of the pointer readings is the same as that obtained from the differences between the equilibrium posi- tions. Therefore the extra step to get the equilibrium positions is unnecessary and is omitted. Test Questions. 1. Explain in detail the method of weighing by vibrations and reversals. How does it differ from the method used in Exp. W — 1 ? Explain what advantages it has over that method. 2. Why is it more convenient to test the sensitiveness of the balance with a 2 eg. mass rather than with a 1 eg. mass? Explain why the re- sult would be the same whether we added 2 eg. to one pan or transferred 1 eg. from one pan to the other. 3. Explain how you can determine which is the greater mass, M or X. How is the difference between M and X computed? 68 MECHANICS 4. Let A be the area of cross-section of the bore; d, the density of mercury; I, the length; V, the volume; and m, the mass of mercury. A = f in terms of m, d, and I? r=? in| terms of m, d, and I? 5. Explain how you calibrated the tube and how you might use the curve you have drawn to determine the mean diameter of the tube. 6. Explain the use of the mirror used in connection with measuring the length of the mercury thread. 7. In which measurement is the percentage error probably the smallest? Estimate its value in this measurement. 8. From your curve what do you judge to be the general shape of the bore? experiment w-3 Mea-surement of Volume and Density of an Irregular Body Use of, the Burette and Pipette In Exp. I — 2 the volume of a regular body was determined by making accurate linear measurements of its dimensions. But it would be impossible to determine accurately the volume of an irregular body by this method. A very convenient, and at the same time a fairly accurate method of measuring the volume of an irregular solid is to immerse the body in a liquid and measure the volume of the liquid which the body displaces. A property of matter called impenetrability is that no two portions of matter can occupy the same space at the same time. Therefore the vol- ume of liquid which a solid will displace is equal to the volume of the material in the solid itself. To determine the volume of a body by this method, then, it is necessary only to secure some means of measuring accurately the volume of the liquid which the body will displace. It may be said that this is the only practical method of measuring the volume of a solid which is so irregular that its linear dimensions cannot be measured directly. There are several ways in which that volume of a liquid may be measured. Perhaps the most common method is to. use a graduate, which is a vessel having a scale etched upon it to in- dicate the volumes it contains ; or a standard flask which is a flask with a long, narrow neck upon which a line is etched to in- dicate accurately a given volume. These are convenient to use but the ordinary graduate is not very accurate and the flask can be used for but a single rather large volume at a time. A small bottle, called a specific gravity bottle or " pyknometer " MASS AND VOLUME 69 may be used accurately to measure out a definite volume of a liquid. It has a ground glass stopper, lengthwise through the center of which is a tiny hole through which the excess liquid is squirted when the stopper is fitted into the filled bottle. Thus each time the bottle is filled at a given temperature, it contains very accurately the same volume of liquid. The volume of a liquid displaced by a body may also be deter- mined by measuring the apparent loss of weight of the body, when it is weighed suspended in the; liquid. The loss in weight is equal to the weight of the liquid displaced by the body. Prac- tice in this method and in the use of the specific gravity bottle will be given in later experiments upon the determination of dens- ity- A pipette is frequently used to measure a small volume of a liquid. It is simply a glass tube with a portion expanded into a bulb and with a mark upon the stem or upper portion. The lower end, which is drawn down smaller, is dipped into the liquid which is sucked into the tube by placing the mouth over the opposite end. The liquid is drawn a little above the mark on the stem and the finger quickly placed over the top of the tube. The liquid is then allowed to fall gradually until its surface is exactly opposite the mark. In this way a definite volume (determined by the size of the pipette) of liquid may be measured very accurately. Another very convenient and fairly accurate means of meas- uring any volume of a liquid is by the use of a burette. This is a long, rather narrow tube, the lower end of which is drawn down to a small bore and provided with a stopcock, or rubber connection and pinchcock, to control the flow of liquid. A scale is engraved along nearly the whole length of the tube. The graduations are usually tenths of cubic cm., beginning at the top and ending with the 50 cc. division near the bottom of the tube. In use the tube is held in an upright position by a clamp. By the aid of the stopcock the liquid may be run out in a stream or in single drops^ and stopped at any instant. Because of an action between the surface of a liquid and the inside of the tube, called capillary action, the surface of the liquid is not plane, but is curved. This curved surface is called 7° MECHANICS the meniscus. In the case of water in a glass tube the surface is concave upward, so in reading the burette the position of the bottom of the meniscus is usually observed. The purpose of this experiment is to measure the volume of an irregular body such as a stone or a piece of coal; then by weighing it, to determine its density, that is, the number of units of mass in unit volume. Procedure. Weigh the dry body upon the laboratory scales. Then, using a burette, measure as accurately as possi- ble the volume capacity of a beaker, which has its rim ground flat so that a glass cover will fit tightly upon it. When the beaker is nearly filled, place the glass cover upon it so that just enough space is left at one edge to run in the water from the burette. Then carefully fill the beaker until the water is in contact with the glass cover as nearly over the whole surface as possible. Especial care will be necessary to avoid dropping some water on top of the cover or running it over the edge of the beaker. As it will take several burettes full to fill the beaker the fol- lowing method is advised. Fill the burette nearly full and read the position of the meniscus. Run the water nearly out of the burette and again read the position of the meniscus. The dif- ference between these two readings will give the volume of water run in from this burette full. Repeat as many times as is -necessary to fill the beaker up to the glass cover. Carefully Tecord the exact position of the meniscus each time. Now empty the beaker and dry it. Place the body in the beaker and, proceeding as before, find the volume of water neces- sary to fill it. From the two volumes measured, the volvune of the body may be determined. Also, since the mass has been determined, the density may now be computed. Repeat the experiment the same as before, but use a pipette instead of the burette to nearly fill the beaker. Two sizes of pipettes may be used, but careful count must be kept of the nimi- ber of times it is necessary to fill each pipette. As the ordinary pipettes have no scales upon them, it will be necessary to meas- ure with the burette the final small amount of water used. A given number of pipettes will probably not exactly fill the beaker. MASS AND VOLUME The record may be arranged as shown in Table XVII. TABLE XVII 71 Volume of Burettes Full Position of Meniscus Volume Run Out Total Volumes Results Number Upper Lower 1 2 etc. (body in beaker) (body not in beaker) Volume of body Mass of body Density of body Pipettes Full (body in beaker) (body not in beaker) No. Size Volume of body Density of body Mean density Burette Remarks. In general it is better not to try, to adjust the meniscus exactly to the zero of the burette before reading its position, nor to run the liquid out exactly to the lowest mark. Simply fill the burette to a point a little below the zero mark and run the water out nearly to the lowest mark. Then make a record of each reading as it is taken. This method will lessen the liability to mistakes. In reading the burette care must be taken to place the eye on a level with the surface of the liquid. If this is not done the true reading is not obtained due to what is known as parallax. To determine when the eye is on a level with the bottom of the meniscus, a metal clip, fitted to slide easily up and down the burette, may be used. When the bottom of the meniscus and the edges of the clip appear to be in line, the eye is at the same level and the reading will be indicated by the edge of the clip on the scale. Another method which may be used is to fit a strip of paper or cardboard loosely around the tube. This can be slipped along the tube and its upper edge placed level with the meniscus the same as with the metal clip. Still another method used to 72 MECHANICS remove this difficulty is to use a hollow glass float inside the tube, a line etched on the float indicating the reading to be taken. If bubbles of air cling to the body when it is covered with water, they must be carefully removed or a considerable error will be introduced. After the water has run out of the burette the reading should not be made within at least a minute after the flow is stopped. (Whyf) When the burette is filled see that there are no air bubbles at any point in the tube. If there are any they must be removed before readings are taken. Test Questions. 1'. Explain the general property of matter upon which this method of determination of volume depends. 2. Explain why each of the precautions mentioned above are necessary. 3. How could you find by the above method, the volume of a body, (a) which floats upon water, and (b) which is soluble in water? 4. What is meant by parallax and how was it avoided in this experi- ment? 5. Estimate what percentage of error is likely to occur in measuring the volume by the above method. Therefore how close, that is to what fraction of a gram, is it necessary to measure the mass? 6- How could you tell that the beaker was filled to the same point each time? ACCELERATED MOTION 73 GROUP IV — A. ACCELERATED MOTION experiment a-1 Study of the Motion of a Falling Body Uniformly Accelerated Motion It is assumed that the subject of uniformly accelerated motion has already been discussed in the class room and that the student is familiar with the various formulas and terms pertaining to this topic. The object of this experiment is simply to verify those formulas, and to obtain a more thorough knowledge of ac- celerated motion. A common example of a uniformly accelerated motion is that of a body acted upon by the force of gravity and free to fall. Since it is acted on by a constant force (gravity) it moves down- ward with a constantly increasing velocity, that is, with a con- stant or uniform acceleration. In order to study its motion it is necessary to measure only the distances fallen in successive equal intervals of time. A freely falling body gains in velocity so rapidly that it is im- possible to measure the time of its descent by the ordinary methods, as, for example, by a stop watch. A convenient and fairly accurate method of finding the distance fallen in a unit of time is by means of a tuning fork of known pitch (i.e., known number of vibrations per sec). The fork is attached to the fall- ing body in such a manner that a trace of its vibrating prong is made on a stationary plate. The apparatus consists of two upright parallel rods mounted upon a tripod base. The falling body is a frame fitting loosely between the two rods and carrying a tuning fork. The frame falls with very little friction on the guides, a distance of about 120 cm. and is stopped by dash pots at the bottom. Behind the falling frame is clamped a glass plate the surface of which is covered with window polish. A stylus attached to one prong of the fork causes a wavy line to be traced upon the coated surface. The frame carrying the forkj is held at the top by a catch and is released by turning a lever so constructed that it sets the fork in vibration at the same time that it releases it. 74 MECHANICS Procedure. Clamp the coated glass plate in place and see that the stylus just grazes its surface. By moving the plate sidewise after each fall of the fork several traces may be ob- tained. Secure at least two good traces, then remove the plate and place it upon the table. Selecting the best trace, lay a ruler very close to one side of the trace and, omitting 3 or 4 cms. at the upper end of the trace, with a fine pointed pencil mark as accurately as possible points, five vibrations apart, that would be points of tangency if the ruler were close enough. By the aid of a try square draw lines through these points at right angles to the axis of the trace. Place one end of a meter scale even with the upper end of the plate, and take a scale reading at each line without moving the scale, excepting for readings beyond the first meter. Using simply the distances fallen by the body in successive equal intervals of time, the following quantities may be easily computed: the velocity at the middle and at the end of each time interval ; the acceleration ; and the total distance fallen after any given number of time intervals. The velocity at the middle of any time interval is numerically equal to the distance fallen in that interval. For the actual velocity at the middle of the interval is the average velocity for the whole interval, the gain in velocity being uniform. The aver- age velocity for a given time multiplied by the time gives the distance traversed during that time. The time in this case is one interval, therefore the velocity at the middle is numerically the same as the distance. Likewise the velocity at the end of any interval is the mean of the velocity at the middle of that interval and the velocity at the middle of the next. The acceleration is the gain in velocity in each equal interval of time. It is equal to the added distance covered in any interval over that covered in the preceding interval, since the distance covered in any interval is numerically equal to the velocity at the middle of that interval. If the value of the time interval is known in seconds, the values of any of the above quantities may be computed and expressed with the second as the unit of time. ACCELERATED MOTION 75 Record the data and compute the results indicated in Tables XVIII and XIX. Some of the columns in these tables are in- serted to assist the student to a more complete understanding of accelerated motion. TABLE XVIII Interval Number Readings Distance Each 5 Vib. Vit Middle of Intervals Cm. /Interval Kat End of Intervals Cm./Interval Acceleration Cm. per Interval per Interval TABLE XIX Interval Kat End of Intervals Cm. /Sec. Acceleration Cm. per Sec. per Interval Total Distances Total Distances from Rest Intervals fromlRest Number Computed Observed No. No.3 Rate of fork vib./sec. Time of 5 vib sec. Mean acceleration in cm. per int. per int ; from curve " " " " " sec. per int " " " " " " " sec ; from curve . Difference from accepted value per cent. Initial velocity from curve, Vo= Time required to attain velocity Vo = Distance passed over in this time = - In Table XVIII each of the columns of computed results is ob- tained from the column preceding it. In Table XIX the ac- celeration is obtained from the preceding column and this is ob- tained from the fifth column of Table XVIII. The total distance in the fourth column in Table XIX is computed by using the formula d = V^t + ^ at^. Compare this column with the follow- ing one as a verification of the equation just given. Plot three curves on the same sheet of cross-section paper as illustrated in Fig. 14. For the first curve use values of the velocity at the ends of the intervals as ordinates and values of the time as abscissas. Extend this curve, determine the inter- cepts, and find the time elapsed from the time the body started from rest to that taken as zero time in your observations. Then by computation, using the average velocity for this time deter- 76 MECHANICS mined from the curve, find the distance traversed in this time. For ordinates of the second curve use the total observed dis- tances, adding to each the distance the body moved from its starting point to the point where measurements began; and for abscissas use values of the time elapsed after the body started from rest. prn- ?1!5 «| ft ri" ^i"! Tii:^ M Si -II ME -.'Jr'-' 1* i::: =rrr : ■ •■"■■ -tth: -Mr; ": ''r i- ".: i ■=! ""!i=j .■.f=' — rii; m tShi .«■: ^dc !..- T 1 - 1.. N " W- -IS .! ■'.—: '^ ':..-ii" -:f;ai ■ JLENi I TffNiiijc- fjoHrt| ■= ■.i ■ -=: .1 ' iii i ■■■!;■ ^ ;■ !■■ - ii« li .:;■ :.- -;■ ■y /, :^^ / ■■'.I .1^ 1- E '-r- ■ ■■ -■■■ .=r-a /^; ■■■ /* y^ ;: 'J- c<> -f ■T~ . - . "■:■ /^ :-■■■ /■ tHJ ^■% ..--■ _ 1 ' A. ■ri- =■::;. =-"■ .^ ' .ii / -^ : '■"■ .1;:. "■ .[ a li ;;;;|;:: -■/ r. ■ ^- /■■- r^: ... ! .. -■■ .: ...^ 1. .tc Z": 4 _jL-: :" :; . ^■r :s W- // 1- ■■ . ^. :.■■■; — - i; ^'ii ^ ^* . _: : ..; 2: ^ - 1* r y / ..r ■''h- i-^l K J / ^ ;■.; ■■ " ::.f::i:>. i^n :-■■ ■A L.~B ;. .,> ^- T-^ ■■=--:- ■~-' . .■d".;4^^ =e:^ '^lI TS. ^■,; ^ a] ■=■■, rir: - 7 o-.-J -:.i<-.:.a.=r= :~ ! ill sIe ^.-i ^■;=. ■P '■'TT ..:■ . 1 I.-- XTiSi kLS- ■■■ lni« !-" ■■~P.'' i^.iij. :;_ ^:^ =hL IN .s'AFhE if U=j::: V "rtF Fn-HW-tt -33 1 :b-~ yaSEf. : - ^n:! jT 111 iliiiCEyia.'fiii'H a EfiiloH./^iiri' ::Mt QnutSEC ... i;! ,i=±iF "^,' ■::.: ::.... LJ^;^... "Tl" . ■ 1 'iM-. biL '.jA.I H%-:- I:- . ■ Fig. 14. For the third curve use the same ordinates used in the second curve, and for the abscissas use the squares of the values of the time used in the second curve. When you have completed Table XIX, you will then have all the data used in plotting the last two curves. If this last curve proves to be a straight line, the equation d = ACCELERATED MOTION TJ \ at^ will be verified, since the values of t begin with t=o, the time at which the body starts from rest. This is the reason for not using the other distances and times given in the table. Find the acceleration by determining the slope of the velocity- time curve as illustrated by the dotted lines. Find the accelera- tion also from the slope of the distance-time^ curve. Explain. Remarks. Before beginning the experiment proper see that the axes of the two upright rods are in the same plane. This can be done by sighting across them. Also by the aid of a plumb bob set the stand so that the rods are vertical. If these precautions are neglected there may be a variable friction which would result in a variable acceleration. Although it would ap- pear that the fork has zero velocity at the instant it is released from the spreader, it is very probable that it does not. The prongs of the fork are sprung outward by the spreader and when they spring back upon being released they probably give the lower side of the spreader a slight push which starts the fork off with a small initial velocity, so that the point from which the fork starts to fall is not the point at which it has zero velocity. In other words, if the body started from rest, it would have to fall further than it actually did to attain the velocity it had at the point where readings began. Test Questions. 1. Explain the method used in this experiment to mark equal intervals of time. 2. Does the body move with uniform acceleration? Explain. If the acceleration is uniform, what makes it so? 3. Explain fully how you determined the distance the body descended each interval of time and the velocity at the middle and at the end of each interval, and state the principles upon which each computation is based. Explain how each column in your table was obtained. 4. Tell how you determined the acceleration and explain fully. How did you determine in what time units the acceleration was expressed? Explain how acceleration was found from the curve sheet. 5. :^plain how you computed the distance covered to attain the velocity, Vo. T 1 i 6. Explain how each of the curves was drawn. Interpret each one oi the curves showing how they agree with the formulae V=V„-^at, V = at, d = Vot + hat^, and d = i at^. Explain the meaning of each intercept of the velocity-time curve. r .. « j-a r ^u 7 Give some possible reasons why your value of g ditters trom the accepted value. How could you compute the retarding force due to fnc- Topics for Conclusions. 1 Meaning of accelerated motion and some relations verified. 2. General method of the time and distance measurements. 3. General character of curves obtained. 78 MECHANICS experiment a-2 Study of Acceleration of Car on Horizontal Track Relations of Force, Mass and Acceleration Whenever a body is moving with an accelerated motion in any direction there must be a resultant force acting in the same direc- tion to produce the acceleration. Also the magnitude of the ac- celeration is directly proportional to this force and inversely pro- portional to the mass of the body (Newton's Second Law). Stated algebraically, P r a ^ — , ot r = ma , m if all the units are in the same system. The purpose of this experiment is to verify the above equation. To do this, the general method is to cause a known force of F dynes to act on a known mass of m grams, giving to it a con- stant acceleration of a cm. per second per second. This accelera- tion a can easily be measured and thus the product my. a de- termined. If this product equals the value of the known force the law is verified. Also, in addition, a constant force is caused to act on various masses and the resulting acceleration measured as before, thus testing the relation between the acceleration and the mass accelerated when the force is constant. Fig. 15. ACCELERATED MOTION 79 The apparatus used is a two- wheeled car (C, Fig. 15), which can move with very little friction along a single horizontal rail. A cord attached to the car passes over a pulley (P) at the end of the track and is attached to a pan (E) into which the weights, which are to be the accelerating forces, are placed. Hanging down from the car on either side are platforms (D) on which weights may be placed to change the mass of the system. At- tached to the top of the car is a light wooden beam (B). Upon this may be laid a long metal strip (B) upon which a thin coat- ing of whiting and alcohol has been evenly painted. Above the beam swings a short pendulum (A), so adjusted that a short bristle (H) in the bottom of it just touches the metal strip and will make a trace in the coating upon it when the pendulum swings. As the pendulum vibrates, equal time intervals will thus be marked off as the car moves forward. The period of the pendulum may be determined by counting a number of vibrations (say 50 or more) and taking the total time with a stop watch. Procedure. To verify the relation between forces and the resulting acceleration when the mass is constant, place on the car a fairly large mass, say 800 to 1200 grams. Then by trial add sufificient weights to the pan to give the car and its load a small acceleration. The pendulum should be adjusted so that the bristle just brushes the metal strip. With the metal strip coated and in place the pendulum is set swinging, the car being held at the end of the track. Give the car a slight push, just enough to set it in motion. Close beside each trace of the bristle, mark with a sharp pencil a symbol (as 1) so that the traces may later be identified. Add some weights to those in the pan and take another record, marking each of these traces with another symbol (say 2). Re- peat at least twice more. After the records have been made, carefully lift the metal strip from its support and lay it upon the table. The distances be- tween traces may then be measured by the use of a beam tram- mel, or a flat scale may be laid upon the strip. From these meas- urements and the time, the accelerations are obtained. 80 MECHANICS To test the relation between masses and the resulting accelera- tions when the force is constant, place such a weight in the pan that at least three traces may be obtained with different masses on the car. Recoat the metal strip and proceed as before. The force which has been used, that is, the weight of the pan and the masses in it, has accomplished three things: (1) It has accelerated the car and its own mass ; (2) it has accelerated the rotation of the pulley (P) and the wheels of the car; and (3) it has overcome the friction in the pulley and in the wheels of the car. The first is the only effect we wish to consider in this experiment. Hence, the other two must be eliminated as far as possible. This is done as follows: In the case of No. 2 what is known as the " equivalent mass " of the pulley and wheels is given and this is added to the mass of the car. The equivalent mass is that mass which would be given the same acceleration by the same force as that which ac- celerated the rotation of the pulley and wheels. In the case of No. 3 the force required to overcome friction is very, nearly constant for a given load. Its value is found by plotting a curve, using values of the acceleration as the abscissas and total weights on the cord as ordinates. This curve will not pass through the origin, for when the acceleration becomes zero, there is still some force acting, — the amount to overcome fric- tion. Thus the F-intercept will be this force, which is to be sub- tracted from each of the total forces, to find the force accelerat- ing the car. In the second part of the experiment the elimination of the frictional force is not so easy since the load on the car varies, which causes the friction to vary. It may be done, though not as accurately, experimentally as follows. With each different load place enough weights in the pan to just cause the car to move the length of the track with uniform motion. This can be judged by the eye, or a trace may be made on the coated strip. The weight on the cord is then approximately that necessary to overcome friction. ACCELERATED MOTION The data may be put in tabular form as follows : TABLE XX 8i First Trace »»i = Second Trace /W2 = Third Trace OT8 = Fourth Trace OT4 = Time Dist. betw. Traces Time Dist. betw. Traces Time Dist. betw. Traces Time Dist. betw. Traces TABLE XXI Trace No. Total Mass on Cord Accelerating Mass Accelerating Force, F (dynes) Acceleration mx a Per cent. Diff. Mass required to overcome friction (from curve) . Total mass of moving systefti (m) Equivalent mass of the pulley, PART II Trace No. Mass Accelerated Force to Overcome Friction Total Wt. on Cord Acceleration na Percent. Diff. Accelerating Force F (dynes) To show graphically the relation between mass and accelera- tion plot a curve using values of \/m (»M = mass accelerated) as abscissas and values of the acceleration as ordinates. Remarks. Since the bristle bends back and forth as it comes in contact with the metal strip, more accurate results will be obtained if measurements are taken between every other trace rather than between successive traces. It should be noted that the weights on the cord cause the acceleration of their own mass. Hence this should be included in the total mass accelerated. The changing of the weights oh the cord in the second part, will cause an error in the results since the mass accelerated is supposed to 82 MECHANICS be constant. This error, which is not large, could be avoided by placing on the car, the weights removed from the pan. The acceleration in each case could be computed by drawing a speed-time curve. The distance in each interval is numerically the speed at the middle of that interval (See Exp. A — 1). The slope of this curve is the acceleration. If this method is used care must be taken to obtain the acceleration in the correct units. The Atwood's Machine described in Ames and Bliss, p. 98, may be used in a way very similar to the method described above. In this case the acceleration is produced in a vertical direction. The mass accelerated is the mass of the two suspended weights, and the force is the weight of the small riders placed on one of the large masses. The equivalent mass of the pulley and the mass of the riders must be included in the mass accelerated. The ac- celeration is determined by measuring the distances the masses move in 1 sec, 2 sec, etc, respectively. Another way in which friction may be eliminated is by inclin- ing the track just enough so that the car will move down it with uniform motion, that is without acceleration. In the second part of the experiment it would be necessary to incline the track at a different angle for each load. Test Questions. 1. State Newton's Second Law in terms of the ac- celeration and explain how you have verified the law in this experiment Which columns of your data prove the law? How do the curves prove it? 2. How did you correct for friction in this experiment? We may re- gard the force of friction as producing a negative acceleration. Show how, by extending the first curve, you can find the value of this negative acceleration. What is the significance then of the X- and F-intercepts of this curve ? 3. Is it necessary to weigh the car and its contents as carefully as you weigh the masses on the cord? Why? What percentage error would be introduced in your equation by an error of 1 eg. in the total mass? By the same error in the mass on the cord? 4. Explain why the force producing the acceleration is the accelerating mass times 980 dynes. 5. In which parts of the apparatus does friction occur? experiment a-3 Study of Uniform Rotation Force and Acceleration Toward the Center By Newton's laws of motion a body in motion continues in a straight line unless a force acts upon it so as to change the di- ACCELERATED MOTION 83 rection of its motion. Conversely to make a body move in a curved path a force must be applied to it at an angle to its path. If a body is to move in a circle the force must be continuously applied, and if the speed is uniform, the force must be constant. In this experiment, distinction must be made between the terms speed and velocity. Velocity means the rate of displace- ment in a definite direction, while speed refers simply to the magnitude of the velocity without any reference to the direction. Thus, velocity is a vector quantity while speed is not. We are to consider the motion of a body in a circle in which the speed is uniform along the circumference. The velocity, however, is by no means uniform, for it is changing in direction at every instant. Suppose the body is at the point P (Fig. 16) in its circular motion about the point O, in the direction shown by the arrow. When the body was at Q a short interval before it reached P, the velocity is in the direction QT, and could be Fig. 17. Fig. 16. represented by the vector v^. At R, a short interval after pass- ing P, the velocity is in the direction RS, and is represented by the vector v^. The change in velocity while the body is passing from Q to R would be the vector difference between these two velocities. This subtraction is shown in the small figure, from ■which we see Note • A negative sign placed before a vector quantity indicates that it is reversed in direction. Subtraction of vectors is performed by making negative the quantity to be subtracted and proceedmg as m addition. z/j-z/i, then, represents the change in velocity in a period of 84 MECHANICS time, t, supposing that it required t seconds for the body to move from to R. The triangle ABC (Fig. 16) is similar to the triangle ROQ. (Why?) Therefore, AC chord QR AB QO Since Q and R are any points on the circumference excepting P, they may be as close to P as we like. If they are taken very close to P, the difference between the length of the chord QR and the length of the arc QR will be very small, and in the limit will be equal to it. No step in our proof will be altered by do- ing this. Therefore, A£^ chord QR arc QR AB QO QO The side AB is the vector v^, and the side AC is the vector difference between v^ and v^, or v^ - v^. Let v be the speed of the body along the circumference. Since this speed is uniform, the vector v^ must have the same magnitude as the vector v^^. Therefore z/j = z'j = v. Also arc QR = vt. Substituting these values in the above equation, calling the radius RO, r, we have V^ — Vi __ vt V r Therefore the acceleration, a, which is the change in velocity per unit time, is ^^v^^v^^l^ (1) / r From the figure we see that the vector AC is in the same direction as PC, that is, toward the center of the circle. If the space QR is covered in unit time, this vector represents the ac- celeration, a. Hence, in uniform circular motion the cccceleration of the body is toward the center of the circle, and its value is the square of the speed along the circumference, divided by the radius. Also, since the change in velocity in equal intervals of time is the same at all parts of the circumference, the acceleration is uniform. Since the acceleration is uniform and directed toward the center, a constant force must act toward the center to produce ACCELERATED MOTION 85 this acceleration. This force is called centripetal force. As in other cases it is measured by the product of the mass of the body, and the acceleration produced. Hence, 2 F = ma = m — (2) r By angular velocity is meant the number of units of angle swept out per unit time. If T is the period, or the time required to make one complete revolution, the angular velocity, _ 27r _ 27rr Zirr/T V T Tr r r '\A''e may now express the centripetal force in terms of the angu- lar velocity ; for since, « = — = = lo^r (3) r r we have, F = ma = nitu^r (4) Centripetal force, then, is the force which causes a body in motion to move in a circle, and it is measured by the product of the mass of the body and the square of its speed divided by the radius of the circle. It is also expressed as the mass of the body times the square of its angular velocity times the radius. The acceleration of the body toward the center is opposed by the inertia of the body. This reaction against the centripetal force is called centrifugal force. A diagram of the apparatus used to verify the laws of cen- tripetal force is shown in Fig. 17. The frame is attached to the whirling table so that it may be rotated at any desired speed. It is only necessary to measure the weight, W , which measures the tension in the cords, cc, and hence the centripetal force; the masses m; the distances of the masses, m, from the center; and the speed of the frame when the weight W is just lifted, that is, when the action of the weight W is just balanced by the reaction of the masses m. Procedure. This experiment is most conveniently per- formed by four students, one to record the data, one to observe the time, one to observe the number of revolutions by holding a speed counter on the frame at the point A; the fourth should 86 MECHANICS rotate the frame as near as possible at such speed that the weight W is just lifted from the collar K. He should watch W to keep it " floating " so that he can just see through underneath W and above the collar K. If desirable, the experiment may be performed by two students by using a stop watch with two second hands (split second hands). One student turns the machine while the other takes and records the speed. The speed counter is held with one hand, while with the other the watch is manipulated and the record made. The watch should be laid in a good light within easy reach and view. Both hands of the watch are started by press- ing the stem just as the index point on the dial of the counter passes the zero of the scale. Then as the index point again passes the zero, the stop is pressed, which halts one hand. Its position is quickly read and recorded, and the same stop pressed again, allowing the hand to move on with the other one. Whether two or four persons are performing the experiment, by changing places, each one should perform all parts of the experiment. Make two separate runs of at least 1000 revolutions each. (Jb- serve the time at the end of each 1(X) revolutions. Plot a curve using number of revolutions and times as coordinates. The slope of this curve will be the number of revolutions per second. Now change the masses, m, and repeat. Data and results may be recorded as follows : TABLE XXII First Trial Second Trial No. of Rev. Time 1st Run Time 2nd Run Results No. of Rev. Time 1st Run Time 2nd Run Results Masses, m Mean Speed from Curve - - Radius W, ... gm.wt. Ang. vel.ra . ■ . Force (obs.)..- "(comp.)... Diff. . .percent. Masses, m Mean Speed from Curve . . Radius W, ... gm.wt Ang. vel.w . . . Force (obs.)... "(comp.).... Diff. ..percent. ACCELERATED MOTION 87 Kemarks. The most convenient way to measure the radius is to measure the whole distance between the two masses, m. Make two measurements : one from the outside edge of one mass to the inside edge of the other, and the second measurement from the inside edge of the first to the outside of the other, or one measurement may be made between the inside edges of the masses and the other between the two outside edges. In either case the mean of the two measurements is used. It is best to adjust the cords so that the masses are nearly the same distance from the center. In computing the centripetal force, both equations should be used for a check on your computations and for practice in the use of the equations. Note that in these equations all the quan- tities should be expressed in absolute units. Test Questions. 1. What causes a body to move in a circle? How does a body in motion tend to move? Why? Define centripetal and cen- trifugal force. 2. If the speed is uniform explain why there is acceleration. In which direction is the acceleration of a body which moves in a circular path? Explain. Derive the algebraic expression for the acceleration. What is the cause of it? 3. Derive the algebraic expression of the centripetal force, first in terms of linear velocity, then in terms of angular velocity. In which direction does the force act? Explain. 4. Why is it that the weight JV, as soon as it is lifted from the collar K, shows such a sudden tendency to rise to the highest possible point? 5. Is it correct to measure the radius When the weight W is resting on the collar K? .Explain. 6. Explain why it is that when the masses m are decreased, the speed must be increased, but not in the same proportion. 88 MECHANICS GROUP V — F. COMPOSITION AND MOMENTS OF FORCES experiment f-1 Determination of the Resultant of Concurrent Forces The Parallelogram and Polygon of Forces Suppose that two forces represented by the vectors A and B ((1) Fig. 18) act at the same time on the point P. P will in general tend to move, but neither in the direction A nor B. It will- tend to move the same as though only a single force, lying somewhere between A and B, were acting upon it. The direc- tion and magnitude of this single force, called the resultant, may be determined by constructing the parallelogram Pmno upon A and B as sides. The resultant of the two forces is then repre- sented by the diagonal (R) of this parallelogram. The vector {-R), which is drawn equal to R but in the opposite direction, is called the cmti^resultant (or, equilibrant), because it is drawn op- posite to the resultant and represents the single force which would hold P at rest when it is acted upon by the forces^ and B. Another method of obtaining the same result is to obtain the vector sum oi A and B as shown in ((2) Fig. 18). The resultant R would be obtained the same as before. This process ex- pressed in symbols is A+B = R. This is known as the polygon method and is simply the addition of vectors, which in the diagram are joined butt to point instead of butt to butt as before. The former is known as the parallelogram method. If more than two forces act at the same time on a point, the resultant of them may be obtained in the parallelogram method by first finding the resultant of two of the forces, then the resultant of this and a third force, and so on until all of the forces have been included. The final resultant is the resultant of all the forces. By the polygon method, the vectors are joined butt to point in any order. The line from the butt of the first vector to the point of the last is the resultant of all the forces. If this line is drawn from the point of the last vector to the butt of the first (that is, in the direction opposite to that of the resultant) it is the anti-result- FORCES 89 ant. Therefore the principle: // the vectors representing forces acting upon a point form a closed polygon when drawn in the vec- tor diagram, the forces are in equilibrium. From the above paragraph it will be seen that when three forces hold a point in equilibrium, the resultant of any two is equal and opposite to the third force. Thus in Fig. 18, if -R were a third force and point P is held in equilibrium, -R must be equal and opposite to the resultant {R) oi A and B as it is shown. Also when four forces hold a point in equilibrium, the resultant of any two is equal and opposite to the resultant of the other two. Procedure. To verify this law the following method is used: Around the edge of a large board, mounted vertically, holes are bored, in any of which a brass pin for supporting a pul- ley may be inserted. Two or more pulleys (according to the number of forces being tested) are thus hung at different points on the board. Over each pulley a cord passes supporting at one end a weight. The other ends of the cords are attached to a ring, from which another weight is suspended directly. Call the center of the ring P. By means of weights (not less than 200 g. on each cord) cause three forces, each having a dif- ferent value, to act on the point P. If the weights and their re- spective angles to each other are suitably chosen, the point will finally come to a condition of rest. On account of friction in the pulleys, the position of this point of rest will vary somewhat. To eliminate this difficulty, pin a sheet of paper back of the movable point, raise the weight on the left and gradually let it come back to the equilibrium position. Mark with a pencil (or pin) the position of the point P. Now pull down the same weight and, as before, gradually let it come back to equilibrium. Again mark the position of the point P. Do the same with each weight. There will result on the paper six points, grouped irregularly. Estimate the mean position of the six points, i.e., the middle of the irregular figure. Place P at this point. To draw the vectors representing these forces a very conven- ient and fairly accurate method is to place a straight edge so that it is parallel to each of the cords in succession, and draw lines along it on the paper underneath. This will give the direction of 90 MECHANICS the force. After a line has been drawn for each force, remove the paper and extend the lines. In general they will not meet in the point P Draw other lines parallel to those you fiave drawn, that do meet in P. Call the angles in order a, h, c, and the forces to the left of the respective angles, A, B, and C. Make in all four trials, each with a different set of weights; also in each case make each force different. In two of the trials use three forces and in the other two use four forces. The data may be arranged as in the following table. TABLE XXIII Trial Forces Resultant Force Values Anti- Resultant Per No. Down (A) Left (B) Right (C) Between (D) Diff. BandC (( (C (< B.c.D.:.;;: (t <( it Place a sheet of report paper under the paper used on the board and with a pin prick through points in order to copy ac- curately the relative directions of the various forces. Draw the vectors to represent the three forces used in the first trial. By the parallelogram method determine the resultant of any two forces and see if it is equal and opposite to the third force. Do the same with the forces used in the second trial. Represent the four forces used in each of the other trials, by vectors as before. Find the resultant of any two and combine it with one of the other forces. Then see if this second resultant is equal and opposite to the fourth force. In each of the above cases determine the percentage difference between the resultant and the observed force which is the anti-resultant; also find the angle between them if they are not parallel. In the last two figures mark in red those lines or vectors which represent the forces joined " butt to point," and thus show how the polygon method for the addition of vector quantities results directly from the parallelogram method. The lines you have drawn in red should form a closed polygon. FORCES 91 Remarks. In your report see to it that each figure is drawn neatly and acxurately. Show on each diagram whether the re- sultant and anti-resultant lie in the same straight line. On each vector write the actual length of the line as well as the vjdue of the force which it represents. Also indicate the direction of each force by an arrow head and state the scale you have used. Instead of the apparatus used above the iron " force table " may be used. It consists of a heavy, circular iron plate, mounted horizontally on a vertical iron rod. The rim of the plate is faced off. On the upper face the angles are graven. Several pul- leys are so mounted that they may be clamped at any point on the rim of the plate. The cords supporting the weights run over these pulleys and are fastened to a ring which is brought to the center of the plate. With slight modifications the above in- structions will apply to this apparatus. Take especial care in scaling ofif your diagrams. Choose a convenient scale to use, and one which will make your diagrams of large size, and always estimate to tenths of the smallest divi- sions on the scale. Test Questions. 1. Explain the parallelogram method for finding the resultant of a number of concurrent forces. Also explain the polygon method and show its relation to the parallelogram method. Show how you have used each of these methods in this experiment. 2. How did you eliminate friction in this experiment? Explain the method. 3. Define in your own words, the terms resultant and anti-resultant. 4. Prove that the following proposition is true: If three forces are represented by vectors, and these vectors taken in order form the sides of a triangle they are in equilibrium'. 5. Explain why, when three forces hold a point in equilibrium, the re- sultant of any two is equal and opposite to the third. State and explain the similar proposition regarding four forces. 6. What are the principal differences between the parallelogram method and the polygon method? experiment f-2 Determination of Various Forces in a Truss Application of the Polygon of Forces The experiment is performed by suspending three springs from a rigid bar in the form of a truss as shown in Fig. 19. At one of the junctions of the springs hang weights amounting to about one kilogram or more, and at the other junction about 200 Fig. 19. 92 MECHANICS g. or 300 g. less. Now take such measurements as are necessary to draw accurately to scale in your note book a diagram showing the direction in which the forces involved are acting. By a law of equilibrium ( See Exp. F — 1 ) the forces acting I at each of the junctions of the /b springs when represented by / vectors form a closed poly- ' gon. Therefore, since the magnitude of one of the forces in each case is known and the relative direction of each force is shown on your diagram, it is possible to find by the polygon method the magni- tude of the others. On your diagram represent the forces by vectors, using some convenient force scale. Then find graphically the tension in each pair of springs. Before taking down your apparatus, as a check upon your re- sult, carefully measure between two points on each spring to determine its length when it is extended by the weights. Select some point or line near each end of the spring between which measurements can conveniently be made. Now proceed to calibrate each spring by the following method : flang the spring on a peg in one of the clamps of the meter stick, and hang upon the spring a known weight (say 600 or 800 grams). Measure the length of the spring, using the same two points used before when they were in tension in the truss. Then hang say 500 or 600 grams more on the spring and again meas- ure between the same two points. From this data determine the constant of the spring, that is, the number of grams required to stretch it one cm. Using the constant of each spring just determined, compute the tension in each spring, when it was set up in the apparatus. Compare this result with that obtained graphically, and deter- mine the percentage differences. Remarks. In all cases see that when a spring is being used sufficient force is caused to act upon it, to separate its ad- FORCES 93 jacent coils, otherwise its tension cannot be determined by cali- brating it as described above. Before taking down the apparatus be sure to observe some mark on each spring by which it may be identified when cali- brating it. Record your data taken in calibrating in some con- venient tabular form, lettering the springs to correspond to the lettering on your diagram. It is not necessary to measure the lengths of the springs when they are not stretched. Notice that in your diagram two scales are necessary: a dis- tance scale, used in drawing the vectors in their proper direc- tions; and a force scale, used in your graphical solution. If it will be clearer, two separate diagrams may be drawn. Test Questions. 1. Explain the principle upon which the tension in each spring was determined graphically. 2. When several forces hold a point in equilibrium, and they are repre- sented in a vector diagram, does some complete figure result? Explain why. 3. Explain how you calibrated the springs, and from the calibration compute the tension. 4. Why should the coils of the spring be separated when it is being tested? Why is it not necessary to obtain the normal length of the springs ? 5. If the weights were replaced by braces from the same points to the beam above, would they be in tension or comjjression ? What sort of resistance would be required in them? Compare the resistance required in them to that required in the other members. In a truss of this shape, which members would need to have great tensile strength? Which mem- bers would need to have rigidity, i.e., resistance to bending? 6. Two parallel trolley wires 8 ft. apart are supported by span wires to poles 20 ft. apart. The point of attachment of the span wires to the poles is 1 ft. above level of the wires. If the downward pull of each trolley wire is 20 lbs., find the tension in each span wire due to the weight of the trolley wire. experiment f-3 The Law of Moments of Forces Conditions Essential for Equilibrium The moment of a force is the measure of its ability to produce rotation. We know that it is easier to turn a shaft with a long crank than it is to turn it with a short crank ; also it turns easier when the force is at right angles to the crank. The least force is required to rotate the body when its moment is greatest; the force has the greatest moment when its distance from the axis 94 MECHANICS about which rotation takes place is greatest. Evidently, too, the ability to produce rotation would be greater if the force itself were greater. Hence, the measure of the moment of a force IS THE PRODUCT OF THE FORCE AND THE SHORTEST DISTANCE FROM ITS LINE OF ACTION TO THE AXIS OF ROTATION. (HoW Can yOU locate the shortest line that can be drawn between a point and a line?) As an illustration, suppose the three forces F, F', and F" are acting on the irregular body shown (Fig. 20), free to rotate about the point O. The moment oi F h Fx p; of F', F' x p' ; of F", F"xp." Although F" is not the largest force it evidently has the largest moment. (Why?) F and F' tend to rotate the body in the same direction while F" tends to rotate it in the opposite direction. The moments of forces which tend to produce rotation in K one direction are called positive and '^)^ ^ %,^^ those which tend to produce rotation in / \. \ the other direction are negative. It is j pV ,\ common to call those moments positive / '"; , ^" } j which tend to produce rotation opposite I — -__i___^ Jt' *° ^^^ motion of the hands of a clock. \ ^\__^y^''^ I Suppose that the three forces in Fig. „. 20 20 hold the body so that it does not turn, that is, in equilibrium. Then the sum of the moments of F and F' must equal the moment of F" ; or, put in another way, if we give to each of the moments its proper sign, the algebraic sum must be zero ; i.e. Fxp^-F'xp'-F" xp" ^Q This principle is known as the law of moments. It may be stated as follows: When several forces in the same plane houj a body in equilibrium, the algebraic sum of their moments is zero. Or, ^Fp = (% means "the sum of such terms as"). In other words the sum of the moments tending to produce rotation in one direction equals the sum of the moments tending to produce rotation in the other direction. The axis is called the center of moments. FORCES 95 The principle of moments is used extensively in the cases of bodies which apparently are not free to rotate. A case like this is that of a beam in a floor, say, which has various loads upon it and may be supported in several places. Of course, in general, the beam would rotate about some axis, perpendicular to the plane of the forces if it were not held in equilibrium by the forces. So the principle of moments applies. In a case like this a center of moments may be chosen at any point in the body, or even at some point outside the body itself. Rotation is assumed to be possible about this imaginary axis, the distances are taken^ from it, and the moments are positive or negative with reference to this point as the center of moments. Equilibrium. There are three kinds of equilibrium; stable, unstable and neutral. A body is in stable equilibrium when it returns to its original position upon releasing it after it has been rotated slightly out of that position. A transit upon its tripod, a cube resting on one face and a chair in its ordinary position, are examples. After the small rotation there is a moment tending to return the body to its former position. A body is in imstMe equilibrium when it tends to' move farther away from its original position after it has been rotated slightly out of that position. A cone resting on its point and a bicycle wheel with its valve-stem at the top are examples. There is a moment tending to turn the body farther from its original position. A body is in neutral equilibrium when upon rotating it and re- leasing it it remains in the new position into which it has been turned. A perfect sphere resting on a smooth horizontal plane and a perfectly balanced pulley are examples. Rotation of the body does not change the sum of the moments. Procedure. The apparatus used in this experiment and the manner of setting it up are shown in Figure 20 b for a sample case. Various forces may be applied at many different points on the disc and may be caused to act in several dif- ferent directions. The weights upon the cords may be taken as the forces. The distances of the oblique forces from the axis are 96 MECHANICS measured by the aid of a try-square and scale. The distances of the forces which act vertically may be read off directly from the meter scale, the plumb line indicating the position of the axis. In all cases estimate to tenths of mm. Fig. 20b See that the disc rotates freely and is fairly well balanced. Level the meter stick with a spirit level. The object of the ex- periment is to test the law of moments for several cases. Also, incidentally, to observe the conditions for the three kinds of equilibrium. Test the law in the following four cases: (I) One vertical force on each side of the axis; (II) Two vertical forces on one side and one on the other; (III) One vertical force on one side and a vertical and an oblique force on the other; (IV) A ver- tical and an oblique force on each side. In each case hang dif- ferent weights on the cords. (I) First place the pegs in holes on the same diameter of the disc and adjust until equilibrium is secured; then move one so FORCES 97 that the line joining the two pegs passes below the axis; finally, place one peg so that the line joining the two is above the axis. In each case test the kind of equilibrium. Change the weights on the cords if it is necessary. 1. Take measurements to test the law of moments when the disc is in stable equilibrium. 2. Move both pegs to another position which will bring the disc into stable equilibrium and take your measurements. 3. Hang different weights on the cords and bring the disc into equilibrium by moving the pegs. (II) Suspend weights from two points on one side, and one on the other, varying the weights and their points of application to procure stable equilibrium. Make another trial changing weights and points of application. (III) Run one of the cords in (II) over a pulley near the end of the meter stick. In the first trial use the same weights as before. In the second vary the weights and the position of the pegs. In one of the trials move the point of application of the oblique force to another point in the line of the cord. (IV) Add another force passing the cord over the pulley near the other end of the meter stick. Make two trials, varying the weights and points of application. Vary the following table where necessary to apply to the various cases. TABLE XXIV Forces Distance Moments 1; 1° 3 CO a> Vertical Oblique From Axis to Pos. Neg. a ^ Ut(F) Rt(fi) UtiFi) Rt(Fa) F(=p) fl (=/>!) F2i= pi) Fs(= ps) Remarks. In computing moments the units of force may be in a different system from the units of distance, but in a given equation the units of all the forces must be the same ; likewise the units of all the distances. Some difficulty may be encountered in locating the exact posi- 98 MECHANICS tion of the axis of the disc. Use any plan you may think of but try to locate the center as accurately as you can. Before taking measurements, tap the disc lightly, for friction in the bearings may cause it to come to rest in a false position. Test Questions. 1. Explain how you illustrated the three kinds of equilibrium. Explain how you produced each of the three kinds of equi- librium in the disc. 2. Define moment of a force and state the law of moments. Explain how you have verified this law in your experiment. What is the essen- tial condition of equilibrium? Upon what does the effect of a force to produce rotation depend? How have you illustrated this in your ex- periment ? 3. Is the sum of the forces acting on one side equal to the sum of those acting on the other in any case? Why do they balance? 4. Is the eilect of a force changed by applying the force at a different point on its line of action? Explain. 5. In the case where two forces are acting about a fixed axis and hold a body in equilibrium, form a proportion between the forces and their shortest distances from the axis. Express it in words. 6. If it is wished to upset a tall column by means of a rope of a cer- tain length pulled from the ground, how can it be done with the greatest advantage? Illustrate by a diagram. 7. Prove by the law of moments that if a balance has unequal arms and a body is weighed first in one pan, then in the other, that the true weight is the square root of the product of the two weights. 8. Why is it not necessary to weigh the cords and pegs? EXPERIMENT F-t Determination of Various Forces in a Derrick Application of the Law of Moments For a discussion of moments and the law of moments read the introduction of Exp. F — 3. The apparatus to be used, and the way to set it up, are shown in the photograph given out with the apparatus. A bar pivoted at one end supports a load at the other end and is held in an in- clined position by a cord from a point near the loaded end to a hook in the standard of the laboratory support. (Cf. Fig. 21.) ~ The problems to be solved experimentally and graphically are : 1. To find by the law of moments the force F-^ (in grams weight) exerted by the cord on the bar at c. 2. To determine graphically (by the addition of vectors) the direction of the force F^ against the pivoted end of the bar at A ; also its magnitude. 3. To determine if %Fp = about any point X in the plane of the forces, (S means "the sum of such terms as"). FORCES 99 4. To test whether the three direction lines of Fi, F^, and the resultant of the two vertical forces (the load L and the weight of the bar F^) intersect at the same point when produced. (See Remarks below.) 5. To find graphically the vertical components of all the forces and to show that their algebraic sum is zero ; also to do the same with the horizontal components. In No. 3 the perpendiculars from X are to be measured on a full page diagram drawn to scale. (See Fig. 21). Specify clearly the scale used for distances and the scale used for forces. Choose a distance scale so that your diagram will occupy nearly a whole sheet of report paper. A convenient distance scale is 1 cm. to 5 cm. and force scale 1 cm. to 100 gm. wt. Nos. 3 and 4 will serve as checks on Nos. 1 and 2. Procedure. Set up the apparatus as shown and level the base of the laboratory support. Take the measurements which you will require to make a full page diagram on plain papier that will show to scale the length and the angular position of the axis of the bar, relative to a horizontal line through the pivoted point A, and the positions (on the axis of the bar) of the points of application of the four forces acting on the bar. Let b denote the point where the load L is attached, and a the center of mass . of the bar (found by balancing it on a knife edge). Begin the construction of the diagram by laying off to scale on a horizontal line the distance from the point ^ to a point m.on the cord sup- porting the load, m being at the same level as A ; then locate h by plotting upward from m, and draw the axis through A and h. The rest of the diagram is then easily constructed. Fig. 21 shows a diagram for a case differing from the one shoum in the print by having the points of application of L and F^ interchanged. Take also all the measurements that are needed to compute the magnitude of F^ by the law of moments. Compute Fj and plot it to scale. After all the necessary measurements are taken and the value Fi is computed, verify this force by inserting a spring balance. Clamp a rod to the standard of thfe laboratory support to locate the position of the bar, in order that it may be brought back to lOO MECHANICS the same position when the spring balance is used. Find the remaining unknown force F^ by the use of the force polygon. (This step is illustrated in Fig. 21 by the triangle drawn in dots and dashes in the lower right hand corner.) Draw F„ to scale from A. itp= 27. S Error ^..05<; Force Scale - Gm-Wt 200 400 £00 BOO WW! 1 1 1 ■+■ 1 1 1 1 h Distance Scale - Cm i = izoo m-^ Fig. 21 In your report arrange your results and computations as fol- lows: 1. Fj X 64.0 - 1200 X 50.0 - etc. = 0. F^= ; Spring balance reading 2. (State clearly how this is shown on your diagram.) 3. Fp = 955 X 40.0 - 1200 x 44.6 - etc. = ' Error in per cent. 4. (Explain as in 2.) 5. (Explain as in 2.) FORCES lOI Remarks. The numerical values of the vertical and hori- zontal components required in No. 5 can be obtained most easily from the force polygon drawn on your diagram, since all the forces that act on the bar are there represented both in magni- tude and direction. The resultant of the two forces F^ and L is found as fol- lows : the magnitude of the resultant is the algebraic sum of the forces; its line of action divides the perpendicular line joining the two forces into two parts which are in the inverse ratio to the magnitude of the adjacent forces. In measuring the moment arm of F^ on the apparatus a tri- angle or tri-square may be used to set the meter stick perpen- dicular to the cord. Test Questions. 1. Make a full statement of at least three principles regarding forces holding a body in equilibrium which are verified or used in this experiment. Mention also any other principles used. 2. Explain fully how you constructed your diagram; also how the com- putations were made to determine jFi and Fi, and how the results were checked. 3. Explain how the results of problems 3 and 4 serve as a check upon problem 1 and upon problem 2. 4. What are the two essential conditions of equilibrium? 5. What is the resultant of two parallel forces? How is the position of its line of action determined? 6. Would the bending of the beam affect your results? Explain why. 7. What does the force Fa represent? What does the equal force in the opposite direction represent? 8. Why is the center of moments taken at Af Why is it not necessary to consider the force Fz in applying the law of moments, when the point A is taken as the center of moments? 9. How many signiiicant figures are there in your spring balance read- ing? How many in your computed value of Fi? Which then should be the more reliable? experiment f-5 Equilibrium of Parallel Forces in the Same Plane Resultant of Parallel Forces If a body is in equilibrium when it is acted on by any number of parallel forces in the same plane: ( 1 ) The sum of the forces acting in one direction must he equal to the sum of the forces acting in the other direction; in other words, the algebraic sum of the forces must be zero. (2) The algebraic sum of the moments of all the forces acting about any point is zero. 102 MECHANICS A very good way to test this principle is to hang weights on different parts of a bar, the bar being suspended by two spring balances. When a greater force is exerted upon one balance than upon the other, the spring of the latter is stretched more and hence the bar does not remain level. This difficulty may be overcome by suspending this balance by a cord which passes over a pulley so that it may be adjusted to keep the bar level. In order to lessen the errors, ( 1 ) the forces should be made to act as nearly parallel as possible; (2) mean readings must be taken similar to those in Exp. F — 1, on account of the friction in the spring balances; (3) the zero reading of the spring bal- ances must be taken very carefully; (4) the bar must be kept level. Procedure. Use an ordinary meter stick. Weigh it, then find its center of gravity by balancing it on a knife blade, being careful not to mark the stick in any way. Support the bar at one end by a spring balance and cord hung from a laboratory support, and at the other end by a spring balance attached to a cord passing over a pulley hung from another laboratory sup- port. Support the bar edgewise to avoid bending as much as possible. Hang two unequal weights (250 g. or more) from points on rthe bar between the spring balances and adjust the cord passing ■over the pulley until the bar is level as determined by a spirit 3evel. Also adjust the loops of cord around the bar until all the forces are acting vertically. Take the readings of the balances and on the meter stick read the positions of the points of appli- cation of all the forces. Make four trials, varying as much as possible the positions and magnitudes of the weights and the location of the spring balances. In one case have one of the balances between the two weights. Consider forces acting up as positive and those acting down as negative. The balances read in ounces, hence it will be necessary to record these readings in ounces, converting the ounces to grams for your computations. The data and results may be tabulated as shown in Table XXV. The center of moments may be taken at any point desired. In each of the four trials choose the center of moments at a diflfer- FORCES 103 ent point. Make a table like Table XXV for recording the data obtained in each trial. Immediately following each set of data make a force diagram to scale, indicating upon it the center of moments, the value of each force and its distance from the center of moments. TABLE XXV Forces Scale Readings on Bar Distance from Center of Moments Moments Ounces Grams Left Bal. Right " Left Wt. Right " Bar (Cent, of Grav.) Sum of + Forces . . . . Sum of — Forces . . . . Difference in per cent. Sum of + Moments . . Sum of — Moments . . Difference in per cent. Remarks. The first principle given above, relating, to the equilibrium of parallel forces might be stated as follows: If a body is held in equilibrium by the action of parallel forces in a plane, the resultant of the forces acting one way is equal in mag- nitude and opposite in direction to the resultant of the forces acting the other way; and the two resultants are in the same straight line. The magnitude of the resultant of two or more parallel forces is their sum. The line of action of the resultant may be found by first considering two of the forces only. The resultant of any two parallel forces is parallel to the lines of the forces and lies in a line so situated that the ratio of its distance from the first force, to its distance from the second is equal to the ratio of the magnitude of the second force to the magnitude of the first. When this resultant has been found, it is combined with a third force, and so on until all the forces have been included, the final resultant being the resultant of all the forces. The line of action of the resultant of several parallel forces is more easily located by applying the principle of moments as ex- pressed in the equation, ^Fp = RX, or X=%Fp/%F I04 MECHANICS in which X is the distance of the resultant (i? = S F) from the center of moments. The first part expresses the general prin- ciple that The moment of the resultant of several forces equals the summation of the moments of the forces. This holds true for both parallel and nonparallel forces. It is involved in the definition of a resultant. Test Questions. 1. State and explain the two principles applying to the equilibrium of a body when acted upon by parallel forces only. 2. Consider a body as made up of a great number of tiny particles and the force of gravity acting on each one of them. Through what point would the resultant of all these forces pass? 3. Why is it necessary to keep the bar level? Ejcplain why you took the precautions you did in performing the experiment. 4. Give as many reasons as you can why the sum of the positive forces is not the same as the sum of the negative forces, and the sum of the positive moments is not just the same as the sum of the negative moments. 5. How would you find the resultant of all the positive forces and where its line of action is located? The resultant of the negative forces? How are these two resultants and their lines of action related? 6. If two equal forces were acting in opposite directions at opposite ends of the bar, the arrangement is called a couple. What would be the resultant of two forces forming a couple? What would be the sum of their moments? UNITS GROUP VI — U. UNITS AND PERIOD 105 experiment u-1 Work Against Gravity and Friction Comparison of Systems of Units Before proceeding with this experiment the student should understand the various quantities he is to measure and deter- mine, and the units used to express these quantities in the differ- ent systems. This knowledge may be obtained by studying care- fully Sections 41, 42, and 43, Chapter I. This experiment is designed to give practice in expressing work and power in several units of the different systems of units ; and to illustrate a method of measuring work and power by means of a friction dynamometer. Part A. Work Against Gravity. Procedure. Determine what data must be observed in or- der to compute the work done against gravity in climbing from the level of one floor to that of the second or third floor above; also the additional data required in order to find the average power. Find (if possible) winding stairs around a " well." Qimb the stairs twice at different speeds. State whether they are climbed very quickly, quickly, moderately, slowly or very TABLE XXVI Abs. Met. Abs. Engl. Grav. Met. Grav. Engl. Multiples Quantities or or or or of C. G.S F. P. S. Fr. Engin'g Eng. Engin'g C. G. S. Mass Grains Pounds *G. M. Units Slugs Kilograms Weight Dynes Poundals iKg. wt. Lbs. wt. ^Megadynes Vert. Dist. Cms. Feet Meters Feet Meters Time Sec. Sec. Sec. Sec. Sec. Work Ergs. Ft. -poundals Kg.-meters Ft. -lbs. Joules Power Ergs./sec. Ft,-pdls/sec. 8Kg.-m./sec. Ft.-lbs./sec. H. P. Watts Kilowatts iThe Rnu.wt. is sometimes used as a gravitationil unit of force and corresponding to this we hav« the gm-cm as the unit of work and the gm-cm/sec as the unit of power. 2 1 megadsfne = 10' dynes. ^11 Kg-in/sec = l Force de Cheval or French Horse-power. *See Sec. 43, Chapter I. I06 MECHANICS slowly. Compare the work done and the power in the two cases. Tabulate the data and results indicated in Table XXVI. Each partner should use only data obtained by climbing the stairs him- self. In this table take especial care to write none but significant figures in any of the numbers. Mark the doubtful figure. Of course use ciphers to locate the decimal point; or if several of these ciphers are required, they may be expressed by indicating multiplication by the proper power of 10. Since only a small number of significant figures can be obtained it will be observed that logarithms may be used to advantage for these computa- tions. (See Sect. 23, Chap. I.) Do not convert values from the English to the metric system or vice versa except to check results. In your report be careful to explain how you obtain each value in the above table. Part B. Work- Against Friction. In computing the work against friction the force used is the average force required to overcome friction, and the distance through which the force acts is the relative displacement of the two surfaces against friction. A large pulley is mounted on a shaft and is turned by a crank against the friction of a piece of leather belt placed over the pul- ley and fastened directly to a small hanging weight on one side, and by a piece of chain to a large weight on the other side. For a given pressure the friction between the belt and the pulley is much greater than the friction between the chain and the pulley. Therefore as the pulley is turned toward the belt the chain is drawn up on the wheel so that the friction of the belt on the wheel is decreased as the friction of the chain uf>on it is in- creased. Thus the total friction is decreased. This movement continues until the friction of the belt plus the friction of the chain on the pulley just equals the difference between the weight on the belt and that on the chain. If the total friction is too small the chain and belt will move back slightly, automatically adjusting themselves so that if the two weights are properly chosen they should ride free from the floor. The weights will ride most steadily and with least vibration if they are so ad- UNITS 107 justed that when the wheel is being rotated the junction of the chain and belt remains near the top of the pulley. Let F^ and F^ represent the tensions on the ends of the belt produced by the masses hung- thereon. The torque is the mo- ment of the force producing rotation. When the wheel is at rest with no torque applied to it, part of the weight of the heavier mass is supported by the floor so that F^ may equal ■ Fj ;■ but when sufficient torque is applied to the shaft to cause the belt to slip the friction exerts a tangential force (F) on the rim of the pulley in opposition to the motion of the pulley. When the masses hung on the belt ride free from the floor the force of friction equals the difference between the weights of the masses or F = Fj - F,. The distance (s) through which this force acts is 2irRN for N revolutions. The work done then is as follows : Work iW)=Fs^iF^-F,)27rRN (= F.C.N.) (1) By rearranging the symbols in this equation it will be evident that the work may be expressed also as the product of the torque and the angular displacement. Thus W={F^-F^)Rx2irN (2) in which {F^-F.,) R is theresultant torque (T) due to the ten- sions at belt ends; and lirN is the angular displacement {6)- in radians. Therefore W=Te (3) The power developed may also be expressed in terms relatiiig to the rotation of the wheel. The familiar expression for power is P=W/t and the expression for angular velocity {•»), which is the number of units of angle described per second, is m = e/t from which 6 = iot Combining these expressions with equation (3) we obtain P = To) (4) Another simple expression for the power which is sometimes useful involves the average linear velocity {v) of the rim of the io8 MECHANICS pulley. The term 2irRN in equation (1) expresses the actual distance (s) through which a point on the rim of the pulley moves. Therefore since s/'t = v we have P = Fs/'t = Fv Turn the pulley as uniformly as possible for two minutes. Count the number of revolutions and record the number at the end of each half minute. Reduce the data to consistent units. Be sure to name the unit of each result. Compute and tabulate the data and results as indicated in the following table. TABLE XXVII Revolu- Time t Angle e Ang.vel. at Work Done Power tions (M W= F.C.N. W'=TB Ft.-lbs./sec. H.P. Watts Tu> •g;- .lb. Radius of Pulley, R= cm.; ft. (F1-F2) =F=. Circumference of Pulley, C= cm. Torque (T) = Test Questions. {Part A) \. Define the terms work and power. 2. In climbing the stairs in which direction does the force considered act? Why is no account taken of the movement horizontally? 3. Does the time required in climbing the stairs make any difference in the amount of work done? Does it affect the power? Explain. 4. In changing from one unit to another, what determines how many significant figures to use? If the unit in which a quantity is expressed is large, is the number expressing it relatively large or small ? {Part B) S. Explain the purpose of the chain attached to the leather belt. How does the amount of friction of the belt upon the pulley change when the amount of surface in contact changes? Why? 6. In which direction relative to the rim of the wheel does the force F act? Explain how the relative displacement in this direction is com- puted. 7. What is meant by the term torque? How is it computed in this ex- periment? 8. Explain how the expression for work is found in terms of torque and the angular displacement. 9. Explain how to obtain the expression for power in terms of torque and the angular velocity. In terms of the force and the linear velocity of the rim of the wheel. 10. Which columns in the table should check each other? il. Explain how you determined in what units the values of Tw are ' expressed. PERIOD 109 experiment u-2 Relation of the Period of a Pendulum to its Length Method of Determination of Period An ideal simple pendulum is supposed to consist of a mass concentrated at one point at the end of a weightless thread sus- pended from a pivot about which it is perfectly free to swing. Although a simple pendulum exists only in the imagination, we can approximate it by suspending a heavy ball by a thread. The length of this pendulum is from the point of support to the center of the ball. The important quantities relating to the vibration of such a pendulum are its period, (T) or the time required for a complete vibration, that is from the passage in a given direction through a given position to the next passage through the same position in the same direction; its length, (/) or distance from the point of suspension to the center of mass of the bob ; and g, or the ac- celeration of gravity. The relation between the three quantities above mentioned is expressed by the equation J-^ (1) r= 2 9 Since the period and the length of any given pendulum may be quite easily measured with considerable accuracy, this rela- tion may be and frequently is used for an accurate determination of the value of the acceleration of gravity, g. Solving equation (1) for gr we obtain Hence it is necessary simply to measure the length of a given simple pendulum and its corresponding period and substitute these values on the right hand side of equation (2) to obtain the value of g at that place. It should be noted however that equation (1) is strictly true only when the amplitude of vibra- tion is infinitely small. Sufficient accuracy is obtained if the amplitude is less than 3° of arc. Procedure. Measure the diameter of the ball as accurately no MECHANICS as required with a caliper. Pass the thread suspending the ball through the slot of a screw head in the top of a laboratory- stand, adjusting it so that the pendulum will be somewhat less than 2 meters in length, and fasten the free end. Suspend a plumb bob directly behind the pendulum to serve as a marker, the time of the transit of the pendulum past the plumb line be- ing recorded. Set the pendulum swinging over an arc of not more than about 6°. One observer takes his place directly in front of the pendulum and gives some signal at the instant when the pendu- lum passes the plumb line. The second observer notes the time of the signal on a watch and records it. On the next trial the observers change places, and so on. Record the time (hour, minute and second) at which each 10th transit, up to the 100th, occurred, reading the watch to half sec- onds. Find the time required for 90, 70, 50, etc. vibrations by subtracting the time of the 10th transit from that of the 100th, and the time of the 20th from that of the 90th, and so on. (In interpreting the formula be careful to distinguish between the period of a single vibration and that of a complete vibration.) In this manner obtain the period for five different lengths of the pendulum, varying by about equal intervals from a little less than 2 m. to about, 40 cm. Record the data for each length and summarize for the five lengths as shown in Table XXVIII. TABLE XXVIII Length of Pendul Number of Transits Time of Transits Duration of Length / Period T 72 / 2-2 "?■' 10th 20th 30th h. m. s. h.m.s. 90 vib. = 70 " = 50 " = 30 " = 10 " = 250 " = etc. Average value of g Accepted value of g Deviation in per cen 1 " = =T t Plot two curves using lengths as abscissas in each. In one use values of T as ordinates, and in the other use values of T*. PERIOD III Remarks. See that the minute and second hands of the watch indicate the even minutes at the same time, or nearly so. It is well for the observer to count aloud three or four transits before the one for which the time is recorded. If you have to make the observations alone it is best to use a stop watch. Test Questions. 1. Explain why the pendulum used is not an ideal simple pendulum. 2. What distance is taken as the length of the pendulum? To what fraction of a cm. is it necessary to measure the diameter of the ball? Explain. 3. What is meant by the period? Why is 2"" used in the equation in- stead of "■? Explain how you measured the period in this experiment. 4. What does the symbol "g" represent? How is its value determined in this experiment? 5. What do your curves indicate as to the relation between the length and period of a pendulum? How is the same relation shown in the formula? What is the significance of the column l/T^f 6. Explain what advantage there is in determinirig the period as you did. Would it have been as well to time, say, 10 vibrations only? Why subtract as you did? 7. A seconds pendulum loses 18 sec. per day when carried__to another station. Compare the values of " z" at the two places. 8. What will be the effect on the period of a pendulum: if carried up- ward with an acceleration equal to " g " ? If carried downward with acceleration equal to " s"? 1 12 MECHANICS GROUP VII — M. MACHINES AND FRICTION experiment m-1 Study of Fixed and Movable Pulleys Mechanical Advantage and Efficiency The mechanical advantage of a simple machine is the ratio of the force overcome (load) to the force applied that will ex- actly balance the load, if friction and the weight of parts are eliminated. Efficiency is the ratio of the useful work done to the total energy transformed in doing it. Arrange the pulleys to be tested in two blocks, one to be the fixed block and the other the movable block. Fasten one end of the cord to one of the blocks containing several sheaves and pass it in turn over the fixed and the mov- able block. Apply the force to the free end and the load to the movable block. Procedure. Use for the force copper burrs and sand in a small pan and determine the amount of the force by weighing the pan and its contents. (In this experiment the term force will be taken to mean " force in grams weight.") Use loads of 0, 100, 300, 500, 700 and 1100 grams. One difficulty in finding the force required to balance the load is that friction in the pulleys interferes with their motion. To eliminate this difficulty proceed as follows : Load the pan with burrs and sand until the pan j.ust moves up uniformly, i.e., without acceleration. The pan then just balances the load (including the weight of the-pulleys) minus the force of friction. Call this weight F^. Now add burrs and sand to the pan until it moves down as nearly as possible with the same motion that it had when mov- ing up. The pan now balances the load plus the force of fric- tion. Call this weight F^. Evidently the mean of F^ and F^ is approximately the force that will produce equilibrium if friction is eliminated. Call this force F. There is still the weight of the movable pulley block to be MACHINES "3 considered. This is corrected for by subtracting the force re- quired to balance the block with no load upon it from that re- quired to balance both block and load. Thus suppose 58 g. just balances the movable block with no load ; and with 300 g. load, 108 g. is necessary. The force balancing the load is then 50 g. (108-58) and the mechanical advantage is 300/50 = 6. In finding the efficiency for the above case the useful work is the work done in raising (not balancing) the 300 g. load. To do this necessitates a force of something over 108 g. weight (since friction had to be overcome) call it 114 g. The mechan- ical advantage is 6 (supposing the load to be supported by 6 cords) and the free end of the cord moves 6 times as far as the load. In this case the efficiency is computed as follows : 300.^ _ (Useful work done) ^ 114 X 6rf (Total energy expended) ' " " ^^^'^^Y Here d represents the distance moved by the load. Using the data you have taken corresponding to the above, compute for each load the mechanical advantage and efficiency. Record data and results as indicated in Table XXIX. Plot a curve using loads as abscissas and per cent, efficiencies as ordinates. TABLE XXIX Load Fi F2 Force Balancing Load Mechanical Advantage Efficiency in Per cent. Graphical Solution. The analytical method given above for finding the data for the mechanical advantage and efficiency is complete ; but the graphical method described below gives prac- tice in the use and interpretation of curves, shows the variation in friction, and serves as a check on the analytical method. Plot two curves using F^ and F^ as ordinates and correspond- ing loads as abscissas. Locate points (F) which bisect the dis- tance between the two curves along several ordinates. Draw a straight line through these bisecting points. Call these curves the P'l - load curve, the F^ - load curve and the F - load curve respectively. 114 , MECHANICS What is expressed by each intercept? Explain why one of the intercepts is the same for all three curves. Draw through the origin a curve (F'-load) parallel to the F - load curve. What do the ordinates and the reciprocal of the slope of the F'-load line represent? Explain. Find values of efficiency, for loads intermediate between those used in the ex- periment, from data taken from the curve sheet. Mark the points on the efficiency curve that are obtained by this method so as to distinguish them from the points obtained by the first method, and draw conclusions. Plot a line showing the variation in friction. What indicates whether the coefficient of friction is constant or variable? Test Questions. 1. What is the mechanical advantage of a simple machine? What determines the mechanical advantage of a system of pulleys? Explain fully why the mechanical advantage of the set of pul- leys you used had the value it did? 2. How many times farther does the free end of the cord move than the movable pulley block? Explain fully. 3. How did you eliminate friction in the pulleys ? How did you take into account the weight of the pulleys? 4. What is meant by efficiency? How was it computed in this experi- ment? Explain clearly the difference between efficiency and mechanical advantage. 5. Why does the efficiency change in general when the load changes? From your curve what general change takes place in the efficiency when the load changes? Does tke efficiency curve pass through the origin? Explain. 6. What is the limit which the efficiency of a_ machine cannot exceed ? Explain why an " efficiency curve " is not a straight line. 7. It is desired to raise a load of 900 g. with your apparatus. Use your efficiency curve and compute what force must be applied to the free end of the cord to just raise this load. experiment m-2 Differential Wheel and Axle Mechanical Advantage and Effiency A differential wheel and axle is a machine for lifting heavy masses. It is a combination of two simple machines. The load to be lifted is suspended from a pulley in a loop of rope. One end of this loop winds on the large part of the axle, while the other unwinds from the small part. The axle is turned by MACHINES 115 unwinding rope from a wheel rigidly attached to the axle. When the wheel and axle make one turn, the length of rope un- wound from the wheel is the circumference of the wheel {2irR'). The length of rope wound on the large part of the axle is its circumference {lirR). The length of rope unwound is the cir- cumference of the small axle (27rr). The rope forming the loop is thus shortened {2-kR - Zttt) and the load lifted half as much, or 7r(i?-r). The work done on the load (L) is w{R-r)L. The work done by the lifting force (F) is 2wR'F. If there were no friction AR-r)L = 2^RF,or~= '^"^ 2^' F ^(R-r) R-r This ratio L/^F is called the " Mechanical Advantage." In an actual case a force F' must be used in order also to over- come friction, so that the work done by it always exceeds that done on the load. The ratio, work done on the load to work done by the working force, is called the efficiency. It is usually expressed in per cent, so if the ratio were % it would be given as 75 per cent. „_ . AR - r)L {R - r)L Efficiency = 2-i2'i^ ^ IRIF Prove also that the Efficiency = F/F' or the total force actually applied divided into the force use- fully employed. Sketch the arrangement of cords by means of which the loads are raised. By applying the principle of moments, show what force F would be required to raise a load L if friction were not present and if the lower block were weightless, and thus obtain the mechanical advantage. Procedure. Experiment with the machine and find forces (F) required to raise loads (L) of 300, 500, 700, .. . 1500 grams. Measure the distance moved by F' and by L dur- ing one turn. How do they compare with those calculated from Tthe radii R', R, and r? ii6 MECHANICS Data may be tabulated as follows : TABLE XXX Load (I) Applied Force Useful Force Work On Load By Force (FO Efficiency Radius of wheel Distance moved in 1 revolution Radius of large axle '. By Load Radius of small axle By Force Mechanical Advantage Curves. Take as the origin a point a few squares to the right of the left margin, and draw three curves on the same sheet and to the same scale, one showing the relation of F to L in the ideal case (no friction and no weight of moving parts), and one showing the actual relation as determined by experi- ment. Draw the third curve from the point where the experi- mental curve cuts the axis of loads and with the same slope as the ideal curve. Find from the curves and explain fully how they are found: (a) The unbalanced weight of moving parts commonly called " dead load." (b) The force required to raise the " dead load." (c) The force required to overcome the friction caused by weight of parts of machine: (d) And for a load of 1000 grams. ( 1 ) The force usefully employed. (2) The force for friction due to load. (3) The efficiency. (Compare with value found on efficiency curve.) Is the efficiency the same at all loads? Draw an efficiency- load curve. Test Questions. 1. Define mechanical advantage. 2. Show that the mechanical advantage of a differential wheel and axle is 2R'/'R-r. 3. Explain how the data for drawing each curve were obtained, and the meaning of each curve. Why do the second and third curves meet on the X-axis? MACHINES 117 4. Define efficiency. 5. Calculate the efficiency of the machine you use4 for a load of 2000 grams. 6. If the two drums were of almost the same diameter, would the mechariical advantage be large or small? Explain. experiment m-3 The Inclined Plane Mechanical Advantage and Efficiency The mechanical advantage of the inclined plane, when tt^e force is parallel to the plane is equivalent to the ratio of the length of the plane to its height; or W^ L F h with which the student is assumed to be familiar. Apparatus. An inclined plane with "ways," pulley, blocks, string, copper burrs, set of weights and a small nearly friction- less car. With this apparatus it is left as an exercise for the student to prove the above law experimentally, using whatever method and taking whatever data he may deem necessary. In any case con- sider the car to be a part of the load, not a part of the machine. The method chosen miist have the approval of the instructor before the experiment is performed. Make a complete report of the experiment, describing briefly but thoroughly the apparatus used, and proving all formulas used in the experiment. The equation W /F = L/h should be proven graphically. Show as in other experiments your complete data neatly tabulated. Exercises. (1) Compute the efficiency and the mechanical advantage of the plane for each load. (2) Find in each case what force acting parallel to the base is necessary to just balance the car. Record these items as a column of your data table. (3) Plot one efficiency curve. (4) Plot curves from which may be found the mechanical il8 MECHANICS advantage, the friction for any giver^ load and the force required to overcome it, the force usefully employed and the efficiency (cf . Experiments M — 1 and M — 2) . Interpret these curves. Test Questions. 1. Express in words the mechanical advantage of the inclined plane in the case when the force is applied parallel to the plane. Also in the case when the force is applied parallel to the base. 2. Explain the method by which you eliminated friction. 3. How did you compute the efficiency? In general how does the ef- ficiency vary with the different loads? Explain. 4. If the car were drawn up the full length of the plane, how would you compute the work done against gravity? How compute the work done against the tendency which the car has' to move down the plane? Compare the two values. 5. In each case, is the force required to draw the car up the plane less than the weight of the car and its load? If so, through how much greater distance must the car be moved to compensate for the diminished force? 6. If a man by the aid of a winch can exert a pull of 900 lbs. wt. how long a plank would he need to pull an iron safe weighing 2 tons into a wagon 30 in. in height, friction amounting to a force of 150 lbs. wt. ? 7. How does the inclination of the plane affect the friction? The ef- ficiency? The mechanical advantage? Explain. experiment m-4 Laws of Friction Determination of Coefficients The resistance which opposes the motion of one surface on another is called friction. This resistance is called sliding fric- tion when one surface slides on another, and starting friction when it relates to the resistance that must be overcome by the force required to start relative motion from relative rest. When one surface rolls upon another without sliding, the resistance is sometimes called rolling friction. Friction is caused in part by the interlocking of projections and depressions on the surfaces in contact. For unpolished sur- faces this is probably the principal cause. The laws of friction may be stated as follows: (1) Friction is proportional to the total pressure upon the surfaces in contact. (2) Friction is independent of the area of the surfaces in contact. (3) Friction is independent of speed for medium speeds. MACHINES 119 (4) Starting friction exceeds sliding friction and is gen- erally quite variable. (5) Rolling friction is directly proportional to the pressure and inversely proportional to the radius of the roll- ing surface. By the first law, the ratio of the frictional resistance (Ft) parallel to the surface, to the force (Fp) perpendicular to the surface is a constant for two given surfaces. This constant (n) is called the coefficient of friction. The defining equation is « = F,/Fp. The value of the coefficient depends upon the materials, the molecular forces, and the conditions of the surfaces, and in some cases may vary somewhat with the speed. When testing the relation of two variables it is important that any additional variable be excluded ; therefore the surfaces used to verify the first two laws should be such that their condition is not easily changed; and then especial care should be used to guard against change. Some surfaces will be changed by touch- ing them with the hand. Favorable conditions may be obtained by using polished wood on a smooth surface of hard unglazed drawing paper. With proper care surfaces of wood, metal, glass, etc., will give reliable results. Procedure. Part (A) Friction Plane Horizontal. Ad- just a long glass plate, covered with drawing paper, to a horizontal position, on blocks, by the aid of a spirit level. Place on the paper the sliding block with the broad polished surface down, and from it run a cord over a pulley, clamped on the edge of the table, in such a manner that by adding small weights to a hook on the free end of the cord the block may be drawn along the surface of the plate. The height of the pulley must be so adjusted that the cord draws parallel to the plane of the glass. Now hang weights on the hook sufficient to cause the block to move, when started, with a uniform motion, adjusting the weights closely with small copper burrs or any other convenient means for close adjustment. From the weight on the cord and the weight of the block, the coefficient of friction may be found by the use of the defining equation. I20 MECHANICS Repeat the observations with successively added loads on the block until five or six different loads have been used. Tabulate as indicated below. TABLE XXXI Weig ht of Block . . Weight of One Burr Load on Block Normal Force (Fp) Weights on Cord Number of Burrs on Cord Tangential Force (Ft) Coefficient of Friction Take and record in like manner a set of data with the narrow polished surface on the plate. Plot a curve for each set of data using Fp for abscissas and Ft for ordinates. Both curves may be plotted on the same sheet with the same scales. The curves may be separated by taking the origin of the second curve one or two spaces to the right of the first and writing another scale of abscissas. Determine the coefficient of friction from the slope of these curves, and show how the first two laws are verified by the curves and table. Compare the coefficient obtained from the curves with the mean value from the table. Procedure. Part (B) Friction Plane Inclined. Sup- port one end of the plane on a block and change the inclination until the sliding block placed on the plane slides down the incline with a slow uniform motion. Take measurements to determine the tangent of the angle of inclination from the horizontal. Compare the tangent of this " slipping angle '' with the coefficient of friction determined in part (A). Make at least two trials with different loads. Procedure. Part (C) Friction of a Pulley. Clamp a pul- ley to the top of one of the laboratory stands. Hang equal weights on the ends of the cord over the pulley ; then put small weights or copper burrs on one side until that side moves down with a uniform motion. From the added weight and the radii of the wheel and pin find the friction, using the principle of mo- ments. Tabulate the data and compute the coefficient of friction for MACHINES 121 the two metals of which the wheel and pin are made. Repeat with the equal weights successively two, three, and four times as large as the first ones, and find the friction and coefiicient in each case. Test Questions. Part {A) 1. Upon what does the friction between two surfaces depend? What three general types of friction are there? How do they compare in magnitude for the same surfaces ? 2. Define the coefficient of friction and explain how its value is com- puted. Upon what does its value depend? 3. Why is it necessary to avoid touching the surfaces while determin- ing their coefficient of friction? 4. Explain why the cord drawing the block should be parallel to the surface of the paper, and be attached near the surface. 5. Explain why the slope of the curve is the coefficient of friction. 6. What is the effect upon the friction of changing the amount of sur- face in contact? How is this shown by your curves? Part (B) 7. Prove theoretically that the tangent of the slipping angle is the coefficient of friction. 8. A plane like the one used in the experiment is inclined 30° to the horizontal. A 250 gram block of wood resting on the plane is attached to a cord which passes over a fixed pulley at the top of the plane and has a piece of iron suspended from the end below the pulley. Taking the coefficient of friction as determined from your data, which way and with what acceleration will the piece of iron move if the mass of the iron is 25 grams; if it is 250 grams; 100 grams? Part (C) 9. How is the friction related to the load in this- case? Is the coefficient the same for the different loads? Explain. 10. Explain clearly what probably would be the effect if the diameter of the pin were doubled; also if the length of the bearing were doubled. 11. Sum up the laws of friction you have verified in this experiment and explain how each one was verified. 122 MECHANICS GROUP VIII — P. PRESSURE IN FLUIDS experiment p-1 Verification of Laws of Fluid Pressure Dependence of Pressure Upon Depth and Density It is proven in the text book that the total pressure Ft o^ * fluid of density £> on a surface having an area A placed at a depth h below the surface of liquid is (in gravitational units) P^^DhA (1) and the pressure per unit area is P = DhA/A = Dh (2) The apparatus used to experimentally test this equation is shown in Fig. 22. The cylindrical chamber C is closed water tight at the top by a sheet of thin rubber R. A metal disc D rests upon the rubber but is prevented from depressing it below its normal position by the Flange F running around the rim of the disc. The weights are placed upon the center of the disc, the center being located by a series of concentric circles graved upon the surface. A tube T leads from the bottom of the cham- ber and is connected by a flexible rubber tube to the glass tube G. This glass tube passes through a hole in a hollow fixed bar B of metal loaded at one end L. When raised or lowered the tube will be held in any desired position by a flat spring S set inside the bar. The chamber and tube are filled with water so that when the top of the glass tube is only a little above the level of the top of the chamber the meniscus in the tube is on a level with the sheet of rubber R. Air bub- bles must be carefully excluded dur- ^'^- ^ ing the process of filling the chamber. If any air remains, the bubble can be seen through the thin rubber. When the water in the glass tube is on a level with the sheet of rubber, the disc rests with its whole weight upon the flange PRESSURE IN FLUIDS 123 F. Then if there is a weight W grams on the disc and the weight of the disc is W grams, the total downward pressure will be (in dynes) If the glass tube G is now raised the free surface of the water in it will be elevated above the level of the sheet of rubber and there will result an upward hydrostatic pressure upon it. When the disc is just lifted by this pressure off the flange upon which it rested, evidently the upward fluid pressure is just equal to the downward pressure due to the weight of the disc and the weights upon it. The upward pressure in dynes from equation (1) then is Pt=DhAg Therefore, {W+W')g = DhAg (3) which is the equation to be tested in this experiment. The dif- ference in level of the two surfaces of water, (or head), h, is ob- tained by resting one end of a meter stick upon the metal bar and reading the position of the meniscus in the glass tube, first when the disc has no weights upon it and again when it is loaded. The difference is the head supporting the weights alone. The area A is obtained by measuring the metal disc. Procedure. Place at least 200 grams at the center of the metal disc. Slowly and as smoothly as possible raise the glass tube, meanwhile watching the meniscus closely. A point will finally be reached where the water apparently ceases to follow the tube and seems to sink back on account of the sudden lifting of the disc. As soon as this is noticed, stop raising the tube. If now the weight is gently pressed down and released alter- nately, there will be a large movement up and down of the meniscus provided the disc is held up by the pressure of the water; but only a very small movement will result if the disc rests upon the rim of the chamber. Lower or raise the tube until the movement of the meniscus when the weight is pressed down exceeds the minimum by a definite small amount. When the. water level is at this point we may assume that the hydrostatic pressure just balances the pressure of the weights. By repeated trials, using a rubber band as an indicator, this point can be de- 124 MECHANICS fermined with a fair degree of accuracy if the work is carefully dcine. Read the position of the meniscus on the meter stick. Now lower the tube and remove the weights from the disc; and, in the same manner as before, find the height of water nec- essary to balance the disc alone. Place the meter stick in the same position you did previously and read the height of the water in the tube. The difference between this reading and the pre- vious one will be the height of water necessary to balance the weights alone, so that it will be unnecessary to weigh the plate, and capillary action will be corrected for. Proceed in this way for loads of 100 grams, 200 grams, etc., up to about 700 or 800 grams. The area of the disc may be taken as the area pressed upon and should be measured by the vernier calipers. Data and results may be tabulated as shown below in Table XXXII. Plot the two variables and interpret the results. TABLE XXXII Diameter of Disc. Area of Disc. Load on Disc (W) Down. Pressure of W (Dynes) Read, of Men. for Disc alone Read, of Men. for Disc + W Head to Balance W(h) Upward Press. iP) dynes/cm. 2 Upward Force to Bal. W (dynes) Per cent. Difference Test Questions. 1. Prove equations (1) and (2) theoretically. Ex- plain the basis for equation (3). What changes are made in these for- mulas if pressures are to be in absolute units? 2. Explain how you have verified tlie law of fluid pressure. Is it nec- essary to include "s" in the equation tested? Why? 3. If a disc with twice the area were used, how would the height of the column of water required to support a given weight be changed? 4. Would the pressure per unit area be changed by changing the size of the disc only? By changing the diameter of the tube? 5. Explain how the efiect of capillarity is eliminated. Why is it not necessary to weigh the disc? 6. Assuirie that the temperature of the water is the same as that of the room and find from tables what the density of water is at this tem- perature. Explain why you are justified in taking the density as unity in this experiment? 7. Could the experiment be performed with another liquid? What changes in computation would be necessary? If the liquid had a density of 1.5 g/cc, what heads would be required to balance each of the loads used in this experiment? What other law of fluid pressure does, this il- lustrate? ' \ pressure in fluids i25 experiment p-2 . Study of the Barometer Method of Measuring Elevation If a glass tube about a meter long, closed at one end, is filled with mercury, all the air being carefully excluded, then if the open end is closed, the tube inverted and set in a vessel of mer- cury and the stopper then removed, the arrangement is known as a Torricellian Tube. The mercury in the tube will fall until it is at a height of about 76 cm., leaving a vacuum in the top of the tube as shown in Fig. 23. There is no pressure then on the top of the mercury in the tube; therefore, the column of mercury is supported by pres- sure upward at the bottom of the tube, which is just equal to the pressure of the mercury downward. This upward pressure is caused by the pressure which the atmosphere exerts on the free surface of the mercury in the vessel ; and since pressure in a fluid is transmitted without loss in all directions, this upward pressure at the bottom of the column of mercury in the tube is exactly equal to the pressure of the atmosphere upon an area of the free surface equal to the area of cross-section of the bore of the tube. From equation (2), Exp. P — 1, this pressure is h D per unit area. Therefore the height of the mercury column, that is, the difference in level of the surface of mercury in the vessel and that in the tube, is a measure of the atmospheric pres- sure. The pressure of the atmosphere is constantly changing, so the height of the column of mercury changes proportionally. A Barometer is a Torricellian Tube arranged with a scale and adjustments for accurately reading the height of the mercury column. The form used in the laboratory is known as the For- tin barometer. The scale is provided with a vernier, the lower edge of which serves as an indicator for the top of the column. The tube and cup are enclosed in a protecting case, a small openr ing in which admits air to the reservoir. A diagram of the lower part of the barometer is shown in Fig. 24. The upper portion of the reservoir is made of glass through which the sur- face of the mercury can be seen. An ivory pointer (P) is set above the mercury in the reservoir, the tip of which is the zero 126 MECHANICS of the scale. The bottom of the reservoir (B) is made of flexi- ble material, usually buckskin, and by means of a screw (S) in the bottom of the case this can be adjusted until the surface of mercury is in contact with the ivory pointer. 0. hi Fig. 23 Fig. 24 Aneioid baiometer. Fig. 25 Procedure. Part A. Method of Reading the Barometer AND Making Corrections. Take readings upon the standard barometer in the laboratory and upon the portable Fortin barom- eter to be used in Part B. The upper part of the scale of the barometer is the only part ordinarily used, so the rest of the scale is omitted. Study the verniers carefully to determine the least count of each. Adjust the level of the mercury in the cup by means of the screw (S) until its surface is just in contact with the ivory pointer (P) in the chamber. This is best ac- complished by observing the image of the ivory point reflected from the mercury surface. When the image of the tip and the tip itself appear to just come together, the adjustment is com- plete. Move the slider which has the vernier engraved upon it down until upon looking through above the top of the mercury the light is shut off just at the uppermost point of the meniscus. Then read each scale by the aid of its vernier. Also immediately read the thermometer attached. Repeat the readings twice, making the adjustments all over again from, the beginning. The most important corrections necessary to reduce the read- PRESSURE IN FLUIDS 127 ing you have just taken to conditions agreed upon as standard, are (1) For temperature. The standard temperature agreed upon is 0° Centigrade. If the temperature is not 0° C, the reading is changed, due to the expansion of the mercury and the expan- sion of the metal tube upon which the scale is engraved. The expansion of the mercury decreases its density and hence makes the reading too large. For each degree Centigrade rise in tem- perature each cm.3 of the mercury expands 0.000182 cm.^ Therefore, if your reading is H and the temperature is t, the reading, h, at zero would be h = H/{\ + Sm\%2t) = H - .000182 t H approx. The expansion of the case and scale makes your reading too small. If the metal is brass, each cm. expands 0.000019 cm. for each degree C. rise of temperature. So the corrected reading at zero would be f/ + .000019 t H. Combining those two corrections, we have as the total temperature correc- tion (- .000182 + .000019) Ht = - .000163 H t. The correspond- ing reading with the mercury and scale at 0° C. would then be h = H (1 - .000163 t) for the metric scale if adjusted for correct readings at 0° C. It is customary to adjust the inch scales of the Weather Bu- reau barometers for correct reading at 62° F. instead of 32° F. In this case the corrected reading is given by the equation A = //[l_. 000101 (^-32) + .0000105 (f-62)] for readings in inches and temperatures t° F. (2) For Altitude. The standard reading is what it would be at sea level. For each 90 ft. rise above sea level the barometer reading is reduced approximately 0.1 inch (or 9 mm. for each 100 meters elevation). The height of the laboratory above sea level is 70 ft. (3) For Latitude. For the same air pressure the mercury in the barometer stands about 4 mm. higher at the equator than at the poles because the force of gravity is slightly less and the centrifugal force greater at the equator. The standard reading is at 45° latitude. This correction may be taken from the Smith- sonian tables found in the Reading Room. (4) For Capillarity. The amount the mercury is depressed in the tube due to capillarity depends upon the size of the tube, be- 128 MECHANICS ing less for larger tubes. The correction is obtained from tables and the diameter of the tube. (5) For Instrumental Error. If the vacuum above the mer- cury is incomplete the reading is too low. Correction for this error is made when the instrument is standardized by lowering the scale until the reading equals the height to which the mercury would rise if the vacuum were complete. The constant error due to capillarity may be included in the instrumental error. Upon the readings taken make corrections (1), (2), and (3). Omit (4). Compare the readings on the two scales of the stand- ard barometer. If they agree, assume that they are correct and find the scale error of the portable barometer. Do not move the scale, or make any attempt to readjust it. Data and Results may be tabulated as indicated in Table XXXIII. TABLE XXXIII Least Count of Vemi ers • . Date and Tempera- ture of Barometer Barometer Read ng Corrections ( + or - ) Corrected Height sj Temperature Elevation Latitude Is In. Cm. In. Cm. In. Cm. In. Cm. In. Cm. ua Procedure. Part B. Measuring Altitude by the Mer- cury Barometer and by the Aneroid Barometer. The aneroid barometer (Fig. 25) is essentially different in principle from the mercury barometer. It consists of an air tight metal chamber (A) from which most of the air has been exhausted. It has a corrugated metal cover which is flexible. As the at- mospheric pressure varies this cover is moved slightly. By a sys- tem of levers (D) and a tiny chain (E) any motion of the cover is multiplied and transmitted to a pointer (H). This pointer swings over a scale which is graduated to correspond with read- ings on the mercury barometer. The whole is enclosed in a metal case for protection. The aneroid barometer is a less accurate instrument than the PRESSURE IN FLUIDS ISQ mercury barometer but is much more convenient for carrying about. By frequent comparison with the mercury barometer it is made fairly reliable. It is used principally for measuring al- titudes. The fact that the barometer i:eading varies with altitude may be used in measuring the difference in altitude between two places. (See correction 2.) With the instructor, carry the barometers to the top of the building and there take three obser- vations on the mercury barometer and three on the aneroid. Then go down to the basement and again read the barometers. Temperatures should be recorded in each, place and suitable cor- rections made. From these observations compute the difference in level between the two stations. Tabulate data and results neatly. Remarks. The aneroid barometer is supposed automatic- ally to adjust itself to different temperatures so that no correc- tions for temperature are necessary upon these readings. The readings of the barometers should be compared with that on the recording barometer and also on the standard before they are taken to the stations. If upon returning it is found that a noticeable change has occurred in the atmospheric pressure, a suitable correction should be made. It is advisable to take nearly simtiltaneous readings on the four barometers at the beginning and end of observations for Part B. Test Questions. Part A. 1. What is the fundamental principle of the mercury barometer? Describe its construction. Explain why the upward pressure is equal to the pressure of the column of mercury. How does the barometer measure atmospheric pressure? 2. Why do the readings of the mercury barometer need to be corrected? Explain the principal corrections. 3. What is the volume and weight of the column of mercury if the area of cross-section is one sq. cm.? (Density of mercury = 13.6.) What is the pressure in dynes of the column of mercury? Of the atmosphere? 4. What change would take place in the height of the column if the diameter of the tube were doubled? Explain. ■ S. What would be the height of a water barometer ? 6. Is it necessary to keep the barometer vertical when making readings? Why? How would you read a barometer held somewhat obliquely? Part B. 7. Explain the principle of the aneroid barometer and de- scribe how it is constructed. 8. Upon what principle does the measurement of height by the baro- meter depend? Explain the method. 9. 'Explain why corrections for temperature only are necessary when the mercury barometer is used to measure heights. 130 mechanics experiment p-3 Verification of Boyle's Law for Gases Pressure and Volume at Constant Temperature Robert Boyle, an Irishman, was the first to investigate the law which governs the change in volume of a confined gas when the pressure upon it is changed. By his experiments he showed that for pressures greater than one atmosphere and at a given tem- perature the volume of a certain mass of gas is inversely propor- tional to the pressure upon it. Mariotte, a Frenchman, later showed that the law also held for pressures less than one at- mosphere. Thus, it is sometimes called Mariotte's Law. More recently it has been found that the law does not hold exactly for all gases at all temperatures or pressures. Yet for most cases at ordinary temperatures the law holds, and is very useful. If into a long U-shaped tube with the top of one branch closed, mercury is poured until it is at the same level in both branches, a certain mass of air is entrapped and it is then under a pressure of one atmosphere. Then if more mercury is poured into the open branch until the level in it is at the same height as the barometer, the air in the closed tube will be under a pressure of two atmospheres. The volume of the air will be found to be Y2 its original volume. Also, if the pressure were tripled, the volume would be % as great, and so on. Stated algebraically, if under a pressure P a given mass of gas has a volume V, and under a pressure P', it has a volume V, we have V : V'-.-.P' : P or VP=V'P'=V"P"'= =K (a constant) The law can easily be verified by enclosing a certain mass of gas in a receptacle, the volume of which can readily be meas- ured, and subjecting it to various known pressures. The prod- ucts of each volume and its corresponding pressure (i.e., PV) should within experimental error, be the same. A convenient form of apparatus for this purpose consits of two glass tubes about one cm. cross section and 30 cm. long, mounted side by side upon clamps which slide along vertical rods. The tubes are joined at the bottom by a rubber hose. PRESSURE IN FLUIDS 131 Between the tubes is a centimeter scale upon which slides a vernier attached to a mirror long enough so that it extends back of, each tube. Across the mirror is ruled an index line. When a reading is to be made the mirror is set so that the top of the meniscus in the tube, its image in the mirror, and the index line appear to be in the same plane. The scale and vernier are then read without parallax. One of the tubes is open at the top ; the other is closed by a stop cock and is graduated to read in cu. cm. The relative levels of the two columns of mercury may thus be changed and differences of level read off from the scale. Procedure. Have the instructor see that the apparatus is properly filled with mercury. " Trap " some air in the graduated tube by closing the cock and then adjust the two columns so that the mercury in the open tube is about 20 cm. below that in the closed tube. Read the position of the top of each column of mercury on the graduated bar by the aid of the sliding mirror. Lower the open arm without changing the position Oi the slider. This will lower the mercury in the closed tube out of the way so that the image of the scale on the tube can be seen in the mir- ror. Thus by the aid of the mirror and scratch read from the scale the volume of air in the closed tube. Now raise the open tube until the mercury column is about 10 cm. below that in the closed tube, and as before read the po- sition of the top of each column of mercury and the correspond- ing volume of air. In this way, obtain readings by 10 cm. in- tervals until a difference of level of 75 or 80 cm. is obtained. Read the barometer before and after the experiment. Take the mean as the true barometer reading. Repeat the readings, using a different mass of gas. Tabulate as follows: TABLE XXXIV Room Temperature Barometer before . . . .; after ; mean {£) Vol. of Air Readings of Meniscus in Diff. in Level Ho- Ho Total Pres. P=(Ho-Hc) + B PX V = K HP Enclosed (V) Closed Tube He Open Tube Ho 132 MECHANICS \/P should be expressed as a decimal. If no gas has es- caped during the experiment, and care has been taken to see that the temperature has not changed, the product PxV should be constant (nearly). Now since from your experiment PxV^K where K is a constant, it follows that V = K{\/P) That is, the volume varies directly as the reciprocal of the pres- sure, or Foe \/P Plot a curve using values of l/P as abscissas and corre- sponding values of V as ordinates. Remarks. Since the temperature of a given mass of gas is increased when the gas is compressed, it is necessary to allow the gas to stand several minutes after each compression so that its temperature may come back to room temperature before the readings are made. The same precaution is necessary when the gas is expanded. Care must be taken all through the experiment to avoid changes of temperature. Never open the cock at the top of the closed tube unless the two columns of mercury are first brought to the same level. Otherwise the sudden change of level when the cock is opened may cause some of the mercury to be thrown out of the tubes. Test Questions. 1. State Boyle's Law and describe your method of testing it. Explain what is meant by an inverse proportionality. 2. How does your data and the curve verify Boyle's Law? Should your curve pass through the origin? Why? (What is the value of the constant K? 3. Express in words the pressure upon the enclosed air when the level of mercury in the open tube is above that in the closed tube; when it is below that in the closed tube. 4. What is the relation between the density of the air (D) and the pressure upon it? 5. Explain the use of the mirror attached to the slider. DENSITY 133 GROUP IX — D. DENSITY OF LIQUIDS AND SOLIDS experiment d-1 Determination of the Density of Solids Method by Weighing in Water Relative density or specific gravity of a substance is the ratio ^ of the weight of a certain volume of the substance to the weight of an equal volume of another substance taken as the standard. The standard usually taken for liquids and solids is water at 4° C, which then has unit specific gravity. „ .r r^ Density of Substance _ D Specihc Gravity = :Fr : 7-^; ; — r ; 01 S = — Density of Standard a Absolute density is the number of units of mass in unit volume. A cubic cm. of pure water at 4° C. weighs one gram. Hence its density is one gram per cubic cm. Its specific gravity is also one. So in the C. G. S. system of units specific gravity and density of a substance is expressed by the same number. In the English or F. P. S. system, however, this is not the case. A cubic ft. of water weighs 62.397 lbs. Its specific gravity in any system of units is one. Hence a substance which has a specific gravity, 2, will have a density of 124.79 Ibs./cu. ft. A very general method of determining specific gravity is based upon the principle of Archimedes, that a body immersed in a liquid is buoyed up by a force equal to the weight of the liquid displaced. In other words, the weight of a body when im- mersed in a liquid is apparently decreased by an amount equal to the weight of a volume of the liquid which is the same as the volume of the body. Thus the specific gravity of a body may easily be obtained by weighing the body in air and again sus- pended in water at a known temperature, the buoyant effort be- ing the difiference between these two weights. The volume {V) of the liquid displaced, and hence the volume of the body, is d in which (w) is the mass of the liquid displaced and (d) its density. The density of the body (Z?) is M M . Md V mid m 134 MECHANICS (M being the mass of the body). Therefore, by weighing the body in air and again in a liquid of known density, we will have the data necessary to determine the value of D. Procedure. By this method determine the density of the brass plate used in Experiment W — 1. For the weight in air the weight found in Experiment W — 1 may be used. For con- venience in weighing the plate in water a small stand is placed over the balance pan in such a way that it does not touch the pan at any point. Upon this stand is set the vessel containing dis- tilled water, the whole being entirely free from the moving parts of the balance. The brass plate is to be suspended in the water by a fine copper wire attached to the hook above the balance pan. Adjust the length of the wire so that as little as possible of it will be under water, and only one strand of it will pass out through the surface. This will make the error due to capillary action on the wire as small as possible. The plate is to be weighed sus- pended by the wire so that it is completely immersed in the water. The weight of the plate in water should be taken as carefully as possible, but on account of the capillary action on the wire the sensitiveness of the balance is greatly decreased and it is imprac- ticable to weigh by the method of vibrations. Therefore pro- ceed as follows: Find the zero point of the balance with the wire dipping in the water and sufficient mass in the opposite pan to bring the balance in equilibrium. With the plate suspended in Ihe water, add weights to the opposite pan until the pointer is ibrought as near the zero point as possible. If it cannot be brought just to the zero point, the weight necessary to carry it to the zero point should be estimated. In order to find the density of the water, its temperature must be taken. Make two readings of the temperature, one just be- fore and one just after the plate is weighed. Take the mean as the temperature of the water, and from the tables find the density of water at this temperature. Record your data as follows: Plate No Zero Point (With wire counterbalanced) Weight of Plate in Water grams. Weight of Plate in Air " DENSITY 135 Temperature of Water at starting ° ; at end ° mean Density of water at ° C Weight of water displaced Volume of the Plate Density of Brass Plate in Metric Units ; in English Units Remarks. Water wets the wire which supports the plate and hence adheres to it. So when the wire moves upward the water at the surface tends to rise with it and so acts as a retard- ing force. Likewise when the wire moves downward its motion is retarded by the adhesion of the water to it. Hence the vibrations of the balance very quickly die out. So instead of using the method of vibrations the small weight required to just bring the pointer to the zero point could be estimated by allow- ing the pointer to come to rest first to the left then to the right, and finding the amount the pointer is deflected by adding a small known weight, as in Exp. W — 1. Test Questions. 1. State Archimedes's Principle.; Explain clearly in your own words how you determined density by the use of this principle. 2. What is the difference between density and specific gravity? 3. Estimate in per cent, about how accurate your results are. How many significant figures in your value of the density? Explain whether your deterxnination of the weight of the plate in air or in water is more accurate. 4. What is the value of the buoyant force upon the plate. 5. What is the total weight (expressed in words) on the stand sup- porting the beaker when the plate is suspended in the water? 6. How would you find by this method the density of a body which floats on water? 7. What would be the error in your result in per cent, if the density of water is taken as unity? Is this error smaller or larger than the other errors entering the experiment? experiment d-2 Determination of Density of Liquids and Solids Use of the Specific Gravity Bottle The specific gravity bottle may be used to determine the density of either solids or liquids. When used to determine the density of solids, it is called a pyknometer. The specific gravity bottle is made usually to hold a certain integral volume (as 25 cc, 50 cc, 100 cc, etc.) of a liquid at a given temperature. It 136 MECHANICS Fig. 26 has a ground glass stopper accurately fitted so that it may be pressed in to the same depth each time. Through the center of the stopper is a fine capillary hole through which the excess liquid is squirted when J\\ the stopper is set in place. These provis- J\^ ions enable one to fill the bottle completely with a liquid and hence at a given temper- ature it will contain very accurately the same volume of any liquid. Two forms of the specific gravity bottle are shown in Fig. 26. Frequently bottles are marked 25 grams, 50 grams, etc., to designate the weight of pure water which the bottle will contain at the given temperature. Procedure. Part (A) Liquids. The bottle is first weighed, perfectly dry, then filled with distilled water and weighed, and finally weighed filled with the liquid whose density is to be de- termined. The weights of equal volumes of water and of the liquid are thus determined, and hence the density of the latter can be computed. The temperature of the water must be taken carefully in order to determine its density. Using this method, find the density of alcohol and the density of a salt solution made by dissolving 15 g. of salt in 85 cc. of distilled water. Be careful in inserting the stopper not to force it in too hard and see that there are no air bubbles left in the bottle. Also be sure that there is no liquid on the outside of the bottle when weighing. Handle the bottle as little as possible to avoid heating the water. Carefully dry the bottle each time before filling it with the liquid. Record the data as follows, making a separate table for each liquid: Weight of the bottle filled with water Weight of bottle alone Weight of water alone Temperature. . . . Weight of bottle filled with liquid Weight of bottle alone Weight of liquid alone Specific Gravity = DENSITY 137 Density in metric units ; in English units Tabular density Per cent, difference Procedure. Part (B) S0LID9. The solid is first weighed in air. It is then placed in the specfic gravity bottle, the bottle is filled with water and the whole is weighed. The weight of the bottle filled with water without the solid inside is obtained. The method is useful when the solid can be obtained only in small particles. When the solid is placed in the bottle filled with water, it pushes out, as it were, an amount of water exactly equal to its own volume. The weight of this water displaced is the differ- ence between the weight of the bottle filled with water plus the weight of the solid, and the weight of the bottle and water with the solid inside, divided by the density of the water (d). Hence dividing the mass of the solid by this volume the density re- quired is obtained. Find, by the pyknometer method, the specific gravity of some brass chips. As usual take the temperature of the water and find its true density. Record your data as indicated below. Weight of chips in air Weight of bottle and water with solid inside Temp Weight of bottle filled with water (from part A) Weight of water displaced Density of brass in metric units ; in English units Tabular density Per cent, difference Remarks. The tabular density of alcohol, of salt solutions of different strengths, and of the cast brass chips may be found in the Smithsonian Physical Tables. Test Questions. 1. Describe the specific gravity bottle. Explain the purpose of the capillary hole in the stopper. 2. Explain clearly the principle which you used to find the density of a liquid. Could some liquid of known density be used in place of water? 3. What would be the effect on your results if the temperature of the liquid increased during the experiment? 4. Explain how you determined the density of a solid by the use of the specific gravity bottle. What is the volume of the solid? Why is it not necessary to weigh the bottle empty? 5. What error in per cent, would be introduced by assuming the dens- 138 MECHANICS ity of water to be unity? Should this assumption be made in this experi- ment? 6. Mention some of the precautions required in this experiment. 7. How many significant figures in your result for the density of the liquids and of the solids? Does this entirely account for the difference between the tabular densities and your values? experiment d-3 Density of Liquids Hare's Method by Balanced Columns Hare's method for comparison of densities depends upon the principle that when two liquids rise in vertical tubes by the same difference in pressure their heights are inversely proportional to their densities. Thus the method is used to obtain the relative densities of two liquids. If one of the liquids is pure water at standard temperature the specific gravity of the other liquid will be obtained directly. The ordinary simple form of apparatus consists of two glass tubes mounted vertically side by side, joined at the top to a single tube. At the bottom the two tubes dip into the liquids to be com- pared. A scale or other means is provided for measuring the heights of the liquid columns. When the air is partially exhausted from the tubes the liquids rise in them, but not to the same height, although the air pressure on the top of each column of liquid is the same since the tubes are connected at the top. In this case the total pressure acting downward at the bottom of each column of liquid is equal to the sum of the air pressure on the top of the column and the pressure of the liquid. The pressure of the liquid (in grams per cm.2) is equal to the prod- uct of the area of cross section (A) in cm.^, the density of the liquid (D) in g/cc. and the height of the column (h) in cm.; or ADh. Call the pressure of the column in the other tube A'D'h', and the pressure of the air above the liquids p. Then the total downward pressure at the bottom of one column is Ap+ADh and of the other A'p + A'D'h' Each of these pressures downward is balanced by an upward DENSITY 139 pressure at the same point due to the pressure of the atmosphere upon the surface of each liquid in the vessels. Call the pressure of the atmosphere (per unit area) P Then the upward force upon one column will be AP and that upon the other, A'P. Since each of these is equal to the downward force we have for one column, AP = Ap + ADh (1) and for the other A'P = A'p + A'D'h' (2) Combining equations (1) and (2), ^, = ^' (3) That is, the densities are inversely proportional to the heights of the two columns. If the air is still further exhausted from the tubes so that its pressure is reduced to p', the liquids will rise to a greater height, say to H and H' respectively. The total pressure at the bottom of each which is still equal to the atmospheric pressure becomes Ap' + ADH and A'p' + A'D'H'. Combining these expressions with equations (1) aiid (2) we have P = p + Dh = p + D'h' = p' + DH = p' + D'H' (4) Subtracting alternate members of this equation we obtain D {H-h)=D' (H'-h') and for the ratio of the densities of the two liquids, ^ = ^^^^' (5) D' H-h ^^> Equation (5) is the more convenient form to use in this experi- ment. Also, used in this form, the error due to the difference in capillary action of the two liquids is eliminated. If scale readings are denoted by R, R' and r, r', and the fixed distances from the zero of the scale to the levels of the two liquid surfaces in the vessels are expressed by a and h respectively then H'-h' ^ (R' + a)- (r + a) ^ R' - r . ,. H-h (R + b) - (r + l>) R- r ^ ' This expresses the fact that scale readings may be used in- 14° MECHANICS stead of actual heights even when the surfaces in the vessels are at different levels, provided the levels do not change, and D ^ K^lA (7) D' R-r ^ ' Procedure. Two glass tubes about a meter long and 5 mm. in diameter are connected at the top by two branches of a three-way connector attached by pieces of rubber tubing. To the third branch of the connector is joined a piece of rubber tubing provided with a pinchcock. A meter scale is placed between the two tubes. The whole, bound together, is mounted vertically so that the lower ends of the tubes dip into separate beakers contain- ing the liquids whose densities are to be-compared. Fill one of the beakers with pure water and the other with the liquid whose density is to be determined. With a thermometer take the temperature of each liquid. Now exhaust sufficient air to cause the liquids to rise a short distance in the tubes and close the pinchcock. Observe whether the liquids maintain their level in the tubes. If the columns slowly fall search for leaks in your apparatus. Unless the connections are air tight the experiment will be unsuccessful. Read carefully on the scale the position of the liquid surface in each tube, always reading at the lowest part of the meniscus. Raise the liquids farther up the tubes and again take the readings. Take the observations thus when the columns are at five or more different heights in the tubes. In the last set have the higher column near the top of the tube. Consider the first reading taken as the zero reading and use it as the r and / in equation (7). The density of water at the temperature found may be obtained from tables (see Smithsonian Tables). Then from the known density of the water and the relative heights of the two columns in the tubes compute the density of the other liquid. Make two sets of readings, comparing first water and alcohol and then water and a solution of salt (15 g. of salt in 85 g. of water). All through the experiment care must be taken not to mix the liquids in any way. The tube must be carefully washed upon changing from one liquid to another. After rinsing with water it is a good plan to rinse the tube with a little of the liquid that is about to be used in it. DENSITY 141 TABLE XXXV Temperature of liquids Density of water at this temperature . . . Liquids compared Top of Column Increase in Heights Ratio of Increases H' -h' H-h Density of Liquid Water Liquid Water Liquid Metric English Plot the scale readings in the first two columns of Table XXXV, using the first column as ordinates and the second as abscissas. Show that the density of the liquid (£>) equals the density of the water (£>') multiplied by the slope of the curve. Check the values of density found from the curve and from the table. Remarks. The height of column {h) is the distance from the level of the liquid in the vessel to the top of the liquid in the tube. We have assumed that the surfaces of the liquids in the vessels remain at the same level as the liquids are drawn up the tubes, which of course is not strictly true. However, if the diameters of the vessels are large compared with that of the tubes the error introduced by neglecting the fall of liquids in the ves- sels is small. More reliable results may be obtained if the ves- sels are placed on platforms adjustable vertically so that the liquid surfaces in the vessels may be brought, before each reading, to their first levels by use of " hook gauges " attached to the tubes or scale. No error is introduced if the two liquids have different first levels. Why? The fact that the expressions for the areas of cross section of the tubes cancel out of each equation shows that the results would not be affected by using tubes of varying bore, or even tubes of different sizes. However capillary action would vary in this case. Test Questions. 1. Explain Hare's method and how it is used. Develop the formulae. What advantage is there in using equation (S) rather than equation (3) ? 2. Is it necessary to know how far the liquid surfaces are below (or 142 MECHANICS above) the zero of the scale, (a) if equation (3) is used, (b) if equa- tion (5) is used, (c) if equation (7) is used? 3. Estimate the accuracy of your results. In other words, how many sigtiificant figures in your values of the density? 4. What precautions are necessary in performing this experiment? 5. Which column measured from the level of the liquid in the beaker has the greater weight? Explain. 6. Express in words the pressure causing each liquid to rise in the tube. 7. Is it necessary to keep the tubes vertical? Why? 8. How will the results be affected if the tubes are not of the same diameter? Must the tubes be of uniform bore? Does capillary action affect the results? Why? 9. When the liquids have risen in the tubes what is included in the total weight supported by the clamps? experiment d-4 Density of Solids and Liquids Density by the Jolly Balance For the theory of this experiment refer to the introduction of Experiment D — 1. The Jolly balance used in Experiment E — 2 is very convenient and fairly accurate for the determination of specific gravity by the " loss of weight " method. Usually an extra glass pan is sus- pended from a hook on the under side of the regular pan of the balance. The water is contained in a beaker placed upon an ad- justable shelf at such a height that the lower pan is completely submerged in the water. This pan is kept immersed to the same depth throughout the experiment. Procedure. Part A. Density of a Solid. Place a beaker of distilled water upon the shelf, adjusting it so that the lower pan is immersed to some definite mark upon the suspending wire. Calibrate the balance by finding the elongation produced by a 10-gram weight and drawing a calibration curve. (See Experi- ment E — 1 ) . Take the temperature of the water. Place a small glass stopper in the upper pan, determine the elongation and from the calibration curve find the weight. Transfer the stopper to the lower pan and again determine the weight, the pan being submerged to the same depth as in the other readings. Make at lea^t four determinations and take the mean result. From this data determine 1. the volume of the body^ DENSITY 143 2. its specific gravity, 3. its density in metric and English units, using the correct density of water at the given temperature found in tables. Procedure. Part B. Density of a Liquid. The method of Part A may be used to determine the density of a liquid. A glass sinker is first weighed in water, then in the liquid the density of which is to be determined. The ratio between the weights of equal volumes of the liquid and of water will be the ratio be- tween the specific gravities of the two liquids. Since the specific gravity of the water is known that of the other liquid may be computed. By this method determine the density of alcohol and the salt solution used in Experiment D — 2, making as before at least four readings. Use as a sinker the glass stopper used in Part A. Express the density in both the metric and the English sys- tems. Be careful not to mix the liquids in any way. Arrange data and results in a neatly tabulated form. Remarks. If there are not two pans for the balance the glass stopper may be suspended by a fine wire from the lower side of the regular pan and in this position weighed both in air and in the water. If necessary for accurate work, the balance may be calibrated with the wire dipping into the water to a cer- tain depth and proceeding as if an extra pan were being used. If the wire is very fine, however, the error due to its immersion may be neglected. Test Questions. Part A. 1. Explain the principle used to determine the density of a solid in this experiment. 2. Explain fully why the lower pan must be kept at the same depth in the water during the experiment. Part B. 3. Explain how the density of a liquid is determined by the Jolly balance. Upon what principle does the method depend? 4. What is the buoyant force of the salt solution upon the solid? 5. What is the weight of water displaced? Of alcohol? Of the salt solution? What volume of liquid is displaced in each case? 6. Mention the principal sources of error in performnig this experiment. 144 HEAT GROUP X — H. HEAT experiment h-1 Testing the Mercury Thermometer Determination of Errors at the Fixed Points The ordinary mercury thermometer consists of a thick walled capillary tube closed at both ends, a bulb being blown at one end. The bulb and tube are completely filled with mercury at a tem- perature higher than that for which the thermometer is designed, and the end sealed off. As the thermometer cools the mercury contracts much more than the glass containing it and hence moves down the tube ; also when the thermometer is heated the mercury expands more and moves up the tube. This apparent contraction and expansion of the mercury is used as a measure of the temper- ature. The wall of the bulb is of comparatively thin glass so that heat will readily be conducted to the mercury. The usual method of graduating a thermometer is to select and mark two fixed points of known temperature and divide the length between these points into an arbitrary number of equal spaces or " degrees." The fixed points agreed upon are the temperatures of melting ice and of steam under a pressure of one atmosphere (760 mm. of mercury). These temperatures are fairly constant and easily obtainable. The space on a thermometer between the two fixed points has been called the fundamental interval. On the Centigrade thermometer the melting point of ice is called 0°, and the space between this and the temperature of steam is divided into 100 equal parts, called degrees. On the Fahren^ heit thermometer the lower point is marked 32° and the upper 212°. In using the thermometer the following precautions should be observed : 1. Since the bulb is made of very thin glass great care must be taken not to strike it against any hard object. The thermometer should not be used as a stirring rod unless great care is taken that the bulb does not strike the vessel. 2. The thermometer should never be subjected to sudden changes of temperature. Do not take it from a hot liquid and plunge it immediately into cold water, or vice versa. THERMOMETRY I45 3. In reading the thermometer care must be taken to place the eye upon a line perpendicular to the stem at the top of the mer- cury column to avoid the error of parallax. 4. When the temperature of a substance is being taken suf- ficient time must be allowed for the whole of the mercury in the bulb to come to the temperature of the substance. Procedure. Part (A) To Determine the Error of the Freezing Point. Fill a small test tube nearly full of distilled water. Place in it a small stirrer and the thermometer to be tested, passing both through a cork fitting the top of the tube. Adjust the thermometer so that the zero mark can be plainly seen. Make a freezing mixture of ice and salt water in a calorimr eter. Into this place the test tube above. Stir the water con- stantly until it is frozen; when the thermometer reads about 1° begin to take readings about every half minute until it has re- mained stationary for several readings, estimating carefully to tenths of degrees. Take the stationary readings as the reading for the freezing point. Several trials should be made and the results averaged. The readings on the thermometer are marked + when above 0° and - when below 0°. The error is marked + when the reading of the thermometer is above the true temperature and - when it is below. Procedure. Part (B) To Determine the Error of tjie Boiling Point. The reading of the boiling point is determined by placing the thermometer in steam under a known pressure. For this purpose a copper boiler upon a stand is used. Screwed to the top of the boiler, and closing it, is a long tubular piece with a cork fitted to its upper end. Just below the top a small opening allows the steam to escape. On one side of the boiler a glass side tube is connected to serve as a water gauge and on the op- posite side is a water manometer. The apparatus is called an hypsometer. The boiler is about two-thirds filled with water and set over a Bunsen flame. The thermometer to be tested (two thermometers may be tested at the same time) is inserted through a hole in the cork so that the boiling point is just visible. Allow plenty of 146 HEAT time for the thermometer to assume the temperature of the steam. While waiting, read the barometer and measure the difference in level of the two columns in the manometer tube. Read the ther- mometer, estimating to tenths of degrees. The temperature at which water boils depends upon the pres- sure upon it, in general, the greater the pressure, the higher is the boiling point. For temperatures near 100" C. water boils at 0°.037C. higher for each mm. of mercury increase in pressure. At 760 mm. pressure it boils at 100° C. When the water is boiling in the hypsometer the pressure upon it is atmospheric pressure indicated by the barometer combined with the pressure of the steam indicated by the manometer. Com- pute the true temperature of the steam in the boiler by making the correction for pressure. Allow the thermometer to cool to the temperature of the room, then again test its freezing point as before, to observe whether it has changed by heating. By the above method test two thermometers, one with a cen- tigrade scale upon it and one with a Fahrenheit scale. Obtain the data and results indicated by the following table. TABLE XXXVI Thermometer No. Readings at Freezing Pt. (Obs.) Boiling Pt. (Obs.) Divisions F. P. to B.P. ... Freezing Point Steam Pressure (Manom.). .mm. 1st Trial 2nd Trial True Boilmg Point » Degrees per Division . . Fundamental Interval of Error at Freezing Point Error at Standard Boilin Scale + or-) S Point ( + or - ) In the above table by " divisions from freezing to boiling point " is meant the number of divisions or spaces representing whole degrees from the freezing point to the boiling point as observed on your thermometer. The " fundamental interval of the scale " is the number of divisions, on the thermometer tested, between the reading in ice and the reading in steam at standard temperature (100° C.) It may be computed by using the num- THERMOMETRY I47 ber of " degrees per division." The " error at the standard boil- ing point " must include the correction at the freezing point in connection with the degrees per division. Procedure. Part (C) To Apply the Corrections for Er- rors AT the Fixed Points. Take the room temperature with the thermometers you have calibrated, apply the corrections found, and determine the true temperature. Compare your re- sult with the room temperature taken at the same point with an accurate thermometer. To correct any reading on a thermometer for the error due to the errors in the fixed points, the following rule may be used : (1) Find the the value of one division in degrees; (2) Find the number of divisions between the reading in ice and the observed reading at the temperature; (3) multiply (2) by (1) and add to the temperature of melting ice. Remarks. If the apparatus described in Part A is not available the following method for determining the freezing point may be used. It is convenient and will yield fairly reliable results. With a twist drill bore a hole into a block of ice large enough to admit the bulb and stem up to the freezing point of the ther- mometer to be tested. Set the thermometer in this hole and fill the remaining space with distilled water. Read the ther- mometer when the mercury has become stationary. In this case the lowest point reached by the mercury indicates the freezing point. In applying the corrections in Part C it is assumed that the fundamental interval is divided into equal divisions; and that the bore of the thermometer does not vary in size. In general the bore of a thermometer is not uniform and for accurate tempera- tures corrections must be made for this variation. A common method of determining these corrections is to detach a short thread from the main mercury column and observe its variation in length as it is moved along the stem. (See also Part IV, Exp. T — 41. The errors thus obtained must be combined with the errors found at the fixed points. Test Questions. 1. How is a mercury thermometer constructed? Upon 148 HEAT what principle does the thermometer measure temperature? What is meant by the freezing point? Boiling point? Fundamental interval? 2. How do you determine when the temperature of the thermometer is at the freezing point? 3. Explain how you determined the temperature of the steam into which the thermometer was placed. 4. Explain how the errors in degrees at the freezing and the standard boiling points are determined. Explain in detail how to find the true temperature from a reading on a thermometer, the erors at the freezing and boiling points of which are known. 5. Explain the distinction between a Centigrade degree and a scale division on a Centigrade thermometer. 6. How would you proceed to determine whether or not the bore of the thermometer is uniform and make suitable corrections? experiment h-2 Linear Expansion for Solids Determination of the Coefficient When a body is heated it expands in all directions. In the case of solids however, it is usually necessary to consider the ex- pansion in one direction only, that is, the linear expansion. The amount that unit length of a substance increases when its tem- perature is raised 1° C. is called the coefficient of linear expan- sion of the substance. Strictly, the increase should be measured when the temperature changes from 0° to 1° C. But between 0° and 100° the increase for each degree is very nearly constant, and for most purposes may be so considered. Hence, if at a tem- perature t the length of the body is /, and at a higher temperature t' the length is /', then for a change of temperature f -t there has been an increase of length /' - /. Therefore the increase for each degree is and the increase in each unit of length for each degree is J ( • _ \ = a = the coefficient of expansion E . This equation is frequently written a = — if E is the elongation for a change of temperature T. The coefficient of cubical expansion may be taken as three times the coefficient of linear expansion. The general method used to determine the coefficient of linear EXPANSION 149 expansion is to change the temperature of a rod of the substance from one known value to another known value and measure the small increase of length. There are several devices for measur- ing the very small change in length. The one given below is quite convenient and fairly reliable. + — c:)0666(:)6 Lamps To LIGHTING Circuit 0000000 ^ h Itiv.Tj^n., ■T-r^^.^r.r....o^..^a^^ L ,&„ sil zL> '^ ^^|-^AAAAAAAAAA/* ixi Fig. 27 A diagram of the apparatus with a horizontal section of the upper part is shown in Fig. 27. The metal to be tested is in the form of a tube (T) about 75 cm. long and 5 mm. in diameter. Wound in a close spiral about the tube and electrically insulated from it is an iron wire (c) through which an electric current may be passed in series with a bank of lamps used to vary the cur- rent. When a current is passed through this coil of wire, the wire is heated and in turn heats the tube. Outside the coil of wire are wrapped several thicknesses of asbestos (A) to re- duce the loss of heat to the air. A thermometer (t) is inserted in one end of the tube. The tips of the screws (S,S) enter the metal of the tube and serve as a rigid stop at that end. Near the other end a flat steel spring (I) about 2 cm., long is inserted at right angles to the axis of the tube. This serves as an in- dicator, a micrometer screw (M) being mounted so that its tip may be brought into contact with the spring, thus completing an electric circuit, and causing the needle of the galvanoscope (G)- to be deflected. Procedure. It will first be necessary to obtain the as- sistance of the instructor in charge, who will see that the electric connections are properly made and the resistances properly ad- ISO HEAT justed and also explain how to vary the current through the heating coil (C) and how to manipulate the micrometer screw. Before the electric current is turned on the micrometer screw is turned until it just touches the indicator (I). The instant it comes into contact the needle of the galvanoscope will be de- flected. At the same time the thermometer (t), which indicates the temperature of the tube, is read then the reading of the micrometer is taken. A small current is now sent through the coil (c) and allowed to run until the thermometer has remained practically stationary for two or three minutes. The micrometer is again adjusted and the readings taken. Evidently the difference between the two readings of the micrometer will be the elongation for the cor- responding change in temperature indicated by the thermometer. Increase the current by about three more steps until a tempera- ture of about 150° C. is reached. The adjustment of the mi- crometer and the reading of the thermometer should be nearly simultaneous. The total length of the tube expanded may be measured with a meter stick. Note. When the readings are completed, back off the micrometer screw so the spring will not be bent when the tube contracts. Data should be tabulated as shown in Table XXXVII. TABLE XXXVII Least Count of Micrometer Screw = Cm. Length of Tube = Cm. Temperature of Metal Micrometer Readings Total Elongation Total Change in Temperature Values of a Plot a curve using total change of temperatures and values of the elongation as coordinates. Find from the curve the total elongation for 100° C, and find from this the value of "a!' Compare with the average of the values in the table. In all your data be careful to retain the proper number of significant figures. Test Questions. 1. Explain what is meant by the coefficient of linear expansion and show how the algebraic expression for it is derived. Of what metal did you determnie the coefficient? 2. Explain the method you used to measure the very small increase in the length of the metal. 3. Do you consider that the whole length of the rod is at the same temperature as the thermometer when you read the micrometer screw? Explain what reason you have for so considering it. SPECIFIC HEAT 151 4. In the formula a = E/IT, I is a variable as found from your experi- ment. Why is not this variation taken into account in computing "a"? With about what precision are you able to read the elongation of the rod? What percentage error, then, is allowable in measuring the total length of the rod? In reading the temperature? How accurate then is it necessary to read these quantities? 5. A surveyor's steel tape is correct at 15° C. If one measures the area of a rectangular field IS m. xlOni. when the temperature is 35° C. and without correcting his tape calls it 150 m.^, compute the actual area of the field and also his percentage error. What percentage error does he make on one side? 6. What do you learn from your curve about the expansion of metals? experiment h-3 Determination of Specific Heat Method of Mixtures for Solids and Liquids When two substances at different temperatures are placed in contact, heat is transferred from the warmer body to the colder one until both are at the same temp^erature. The amount of heat given up by the warm body equals the amount absorbed by the cool body, excepting that some heat is lost by radiation. The temperature resulting after the transfer is completed depends upon the masses of the bodies and a property of the material known as specific heat. Specific heat of a substance may be de- fined as the number of calories required to heat one gram of the substance through \° C. Since one calorie is the amount of heat necessary to raise the temperature of one gram of water 1° C, specific heat may also be defined as the ratio of the amount of heat necessary to raise the temperature of a certain mass of the substance through a certain temperature to the amount required to raise the temperature of a/ii equal mass of water through the same temperature. That is, water is taken as the standard and its specific heat is unity. Procedure. Part (A) Solids. Suspend in boiling water in a calorimeter a small block of the metal whose specific heat is required. See that the water is boiling briskly and allow the metal to remain in it until you are sure that it has attained the temperature of the water, which is found by a thermometer. Weigh another calorimeter with its stirrer, carefully. Fill it about two-thirds with water and weigh again. When all is in readiness, stir the water thoroughly and take its 152 HEAT temperature, estimating tenths of degrees; also take the tem- perature of the boihng water. Immediately transfer as quickly as possible the metal from the boiling water to the cool water. Keep the water well stirred and watch the temperature closely. When it has reached its highest point, read the thermometer again estimating tenths of degrees. Arrange your data in neat tabular form. In this case the heat is supplied by the metal and the amount of heat which the metal gives up equals the amount absorbed by the calorimeter and water. Suppose that M grams of the metal of specific heat S, at T° C. was plunged into m grams of water at t° C, contained in a calorimeter weighing m' grams of specific heat S', and that the resulting temperature was fj" C. Answer the following questions using these values; use also the first set of data obtained in your experiment. You will thus have two answers for each question. In each case either state the question or so word your answer as to indicate what the question is. 1. What was the rise in temperature of the water? 2. How much heat was absorbed by the water? 3. What was the rise in temperature of the calorimeter and stirrer? 4. How much heat was gained by the calorimeter and stirrer? 5. What was the total amount of heat absorbed by the water, calorimeter, and stirrer? 6. Therefore what was the total amount of heat given up by the metal? 7. What was the decrease in temperature of the metal? 8. How much heat was given up by the metal in cooling 1°C.? 9. How much heat was given up by one gram of the metal in cooling 1° C? 10. Hence what is the specific heat of the metal? Make an equation using the literal terms above. On one side express the heat given up; on the other side express the heat ab- sorbed. Substitute in this equation and compute the specific heat of the metal from your data. A quantity called the " water equivalent " is often used in the above computations. It is the number of calories required to raise the temperature of the calorimeter and stirrer 1 ° C. It may SPECIFIC HEAT 153 be found experimentally or may be computed from the specific heat and mass of the calorimeter. When used it is added to the mass of the water. One of the greatest sources of error in this experiment is radiation to or from the calorimeter when its temperature is lower or higher than that of the surrounding air. This error may be almost wholly avoided by adjusting the temperature of the water in the calorimeter before the metal is introduced, just as far below room temperature as the final temperature will be above. A preliminary trial will be necessary to ascertain approximately how much the temperature will change. The initial temperature is then adjusted accordingly. Make two complete determinations of the specific heat of alu- minum and compare your value with that found in tables. Procedure. Part (B) Liquids. Working the above process backward, from the now known si)ecific heat of aluminum, de- termine the specific heat of a salt solution (strength 15 parts salt to 85 parts water). Make use of an heat equation similar to the one used in part (A). Remarks. The specific heat of a substance varies some- what as the temperature changes. For wide ranges of tempera- ture this change is considerable in most cases. But between 0° and 100° the change is so slight that in this experiment the specific heat may be considered constant. Strictly, specific heat is defined as the number of calories re- quired to raise the temperature of a gram of the substance from 15° to 16° C. Test Questions. 1. What is a calorie of heat? What is specific heat? Derive an algebraic equation for specific heat from the " heat equation " used to compute its value. When a certain amount of heat is applied to a body, upon what does its final temperature depend? 2. What is the basis for the heat equation used to determine the specific heat? Explain each term of the heat equation. 3. What is meant by the water equivalent of a calorimeter? What is the algebraic expression for it? 4. Explain how during the experiment some heat is lost of which you have taken no account. How did you lessen the errors due to these losses? Explain fully. 5. If one gram of water and one gram of mercury, each at C, are heated in turn over the same burner, find by computation which will boil first? (Specific heat of mercury = 0.033 ; boiling point, 350° C.) 6 Compute the error in per cent, which would be introduced «ito your 154 HEAT result if you made an error of .1°C. in reading the final temperature of the water. How close then should the temperature be read? 7. A British Thermal Unit (B. T. U.) is the amount of heat necessary to raise the temperature of one lb. of water 1° F. Express the specific heat of aluminum in B. T. U. (Refer to the definition of specific heat given above.) H'ow many B. T. U.'s would be required to change the temperature of 1 g. of aluminum' 1° C. ? experiment h-4 Heat of Fusion Determination for Ice When a solid reaches the temperature of its melting point, even though heat is still continually applied, there is no further in- crease in temperature until the whole solid is melted. The kine- tic energy applied as heat during the stage when there is no change in temperature is used to accomplish the change of state from a solid into a liquid, and hence is converted into the potential form; it does the work of giving to the molecules a greater range of motion. The heat energy absorbed in the proc- ess of melting has been called " latent heat," but it is now called the heat of fusion. The heat of fusion is the number of calories of heat required to change one gram of a substance from the solid to the liquid state at the same temperature. Procedure. Weigh a calorimeter and stirrer dry, then fill with water to within two or three cm. of the top. Heat the water to about 40° C. Weigh the calorimeter and its contents to determine the mass of water taken. Break up a quantity of ice into fairly small pieces and lay them upon a towel so that the water resulting from the melting ice will be absorbed as much as possible. Stir the water in the calorimeter and carefully take its temper- ature, estimating to tenths of degrees. Immediately begin to add dry pieces of ice, one at a time, stirring constantly. Never dur- ing the run allow one piece of ice to entirely melt before another is added. Continue to put the ice in until the temperature is about as far below room temperature as the water was originally above room temperature. Now weigh the calorimeter and con- tents again to obtain the mass of ice added. Repeat the experiment the same as before only use a single HEAT OF FUSION 155 large piece of dry ice instead of several small pieces. Arrange data in tabulated form. Suppose M grams of ice have been added to m grams of water at a temperature t° in a calorimeter of mass m' and specific heat S', and that the resulting temperature was t^° . Call L the heat of fusion. Using these literal values, and also the values ob- tained from your data, answer the following questions. Indicate in your answers what the question is. 1. What was the change in temperature of the water and cal- orimeter ? 2. How much heat was given up by the water in cooling? 3. How much heat was given up by the calorimeter in cooling? 4. What was the rise in temperature of the water produced by the melting of the ice? 5. How much heat was absorbed by this water to raise its tem- perature ? 6. Therefore how much heat was used to change^ the ice to water? (Compute from your answers to questions 2, 3 and 5.) 7. How much ice was melted? 8. How much heat was required to melt one gram of ice ? L = ? Using the answers given to these questions, write an heat equa- tion expressing on one side the heat given up by the warmer bodies in cooling, and on the other side, the heat absorbed by the ice to melt it and raise its temperature. Substitute from your data in this equation to find L. Compute the error in per cent, from the accepted value of L. Test Questions. 1. What is meant by the heat of fusion? Explain how you determined its value for ice in this experiment. 2. Explain each term of the heat equation used to compute the heat of fusion. 3. Do you consider it more accurate to use a single large piece of ice or a number of small pieces? Give some reasons why this should be. 4. Why did you start with the temperature of the water as far above room temperature as you ended with it below? Name as many sources of error as you can and explain how you eliminated some of them. 5. Estimate how accurate it is necessary to read the temperatures and explain why. Which is the most important error in taking your data? Explain why it is so. 156 HEAT experiment h-5 Heat of Vaporization Determination for Water When an open vessel of water is allowed to stand, the water gradually disappears, or, as we say, " evaporates." The process causes a lowering of the temperture of the liquid and the more rapid the vaporization the greater the amount of cooling. Upon the molecular theory this is explained as follows: some of the molecules of the liquid, especially near the surface, attain a velocity sufficiently high to enable them to escape from the at- traction of the other molecules and become independent of them. These molecules are practically free and form the vapor of the liquid. Only these .molecules which have an unusually high velocity can thus escape. Hence, since' only the most rapidly moving molecules are re- moved from the liquid, the total kinetic energy of all the mole- cules in the liquid is diminished; that is, the liquid loses heat. This lost kinetic energy is changed into potential energy due to the separation of the molecules against their attraction for one another. When the vapor condenses this potential energy is changed back into the kinetic form, that is, into heat energy; for the most rapidly moving molecules again pass into the liquid and the total kinetic energy of the molecules of the liquid is increased. Therefore, just as much heat is generated when each gram of vapor condenses as was used up when one gram of the liquid vaporized. If heat is continually supplied to the liquid faster than it is used for vaporization, the temperature will rise until a point is reached at which vaporization takes place not only at the free surface but all through the liquid, the vapor forming bubbles which rise to the surface. Vaporization takes place much more rapidly on account of the enormously increased surface oflfered by the bubbles. If more heat is applied, it simply causes more bubbles to form and thus increases the rate of vaporization so that the heat required for the rapid vaporization remains equal to the heat supplied to the liquid. Hence the temperature ceases to rise until all the liquid is changed to its vapor. The tempera- HEAT OF VAPORIZATION 157 ture at which this phenomenon occurs is called the boiling point of the liquid. There is no essential difference between boiling and ordinary evaporation unless it be that in boiling the vaporization occurs all through the liquid. Since bubbles exist within the liquid the vapor pressure must equal the atmospheric pressure on the free surface of the liquid, for if they were different the bubble would change until the two pressures were equal. So the boiling point might be defined as that temperature at which the pressure of the vapor coming from the liquid is equal to the atmospheric pressure upon the free surface of the liquid. If atmospheric pressure becomes greater, the boiling point will be higher be- cause the vapor pressure would not equal atmospheric pressure and thus could not form bubbles unless the temperature increased. But at constant pressure the boiling point is always the same. The heat used in changing i gram of a liquid to its vapor is called the heat of vaporization of the liquid. The apparatus used to determine the heat of vaporization con- sists of a glass flask to contain the boiling water. The flask is closed by a stopper through which a single straight glass tube passes, reaching nearly to the bottom of the flask. The flask is about half filled with water, the stopper and the tube inserted, and the flask inverted and placed on a ring stand. This leaves an enclosed space in the upper portion of the flask connected to the exterior by the tube. A ring form Bunsen burner is ad- justed about the neck of the flask and clamped to the same standard that supports the ring holding the flask. The whole is mounted so as to slide up and down a vertical rod. The tube is of the proper length to just dip into water in a calorimeter, when the flask is lowered. A double walled calorimeter is used to reduce radiation. Very little heat is radiated directly from the burner to the calorimeter and the arrangement of the tube and flask serves as a trap to catch the water condensed in the upper walls of the flask. Better results will be obtained without using the cover to the calorimeter. Procedure. Weigh the inner calorimeter and the stirrer; then fill it to within 3 or 4 cm. of the top with water at least 15° C. below room temperature. Weigh again to find the mass IS8 HEAT of water. Adjust the ring burner so that it is well above the cork in the flask. When the water in the flask is boiling briskly and fairly dry steam is issuing from the tube, stir the water in the calorimeter and carefully read its temperature, estimating to tenths of a degree. Then place it in position under the flask and quickly lower the flask until the end of the tube is about 2 cm. beneath the surface of the water so that steam from the tube will pass into the water. Stir the water constantly and when its temperature has risen as high above room temperature as it was below when you started, remove the calorimeter. Stir the water quickly and thoroughly and read the temperature. Weigh the calorimeter and contents very carefully to find the mass of steam condensed. During this weighing leave the ther- mometer in the water, then weigh the dry thermometer and sub- tract. This will avoid removing water with the thermometer before weighing. Read the barometer to ascertain the tempera- ture of the steam. Arrange data in neat tabular form. Make two complete determinations. Supposing that the calorimeter of mass m' and specific heat S' contains m grams of water at a temperature of t° ; that M grams of steam at a temperature T° was passed into the water raising its temperature to t^. Call the heat of vaporization H. Using these literal values and also numerical values from your data answer the following questions. . (Word your answer so that it will indicate what the question is.) 1. What quantity of heat was absorbed by the water to raise its temperature to that of the mixture? 2. What quantity of heat was absorbed by the calorimeter to raise its temperature to that of the mixture? 3. What quantity of heat was given up in cooling to the tem- perature of the mixture the water produced by condensing the steam ? 4. How much heat was given up by the condensation of all the steam? (Use the answers to the previous questions.) 5. What quantity of heat was developed by the changing of one gram of steam to water at the boiling point? .'. H = ? Remarks. The heat of vaporization is in general less the higher the temperature at which vaporization takes place. At CONDUCTION 159 the normal boiling point its value is very nearly 538 calories per gram. Between 0° C. and the normal boiling point the de- crease is about 0.6 cal. per gram for each degree rise in tem- perature. When the steam is passing into the water, the end of the tube may be 3 or 4 cm. below the surface of the water. This will cause an added pressure upon the steam, which should not be omitted from your coqiputations. Test Questions. 1. What is meant by the heat of vaporization? Is It always the same? How did you determine the heat of vaporization of water in this experiment? What is meant by the boiling point? 2. Explain each term of the heat equation used to compute the heat of vaporization. 3. Why is it necessary to set the calorimeter upon wood rather than upon metal, say? Does any of the steam condense before reaching the water in the calorimeter? Does any of this condensed steam enter the calorimeter ?_ Show how the apparatus is designed to reduce this error. 4. Why did you start with the temperature of the water as far below room temperature as you ended with it above? Explain fully. 5. Why is it necessary to weigh the calorimeter and water so carefully each time? Would it be necessary to determine the mass of the water to the same fraction of a gram? Explain why. Why did you read the tem- perature as closely as you did? 6. Why is steam such an efficient agent in heating buildings? experiment h-6 Transfer of Heat by Conduction Conduction in Different Metals When a body is heated at one point the heat is transferred more or less rapidly to the other portions of the body by the method of conduction. At the point heated the molecules of the body are given a greater kinetic energy and therefore a greater mean velocity. By collision these molecules impart to neighboring molecules an increased velocity and hence a greater kinetic energy. The latter molecules collide with their neigh- bors, and so the kinetic energy is passed along through the body. This is the process by which heat is supposed to be transferred by conduction. The rate of conduction of heat through solids varies con- siderably in different substances. If a solid is heated at one point, the rate at which the temperature rises at another point depends upon the rate of ' radiation ; the amount of heat l6o HEAT " absorbed " by the material of the solid, that is, the specific heat of the substance ; and upon the rate at which the substance is able to transmit the energy from molecule to molecule, that is, its actual conducting power. The relative rates at which different substances transmit heat energy may be compared by the following method: Procedure. A bar of the metal to be tested has several holes drilled into it at regular interv&ls along its length. The bar is supported by two iron tripods. The bar, however, should rest directly upon asbestos or other good heat insulator. In each of the holes in the bar is inserted a thermometer which is sus- pended from a rod and clamp. After all the thermometers are mounted in position, the hole about each thermometer is filled with mercury which will readily take on the temperature of the metal and transmit it to the thermometer. A Bunsen flame is now to be applied to one end of the bar and the thermometers read at regular intervals of time for a given period. The flame must be first adjusted so that it will not heat the metal so highly that the temperature of the nearest hole will rise above the range of the thermometer. Probably the flame itself should just a little more than reach the bar. Now, when everything is in readiness, and before the flame is applied, read all the thermometers. They should all read the same. Take all thermometer readings to tenths of degrees. Exactly on an even minute the flame is applied to one end of the bar and the time recorded. Exactly on the next minute read the thermometers in succession as rapidly as possible, one partner taking the readings and the other recording them. Take read- ings thus each minute for five minutes, always reading the ther- mometers in the same order. Then remove the flame, but con- tinue the readings at one minute intervals for fifteen minutes. When the flame is removed do not turn it out nor change its height. After the readings are completed, remove the bar, replacing it by another bar of a different metal, and repeat the above process. Use the flame at the same height as with the first bar. Thus, approximately, heat is supplied at the same rate to both. MECHANICAL EQUIVALENT TABLE XXXVIII l6l Time Thermometer Readings A B C D E By the method outlined, test a copper and an iron bar. Tabu- late data as in Table XXXVIII. Plot a curve for each thermometer, using times as abscissas and temperatures as ordinates. All the curves should be plotted on the same sheet, using the same scale and starting from the same origin. Draw in those for copper with red ink, and those for iron with black. Study the curves drawn and endevaor to explain all peculiarities observed. Compare each curve with the others. If any rise higher than any of the others, explain it. Explain why the slopes of the curves are different. Compare in general the curves belonging to the copper bar with those of the iron bar. Test Questions. 1. Explain how heat is transmitted by the process of conduction. Upon what properties of a substance does the rate of trans- fer depend? 2. Copper and iron have different specific heats. Do your curves show that this affects conduction? Explain. 3. What is the general effect of the loss of heat by radiation upon the position of your curves? 4. Explain why the slope of each curve changes as it does. Compare the slopes of the curves at corresponding points. 5. How does copper compare with iron in its conducting power? 6. Would the curves reach the same point if carried far enough? Ex- plain. experiment h-7 The Mechanical Equivalent of Heat Determination from Impact of a Falling Body It has been demonstrated that whenever mechanical energy is changed into heat the same amount of heat is always produced by the disappearance of a given amount of energy. Thus when a moving body is stopped, its energy disappears and a certian amount of heat is developed. To develop each calorie of heat a definite amount of energy must be used. The mechanical I 62 HEAT equivalent of heat is the amount of mechanical energy that must be expended to produce one calorie of heat. The object of this experiment is to make a determination of the mechanical equivalent. The method is to determine how much a body is heated when it is stopped after falling a known distance. If a body of weight W falls a height h, the work done is Wh. If this work is transformed into heat and the tempera- ture of the body is raised from t° to t^°, the specific heat of the body being s, the number of calories of heat developed is Ws{t^ — 0- Now if / is the number of units of work required to develop one calorie of heat, we have Wh = JWs{t^ — t) The " body " used in this experiment is about a pound of lead shot. The shot is caused to fall inside a fibre tube closed by corks at each end. An extra cork, with a thermometer inserted through it, is provided. Procedure. Cork one end of the tube and pour in the shot. Close the opposite end of the tube with the cork with the ther- mometer in it. In handling the tube grasp it at the center to avoid heating the end where the shot may be. Mix the shot thoroughly around the thermometer by rolling the tube and after a minute or more read the thermometer carefully. Slowly invert the tube and quickly replace the cork with the thermometer by the solid cork. Start with the tube vertical and with a quick motion invert it completely so that the shot will fall practically as a solid mass to the opposite end of the tube, and with as little friction as possible against the sides of the tube. As the tube is turned rest the lower end on the table to avoid the forcing out of the cork at that end. Continually invert the tube in this way rapidly at least 100 to 120 times, keeping accurate count of the reversals. Quickly remove the upper cork and replace it by the one with the thermometer. Take the temperature of the shot as before. Invert the tube and remove the upper cork. Measure the dis- tance from the top of the shot to the level of the bottom of the upper cork. This will give the mean distance the shot has fallen each time. MECHANICAL EQUIVALENT 163 Repeat the experiment at least three times. Calculate the per- centage difference between your results and the commonly ac- cepted value of the mechanical equivalent. Neatly tabulate your data. Remarks. A very considerable error is introduced into the experiment by what is known as the " lag " of the thermometer ; that is, its inability to come at once to the temperature being measured, due to the fact that not only the glass but also the whole of the mercury in the bulb must be changed in temperature. When the temperature of the body is changing and the amount of heat involved is small as in this experiment, this error may be quite large. Another considerable error is due to the fact that the air in the tube as well as the inside of the tube itself must be heated by the shot. Test Questions. 1. Define the mechanical equivalent of heat and ex- plain how you determined its value in this experiment. Derive the equations. 2. Compute as closely as you can the amount of heat developed which does not heat the shot and see whether it accounts for the error in your results. 3. Explain what three transformations of energy take place during each turn. 4. Why was it not necessary to weigh the shot? Why use a fibre tube? 5. Can you think of some reason for using lead instead of some other metal? (Compare specific heats.) TABLE OF CONTENTS PART II— SECOND TERM CHAPTER III. NOTES ON MAGNETISM AND ELECTRICITY I. Magnetism SEC. PAGE SEC. PAGE 201. .Magnets 201 204. Lines of Force 202 202. Laws of Force 201 205. Magnetization 203 203. Magnetic Field 202 206. Electromagnets 204 II. Electrostatics 207. Electric Potential 204 208. Facts from Experiment ... 204 III. Electrokinetics 209. Current Analogies. .' 206 212. Heating Effect of Current. 210 210. P.D. and E.M.F 206 213. Outline of Laws, etc 211 211. Magnetic Effect of Current 208 IV. Statement of Principles, Laws and Definitions 214. Characteristic Tendencies of Lines of Force (I) 212 215. Relative Directions of Current and Field (II) 212 216. Ampere's Laws of Thrust on a Conductor (Ill) 212 217. Joule's Laws for Heating" (IV) 213 218. Resistance of Conductors. Connections . . . . : (V) 214 219. Measures of P.D., E.M.F. and Power (VI) 214 220. Ohm's Law Modified to Include all Cases (VII) 215 221. Faraday's Law of Induced E.M.F. , (VIII) 216 222. Lenz's Law of Direction of Induced E.M.F (IX) 216 223. Faraday's Law of Electrolysis (X) 217 V. Symbols, Laboratory Methods, etc. 224. Parts of Circuits, Instruments, Apparatus, etc 217 225. Methods of Making Connections 217 226. Reading Instruments 218 227. Descriptions of Circuits, Symbols 218 CHAPTER IV. EXPERIMENTS IN MAGNETISM AND ELECTRICITY Group I — M. Magnetism EXPERIMENT PAGE M — 21. Study of Magnets and Magnetic Fields 220 Determination of Polarity. Comparison of Pole Strength M — 22. The Magnetic Field and Lines of Force 226 Tracing Fields with Compass and Iron Filings X CONTENTS EXPERIMENT PAGE Group II — S. Electrostatics , S — 21. Study of Electrostatic Induction 229 Use of the Electroscope to Test the Charges S — 22. The Faraday Ice Pail Experiment 233 Relationship of Inducing and Induced Charges Group III^O. Ohm's Law 0—21. Ohm's Law for Part of a Circuit 235 Relations of V, E, I and R 0—22. Study of Primary Cells 242 Comparisons of Various Types Group IV^ — C. Measurement of Cxj'rrent by Electrolysis C— 21. The Hydrogen Voltameter 246 The Electrochemical Equivalent of Hydrogen C— 22. The Copper Voltameter 250 Determination of a Galvanometer Constant Group V — R. Measurement of Resistance R — 2 1 . Comparison of Resistances by Fall of Potential 254 Variation with Length, Cross-section and Material R — 22. Determination of Resistances in Parallel and in Series 257 Resistance from Current-and Potential Difiference R— 23. Study of the Slide-wire Bridge. . 259 Resistances Separately and Combined R— 24. Study of the Decade Bridge 262 Calibration of a Resistance Box R — 25. Comparison of Resistances 263 Carey-Foster Method R — 26. Variation of Resistance with Temperature 263 Determination of Temperature Coefficient Group VI — G. The D'Arsonval Galvanometer G — 21. Essential Parts of a Voltmeter and an Ammeter 266 Calibration of a Galvanometer for Direct Reading G — 22. Constants of a Sensitive Galvanometer 270 Resistance and Sensibility Group VII — ^V. Potential Difference and E.M.F. V — 21. Potential Difference in Different Parts of a Circuit 273 Construction of a Potential-Resistance Diagram V — 22. Principle of Potentiometer Measurements 277 E.M.F. of Different Types of Cells Group VIII — X. Miscellaneous X — 21. Theory of Shunts. Experimental Verification 279 X— 22. Verification of Kirchoff's Laws 279 X — 23. Joule's Equivalent by Electrical Method 279 X — 24. The Quadrant Electrometer 279 X — 25. The Ballistic Galvanometer. Comparison of Condensers . . . 279 PART II^SECOND TERM Elementary Course CHAPTER III NOTES ON MAGNETISM AND ELECTRICITY I. MAGNETISM 201. General Properties and Parts of Magnets. A magnet attracts iron (and a few other substances) toward its poles. The poles are the portions toward which iron is attracted. They are called concentrated poles if the attraction is toward (or nearly toward) single points, and distributed poles if toward distributed groups of points. The straight line through pole centers is called the axis of the magnet. If free to turn, one pole of a magnet points northward and the other southward. The north- pointing one is called a north or positive pole, and the other a south or negative pole. Every magnet has at least two poles, a north and a south. If the poles at the ends of a magnet are alike, and the other like poles are together between the ends, the latter are called consequent poles. 202. Laws of Force Between Poles. When two or more magnets are near each other, like poles repel each other and unlike poles attract. Magnets differ in their power of attracting iron, and in their action upon each other; that is, pole strength is something capable of measurement. A pole that will exert a force of one dyne on another pole of equal strength at a distance of one centimeter is called a unit pole. The force between two poles of strengths m and m' varies directly as the product of the pole strengths, and inversely as the square of the distance (d) between their centers. F « mm'/d^ and F = mm'/d' (1) in air if the unit of pole strength is defined as above. aoi 202 MAGNETISM AND ELECTRICITY 203. Magnetic Field. Any region in which a magnet pole, if present, would be acted upon by a magnetic force is called a magnetic field. Magnetic fields may be due to other causes besides the presence of magnets; for example, an electric current produces a magnetic field around it. Since a compass needle always points in some definite direction wherever placed, the whole earth and the space around it is a magnetic field, the cause of which is not known. A field that is capable of exerting a force of one dyne upon a unit pole is called a field of unit intensity or strength; and the intensity of field {H) at any point is measured by the 'number of dynes of force that it would exert on a unit pole if placed at the point. The force on a single pole in a magnetic field is pro- portional to the strength of pole, and to the intensity of field; F = mH, and H = F/m, or H = F when w = 1. From the formulas for F, it may be shown that the field due to a pole is H = m/(P ■ (2) at a point d centimeters from m. 204. Direction of Field and Lines of Force. Lines of force may be used to represent the direction of field. They are the lines along which the force acts in a magnetic field. The direction of the line through any point is the direction of the resultant force at that point. Lines of force never cross and are endless, except where they enter or leave a magnet. (The parts of a magnet where they enter or leave are called the poles.) These lines converge toward a stronger field, diverge toward a weaker field, and are parallel in a uniform field. The positive direction of lines of force, or of a magnetic field, is the direction in which a small positive pole would tend to move if placed in the field. A negative pole tends to move in the negative direction along the lines of force. The lines are directed away from a positive pole and toward a negative one. The intensity of field at a point may be represented both in direction and m'agnitude by a vector, in the same way that a mechanical force is represented. It may also be resolved into components, or the resultant of two or more fields may be found or represented graphically. Since any point in a field may have a line through it, we may MAGNETISM 203 think of the field as being filled with an infinite number of lines ; but it is customary to express the intensity of field by the number of lines per square centimeter of cross section of field. In this latter case the lines represent intensity as well as direction of field. The number of lines passing through any area is called the "magnetic fiux" through that area. (Show that the mag- netic flux ($) across any surface enclosing a pole of strength w is $ = iirtn.) Magnetic phenomena indicate a tendency for lines of force to shorten their length and to repel each other laterally. These two characteristics will be very useful in subsequent study of phe- nomena and principles relating to magnets and electric currents. When a magnet is placed in a uniform field, the forces on the two poles are oppositely directed and each equal to Hm. If the axis of the magnet is at right angles to the field, the perpendicular distance between the forces is the length (Z) between the poles and the moment of the couple is C = F X I = Hml. The magnetic moment (M) of a magnet is the moment of the couple when it is perpendicular to a field of unit intensity ; hence M = ml. When a magnet is placed in a uniform field with its axis at an angle with the lines, the perpendicular distance between the forces at the poles will be I sin d and C = F/ sin = Hml sin B = HM sin d (3) If the magnet is suspended and free to vibrate, this couple will cause the magnet to vibrate with a period (P) that is inversely proportional to ^H, or P^ « l/-ff, which enables one to compare the intensities of field at two or more points by simply obtaining the periods of the same magnet at the difi^erent points; then, H,IH^ = P^'lPi^ (4) 205. Magnetization. Iron, or any other magnetic substance, when placed in a magnetic field becomes a magnet by induction. Jarring or tapping assists a change of magnetic condition. The magnetized condition is partly lost upon removal from the field. The portion retained is called residual magnetism. Long bars retain more than short bars. (Why?) Hardened steel retains magnetization most persistently' and is used for what are called permanent magnets. A piece of soft iron is attracted by either end of a magnet. (Why?) 204 MAGNETISM AND ELECTRICITY When a piece of soft iron, originally unmagnetized, is placed near a permanent magnet, the magnetic field is modified ; for the iron becomes a magnet by induction, so that the field at any point is really the resultant of two fields, one due to the original magnet, and the other due to the magnetized iron. If the lines of force of such modified fields are mapped by iron filings, or are plotted, for several cases; for example, a piece of soft iron near a strong pole; a rod of soft iron parallel to a bar magnet; a ring of soft iron near a strong pole; etc., they show that in all such cases the change in the distribution of lines of force is such as to suggest that "iron is a better carrier of lines of force than air." The lines of force tend to converge into the iron. 206. Electromagnets. The region around a wire in which current flows is a magnetic field. Iron in the neighborhood of a current is magnetized. The magnetic field due to a current has in fact the same properties as a field due to other causes, it is conveniently represented by lines of force, these being defined in the same way as heretofore. The lines of force are endless curves surrounding the current. If a bar of iron is surrounded by a coil of wire (called a solenoid) in which current is flowing, the bar becomes a magnet, and is called an electromagnet. II. ELECTROSTATICS 207. Electric Potential. The condition of an electrified body (e. g., a charged conductor) is not wholly determined by the amount of its charge, but depends also upon another factor called the potential of the body. Roughly speaking, the potential at a point is the measure of the tendency of electricity to pass from the point in question to the earth. The potential of the earth is arbitrarily chosen as zero. A point having a positive potential is a point from which a positive charge (if present) would tend to escape to the earth. If the potential is negative, then a positive charge would tend to flow from the earth to the point. Positive electricity tends to flow from points of high potential to points of low potential, just as water tends to flow from a high level to a low level. 208. Fundamental Facts Found by Experiment. (Shearer's Notes and Questions in Physics, pp. 140-141.) ELECTROSTATICS 205 (a) "All portions of matter appear to be associated with electric charges equally + and — . (b) "In any operations resulting in the production of free electric charges, equal quantities of + and — are invariably produced. (c) "Like charges repel. Unlike charges attract. (d) "Electric charge moves rapidly along certain bodies and with much greater difficulty in others. The former are called conductors. On account of frequently placing the latter, called insulators, between two conductors having unlike charges, insulators are often spoken of as dielectrics. (e) "When an electric charge is brought in the vicinity of uncharged bodies, the charge is able to cause a partial separation of 'the charges in these bodies, attracting charges unlike itself to points as near as possible, and driving like charges as far away as possible. This is named charging by induction. (f) "The entire region through which a charge exerts a meas- urable force is spoken of as the field of the given charge. This is theoretically infinite, but practically is usually rather limited. (g) "The work which the electric force would do in moving a unit charge from one point to another in an electric field is called the potential difference between the points in question. Po- tential difference is here indicated by V. (h) "The mechanical work done in the movement of electric charges is the product of the charge and the potential difference through which this charge is moved. Work = QV (when F remains constant) = Q X average V (when V varies) (It is much more convenient to measure V than to determine forces due to distributed charges.) (i) "The forces encountered, and, therefore, the work done in moving a charge from one point to another, will depend not only on the amount of charge and on the distribution and magni- tude of the neighboring charges, but also on the medium sepa- rating the charged bodies, i. e., the dielectric. (j) " If two charged bodies are separated by air, work done in the transfer of unit charge from one to the other (i. e., the difference of potential) is the product of average force and distance 206 MAGNETISM AND ELECTRICITY moved. If, without changing the position or the quantity of charge, the air is replaced by another dielectric, the work required for the above transfer will usually be different. As average force has been reduced to Ijk of its original value, the work done has been reduced in the same ratio, k is named the specific inductive capacity of the dielectric. (k) "When two conducting plates are separated by a di- electric, the work stored by charging one plate + and the other — will depend on the potential difference betvy^een the plates (F) and on the amount of charge required for this V. The quantity of charge per unit of V is the capacity of the condenser; as such an arrangement is called. If we regard the charging as done by taking + away from one plate and depositing it on the other, starting with uncharged plates, then the work done on the first small charge g is g X 0, or 0, because of no initial V. The work on the last charge q is q X V; and as V increases directly as the charges accumulate, the average work done per unit charge is V/2, or the total work = QV/2 (where Q = Sg)" III. ELECTROKINETICS 209. Analogy Between Electric Current and a Current of Liquid in a Pipe. The electric current was for a long time believed to consist of the flow of a fluid (called "electricity") through the wire that carries the current. This theory is no longer believed. But there are many points of resemblance between the electric current and a current of liquid in a pipe, and it is a- help in getting clear ideas of the electric current to keep in mind the analogy between the two. For example, electrical resistance is analogous to the frictional resistance of the pipe; in each case the resistance leads to a development of heat; and in each case the amount of the resistance is a factor in determining whether the current will be great or small. To maintain a current of liquid circulating in a pipe something equivalent to a pump is necessary; this corresponds to the generator that is necessary to maintain an electric current. 210. General Statements Concerning Difference of Potential and Electromotive Force. (From Nichols and Blaker's Labora- ELECTROKINETICS 207 tory Manual, Vol. I, pp. 268-270.) "The indiscriminate use of the terms 'electromotive force' and 'difference of potential' has given rise to much confusion. The following treatment of the subject, though different from that of many writers, is believed to be entirely consistent with the facts. Moreover, it is hoped that it will make clear to the mind of the student the relation between two idfeas, which, though intimately related, are never- theless entirely distinct. "The difference of potential between two points is that difference in condition which tends to produce a transfer of electrification from one point to the other point. The measure of this difference of potential is the amount of work that would be done by, or against, electrical forces in carrying unit quantity of electricity from the one point to the other point." "Any generator of electricity (whether it be a battery, dynamo, or electrical machine) is capable, when energy is supplied to it, of maintaining a difference of potential between its terminals, even though they are connected by a conductor. It is to this capability of maintaining a difference of potential that we apply the name of electromotive force. The electromotive force of a generator is measured by the maximum difference of potential which it is capable of producing when no current flows. Or, when a current is allowed to flow, it is measured by the difference of potential at the terminals, plus the fall of potential due to the resistance of the generator. ' "From these definitions it follows: " (1) That there is a difference of potential between any two points of a circuit conveying a current. "(2) That the electromotive force of a circuit is always located in the generator. The source of a counter electromotive force may always be looked upon as a negative generator. So far as our present knowledge extends, there is never any electro- motive force in a perfectly homogeneous conductor which is not moving relatively to a magnetic field." "Ohm's law as originally stated, using modern terms, is: The current flowing in a {perfectly homogeneous) conductor {not moving relatively to a magnetic field) is directly proportional to the difference of potential between the terminals of the conductor. If the conductor between any two points is in any way varied 208 MAGNETISM AND ELECTRICITY subject to the above conditions, the current will be equal to the difference of potential between the points divided by a quantity known as the resistance of the conductor between the points. From which we have / = (F. - VB)/Ral, "The statement that current flowing in a circuit is equal to the total electromotive force in the circuit, divided by the total resistance of the circuit, is a deduction from Ohm's law." For a modification of Ohm's law that covers all cases, see Sec. 215, page 220. 211. Magnetic Effect of Current. The lines of force due to current in a long straight wire are circles concentric with the wire. We may imagine that new lines are produced next to the wire when current increases and that the lateral repulsion, mentioned in Sec. 204, causes the old lines to expand to make room for the new; also that all of the lines tend to shorten so that their circles will grow smaller and collapse on the wire if the current is decreased to zero. The field (H) due to a given current (/c) decreases in strength from the conductor outward. At a given distance (r), H = 2/,/r (1) if the wire is \'ery long and straight, and there are no external disturbing influences. The field at the center of a circular coil of n turns of radius r is H = 27rnl,/r (2) These formulae are derived from Laplace's general law for the magnetic force (A/) at a point (p) due to the current in an element of length (AF) of conductor at the distance r from the point. Let a be the angle between Al and the line r from the center of Al to p; then Laplace's Law is expressed by A/ = I,Al sin a/r^ (3) The field due alone to current in a long straight wire is shown in Fig. 51, in which the distance between lines increases as the strength of field decreases from the wire outward. Figure 52 represents a uniform field perpendicular to a conductor in whch there is no current. If current is sent through the conductor away from the observer, so as to produce a field as shown in ELECTROKINETICS 209 Fig. 51, the straight line and the circular component fields will combine to form a distorted resultant field as shown in Fig. 53. _Q_ Fig. 51. Fig. 52. Note that on the side of the wire where the component fields have the same direction, the resultant field is the sum of the component field intensities; and on the opposite side the differ- Fig. 53. ence: i. e., the current causes a distorted resultant field in which the lines are bent around the wire so as to be crowded together on one side and comparatively far apart on the other side. Considering that lines of force tend to shorten (like elastic threads) it is evident that a force must be exerted to maintain the distortion of the lines, and since the current in the wire is the cause of this force, it follows that the distorted lines will react upon the wire carrying ihe current, tending to push it away 210 MAGNETISM AND ELECTRICITY from the strong field into the weak field. This is also in accord with the other characteristic of lines of force, viz.: that lines in the same direction repel each other. Note that the wire tends to move away from the side where the component fields have the same direction. The important fact is : — That a wire in which a current "flows" when placed in a magnetic field, is acted upon by a mechan- ical force tending to make it move in a direction at right angles both to the wire and to the field. When the current is increased, the force becomes greater. The force is proportional to the intensity of the field (H) and to the length of the wire (J), that is, F cc HI. If, by increasing the current strength, other conditions remaining unaltered, the. force has been doubled, the current is said to be twice as great as before: i.e., this mechanical force is used in obtaining a measure of current strength. The C.G.S. unit current is a current of such strength that if the wire carrying it were placed in a field of unit intensity, being everywhere at right angles to the lines of force, a force of one dyne would act on each centimeter of the wire. Let current in C.G.S. electromagnetic units be denoted by Ic- It then follows that F = HIcl when the wire is perpendicular to the field. If the wire makes an angle a with the field, F = HI J, sin a (4) The common or practical unit of current is the Ampere; it is one-tenth of the C.G.S. unit current, and the number (7) of amperes is ten times as great as the number {Ic) of C.G.S. units; or /c = lyio. 212. Heating Effect of the .Current. An electric current develops heat in the wire through which it flows. The rate at which heat is developed {H, that is, ergs per second) in a given wire is found to increase as the current increases, — not, however, in the same proportion, but in proportion to the square of the current. For a given current the rate of heat development is different in different wires, depending upon the length, cross section, and material of the wire. For any given wire a constant R may be found, such that the rate at which heat is developed in the wire is always equal to R multiplied by the square of the current, that is, H* = RP (5) The dot over a symbol indicates that it is a time rate. ELECTROKINETICS 2 1 1 This constant R is called the resistance of the wire. (The resistance is to be regarded as one of the physical constants of the wire, like its length, density, etc. Like these, the resistance always changes somewhat with temperature.) If H in Eq. 5 expresses ergs per second, and I is C.G.S. units of current, R is said to be expressed in C.G.S. units of resistance. The ohm, which is the unit chiefly used in practice, is 10' C.G.S. units of resistance. Therefore, if I (in Eq. 5) is expressed in amperes and R in ohms, H will be expressed in terms of a unit which is equal to 10' ergs per second; that is, in watts. (The watt is a unit of power. 746 watts are equal to one horse power.) The resistance of a wire is directly proportional to its length (/) and inversely proportional to its cross section (A) R = pl/A (6) The proportionality factor p is called the specific resistance of the material. It is the resistance of a wire of unit length and unit sectional area. The reciprocal of the resistance of a circuit is called its conductivity. 213. Outline of Principles, Laws, and Definitions Relating to Magnetism and Electrokinetics. The more important ones relate to I. Characteristic tendencies of magnetic lines of force. II. Relative directions of current and magnetic lines of force due to the current. III. Ampere's Laws relating to direction and magnitude of force on a conductor carrying current in a magnetic field. Practi- cal unit of current (ampere, T) and quantity (coulomb, Q) defined. IV. Joule's Law for heating of a conductor by current. Practi- cal unit of resistance (Ohm, R) defined. V. Variation of resistance of conductors with length and cross-section. Parallel and series connections. VI. Measure of potential diiiference (F); and practical unit (volt) defined. Measure of energy transformed, and practical unit of power (Watt, W) defined. Electromotive force {E or E.M.F.) of a generator. VII. Ohm's Law, — for relations of V, E, I, and R for the part of a circuit between any two specified points; and of SE, /, and Si? for the entire circuit. 212 MAGNETISM AND ELECTRICITY VIII. Faraday's Law of Induced E.M.F. Self and Mutual Induction. IX. Lenz's Law, relating to direction of induced E.M.F. X. Faraday's Law of Electrolysis. Electrochemical equiva- lent defined. IV. STATEMENT OF PRINCIPLES, LAWS, AND DEFINITIONS 214. Characteristic Tendencies of Lines of Force. (I) Mag- netic phenomena indicate that the lines of force in magnetic fields (a) tend to shorten lengthwise and (b) tend to repel each other laterally when they are in the same general direction. Component fields in opposite directions tend to neutralize each other. Magnetic phenomena may be predicted by use of these characteristic tendencies of lines of force. 215. Relative Directions of Current and Field. (11) The positive directions of current and of the lines of force due to it are related the same as the directions of advance and rotation of a right-handed screw. The passage of a current through a conductor always produces a magnetic field around the conductor. The lines of force surround the conductor and are endless, and for a long straight conductor are circles concentric with it and in planes perpendicular to it. [If the conductor is grasped by the right hand with the thumb extended in the direction of current, the fingers will point in the positive direction of lines around the conductor.] 216. Ampere's Law of Thrust on a Conductor. (Ill) If a conductor carries current perpendicular to a magnetic field H (not due to the current), a force is exerted upon the conductor tending to push it sidewise away from the side where the circular component field H', due to the current, has the same direction as the straight component field H through which the conductor passes. This force, perpendicular to the conductor and to the field H, is proportional to the strength of field H, to the current I', and to the length L of the conductor if perpendicular to the field, or to the perpendicular component (L) of length if the conductor is not perpendicular to the field. Hence, F : Q, or M = zQ = zit . (19) z oc o/i; or z = Hajv = 10.36 X 10"^ ajv (20) M = 10.36 X 10-« Ita/v (21) in which M is in grams, J in amperes, and t in seconds. H is the mass of hydrogen per coulomb. V. SYMBOLS, LABORATORY METHODS, DIRECTIONS, ETC 224. Parts of Circuits. The symbols in Fig. 54 are convenient for representing connections and apparatus used in experiments. 225. Methods of Making Connections. In planning con- nections, make a preliminary trial sketch as follows: Represent the apparatus without connections. If there are both current and potential measurements to be made, trace the current circuit first from the positive terminal of the generator around to the negative terminal, omitting all other connections; then later add the voltmeter and other connections t'o complete the sketch. Study the sketch for improvements, then make the connections in the same order that they were drawn but have an opening in the current circuit. Do not connect to the negative terminal of the battery until you have gone over the connections very carefully to be sure there is no short circuit, and that the measuring instruments are fully protected. If connection is to be made to the lighting circuit, do not insert the plug until all the other connections have been made and carefully gone over. Improve the connections and diagram in whatever ways may seem desirable before making the final drawing that represents the connections used. If the generator has low internal resist- ance or is capable of giving a current that would endanger any of the instruments, be sure to provide protecting resistance 2l8 MAGNETISM AND ELECTRICITY before closing the circuit or connecting sensitive instruments. The connections should be inspected by the instructor in charge before closing the circuit if the student is not perfectly sure that no instrument or part of the apparatus is in danger. 226. Reading Measuring Instruments. It is customary to read tenths of the small divisions. Note carefully the least count and zero reading of each instrument scale. For some scales it is better to read and record the divisions, as the observed data, then place the corrected derived values in another column. 227. Descriptions of Circuits in Reports. In every case where a circuit is used in an experiment, the report should contain a diagram of it. This should not be merely a conventional dia- gram such as many of those given in the Manual, but should show exactly the apparatus used and the details of the con- nections just as they were made in the experiment. Of course the symbols shown in Fig. 54 may be used in many cases where they are sufficiently explicit. For example, if the symbol for a resistance box (Fig. 54, h) is used and it is necessary to indicate -^|— -J^^y^ -j^^y^ " (o) Cell (b) Dynamo (c) Motor (rf) Lines from a generator WwvW WaamAaa [smm) [IB^^ esistance (g) Inductive I JS (e) Resistance (/) Variable Resistance (g) Inductive Resistance (fc) Resistance Box (45 ohms out) #■ (»■) Ammeter (i) Voltmer (ft) Galvanometer (7) Voltameter — O o' O — i.,cw!) 'S^l' (m) D.P.D.T. Switch (n) Reversing Switch (o) Contact Key «>) Reversing Key -<: (a) Condenser (r) Induction Coil (s) A.C. Generator (() Incandescent Lamps J-^ ZL [qH^fcigamg] («) Switch (tj) Ground (w) Fuse {x) Electrometer Fig. 54. LABORATORY DIRECTIONS 21 9 the actual resistance used, it may be done by marking the resistance opposite each plug and indicating which plugs are removed. In addition the circuit should usually be described' in words, and all instruments or pieces of apparatus used for the first time should be carefully described. The circuit through an instrument or special piece of apparatus should be traced or shown by diagram if it can be easily determined. CHAPTER IV EXPERIMENTS IN MAGNETISM AND ELECTRICITY GROUP I— M. MAGNETISM experiment m-21 Study of Magnets and Magnetic Fields Determination of Polarity. Comparison of Pole Strength A magnet is a body such as a piece of iron or steel or a piece of iron ore which has the property of attracting certain substances such as iron or steel. Magnets may be classed as natural or artificial according as they are found to have this property in their natural state, or must be subjected to a certain process in order to become magnetic. The property of magnetism may be imparted to a magnetic substance by stroking it on another magnet, or by jarring or heating it either near another magnet or surrounded by a conductor carrying a current of electricity. Artificial magnets may be either temporary or permanent. Magnets are also characterized by the fact that when sus- pended freely from the center, at some distance from any mag- netic substance, they turn with their longer dimension approxi- mately north and south, the same end always pointing north. If two suspended magnets are approached near together, the north seeking ends will repel each other, and the south seeking ends will also repel each other, but the north seeking ends will attract the south seeking ends and vice versa. There is then a difference between the two ends of an ordinary magnet. Points near the ends of such a magnet which are the centers of attraction are called poles; the north-seeking end is called the north (N) or + pole and the south-seeking end, the south (S) or — pole. The line connecting these poles is called the axis of the magnet and usually coincides with the geometric axis. Some magnets have more than two poles, alternating N and S. Magnets in which like poles are adjacent are called consequent pole magnets. MAGNETISM 22 1 Some poles are not concentrated at one point, but are distributed poles. Poles of different magnets exert different forces of attraction or repulsion on a given pole at a given distance. The relative forces thus exerted express the relative pole strengths of the magnets. A pole is said to be of unit strength when it exerts a force of one dyne on a pole of equal strength at a distance of one cm. in air. Coulomb's Law states that the mutual force exerted by two poles varies directly as the product of the pole strengths {mm') and inversely as the square of the distance between them, or _ mm' H oc d^ In air, using proper units (C.G.S.) F = mm'/d^. The field of a magnet is the space about the magnet through which the force of the magnet may be exerted. The positive direction of the field at a point is the direction that an N pole would ' tend to move if placed at that point. The strength of field (H) at a point is measured by the resultant force in dynes that would" be exerted upon a unit N pole placed at the point. H = m/d^, due to a pole of strength m and at a distance d from the point. From this, the force on a pole of strength w in a field of strength H would be F = Hm. The ordinary methods of adding vectors may be applied, of course, to find the resultant magnetic field due to several component magnetic forces. If a freely suspended magnet, such' as a compass needle, be placed in a magnetic field and displaced from its equilibrium position, it will vibrate similarly to a pendulum in the gravi- tational field. The period of the vibration is the time between two successive swings through the same point in the same direction. As in the case of the ordinary pendulum*, the number of vibrations («) in equal times of the same needle in different magnetic fields varies as the square root of the force, — or the strength of field. Then n «. s/E. or n^ oc H. But the strengths of field at a point due to placing magnets in succession at the same distance from the point vary as the strengths of poles of the magnets, or H ec m. This gives us a method of comparing pole strengths. Let wi and m^ be the pole strengths of two mag- nets placed successively at equal distances from a compass 222 MAGNETISM AND ELECTRICITY needle, and wi and n^ be the corresponding numbers of vibrations of the compass needle in equal times. Then mi _ «i^ If there is a constant field of force present having the same direction as the fields of the magnets, in which field the compass makes n vibrations in the given interval, then n' would represent the force due to this constant field. This must be subtracted from Wi^ or ni^, which represents the sum of the forces due to the constant field and of Wi or ma respectively. The formula would then be OTi _ Hi _ n-^ — n^ m^ Hi n^ — »^ Another method of getting pole strengths is based on the fact that if two poles are brought up on opposite sides of a compass needle so that one just neutralizes the effect of the other, where m, mi, and m^ are respectively the pole strengths of the compass and of magnets one and two. Then v mi _mi jMi _ d^ d^ df m-i d^ A discussion of the above topics is also given in Chap. Ill, Sees. 201-203. Special Directions. Be careful 'in handling the magnets not to strike them together or against other metal objects, as this tends to destroy their magnetism. Keep magnets, other than those being tested, at a distance. Record in your data book, at the time, the results of each step of the experiment. Your report should contain a description and an explanation of the observed phenomena, as well as all data taken and answers to the questions asked. Each section should be taken up in order and the exact procedure described. If some of the Test Questions are answered in the discussion, the numbers of the questions answered should be placed in the MAGNETISM 223 margin. (See Chap. I, Sec. 8 (h).) Free use should be made of diagrams as an aid to clearness. Procedure. 1. Determine which is the north pole of the compass needle and state how it may be identified. Explain why, when displaced from the north-south line, it will oscillate and why it finally comes to rest always pointing in the same direction. Lay out a north-south line on the table with pins and thread. 2. Locate approximately the "poles" of each of the three magnets by noting where the axis of the compass needle inter- sects the axis of the magnet when the compass box is so placed (with its side against the magnet) that the two axes are at right angles to each other. Make full size diagrams locating (and marking) N and S poles of each magnet. Explain any peculi- arities observed. 3. Set the compass needle vibrating. Note roughly its period. Bring a magnet near it and again set the needle vibrat- ing. Explain what you observe. Note that by placing first one magnet pole near the compass, and then another, the relative strengths of the two poles will be indicated by the rates of vibration of the needle. If each magnet is so placed that the earth's field always coincides in direction with the magnet's field at the point, and if the poles being tested are successively placed at the same distance from the compass, the ratio of the pole strengths is m-i. _Hi _ n-? — n^ nii Hi M2^ — M^ where «i, W2, and n respectively are the numbers of vibrations of the compass needle in the same time, when in the field of each magnet and that of the earth alone. Using this relation determine approximately the ratio between the strengths of like poles of two different magnets; and also, for one of the magnets, the ratio between the strengths of its N and S poles. Count as many vibrations as possible and observe the time. Express the ratios in decimal form. Explain the ratio you find between the N and S poles of the same magnet. 4. Place the compass under the north-south line and bring up one magnet end on perpendicular to the direction of the earth's 224 MAGNETISM AND ELECTRICITY field to within a few cms. of the axis of the compass. Then HxjH = tan 6, where Hand Hi are respectively the field strengths of the earth and of the magnet at the point and 6 is the angle it is displaced from the N-S line. Now if the other magnet is brought up end on in a similar manner on the opposite side of the axis until the effect of the first is just neutralized, then the strengths of the fields of the two magnets at the point are equal and mi/m2 = d^fd^; or ratio of the pole strengths equals the square of the ratio of distances from the point. Check (3) by this method substituting values obtained by experiment. 5. Place the magnet in a north-south line with the N pole pointing north. Find the positive direction of the "field" due to the magnet at points in a horizontal line perpendicular to the axis of the magnet at its mid-point, also in a vertical line similarly perpendicular; then state the -|- direction of the magnet's field in a plane similarly perpendicular. What is the direction of the magnet's field along a prolongation of its axis? Indicate these results by a diagram. In each case indicate the direction of the earth's field. 6. The field around a magnet is generally the resultant of two fields: i.e., that of the earth and that of the magnet. With the magnet placed as in (5) where would the two fields be in the same straight line but opposite in direction? In the same straight line and same direction? A neutral point is a point in a field where the resultant field is zero. Locate approximately by the compass the neutral points with the magnet placed as above. Explain. Answer the questions above with the magnet reversed. It is desired to weaken the earth's field at a given point without changing its direction. In what positions with reference to the point would you place the magnet? (Show by diagranjs three positions with the magnet parallel to the field, and indicate in each case the location of the neutral point.) 7. If a unit pole is placed near a magnet it will be acted on by both poles of the magnet, one repelling it and the other attracting it. The resultant of these two forces is the field strength at the point, due to the magnet. We can use this principle to find the direction of field at any point near a magnet as follows : MAGNETISM 225 Make a full-sized drawing of the magnet on report paper. Select a point (Pi) 10 cm. from the center of the magnet on a line perpendicular to its axis at the mid-point. Call the pole strength of the magnet m. From Pi draw to scale, in terms of m, the force acting on unit pole due to the N pole of the magnet. Then that due to the S pole. Find graphically the resultant, also in terms of m. Next take a point (P2) at 10 cm. from the N pole and on the axis produced. Then a point (P3) inter- mediate between these two. Verify the directions at Pi and P2 by comparison with (5). Verify that at P3 by the compass needle, eliminating the effect of the earth's field (see next experi- ment) . Remarks. In (3) the results obtained will be only approxi- mate. If, however, a magnet heavier than the ordinary compass needle be used, the vibrations will be much slower and hence better results will be obtained, due to the greater number of vibrations which may be counted. In most of the above tests a compass with a short needle is, desirable. Test Questions. 1. What is a magnet? How is an artificial magnet produced? What precautions are necessary in using magnets? What is the "pole" of a magnet? The "axis"? How is each pole identified? How are their positions located? 2. What is the period of a compass needle? Define "pole strength." Give two ways by which you can test relative pole strengths. Explain the formulas, mi _ Hi _ »i' — n^ J, Wi _ rf OT2 Hi n^ — ■»?■ mi Ti^ 3. What is meant by the "field" of a magnet? What do we mean by the "positive direction" of the field? When a magnet lies in a N-S direction, what is the direction of the field at all points in a plane perpendicular to the magnet's axis at its center? 4. Under the same conditions where would the earth's field and the magnet's field be in the same straight line and same direction? In opposite directions? What is a neutral point? Where are they located in the combined fields? Are there any such points in a magnet's field if the earth's field be eliminated? How should the magnet be placed so as to weaken the earth's field at a point? 5. In Sec. 7 suppose the magnet to be one of 500 units strength. What would be the resultant force on a pole of strength 60 units at each point. Pi, Pi and Pa? 6. Prove in your diagram that at Pi the' resultant field Hi = mljr^, where m is the pole strength, I the length of the magnet, and r the distance of Pi from the poles. Check with the numerical values in question S. 226 MAGNETISM AND ELECTRICITY experiment m-22 The Magnetic Field and Lines of Force Tracing Fields with Compass and Iron Filings Read first the introduction to Experiment M — 21, and Sec- tions 204-205, Chapter III. ' Part A. To Plot a Field with a Compass When a compass needle is placed in a magnetic field, the needle itself will be tangent to the lines of force. We may take the direction of the needle as the direction of the field at the axis of the needle, and we may consider (if the needle be short enough) that the needle itself coincides with the line of force along its length. If we then lay a compass on a sheet of paper in the field of a magnet, we may plot out the lines of force as follows: Start with say the blued end of the needle on a point near the magnet, mark the position of the bright end, move the compass so that the blued end shall fall over this point and mark the position of the point of the needle again, and so on. In this way a complete line of force may be mapped out, starting from one point of the magnet and always returning to another. The lines of force thus laid out will be the resultant field of that of the earth and of the magnet; for at any point the field is the resultant of two components, the earth's field, and the magnet's field. If they be different in direction, the direction of the needle will be intermediate between the two. To over- come this difficulty it is only necessary to get the two fields in the same direction. This we can accomplish by rotating the magnet until the direction of its field at a point coincides with the direction of the earth's field. Procedure. Determine, with the compass, the direction of the earth's field alone. Set up two pins connected by a thread forming a north-south line. Then fasten the magnet with wax to the center of a piece of paper, directly under this line. Mark off six points on each side about the south end of the magnet, and, starting from these points, lay out the magnet's field. Each time the compass is placed move the paper, com- MAGNETISM 227 pass, and magnet, until the needle, directly under the thread, points directly north. We then have the direction of the magnet's field at the point as well as the direction of the earth's field. Mark the position of the point of the needle and move on, rotating the paper as before. Connect the points in order and we will have twelve lines representing lines of force due to the magnet alone. These lines should all terminate at corresponding points at the north end of the magnet. If the lines run off the paper, start from the corresponding points at the north end and trace the lines until they run off the paper. Thus make a symmetrical diagram of a plane section of the field about the magnet. Plot also one line of force with the magnet stationary in the N-S line. Part B. To Plot Fields with Iron Filings If a sheet of glass is laid over a magnet and iron filings be sprinkled evenly over the plate and the latter be tapped gently, the filings will arrange themselves very nearly in the direction of the lines of force, for each particle itself becomes a temporary magnet in the field. The diagram of the magnetic field thus represented may then be sketched, or blue prints made, showing the character of the field more accurately. Special Directions. If a sketch is made to represent the arrangement of the filings aim to have it show the characteristics of the field in detail rather than a mere picture of the filings. Remember that lines of force cannot cross each other, for at one point the field cannot have two directions. Whether represented by sketch or print, show in your report, in each case, the general arrangement of the filings. Indicate clearly north and south poles of the magnets and where straight lines of force and neutral points occur. Discuss each diagram, calling attention to the characteristic arrangement of the filings and explaining the peculiarities of the field. Give especial attention to important parts of the field such as those around the poles and those between the magnets. Procedure. Give diagrams for the magnetic fields of the following : 1. Bar magnet in a vertical position under horizontal glass 228 MAGNETISM AND ELECTRICITY plate. Explain the radial arrangement of the lines. Are there any indications whether the lines are vertical or horizontal? Do lines of force enter or leave the upper end of the magnet? Test with a compass. 2. Two bar magnets, vertical, separated by about 4 or 5 cm. (a) like poles up, (b) unlike poles up. 3. (a) A two pole magnet horizontal. (b) A "consequent" pole magnet. (c) Two horizontal magnets side by side about 4 cm. apart, with like poles first in opposite directions, then in the same direction . (d) Magnet and soft iron bar. (Take care that both are close to the glass or paper and quite close together, about 2 or 3 cm. apart.) (e) Horseshoe magnet. Remarks. The iron filings will mark out the field quite distinctly if care is used in dusting them upon the glass plate. They should be sprinkled from a height of 8 to 10 in., evenly, and not too thickly. The plate should be tapped very gently and not too many times. If the tapping is continued too long, the filings will draw in and the details of the field will be lost. Test Questions. 1. Define magnetic field and lines of force. 2. What are the general tendencies of lines of force? 3. Outline the method for plotting out the field of a magnet. Show that in general this would give a resultant field. What method is used to get the field of a magnet alone? 4. In what way do you think your map would have been changed if you had not eliminated the effect of the earth's field? 5. Why do the iron filings arrange themselves so as to show the character of the field? 6. How could the intensity of a certain portion of the field of a given magnet be increased? ELECTROSTATICS 229 GROUP II— S. ELECTROSTATICS experiment s-21 Study of Electrostatic Induction Use of the Electroscope to Test the Charges When glass is rubbed with silk, the glass attains the property of attracting light bodies such as cork or bits of paper. The glass is then said to be electrified. Two pieces of glass thus electrified will repel each other, but will be attracted, for instance, by a piece of ebonite rubbed with flannel or fur. Thus there are two kinds of electrification : that like the one possessed by the glass rubbed with silk is called positive ; while the ebonite rubbed with fur is said to have a negative charge of electricity. All neutral bodies are said to have equal amounts of positive and negative electricity. When two bodies of different sub- stances are rubbed together, then separated, one body receives a greater amount of positive electrification while the other body receives a greater amount of negative. Therefore the one is said to be positively charged and the other negatively charged. In the case of metals and other conductors held in the hand, the charges are led off to the earth as fast as they are formed, thus giving no evidence of electrification. Bodies may also be charged by conduction and induction. To charge by conduction, bring the charged rod into good contact with the uncharged conductor and the latter will receive a charge similar to that on the rod. Thus the charge on an ebonite rod rubbed with fur is by definition negative. If this rod be brought into good contact with the knob of the electroscope, the aluminum leaves will also be negatively electrified on re- moving the rod and will diverge because they repel each other. Induction means the separation of the charges in a conductor, when brought near a charged body. The charge opposite in sign to that of the charged body is attracted to the end of the conductor nearest to the charged body and is held "bound," and the charge of like sign is repelled to the opposite end. But when the charged body is removed, the separated charges in the conductor flow together again and neutralize each other, thus leaving the conductor in its original condition. In order to give 230 MAGNETISM AND ELECTRICITY the conductor a permanent charge then, it is necessary to conduct off the repelled charge of same sign as that of the charged body, after which the conductor will be charged oppositely to that of the original charged body. This is most easily done by touching the conductor with the finger while the charged body is held near. Thus, the charge similar to that of the charged body, repelled, will escape to the ground, while the unlike charge will be held bound. Therefore, if the charged ebonite rod be merely brought near, without touching the knob of the electroscope, then the knob touched with the finger and the latter removed before the rod, the charge left on the leaves will be positive, al- ways opposite in sign to the inducing charge. This is called charging by "induction." When the leaves of an electroscope are charged, and a charged body brought near the top, if the leaves contract, the charge of the body is opposite to that of the leaves; if they open wider, the charges are similar in sign. Be able to explain the principles involved in all these changes. Read also Chap. Ill, Sect. 208 (a) to (f) and 207. The apparatus needed is: two electroscopes, an ebonite rod, a piece of fur, and a piece of copper wire about 25 cm. long. Special Directions. Follow carefully the following instruc- tions, noting particularly all the changes which take place in the leaves of the electroscope. Take up the sections in order, noting what happens and recording it at the time in your data book by diagram and brief explanation. In your diagrams indicate clearly the position and sign (-|- or — ) of the charges. Do not proceed to the next section until you thoroughly understand the one with which you are working. If possible study out for yourself the explanation of the observed phenomena rather than obtain help from someone else, as it is only by independent reasoning that a thorough understanding of the subject will be obtained. In your report, letter each section, describe fully ' what was done and the effect, giving and explaining the distribution of the charges and the action of the leaves of the electroscope at each step. The latter can best be shown on the diagram. Remember to discharge the electroscope after each step by touching the top with the finger, when no charge is near. Be ^^ery careful not to have too heavy a charge on the rod when it is brought near the electroscope for fear of injuring the aluminum ELECTROSTATICS 23 1 lea\'es. In general it will be necessary to remove most of the charge from the rod with the hand before approaching the rod to the electroscope. Remember that the charge on the rod developed by rubbing it with the fur is distributed along the rod ; therefore in removing most of the charge, if possible remove all excepting that at the end of the rod, as the results will be better if the charge is located at a point. Bring the rod down from above unless otherwise directed. Procedure, (a) Rub the rod very slightly with the fur and touch the knob of one of the electroscopes with the rod, making sure of good contact, and then remove the rod. (b) Rub the knob with the fur and remove the latter. Explain. (c) Bring the rod, lightly charged, near the knob, but not near enough to hurt the leaves. Touch the knob with the finger and then remove first the rod, then the finger. Repeat, only re- moving this time first the finger, then the rod. Explain the difference. (d) Charging, as in the latter part of (c), bring in turn the rod, then the fur, and the hand near the knob. Notice what happens as the object approaches the electroscope. (e) Recharge with the end of the rod at a distance of about 10 cm. Remove the rod and bring it slowly up from a distance of more than 40 cm., watching closely the behavior of the leaves as the rod approaches to about 5 cm. Is there a change of sign of charge on the leaves? Withdraw slowly and note the changes. Explain especially the condition of the electroscope (1) when the rod is beyond the charging distance; (2) when at the charging distance ; and (3) when it is within the charging distance. Is the final state of charge the same as the original? (f) Connect the knobs of the electroscopes with the wire. Call one A, the other B. Bring up the rod from the left toward A. Give diagram of charges. Holding the rod still to the left of A, touch A with the finger and remove first the finger, then the rod. Give diagram showing distribution of charges at each step. (g) Repeat, holding rod near A, only charge by touching B instead of A with the finger. Is there any difference? Explain. Hold the rod near the middle of the wire and touch A. (h) Charge lightly as before, holding the rod about 3 cm. to 3 232 MAGNETISM AND ELECTRICITY left of A. Then remove rod and bring it up to 5 cm., 3 cm., 1 cm. Compare with (e). Remarks. An insulated brass or copper ball one or two cm. in diameter may be used in place of the rod, and has the ad- vantage of retaining the charge within a relatively small region, which is more favorable to good results. However, the charge developed by rubbing the metal with fur is much less than that developed on the rubber, so that during unfavorable weather conditions it may be impossible to obtain sufficient charge for the tests. Another advantage of the metal is that, being a conductor, the charge upon it may be all removed by grounding the ball at any one point, and if touched to another conductor, the latter will be affected by the whole charge upon the first. If no results can be obtained in this experiment, or if the results obtained do not seem to be what the theory would indi- cate, try to think out the reasons and a way to remove the difficulties. Sometimes weather conditions are so unfavorable that the charges cannot be retained either upon the electroscope or upon the rod, in which case special precautions must be taken. In some cases the electroscope leaves are not sufficiently insulated, or the case may become charged. Also the charge upon the rod may be too strong or too weak. The behavior of the leaves should be watched during the whole time the rod is being brought up, in general the first effect being the one sought. A later effect may be different for some other cause than the one under consideration. (Cf. Sect, e.) Test Questions. 1. State the principles involved in the above tests regarding, (1) the source of the charge on the rod; (2) the action between like charges and between unlike charges; (3) the movement of like and of unlike charges when joined by a conductor, both when no charge is near and when another charged body is near; (4) the production of a "bound" charge; (5) the effect of grounding a conductor; and (6) the way in which the effect produced by a charged body on an insulated conductor depends upon the distance between the bodies. 2. Distinguish between charging by induction and by conduction. What is a conductor? An insulator? Describe the electroscope used. 3. Given an electroscope, an ebonite rod and a piece of cat's fur or silk, tell in detail how you would determine the unknown sign of a charge. 4. Why must the finger be removed before the rod in charging by induction? 5. In charging by conduction what is the cause of the spark sometimes observed when the rod nearly touches the electroscope? electrostatics 233 experiment s-22 The Faraday Ice Pail Experiment Relationship of Inducing and Induced Charges Read over first the introduction to Experiment S — 21. The main conclusions of Faraday's experiments, which it is the object of this experiment to prove, may be summed up as follows: 1. If an insulated charged body is placed inside a closed con- ductor, a charge of opposite sign is induced on the inner surface of the conductor and one of like sign on the outer, the two induced charges being equal in amount to each other and to the charge on the body. 2. Only insulated charges (that is charges on an insulated conductor), or charges bound by insulated charges can remain within a closed conductor. 3. Charges outside of a closed conductor cannot produce an electric field inside of the conductor, or pass, unaided, into it. The apparatus required consists of a large and a small tin can or copper calorimeter, a wire cage, cakes of paraffine, tin plates, two electroscopes, 50 cm. of copper wire, "proof-plane," ebonite rod and fur. ' Special Directions. In performing the experiment keep always in mind the three laws stated above, and note how the laws are verified at each step. Read the special directions in Experiment S — 21 and as in that experiment take up the sections in order, note what happens especially to the leaves of the electro- scope at each step and record at the time in your data book by diagram and brief explanation. In your diagrams indicate clearly the position and sign (-|- or — ) of the charges at each step, and explain their distribution. Procedure. Place the smaller can on the cake of parafifine, which serves as an insulating stand, and connect by tlie wire with the knob of the electroscope, being sure the wire does not touch any other conductor. Charge the other electroscope by induction and use it throughout to test the sign of the charges. (See introduction to S — 21.) An electrical machine may be used to furnish charges for the proof-plane. The sign of the charge may be determined by the free electroscope. 234 MAGNETISM AND ELECTRICITY (a) Completely discharge the can and electroscope. Charge the proof-plane and lower it into the can, being very careful not to touch the can, wire, or electroscope with the hand or proof- plane. Test the sign of the charge of the electroscope leaves with the charged ebonite rod. (b) Remove the proof -plane without touching the can, and test whether its charge has been changed. Also note whether there is any charge left on electroscope or can, and explain the changes, if any, that have taken pjace in the electrical condition of the system. (c) Again lower the charged proof-plane into the can, earthing the latter for a moment, remove finger, then the plane. Explain the changes at each step. Is there any change in charge on the plane? (d) Discharge the system. Lower the charged proof-plane into the can, touching it to the bottom and making sure of good contact, and then remove the proof-plane. It is essential here to observe carefully and explain the action of the leaves at each step. Is there still any charge left on the ball? What con- clusions would you draw as to the relative values of the inducing and induced charges? (e) Place the larger can on a cake of paraffine and the smaller can inside the larger, the two cans insulated from each other by another cake of parafhne. Connect the larger can to the electro- scope by the wire. Discharge the system, then lower the charged proof-plane into the inner can and remove. Describe and explain. Lower again, touch the inner can with the plane, insuring good contact. Remove. (f) Repeat, only "ground" the inner can with the plane touching inside. What would have happened if the can had been grounded before the plane touched the can, and the plane then removed without touching it to the can? (g) Surround the electroscope with a wire cage and place on it a tin plate. Connect the wire cage with the electrical machine and excite the latter until sparks can be drawn from the cage. Note the effect on the electroscope. (h) Remove the plate, discharge the cage, and lower into it the highly charged proof-plane. Ground the cage, then touch OHM S LAW 235 the cage with the proof-plane. Discuss the changes in the cage and in the electroscope. (i) Discharge and place the electroscope top close to the small cage outside. Again highly charge the proof-plane and lower into the cage. Then, ground the cage with the finger, with the plane inside. Remove both. Explain. Test Questions. 1. What are the main conclusions which Faraday drew from his ice-pail experiment? 2. Show in detail how you have proven by experiment the main con- clusions which you aimed to prove. 3. If you wished to protect a body from the effects of a nearby electrical machine or other charged body, what would you do? 4. What is the difference in effect produced by a heavily charged cloud upon a house protected by lightning rods and upon one not so protected? 5. Suppose two equal spheres of radius 2 cm., one charged with 24 electro- static units of electricity and the other with 30 units, to be separated by a distance of 60 cm. What force do they eiert upon each other? If touched together what charge will each then have? What is the potential of each before and after? (See Chap. Ill, Sect. 207 and 208g— .) GROUP III— O. OHM'S LAW experiment 0-21 Ohm's Law for Part of a Circuit Relations of V, E, I, and R Part A. No E.M.F. in Part of Circuit The difference of potential between two points on a circuit, in which current is flowing, is measured by the work done or the energy transformed during the transfer of unit quantity from one point to the other. The practical unit of potential difference is the volt. One joule of energy is transformed for each coulomb transferred between points having a potential difference of one volt; or, the difference of potential in volts (F) equals the joules transformed per coulomb transferred. V = W/Q (1) From Joule's Law the number of joules (W) transformed into heat when I amperes flow through R ohms for t seconds is expressed by W = PRt (2) 236 MAGNETISM AND ELECTRICITY Since unit quantity is transferred each second by unit current, Q = It (3) Substituting (2) and (3) in (1) V = IR (4) The interpretation of this equation is: The fall of potential (F) in any resistance (J?) through which a current (/) is flowing is measured by the product of the current and the resistance between the points where V is measured, if there is no source of electromotive force (£) in that part of the circuit. (See definition of E or E.M.F. in Part B and in Chap. Ill, Sec. 210.) The object of Part A is to illustrate experimentally the relations expressed in Eq. 4. These are V <^ I when R is constant, and F cc i? when I is constant, and V = IR when F, 7, and R are expressed in the same system of units. Procedure. Connect a source of E.M.F., a 100-ohm resistance box, an ammeter, and a switch in series as shown in Fig. 55 (a) . HI _i a_3 .13 r^h-^rt::] (aj (b) Fig. 55. Connect the voltmeter across a part of the resistance as AB in the figure (a). The current may then be varied by changing the resistance not included in the part AB. Be sure to keep enough resistance in circuit to protect the ammeter. Find the least count of the instruments and take readings in divisions. When the data for variable current is completed, adjust for constant current in the second case (b) and measure the fall of potential when the voltmeter is connected across varying amounts of resistance. Use a resistance box provided with a pair of "travel- ling plugs" for making contact with the brass blocks. Sample sets of data are given below. Connections for the second set are shown in Fig. 55 (b). OHM S LAW 237 Volts .178 .305 .560 .780 1.04 1.27 1.57 03 Amps .085 .148 .274 .380 .512 .623 .773 /XR An .300 .556 .772 1.04 1.27 1.57 Volts Ohms Amps IX R .280 1.0 .276 .276 .555 2.0 .552 .840 3.0 .828 1.11 4.0 1.10 1.40 5.0 1.38 1.68 6.0 1.66 1.95 7.0 1.93 — 0' 1 1 1 { 1 1 h 1 in OhiJs ^ :^^4 ) 1 .2 .4 1:6 2 z!* -2^ J in Amps R 1 ■a ■*» <-* > -|- tS +» -l" p. _"S- ^ ^ ^ \ N ^ V -8 l.O- 1.2 1 A. \ N N. ^ <.- \ > N ^ ^ \ ^ ^. ^ V, X> L s S ^ V, t V s \ s 1.8 2.0 \ N N, V \ Fig. 56. Plot curves as shown in Fig. 56 for the variation of fall of potential (a) when / varies, (b) when R varies, and (c) when their product varies. Interpret the results and draw conclusions. Part B. E.M.F. Included in Part of Circuit Difference of Potential (F) exists between any two points of the entire series circuit when current flows. Electromotive Force (£) exists in a generator and is measured hy its ability to maintain a difference of potential ; it equals the potential difference at the terminals when no current flows . E relates to the parts of the generator where other forms of energy are transformed into electrical energy in such a way that it is possible for the current to pass from a point of lower potential where the current enters the generator to a point of higher potential where the current leaves thp generator. Take for example a cell as a generator. Passing through the cell in 238 MAGNETISM AND ELECTRICITY the direction in which the current is flowing, two changes in potential are met: a rise in potential due to the E.M.F. of the cell, and a fall in potential due to the resistance within the cell. The potential difference (F) at the terminals of the cell then is the net result when the fall of potential (IK) in the resistance of the cell is subtracted from the total rise, or E, of the cell; or V = E- IR (5) This equation holds true also for any part of any series circuit. Let V be the potential difference between any two points a and b limiting a part, of which the sum of the resistances is R, through which I amperes ,flow from a to b. If the part includes an electromotive force (£) which tends to send current in the same direction as /, V is the algebraic sum of the rise of potential (+ £) in the generator, and the fall of potential (— IR) in the total resistance of the part. If E is greater than IR, the potential of b is higher than that of a; but if the fall exceeds the rise of potential, the potential of b is lower. The electromotive force is negative (relative to the current) if it opposes the current. The current flows in the direction of the resultant E.M.F. (2£) of the whole circuit. If the part of a circuit includes a negative electromotive force (— £), the fall of potential in that part is increased by that amount, and the total fall is — (S + IR). In this case the voltmeter reading is greater than the E.M.F. For example, in charging storage cells current is forced through them in opposition to their E.M.F. The impressed difference of potential (F) must exceed their E.M.F. (E) by the amount IR. If V is increased, the charging current will increase. If the part of a circuit is extended to include the whole circuit, a and b coincide and F = 0, in which case E = IR (6) for the whole circuit. This is a special case included in the general case expressed by Eq. 5. Another special case is when £ = 0. Then V = IR as illustrated in Part A of this experiment. The object of this part of the experiment is to illustrate graphically, from experimental data, what is expressed alge- braically by the general equation V= E- IR (5') OHM S LAW 239 t V and E may be measured by use of a voltmeter and I by use of an ammeter. Then the total R of the part may be computed from Eq. 5. To check the mean value of R found, and to verify the law expressed by the equation, R is found in two other ways, as described below and illustrated in Fig. 59. For the whole circuit, F = and E = IR (6') Therefore, since £ is a constant, I varies inversely as i?; or R ^ l/I (7) Compare the above discussion with that given in Chap. Ill, Sec. 220. Special Directions. To make the results of the experiment clear, it is very important to keep in mind when taking the data , just what relation is being verified, what the variable quantities are, and which quantities it is necessary to keep constant. If a quantity which is supposed to be constant cannot be kept so, consider what variation in this quantity is permissible. In your report you should explain clearly just how each curve is plotted and why it is plotted in a particular way if different from the ordinary. Also carefully interpret each curve, especi- ally the straight line curves, pointing out some conclusions which may be drawn from each. Show how the different equations expressing Ohm's ^'i[-4l'i|-^-H-^'WW\/ law are verified. '> / ^y Observe the directions given in Chap. Ill, Sec. 227. You should show clearly how you obtained computed results' by giving a sample of the computations. Fig. 57. (See Chap. I, 8 (f)). Procedure. Connect in series two cells of high internal resis- tance, (Ci, Ci, Fig. 57), an ammeter (A), a variable resistance {Rb) , and a switch (S) . Connect the voltmeter ( V) in shunt with the cells and the ammeter. Observe the data necessary to verify equations (5) and (6), one quantity in each equation being con- stant. These data are indicated by the headings of the first, second, third and sixth columns of Table LI. Plot curves similar to those shown in Figs. 58 and 59. Table LI will contain all the 240 MAGNETISM AND ELECTRICITY data required to plot these curves. The coordinates to be used for each are shown in the figures. The unknown resistance in the rest of the circuit outside the resistance Rb is given by the intercept on the resistance axis of the curve (Fig. 59) in which the known box resistance {Rb) and the reciprocal of the current (/) are the coordinates. This is i 5 U La .^ ^ -^ u f tc t- __ _ •>- f^ ._ _ (, l-S 6 a ^ ..-' 1 4 '■". ^ ^ ^ ■^ s ■^ g , ^ T "& K Ri2£|^ ♦ r'a rob —i 2 [ff K HiEdCTAtcE ^^rtt Fig. 58. F;g. 59. practically the same as the resistance between the points a and h (Fig. 57) (if the resistance of the leads and switch is neglected), and is essentially the internal resistance of the cells if the resist- ance of the leads and ammeter is negligible. Find also the resistance between a and h in another way. When the fall in potential in the box equals one-half -E the resistance in the rest of the circuit equals that in the box; or, i?' is practically equal to the value of Rh when F equals one-half R. After the curve in which F and Rh are used as coordinates (Fig. 59) is plotted, find the ordinate which is equal to one-half R. The corresponding abscissa gives the value of R! approximately. Compare the mean of the two values of R' found as above with the average value of R computed from V = K — IR (Table LI), and draw conclusions. Remarks. The observed data for scale readings should be expressed in terms of small divisions of the scales, regardless of the value of a division. The amperes and volts are derived data depending upon the "least count," or value of a small division. In general in one part of any experiment it is desirable to keep all the quantities constant excepting two; that is, we wish to deal with only two variables at a time. (See Chap. I, Sect. 34.) Notice that in Part B of this experiment data are taken with three OHM S LAW 241 TABLE LI Ohms Am- Volt- in meterfimetert Amp. Volts E.M.F. R from 1 Fall in Box Scale ! Scale U) (V) (E) V=E-IR / R' (IR') (^) Div. Div. 30.1 1 1.8 .301 .036 2.16 7.09 3.32 2.140 1 26.5 14.6 .265 .292 7.08 3.37 1.884 2 23.5 24.7 .235 .495 2.16 etc. 4.25 1.671 3 etc. ' etc. etc. etc. 2.17 etc. etc. 4 6 etc. 10 * 7.15 15 20 QO Av. = 2.17 Av. = 7.11 * Connection changed from 1500 to 150 milli-ampere scale. t Least count of 1500 scale = .01 amp. Zero reading = 0. Least count of 150 scale = .001 amp. Zero reading = 0. t Least count of 3-volt scale = .02 volt. Zero reading = 0. quantities variable. Usually it is rather difficult to find from experimental data the relations between three quantities all of which vary at the same time. This complication is avoided in this experiment by plotting the data for a part of the circuit within which only two of the quantities vary. Thus in Fig. 58 the curves are plotted for the part of the circuit between a and b within which E and R are constant, and since the curves are straight lines we can easily obtain the relation between the two remaining variables. In Fig. 59, when the upper curve is extended to the X axis the whole circuit is included, in which the only two variables are / and Ri,. In the lower curve the third variable is not eliminated and the graph is curved because I varies. It is a regular curve because the variation of / is defi- nitely related to the variation of Rb. If I had been kept constant this graph would have been a straight line. Test Questions. 1. Derive and interpret the equation V = IR. Does this equation apply to a part of a circuit including an E.M.F.? 2. Explain the significance of the fact that the two curves in Fig. 58 are parallel 3. Define electromotive force. Does this term apply to any part of a circuit outside of a generator? What is the measure of the E.M.F. of a generator? Define difference of potential. 4. Derive and interpret the equation V = E — IR. What determines whether E and V are positive or negative relative to /? Why is the E.M.F. of the cell the V between the terminals when no current is flowing? 242 MAGNETISM AND ELECTRICITY 5. Explain what the vertical distances between your V — I and E — I curves represent, and the conclusion from the fact that the sum of any one of these vertical distances and the corresponding ordinate of the V — I curve is equal to E. 6. How are current and total resistance related in a series circuit in which E is constant? How is the relation derived? 7. Explain why the resistance between a and 6 is equal to the resistance in the rest of the circuit when V in ab is equal to one-half E. 8. How is the difference of potential at the terminals of a battery affected by increasing the discharging current? Upon what does the internal loss of energy depend? Into what form of energy is it converted? 9. If the E.M.F of a cell is 1.3 volts and the internal resistance is 4.2 ohms, what is the difference of potential at the terminals when the current is .35 amp.? What current will it give on short circuit? What would be the current on short circuit if the internal resistance were .07 ohm? 10. What P.D. must be impressed to send a charging current of 6 amperes through a battery of 10 cells against an E.M.F. of 2.1 volts per cell if the internarresistance is .1 ohm per cell? Explain. experiment 0-22 Study of Primary Cells Comparison of Various Types Definitions, and Theory of Primary Cells. A salt, as copper sulphate (CUSO4), or an acid, as sulphuric acid (H2SO4), in solution is supposed to be separated into parts called ions, each carrying an electric charge, the metals and hydrogen being positively charged and the acid components negatively charged. Thus the ions into which the CUSO4 is separated are the + Cu and - (SO4); the H2SO4, + H, + H, and - (SO4). While ionized the substances do not manifest their ordinary physical properties. But if the electric charge is removed in any way the parts are no longer ions, and the H appears as a gas, or the copper as a metal, etc. A Primary cell is a contrivance for the conversion of chemical energy into electrical energy. It consists of two pieces of unlike metals (or one may be carbon) immersed in either an acid, an alkali, or a solution of a salt. The solution, called an electrolyte, attacks the two metals, called electrodes differently. At each electrode there are three tendencies, existing in different degrees, as follows : (a) The tendency of the metal to go PRIMARY CELLS 243 into solution (called solution pressure) ; let it be denoted by 5. (b) The tendency for the metal to be thrown out of solution and deposited on the electrode with which it is in contact (called os- motic pressure) ; let it be denoted by D. (c) The tendency due to electrostatic action between the charges on the ions, and those on the electrodes (called electrostatic pressure); let it be expressed by e. The tendency 5 opposes D, and when their difference is balanced by e, the three operations (solution, deposition, and transfer of charges) will cease. If, however, anything is done to isturb this balance, one or all of these operations will go on. If we take, for example, Zn and Cu elements in acidulated solutions of ZnS04 and CUSO4 respectively, the solutions being in contact through a porous cup, 5 is much greater than D at the zinc terminal, and the converse is true at the copper terminal. Therefore, solution of zinc will proceed until e' (due to the electrostatic field between the negative charge on the zinc and the positive charges on the neighboring ions) is large enough to balance the excess of S over D; and deposition of copper on the copper electrode will proceed until e" (due to the electrostatic field from the positive charge on the copper) is large enough to balance the excess of D. When this condition of equilibrium is reached, the zinc is negatively charged by the removal, during solution, of ions conveying positive charges, and the copper is positively charged by the copper ions deposited. Their difference of potential is a maximum. This maximum P.D. is the measure of the E.M.F. of the cell. This E.M.F. is due in part to the excess of S over D at the zinc terminal tending to transfer a positive charge from a point of lower potential to one of higher potential, and in part to the excess of D over 5 at the copper terminal, which in turn passes the positive charge to a point of still higher potential. This work is done by the transformed chemical energy. The P.D. at the terminals will be reduced if the terminals are connected through an external circuit so that current can flow from the + Cu to the — Zn, and the processes of solution of zinc and deposition of copper will go on with a rapidity depending upon the reduction of e' and e", or upon the fall of potential at the terminals due to the flow of current. 244 MAGNETISM AND ELECTRICITY The maximum difference of potential on open circuit, called the E.M.F., is characteristic of a given kind of cell, prepared according to formula, and may be obtained approximately by means of a high resistance voltmeter. If the Cu and Zn terminals had been placed in a solution of H2SO4 instead of the salts, a fall in the value of the E.M.F. would be noted, for bubbles of gaseous H will collect on the electrode instead of Cu, and thus not only prevent further chemical action, but by their electrostatic action cause a back E.M.F. Attempts to overcome this polarization effect, as it is called, account for the different types of cells. In general some oxidizing substance, either liquid or solid, is used to get rid of the H. Sometimes impurities in the zinc cause a P.D. between two points on this metal and consequently a useless local current is set up. This is called local action and is largely eliminated by amalgamating the Zn with mercury. It can readily be seen that the E.M.F. of a cell depends only on the kind of electrodes and electrolytes, and not upon the size of the electrodes. Therefore, if several like cells be connected in parallel, i.e., all the Zn terminals connected together and the Cu terminals together, the E.M.F. will be the same as that of one cell. But if connected in series the E.M.F. will be additive. For current, however, cells in parallel will give a greater value than for one cell, for the resultant resistance will be less. By Exp. O — 21 for a complete circuit 7 = 5^ XR For one cell with internal resistance, r, and external, R, E I = {R + r) Write the equations for current through n cells connected (a) in parallel, (b) in series, also show that for m equal branches having n cells in series in each branch, where mn = TV, the total number of cells, the current nE E I = m n m PRIMARY CELLS 245 It can be shown mathematically that we shall have maximum current when Rjn = rim or when the external and internal resistance are made as nearly equal as possible, or i? = ^' = ~N or RN r (2) It can be seen that the connection for maximum current depends upon the relative values of the external and the internal resist- ances. Special Directions. Notice that in each of the following sections some data are to be taken, and that all these data cannot be included in Table LII. A good plan is to take up in your report the sections in order, giving the observed data and dis- cussing the conclusions to be drawn from these results. Procedure, (a) Connect a high resistance voltmeter across the terminals of a cell having dilute H2SO4 for electrolyte and Zn -f Cu electrodes. Take the E.M.F. immediately on plunging the metals into the solution, and note the effect of polarization. (b) Take the E.M.F. of the different types of cells, note whether they are one or two fluid cells, note the elements, the depolarizing agent, and the degree of polarization. (c) Connect the cell, a 10-ohm resistance, and a mil-ammeter in series. Read the current and also the P.D. at the terminals and compute the internal 'resistance, r from V = E — Ir. Check by £ = /(i? -|- r) for the whole circuit, where i? = 10 ohms (-|- the resistance of the ammeter and connections, which should be negligible) . In this way compute r approximately for each kind of cell. Also compute the current on short circuit. Tabulate data as follows: TABLE LII Electrodes Electrolytes Depolarizer E.M.F. r Current Cell Through 10 Ohms Short Circuit 246 MAGNETISM AND ELECTRICITY Do the same with a given cell only using different sized electrodes. (d) In one of the cells, as the Daniell, investigate with Au tips, the P.D. of different points in the solution, Zn to the solution next to it, Cu to the solution next to it, etc. (e) Take six similar cells; test the E.M.F. of each and see if they are approximately equal, then connect them in various ways, — series, parallel, and multiple,- — noticing the combined E.M.F. and taking the value of current in each case both when connected in series with two ohms and with fifty ohms. Before completing the circuit make sure the current will not be greater than that which would give the maximum reading on the scale of the ammeter. Test Questions. 1 Discuss the principle of the primary cell. 2. Define polarization and local action and tell how the effect of each is more or less fully eliminated. 3. Given twelve cells of each variety used in this experiment, compute by (2) how they could be connected in each case for maximum current both for an external resistance of 2 ohms and for 60 ohms. 4. What would you conclude from (d) in regard to the distribution of the E.M.F. in a cell? 5. Compare the values of current obtained in each case in (e) with that computed from Eq. (1). 6. What is the essential difference between a "closed circuit" and an "open circuit" cell? GROUP IV— C. MEASUREMENT OF CURRENT BY ELECTROLYSIS experiment c-21 The Hydrogen Voltameter The Electrochemical Equivalent of Hydrogen Read the first paragraph of Experiment O — 22. The theory stated there is that a compound when brought into solution is broken up into its ions each with a positive or a negative charge upon it ; and that when this charge is removed the particle ceases to be an ion and exhibits the characteristic properties of the substance. It has been found that a solution will not conduct electricity unless there are free ions in it. If two plates, not acted upon by the solution, are inserted in an ionized solution and connected ELECTROLYSIS 247 to the terminals of a generator, one being thus positively charged and the other negatively, current will pass through the solution from one plate to the other. The + ions in the solution will be repelled from the positive plate and migrate to the negative plate, where they will give up their charges and cease to be ions, appearing there as the substance. Likewise the — ions will pass to the positive plate, the other part of the compound appearing at this point. Thus we say the solution is decomposed. The process is called electrolysis and the solution an electrolyte. The transfer of electricity through an electrolyte differs from its transfer through a conductor in a way somewhat analogous to the difference between the transfer of heat by convection in gases and its transfer by conduction in metals. When the two terminals connected to a battery are inserted in an electrolyte, the ions of the decomposed substance convey the charges of electricity between the two terminals. These terminals are called electrodes. The one by which the current leaves the electrolyte is called the cathode (or negative electrode) ; the one by which it enters the electrolyte is called the anode (or positive electrode). The ions carrying the + charge to the cathode are called cations; those carrying the — charge to the anode are called anions. Thus, if a solution of silver nitrate be the electro- lyte, Ag is the cation, and NO3 the [anion. Since the metal carries the positive charges and the acid radical carries the negative charges, the Ag is deposited upon the cathode and the NO3 liberates O from the solvent, water, at the anode. In the so-called electrolysis of water, as in this experiment, it is really the sulphuric acid with which the water is acidulated which is broken up; the H2 being given off at the cathode and the SO4 liberating O from the water at the anode, the quantities being in the same proportion in which, they exist in pure water. Faraday summed up the results of his experiments on elec- trolysis as follows: I. The amount of decomposition is proportional to the quantity of electricity which passes. II. The muss of any substance liberated by unit quantity of electricity is proportional to the chemical equivalent of the substance. The first law may be expressed in the form of an equation, thus: 4 248 MAGNETISM AND ELECTRICITY M o, Q, or M = zQ = zit (1) where M is the mass liberated; Q, the quantity of electricity transferred; I, the strength of current; t, the time; 2, a constant, called the electro-chemical equivalent, which may be defined as the mass of a substance deposited per unit quantity of electricity transferred — usually the coulomb — or per unit strength of current, the ampere, in unit time. For the second law, a Z oc - V where a is the atomic weight, v the valence, and ajv the chemical equivalent of the substance. Therefore if H is the proportion- ality factor, z = H- (2) V and since the atomic weight of hydrogen is practically one, and its valence is unity, H equals the electrochemical equivalent of hydrogen (.00001036 g. per coulomb) and z = .00001036 - = 5T^- (3) V 96500 V for any elementary substance. A gram equivalent of any element carries 96500 coulombs of electricity. It can readily be seen that if z for any substance be known, and the mass M liberated in a certain time, t, be measured, the mean value of the strength of current I can be calculated. Such an apparatus, used for the measurement of current, is called a voltameter. The object of this experiment is (1) to prove, in accordance with Faraday's first law, that when an electrolyte is decomposed by the passage of an electric current (a) the rate of decomposition is proportional to current strength, (b) the amount decomposed by the passage of unit quantity (one coulomb) is a constant; and (2) to find this constant, the electro-chemical equivalent, in the case of hydrogen, and from this to compute the electro- chemical equivalents of other substances by the second law (Eq. (2)). Special Directions. Extreme precaution must be observed in ELECTROLYSIS 249 this experiment to avoid short circuiting the lighting circuit and also to prevent injuring the ammeter. Do not insert the plug until all the other connections have been made and do not close the switch until the circuit has been examined and you are assured that the current cannot pass from one lead to the other excepting through a resistance. Do not change connections while current is flowing nor put the ammeter in circuit until connections have been found by trial to be correct. (See Chap. Ill, Sec. 225.) Also see that the current passes in the right direction through the ammeter. The direction of the current may be determined by a compass. Procedure. Join in series the hydrogen voltameter, an ammeter, and a bank of incandescent lamps which are so arranged that they can be connected all in series, all in parallel, or in any intermediate arrangement. Connect this combination through a suitable switch, after connections have been approved by an instructor, to the 110-volt lighting circuit. By various arrange- ments of the lamps obtain five values of current varying from .3 to 1.5 amperes. In each case note the exact time required to evolve a suitable quantity of the mixed gases (oxygen and hydrogen), collecting the gas in a gradiiated tube or flask over water by displacement. Take ammeter readings every minute during a run and average for the mean value of current during the run. Record data as follows : TABLE LIII Runs II III IV Reading on Tube at Beginning . . Reading at End Time at Beginning Time at End Average Ammeter Reading Height of Water in Tube at End . Gas Evolved (c.c.) Duration of Run Average Current Corrected Pressure Standard Volume of H Mass of H Mass per Second Grams of H per Coulo mb Ammeter No Zero Reading . Barometer Temperature . Vapor Pressure. 250 MAGNETISM AND ELECTMCITY Remarks. The volume of the hydrogen under standard con- ditions would be FX [273/(273 +<)] X/'/760; where p = 2/3{b ± /s/13.6 — e), b being the barometer reading, h the height of water in the tube, e the vapor pressure over the water (about 17.3 mm. at 20° C), p the partial pressure of the hydrogen, and V the volume of the gases read from the tube. 1 liter of H weighs .090 g. If two or more runs are made without refilling tube with water, corrected values for the volume of H under standard conditions should be computed for the total volume of the gas in the tube, at the end of each run, from which the standard volume of the previous run or runs may be subtracted. Test Questions. 1. Define electrolyte, cathode, ion. What is a voltam- eter? For what is it used? 2. What are the objects of this experiment? 3. Express the laws of electrolysis in the form of algebraic equations and interpret. How have you proven these laws? 4. Why are incandescent lamps used in this experiment instead of resistance boxes? Suppose the largest amount of heat that could be radiated from any coil in the box would be that due to 2.5 watts, what would be the highest potential difference that could safely be applied across a 200 ohm coil? A 2 ohm coil? 5. Explain the corrections that must be applied to get the volume of hydrogen under standard conditions. 6. What are some of the principal sources of error in this experiment? Would some of these affect the amount collected of one of the gases more than that of the other? 7. How many standard c.c. of H will be evolved by one coulomb? 8. Compute from formula (2) the electrochemical equivalent of copper and silver from that of H. 9. Show how you could use a hydrogen voltameter to calibrate an ammeter. experment c-22 The Copper Voltameter Determination of a Galvanometer Constant Current strength may be measured by the voltameter, in which use is made of the electrolytic effect of the current as in the last experiment. See Exp. C — 21. Use is also made of the magnetic effect of the current in its measurement by means of ^ c : T U s ELECTROLYSIS 25 1 the galvanometer. The essential parts of a galvanometer of the D'Arsonval type is the coil C (Fig. 60) suspended by a very light conducting suspension between two magnetic pole pieces, iV, S- Note that the coil C carries current perpendicular to the field from N to 5, and that the right-hand side will be pushed toward the observer, and the left-hand side away according to Ampere's law of thrust stated and explained in Chap. Ill, Sees. 216 and 211. These forces form a couple which twists i the fine wire suspension. Evidently the couple, 1 and consequently the deflection of the coil, as y noted by the reflection of a scale in the mirror M, will depend upon the strength of the current sent through C. If the current is reversed, the deflection will be reversed. These are very sen- sitive instruments and must be treated with great care. Only a few millionths of an ampere should be allowed to pass through the coil, consequently pig. 60. ordinary currents must be cut down by large resis- tances or else a shunt or by-path be provided in parallel, so that only a small fraction of the total current can pass through the ga vanometer itself. The object of this experiment is to determine by means of the copper voltameter the current which must .be passed through a sensitive D'Arsonval galvanometer in order to give a scale deflection of 1 cm. This value of current might be called the constant of the galvanometer, and since in such a galvanometer the deflection is (nearly) proportional' to current, we could use this constant to find the value of an unknown current passing through the galvanometer by multiplying the constant by the deflection (in cm.) produced by the unknown current; thus I = hXd (1) where I is the unknown current, /o the galvanometer constant, and d the deflection produced by /. Conversely, by knowing I and d we can find /o. Procedure. Prepare the voltameter as follows: Prepare the CuS04 solution in the following proportions: to 100 gm. distilled water add 15 gm. copper sulphate, 5 gm. sulphuric acid, and 5 gm. alcohol. Make enough of this solution, probably about 252 MAGNETISM AND ELECTRICITY 600 cc, to fill the glass jar to a convenient depth. Polish up the electrodes of sheet copper or coils of heavy copper with fine sand paper, and wash and dry with alcohol, being especially careful of the cathode. Weigh the cathode very accurately, being careful not to touch the surface with the fingers. For a source of current use two storage cells in parallel. Connect this battery in series with the voltameter, a switch, and a standard ohm (using a false cathode for trial adjustment). The current flowing through this circuit is much too large to pass thrgugh the galvanometer. Conse- quently a "step-down" arrangement may be used, as follows: In parallel with the standard ohm S (Fig. 6i) are two resist- ances, i?i (= 1000 ohms) and R2 {= 10 ohms). If the shunt {Ri + Ri) were not there the current flowing through the standard ohm would be /. = F,/l = V; (numerically) (2) where V, is the potential difference between a and b. Since (J?i + Ri) is very large compared with S, this difference will remain practically the same when the shunt is connected, and hence the total fall of potential through the resistance (.Ri + R2) is Vs and the fall through R^ is In parallel with R2 is the galvanometer G and a high resistance R (several thousand ohms). Applying the same reasoning as above and noting that {R + Rg) is very large compared with R^ we have current, Ig, flowing through R and the galvanometer given by '' = R^g (^) Hence, by means of the voltameter, A, I^ may be found. From /, we may obtain V, by (2). Knowing F,, we can find Vi by (i) and from V2 compute Ig by (4). With the apparatus connected as shown in Fig. 61, close the circuit (using false cathode) and determine proper direction of current by compass needle. For a rough check on the value of current, insert an ammeter in the main circuit. Then with R ELECTROLYSIS 253 = 10,000 ohms, close K' . Adjust R until the galvanometer reads 1 2 or 15 cm. deflection. Now remove ammeter, insert the weighed cathode, and let the current run for about 1 hour, noting the exact time when started and when stopped. Take galvanometer read- Fig. 61. ings, both direct and reversed, every two minutes. Take average as the mean value of " d" (equation 1). Finally, wash, dry, and weigh cathode; wash first with distilled water, then with alcohol. Record data as follows: TABLE LIV Galvanometer No Weight of cathode before Weight of cathode after Gain of cathode ( = M) Total coulombs passed Beginning of run End of run Duration of run (seconds) Coulombs per second ( = 7») Fall of potential through 5 ( = V,) Ri= , i?2 = ,R= Fall of potential through i?2 ( = Vi) Current through galvanometer (= Ig) Average deflection of galvanometer Current per scale division Galvanometer resistance R„ will be given by instructor. Electrochemical equivalent of copper, 32.94 (10)~' gm. /coulomb. Test Questions. 1. Describe briefly the D'Arsonval galvanometer. 2. Define the galvanometer constant, and outline the method of deter- mining its value with the voltameter. 3. Describe the preparation of the copper voltameter, including the pre- cautions necessary. 4. Upon what does the rate at which the copper is deposited depend? 5. Where does the copper come from that is deposited upon the cathode? Does the anode lose in weight during the process? 6. What error would be introduced in your result if 5 were .1 ohm greater than the accepted value? If a mistake of .1 g. were made in weighing? 254 MAGNETISM AND ELECTRICITY 7. If the electrochemical equivalent of hydrogen is 1036 (10)"* gm. per coulomb, find how many c.c. of H would have been liberated had you used a hydrogen voltameter instead of the copper voltameter? 8. Find the electrochemical equivalent of calcium (atomic weight = 40, valence = 2). GROUP V— R. MEASUREMENT OF RESISTANCE experiment r-21 Comparison of Resistances by Fall of Potential Variation with Length, Cross-section and Material If we maintain a difference of potential between two points L and S {Vi and Fj), and these points be connected by a con- ductor, a current of electricity will flow from the point of higher potential to that of lower potential. If a constant difference is maintained, we have a constant current (/). If the P.D. is in- creased, the current is increased, and I ^ (F2 — Fi), or, as stated in Ohm's Law {Vi — Vi)/I = a constant for a given part of a circuit, which is called the "Resistance" of the circuit. Or we may say (Fa — Fi) = IR. Now if I is kept constant, as is possible in a given circuit containing a constant source of E.M.F., then (F2 — Fi) «: R. It is possible to measure the P.D. between different points on the uniform wire of the battery circuit, and it is the object of Part A to show that this P.D., and consequently the resistance, varies directly with the length of this uniform wire. For this purpose we make use of a Wheat- stone bridge merely for the sake of the graduated wire. Part A. Relation between Resistance of a Wire and its Length Procedure. For the use of the galvanometer, see Exp. C — 22. Connect the apparatus as shown in the diagram (Fig. 62). B is the Wheatstone bridge. At C are two constant cells in series with a resistance r which is used to help keep the current con- stant. Sw is a switch kept open except while readings are being taken. This main battery circuit then will be completed through the uniform wire LM. Connect the galvanometer (G) terminals through the protecting resistance R to the reversing key RK, and have one terminal of this permanently connected to one end, RESISTANCE 255 L, of the wire to be investigated. The other terminal is to be connected with the sliding contact 5. Now the resistance of the galvanometer circuit, with R equal to about 5000 ohms, will be so great that the variation of the current in the main circuit due to changing 5 will be almost infinitesimal; but the variation in the current in the galvanometer rH]|-Q \—A/mm^ — -g- L B A M 10 20 30 40 50 60 70 80' '90 100 O 6 o o Fig. 62. of high sensibility will be easily measured. The resistance of the galvanometer circuit being practically constant, the deflections of the galvanometer will serve to measure the P.D. between L and 5, and, since the current in the main circuit is practically constant, (F2 - Fi) cc R. Therefore these deflections will show the variation in resistance. Do not start to take readings until your connections have the approval of the instructor. First adjust r so that you get a galvanometer deflection of about 20 cm. for a length LS of 100 cm., and then keep r constant for the rest of the experiment. Take readings direct and reversed, placing 5 successively at 20, 40, 50, 60, 80, and 100, and then repeat in reversed order. Why? TABLE LV R, in galvanometer circuit = ■ ■ ■ ohms; /, in battery circuit = • • : ohms Reading of point where wire is attached to terminal L = ■ ■ ■ cm. Reading of Sliding Contact 5 Length of Wire LS Galvanometer Readings in Cm. Original Zero Right Left Mean Returning Zero Right Left Mean Mean Defl. Draw a curve showing the relationship between resistance and length of wire. 256 MAGNETISM AND ELECTRICITY Part B. Relation between Resistance of a Wire and ITS Cross-Section. Specific Resistance The electrical resistances of equal lengths of uniform wires of the same material vary inversely as their cross-sections (R cc 1/s), or inversely as the square of the diameter {R « l/d^). Interpret R '-\ I [^] l^-www^— -^ E.S. Fig. 63. Evidently the resistances of different wires of equal length and cross-section would vary with the material. The constant p of the above equation, called the specific resistance, depends upon this fact. Specific resistance may be defined as the resistance offered per unit length of a substance of unit cross-section. One system takes p for the length of one foot and a cross- section of one circular mil (having a diameter of .001 in.), called the mil- foot; another system takes p for the cm. length and a sq. cm. cross-sec- tion. One object of this experiment is to determine relative values for this quantity. The method of this experiment is based upon the principles of the preceding part. In this case a board is used, having several different wires of equal length; a single copper wire, a single small copper wire, a double copper wire, an iron wire, and a German silver wire (Fig. 63). Procedure. Connect all the wires of the board in series in the main circuit as illustrated so that the current may be constant. Then connect the galvanometer to LS, LS', etc., in succession, noting the deflection, right and left, each time, then return. Measure the diameters of each wire carefully with micrometer calipers, and find the length of any one. Taking first the large and then the small copper wires, — single and double, — show that the R oc l/d^ or R cc \/s. Having three values and the origin a curve may be drawn to show this relation. Show by formula that if the lengths and cross-sections vary, RESISTANCE 257 pIp' = RIR' X Vslls'. Knowing 5 and s' , that V = I, and finding the ratios of the resistances from your data, compute the ratio o'f the specific resistance of iron to that of copper and the ratio of the specific resistance of German silver to that of copper. Test Questions. 1 . What right do we have to take galvanometer readings in this experiment as proportional to the resistance? 2. \A'hy is it necessary to keep the current constant in the battery circuit? 3. Why do we need to adjust the resistance in the battery circuit first, so as to get about full scale deflection? 4. In what way do we overcome the possible error due to the running down of the battery during a set of readings? 5. Interpret R = p X l/s, and show how your results proye the equation. 6. Define the two units of specific resistance. 7. Show in detail Ijow you find the ratios of specific resistances of the wires. 8. Taking the specific resistance of copper as 1.56 (10)~^ ohms per cm.', find the specific resistance of the German silver and iron wires. In what ways could you account for a deviation from table values? 9. Find the actual resistance in ohms of the German silver wire, taking the specific resistance of copper as 10.8 ohms per mil-ft. experiment r-22 Determination of Resistances in Parallel and in Series Resistance from Current and Potential Difference In this experiment the fall of potential method is to be used. It is based upon the fact that whenever a current passes through a resistance there is a fall of potential (F) in the resistance that is proportional to the resistance (i?) and the current (I), or F oc 7i?. In any one system the units are so defined that the proportionality factor is unity. In the practical system of units V (in volts) = / (in amperes) X R (in ohms). (See Exp. O — 21.) Procedure. Connect in series a cell having a low internal resistance, a ten-ohm standard resistance, and an ammeter. Note the ammeter reading, and find I, then connect a low reading voltmeter across the terminals of the resistance and find F. Test whether V = IR for two or more values of /, obtained by putting additional resistance in the circuit. In this case each of the three quantities is supposed to be known. If the apparatus is not properly calibrated, one or more of the quanti- ties may be incorrect. 258 MAGNETISM AND ELECTRICITY If for the known resistance R = V/I, it is probable that any other resistance (unknown) could be found from the values of V and I found in like manner. Find the resistance of each of the four coils provided for this experiment. (Some sets are made of german silver wire No. 23 B. and S. The lengths are 3, 4, and 5 meters.) Observe the necessary data to show: (1) That the combined resistance of two or more resistances in series is the sum of their separate resistances ; (2) That the reciprocal of the combined resistance of two or more resistances in parallel is the sum of the reciprocals of the separate resistances; (3) How to combine resistances in mixed series and parallel arrangement. Find the resistances when two, three, and four resistances are connected in series, when two, three, and four resistances are connected in parallel, when three in series are in parallel with the other, and when two in series are in parallel with the other two. Compare them with the values computed by applying the principles stated above. Make a list of ten possible combinations in the order of magnitude of combined resistance. Designate the resistance by the letters A, B, C, D, in order from the binding posts. If preferred, the list of combinations may be divided into three parts as follows: (1) series; (2) parallel; (3) mixed con- nections; each set to be in order from large to small. Show each arrangement by a diagram. Note that the combined resistance of two or more resistances in parallel is always less than the smallest one. With some convenient value of current through the four resistances in series, observe the fall of potential through each resistance separately. Add these together and compare it with the observed fall of potential through all. Do the same with a set of four incandescent lamps connected to the same cell. Describe the discrepancy in results and explain. Test Questions. 1. On what fact is this experiment based? Define the units involved. 2. What are the essential characteristics for a voltmeter and for an am- meter? Explain. 3. What are the two general laws of resistance to be proved? How does your data accomplish this? RESISTANCE 259 4. What is ' the combined , resistance of 100 220-ohm lamps in series? in parallel? If connected in parallel on a 110- volt circuit, what total current would flow? What current through each lamp? experiment r-23 Study of the Slide Wire Bridge Resistances Separately and Combined If four resistances, ri, r-i, n, and n are connected as shown in Fig. 64 in an electric circuit, it may be shown that when there is no deflection produced on closing the circuit through the galva- nometer G, Ti : r2 : : rs : n The resistances are so adjusted that there is no deflection of the galvanometer and hence no current through it. The points C and D are then at the same potential. Therefore the potential drop through ri is the same as that through rs, and through fi the same as that through Ti. Then if ii, i^, iz and i^ are the currents in f], r-i, rz and ^4 respectively show that Also show that iiTi = iiH and iiT^ = Uft, ii = ii and ^2 = ii Using these results derive the equation given above. ^WAA MAM 1. ,s 1 I . I . f" , I , i=t F=r jMA^IAK ^^ Fig. 64. Fig. 65. The This principle is applied to the Wheatstone bridge, connections are shown in the diagram, Fig. 65. By comparing Fig. 65 with Fig. 64, the correspondence can readily be traced, r^ is a known resistance, ri the unknown resistance, and the resistances of the lengths of wire, h and h, correspond to the resistance n and u of the diagram. Thus we can show that when there is no deflection in the portable 26o MAGNETISM AND ELECTRICITY galvanometer G, ri = r2 X hlh, from which ri can readily be computed. Make the connections for r-i and r^ as short as possible. Procedure. Part A. To Measure an Unknown Resist- ance. In this experiment find the resistances of two telephone receivers, or other resistances provided. First find the resistance of each, separately, and then when connected in parallel, and then in series. After connecting the apparatus as indicated in the diagram put a large resistance at R during the preliminary adjustment to protect the galvanometer; but when the final adjustment is to be made, R should be reduced until it is cut out altogether. Why? For the preliminary adjustments change Ti, the knowji resistance, until the bridge is nearly balanced with 5 on the middle third of the wire. Then vary S until there is no deflection of the galvanometer. Then ri = rzih/h)- In taking readings, the main. circuit should be closed first before contact is made at S. Why? Make at least two independent determinations of each unknown resistance, using different values for r^. From the values of the two resistances found separately compute what the resistances should be according to the formula, in parallel, and in series, and compare with your values found by experiment. Part B. Manxe's Method of Measuring Resistance of A Battery. The same apparatus is used as in Part A except that the battery replaces ri and a key or switch replaces the battery. A resistance R (Fig. 66) of at least 1000 ohms should be in series with the galvanometer. Be- fore the key is closed, the principal circuit is A CBDA and the galvanometer connect- ing with points at C and D will have cur- Pig_ gg rent flowing continuously in it. When the bridge is balanced, no change in this gal- vanometer current will occur when the key is closed. Closing the key adds a shunt circuit from B to A, thus reducing the current through BDA and also drawing more current through A CB. If no change in the P.D. across CD occurs, it must be because the fall of potential in the resistance DA C is the same as before, that is, RESISTANCE 26l the fall is increased in AC because of the larger current but diminished an equal amount in DA where the current is dimin- ished. Let the increase of current which is the same in ^C and in CB (since none of the excess flows through CD) be denoted by ii, and the decrease in BD and DA by ii. Then the increase of potential difference across ^ C is iifi and across CB is iiri ; likewise across BD the decrease is iiU and across DA is i2?'3- As the increase in ^ C must be equal to the decrease in DA and the increase in CB equals decrease in BD, iiri = izra, and iir^ = i-iri. Dividing one equation by the other rjri = r^lri as in the Wheatstone bridge. It will be noticed that the resistance of the branch added by closing the key does not enter the computation and may have any \-alue, even to a virtual short circuit and may even reverse the current in Ti. In this way find the resistance of two Daniell or gravity cells separately and in series, doubling the resistance in series with the galvanometer in the last case. Test Questions. 1. Explain fully the principle of the slide wire bridge and prove that after adjustment n : r^ : : h : h. 2. Tell in detail how you secure this adjustment. What is the use of the galvanometer in this experiment. 3. Show in detail how your arrangement agrees with the ideal diagram. 4. Is it necessary that the current in the main circuit be entirely constant? Why? •5. Will the resistance of the large leads of the bridge cause serious error in the computation? Why? How about the connections to ri and ra? 6. Suppose the mean cross-section of h were greater than that of h, explain the error that would be introduced in the values of ri that you have computed. 7. What conclusions can you draw from the results of your experiment? 8. What would be the resistance in general for n branches, with m equal resistances in each branch? Show five ways in which 60 220-ohm lamps might be connected, and give the total resistance for the circuit in each case. 9. Explain the use of the Wheatstone Bridge for measurement of battery resistance. What errors are liable to affect' the results? 262 MAGNETISM AND ELECTRICITY experiment r-24 Study of the Decade Bridge Calibration of a Resistance Box Make in your note book a diagram of the whole bridge showing arrangement of the various parts of the decade bridge and the connections. Then make an "ideal diagram" of the Wheatstone bridge (Exp. R — 23, Fig. 64) and letter the points and parts on the decade bridge to correspond to the various points and parts of the "ideal diagram." Study and explain the use and operation of the resistance and ratio coils. The latter are illustrated in Fig. 68. Draw diagrams to show how the four coil's are connected for Points Coils Value Connected Used 5-1 1 2-5 1 2 4-1 2 3 2-* 1,2 * 3-5 1,3 5 1-3 3',2 t 2-3 1,3,2 7 5-4 1,3,3' e 1-2 3,3',2 « l,3,3',2 Fig. 67. Fig. 68. producing the "decade arrangement" (Fig. 67). Assume the plug to be in two or three different positions, and explain what the resistance is, and why, for each case. Procedure. Use the bridge to " check " the values of the several coils of a resistance box. This can be dotie by measuring the resistance through the box with all the plugs in. Call this Ro. It will be the resistance of the leads and plug contacts. Remove the plug from the coil whose resistance is to be measured and RESISTANCE 263 get another reading i?i. Then (i?i — Ro) is the resistance of the coil. Make two determinations for each coil. Note. Ro should be measured before each coil. Use a double contact key so connected that it will close first the battery circuit and then the galvanometer circuit. 10,000 ohms should be placed in series with the galvanometer to prevent too large deflections when feeling for the balance. This should be short circuited when near the balance in order to increase the sensitiveness of the galvanometer. Tabulate results as follows: TABLE LVI Coil Ratio Used, A/B R, Plugged X, Computed Error + or- Per Cent. Error Test Questions. 1. Show how the different resistances are obtained in the decade box by the insertion of a single plug in each case. 2. Also show how the ratios are obtained by the use of two plugs. 3. Explain some advantages of the "decade" arrangement over the ordi- nary arrangement of a resistance box. experiment r-25 Comparison of Resistances by the Carey-Foster Method Perform this experiment as directed in Duff and Ewell, Exp. LII, pp. 183-185; or in Nichols and Blaker, Exp. Ti, Part II, pp. 308-311. experiment r-26 Variation of Resistance with Temperature Determination of the Temperature Coefficient If i?o represents the resistance of a sample of wire at 0° C, its resistance at some other temperature t° C. will, in general, be different, say Rt- The change of resistance will thus be Rt - Ro (1) 264 MAGNETISM AND ELECTRICITY and the change per degree will be -^' ~ ^0 (2) Since the initial resistance was i?o, every ohm of Ro will change (per degree) Rt — Ro /■2\ -RT- = " ^^^ This quantity a is called the temperature coefficient of resistance and it may be defined as equivalent to the change of resistance per degree change of temperature of a wire whose resistance at 0° C. is 1 ohm. Transposing (3) we have Rt = i2o(l + at) (4) which is the general working equation for determining change of resistance with change of temperature. (Cf. equation for linear expansion.) To determine a for a given sample of wire it is necessary then to measure its resistance at 0° C. and at some other temperature t. From these quantities a may be computed, using either equation (3) or (4). The object of this experiment is (1) To prove that change of resistance is proportional to change of temperature. (2) To find a for a given sample of wire. Procedure. The sample to be tested is ordinarily wound on a thin cylinder, the ends being connected to binding posts on the top which supports the cylinder. The whole is placed in an oil bath, oil circulating into the inner part of the cylinder through a hole in the bottom. By alternately raising and lowering the cylinder, the oil can thus be thoroughly stirred, making the temperature of the bath uniform. This oil bath may be then heated to any desired temperature and the corresponding resistance of the wire be measured by a decade Wheatstone bridge. The temperature of the bath may be observed by a thermometer inserted into the interior of the cylinder and held in place by a rubber stopper. With the coil at room temperature take two observations of its resistance, thoroughly stirring the oil bath, and reading the temperature to tenths of degrees when a balance is obtained. RESISTANCE 265 Now heat the oil bath to about 12° or 15° above the first tempera- ture, stirring constantly. Again take an observation as before. Then apply heat for just a few seconds, thoroughly stir and take another observation at as near the other temperature as possible. Repeat these observations at intervals of 12° or 15° up to about 150° C. taking care at each reading to have the oil thoroughly mixed so that the thermometer gives the correct temperature of the coil. Call this resistance R. The resistance observed (i?) includes the resistance of the leads, which must be subtracted from the average of the two nearly coincident values in each case to get the actual resistance of the coil at the several temperatures. This gives Rt in the above formula. Measure the resistance of the leads at the beginning and at the end of the experiment. To get Ro plot a curve with the average of the two nearly coincident temperatures as abscissae (make scale from 0° to 180°) and the average of the corresponding resistances as ordinates. If this curve be extended backward until it intersects the line for 0° C, the resistance at 0° (i?o) can be read directly from the curve. Draw conclusions from this curve.' We know the quantities Rt and J?o, and t being simply the change in temperature from 0° C. up to the observed temperature is therefore numerically equal to the observed temperature > The computation of a is thus rendered easy for each of the observed temperatures. TABLE LVIII Observed Temperature Mean Observed Resistance Mean (R) Resistance of Wire (Ri) Ri — Ra a 20.1 20.5 etc. 20.3 etc. 1.375 1.377 etc. 1.376 etc. 1.335 etc. .099 etc. .00395 etc. Resistance of Leads = .041 Ohms. Ro (from curve) = 1.236 Ohms. Test Questions. 1. Define temperature coefficient of resistance. 2. State the necessity for having two contact keys for the bridge con- nections. 3. Show how you would compute the value of a, without the use of Ro, from any two single observations. How should those two observations be selected so as to give the best results? Prove by trial. 4. What do you conclude regarding the change of resistance with change of temperature? 266 MAGNETISM AND ELECTRICITY 5. How could you find at what temperature the resistance of your coil would be zero, assuming the coefficient to remain constant? 5. How much would the resistance of a 100-ohm coil of Cu. wire be changed if a constant current of .25 ampere be sent through it for 5 min., and if it radiates only the heat due to 2 watts? a = .00428, sp. ht. = .093, m = 4Sg. GROUP VI— G. THE D'ARSONVAL GALVANOMETER experiment g-21 Essential Parts of a Voltmeter and an Ammeter Calibration of a Galvanometer for Direct Reading Portable voltmeters of the D'Arsonval type, such as the Weston, American, and Keystone, are nearly ideal instruments for general use where large currents are used so that the current required to operate them (from .008 to .012 ampere for full scale deflection) is negligible in comparison with the current through the part of the circuit around which the instrument is connected; but, if the resistance in the voltmeter is not many times larger than the resistance around which it is connected, the voltmeter current 'is not negligible and the potential differ- ence when the instrument is connected may be quite different from the value when it is removed. To illustrate this, solve the following problem: Two incandescent lamps, 200 ohms each, are connected in series with a cell of negligible resistance and an E.M.F. of two volts. A voltmeter having a resistance of 300 ohms is connected around one of the lamps at points a and h. Find the potential difference of these points when the voltmeter is connected, and when disconnected. The combined resistance of the lamp and voltmeter in parallel is 40 per cent, less than the resistance of the lamp alone. The potential difference is decreased 25 per cent, when the voltmeter is connected, and the instrument is nearly worthless for such measurements, especially when con- nected around a part of two or more high resistances in series. The difficulty may be avoided by using a sensitive galvanometer in series with a very high resistance, say 10,000 or 20,000 ohms, as a voltmeter. When this is shunted around a part of the main path, the combined resistance differs but slightly from that ot the part alone. GALVANOMETERS 267 The galvanometers used in the laboratory require about .000005 ampere through them to produce the maximum deflection of 25 cm. It would therefore require a resistance of 200,000 ohms in series with the galvanometer to keep the reading on the scale if a potential difference of even one volt were applied. Since this resistance is so large and expensive, some other mode of limiting the amount of current through the galvanometer must be adopted. This may be done for the voltmeter by , arbitrarily fixing upon some suitable value for the high series resistance; then if the current that gets through the high resist- ance is say 40 times the desired galvanometer current, a path for the 39 parts may be placed in shunt with the galvanometer. The one part in the galvanometer will then produce the deflection which will be proportional to the potential difference of the voltmeter terminals. Procedure. Part A. Calibration of a Galvanometer as A Voltmeter. The parts of the apparatus are shown diagrammatically in Fig. 70. The galvanometer is connected through a resistance box R to the reversing switch (RS) and then to the double pole double throw switch (DS) for convenience in connecting the galvanometer to either the voltmeter resistance on the left or to the ammeter shunt on the right. The potential difference to be measured is connected to the terminals VV. Note that the ten-ohm coil is connected in shunt with the galvanometer when DS is thrown to the left; hence only a small part of the current through the two 5000-ohm coils will pass through the galvanometer. If R is increased, the galvanometer current is decreased; thus the deflection may be controlled for calibration by varying R. To calibrate the galvanometer as a voltmeter apply a known potential difference (PD) at the terminals VV and change R to give a galvanometer reading decimally related to the known value of FD. For example, let 10 cm. represent one volt; then if PD is 1.105 volts, the scale reading should be 11.05 cm. Note the value of R and keep it the same whenever this calibration is used. Any PD applied to VV may now be read directly on the galva- nometer scale in volts provided the deflections are proportional to potential differences. This is tested in Part B below. 268 MAGNETISM AND ELECTRICITY For the known PD each partner should prepare a standard Daniell cell as follows: Fill a dry porous cup two-thirds full of zinc sulphate solution and let it stand till moisture appears on the outer surface; then place the cup in the glass cell and put the copper sheet (thoroughly cleaned and polished with sandpaper) around the cup. Carefully fill the space outside of the cup with copper sulphate solution to the level of the zinc sulphate. A well- amalgamated zinc rod is now put into the porous cup and the cell is short cir- cuited for 10 minutes. Let the cell stand on open circuit for at least 10 minutes before using. The E.M.F. of the cell is 1.105 volts within 0.2 per cent. The standard solutions are: 44.7 gm. C.P. ZnS04 crystals per 100 cc. of distilled water; 39.4 gm. C.P. CUSO4 crystals per 100 cc. distilled water. Take readings on the red and black scales for each standard cell separately, then for the two cells in series. Note the devi- ation of the readings for the two cells from the known P.D. Part B. To Test Whether Deflections are Propor- tional TO Potential Difference, send current through several coils (say 1-10 inclusive) of a resistance box furnished with a pair of "travelling plugs" so that the voltmeter may be conveniently connected across parts of the box resistance (for example, across 20-17-14-10-7—5-3 ohms). Or put two boxes in series and shunt one of them with the terminals VV, using the other to keep the total resistance constant. If the current is constant, the fall of potential in the various parts is proportional to the resistance of the parts. Use either a box that has been carefully calibrated to determine the correction for each coil, or one in which the coils have been carefully adjusted to the marked value. Take readings over the whole range of both the red and the black scales. From the two readings nearest the calibration readings (or from the one very near the calibration point) find the constant current, and compute the P.D. for each of the other readings. Subtract each reading from the computed P.D. and plot to a GALVANOMETERS 269 large scale a curve of deviations for both the red and the black scales. Check the results with the check calibration made with the two cells in series, and draw conclusions. Part C. To Caljbrate the Galvanometer as an Ammeter, a known (or determined) current must be sent through the large wire (A A') which carries the sliders {s, s') that connect two points on this wire to the galvanometer when the arms of the switch DS are thrown toward the right. The deflection of the galvanometer is controlled by moving the sliders, which should be adjusted until the deflection bears a convenient decimal relation to the fraction of an ampere; for example, .01 ampere per centimeter. The calibrating current may be measured by using the volt- meter part to measure the fall of potential in a known resistance through which the current flows. If the resistance is 10 ohms, the ammeter and voltmeter readings will be the same when the switch DS is thrown over from one side to the other. Specify in some way the distance between the sliders on the slide-wire, for example, measure from center to center of the contact points. Draw a diagram of connections for the ammeter. A "travelling plug" resistance box or two ordinary resistance boxes in series with two or three cells and the ammeter shunt may be used for the ammeter calibration. The voltmeter termi- nals may be connected to the terminals of one box or across a fixed resistance in the travelling plug box, while the current is varied by changing the resistance in another part of the circuit. Make a table giving ammeter and voltmeter scale readings, also values of current, potential difference, and resistance across which the latter is taken, and also the computed value of current. The deviations from direct proportionality of deflection and current will be the same as those found for the voltmeter. Remarks. The resistance of this as an ammeter is much lower than that of commercial portable ammeters that have a corresponding range, and for that reason is better for some purposes. The resistance of the main line through an ammeter should be as small as possible while that of the voltmeter should be as large as possible. 270 MAGNETISM AND ELECTRICITY Test Questions. 1. What are the essential characteristics of a reliable voltmeter and a sensitive ammeter? Explain. Show how your apparatus illustrates these characteristics. 2. Are your galvanometer readings directly proportional to potential difference throughout the scale? If not, why not? 3. Describe fully each circuit used for this experiment and explain why the connections are made in this way. 4. What is meant by "calibration" of a galvanometer as a voltmeter? As an ammeter? Why is it necessary to shunt the galvanometer? What is the shunt in each case? 5. Why must the current be kept constant in Part B? How are the unknown potential differences determined? 6. How are the unknown currents determined in Part C? 7. With your apparatus connected as in the experiment, compute what resistance would have to be put in series with the galvanometer in order to read 10 cm. for a potential difference of two volts applied at the voltmeter terminals, if the resistance of the galvanometer is 110 ohms and the galvanom- eter constant is 2.2 (10)"' amp. per cm. deflection. Also, with this fixed, find the resistance of the shunt required to cause a deflection of 10 cm. for .2 ampere through the main circuit when the switch is thrown to the ammeter side. experiment g-22 Constants of a Sensitive Galvanometer ResistXnce and Sensibility In the laboratory use of a sensitive galvanometer such as the D'Arsonval there are two characteristic constants of the instru- ment that frequently need to be known with some degree of accuracy. They are the galvanometer resistance and the gal- vanometer constant, i.e., the amount of current passing through the instrument necessary to produce a deflection of one scale division, the reading being taken from a scale at a fixed distance. "The latter is probably most accurately determined by means of the voltameter as in Exp. C — 22, but another method is often useful, especially when it is not convenient to prepare the voltameter. If a cell of constant E.M.F. and low internal resistance be connected in series with a high resistance R and a small \^ariable resistance Si, and the galvanometer together with a fixed resist- ance R\, be joined in shunt with Su then the current in the main circuit will be GALVANOMETERS 27 1 I = ^ where G is the total resistance of the galvanometer shunt; i.e., the galvanometer resistance plus i?i, and SiG/{Si + G) would be the combined resistance of the parallel branches. Then the current through the galvanometer (/i) would be the Si/(Si + G) part of the total, or -^1 = F"7=^ — X ■?; — ^1 — -p^ = lodi (1) where 7o is the galvanometer constant, and di the deflection produced . If, now, Si be changed to some other value 52 such that the deflection produced is di, then I2 = e r' X "e — 1 — n ~ -^"'^^ (2) ^ + 57TG Dividing (1) by (2) and noting that if R be large, i= + ^ = « + ^ 0) approximately, we have (h _ 5i(52 + G) d2 S,{Si + G) Solving for G and taking £^1/^2 = n we have G = (JL-J^^ (4) If n = 2, for example, 0102 G = Si - 2S2 Then the galvanometer resistance Rg would equal G — Ri. Having found G, that value can be substituted in either (1) or (2) and Jo, the galvanometer constant, can be determined. Procedure. Connect a cell of constant E.M.F. and low internal resistance, such as the Edison Lelande, in series with 272 MAGNETISM AND ELECTRICITY two 5000 ohm spools and a box 5 with small resistance coils. Shunt the galvanometer and a box, Ri, about S. Take, say the 10 ohm plug (Si) from S, and, starting with a high resistance in Ri, cut it down until you have about 20 cm. deflection. Take readings left and right and take the average. Then leaving Ri constant, change Si to some other value 52 so that it will give some convenient value for di, say 10 cm. Then in that case « = 2. A third value might be taken so that da = IS cm. In this way two independent values can be obtained for the galvanometer resistance. Three values can be determined for /q. The E.M.F. may be tested with a high resistance voltmeter, which should remain connected while data are being taken. It would be well to repeat the experiment using a different value for i?i, to give, say, 16 cm. with Si and 8 cm. for ^5. TABLE LIX R = ■ ■ ■ ohms £ = • ■ - volts Ri 5 d n G R, /o d, d2 d. , (?2 d. d, d. 1 Test Questions 1. Outline the theory of the experimer It. What approxi- mations are made and how are they justified? 2. Solve (1) and (2) for G without making the assumption stated in Eq. 3 and find the percentage deviation. 3. The resistance of your galvanometer is . Its constant per scale division for current is . A gravity cell has an E.M.F. of 1.08 volts and an internal resistance of S ohms. If the gravity cell, a 25-ohm resistance, and a galvanometer are to be connected in circuit, what would be the resistance of a German silver wire (5i in (1)) that must be shunted across the galvanom- eter terminals to keep the deflection on the scale? Make a complete diagram of the connections. POTENTIAL AND E.M.F. ' . 273 GROUP VII— V. POTENTIAL DIFFERENCE AND E.M.F. experiment v-21 Variation of Potential in a Circuit Construction of a Potential-Resistance Diagram The typical series circuit to be experimented upon, and studied, is shown by diagram in Fig. 71. The E.M.F. of one of the cells is opposed to the current maintained through the three resistances by the other three cells. It is proposed to observe the data indicated in the starred columns of Table LX, and to compute from the observed data the results indicated in the other columns. The last four columns contain the data from which the potential-resistance diagram is plotted. The experiment gives practice in the use of Ohm's law for all three cases stated and discussed in Exp. O — 21. From the theory of the primary cell, given in Exp^O — 22, there is an abrupt rise of potential in passing from the zinc to the electrolyte, and another abrupt rise from the electrolyte to the copper. It is probable that the greater part of the rise from zinc to copper occurs at the zinc; but, since the ratio of the parts is variable, or in general unknown, the parts are arbitrarily plotted equal in the diagrams for convenience, as indicated in Figs. 72 and 73. When the S.P. switch in Fig. 71 is opened, the potentials at the cell terminals a, b, c, are as shown at a, V, c' , in Fig. 72. When the switch is closed, there is 3, fall of potential {F.P. = Ire) due to current through resistance (re) of the cell, so that b' falls to the value b shown in Fig. 73; also c' drops to c, due to the additional F.P. caused by current through i?i. The current in the main circuit is so small that the current required to operate an ordinary low reading voltmeter is not negligible and the P.D. of the points to which the voltmeter is applied is changed so much that the readings are very unreliable or worthless. For this reason it is necessary to use a voltmeter having a very high resistance. A commercial voltmeter having 1000 ohms per volt (instead of 100 ohms per volt) will give fairly good results, but the galvanometer calibrated in Exp. G — 21 as a direct reading voltmeter and ammeter, and shown in Fig. 71, 274 MAGNETISM AND ELECTRICITY is better since it has a resistance of 5000 to 10,000. ohms per volt, and the ammeter part is calibrated for small currents. The galvanometer measures amperes when the D.P.D.T. switch is thrown to the right, and measures volts when the switch is thrown to the left. Procedure. Arrange four Daniell cells and three resistances as shown in Fig. 71, so that current flows from left to right through the cells. Assume the point a to be at zero potential. ^i[AAAAAA/^2|^AAAA/VW^3|]^/WWVw2D*k n^n-f^»° \SM.-A Fig. 71. If convenient, the point may be connected to earth (through a water pipe or gas pipe), thus assuring zero potential; but this is not a necessary procedure. Connect the potential galvanometer or voltmeter to the points o and h, and observe both the magnitude and direction of the deflection. In recording the data place a + mark before the reading when the deflection indicates a rise of potential from left to right; and a — mark when the converse is true. If the galvanometer deflections right and left are not equal for a given P.D., deflections one way only may be used by throwing the reversing key, and determining the rise or fall of potential by the direction of throw if the same voltmeter terminal is kept ahead at each step. Observe in like manner readings when connected successively to he, cd, de, etc., to h and back to a, connecting the terminals in the same order each time. The deflection for current in the main circuit should be observed occasionally on the current galvanometer to see if it remains constant. When the P.D. and current have been read for each part of the circuit, the circuit should be broken, and readings taken for the E.M.F. of each cell. Place the proper sign before each. POTENTIAL AND E.M.F. 275 Tabulate the observed and computed data as indicated in Table LX; then plot a potential-resistance diagram with resist- ances a abscissas. Fig. 73 is a typical potential-resistance dia- gram for one cell and one resistance. u rii, ~i Fl c 1.0 Tli fin Yn nf 1?" b 1 Zi J ir '^\\ r 1 ^.e uT 1 ^ 1 1 ^6 n lA K u ■n' u 1 n" 1 1 1 bi 1 2.2 ai 1 1 1 1 n' Zti 1 '. R es ist i an ce- 4 Oh Ulf t Fig. 72. l.ii — — — If) A 1 I TDi <1fl^ miLfo- •|> ft to .8 CI KP( Ti rni t| £a r-o 1 ' m^c\ 1 ,1 UJ in BU v^ Uj ^ "^A ^ 1 ci? '-/ s^ n Q 2 ^ 1 Cu 1 a Zn i 1 R 2 3 esistance — ot - 5 MIS Fig. 73. For conclusions to be drawn, see Test Questions below. TABLE LX Cur. Galv. Potential Galvanom. Re- sist- ance E.M.F. of Cells Total from a tn Parts of Circuit be 1 ^ JS! to , in ^Sh (0 en 4J Fi l< :^'^ ^^ Ohms ^^u ;^ H ;^ * * * Cell No. 1 ft Ri (b-c) c Cell No. 2 d i?2 (.d-e) e Cell No. 3 f i?3 (f-g) e Cell No. 4 h ' r {h to 0) a Totals 1 For potential galvanometer one scale division = volts For current galvanometer one scale division = amperes * Observed data. 276 MAGNETISM AND ELECTRICITY Remarks. If the galvanometer calibrated in G-21 is used as the ammeter and voltmeter in this experiment, it will be necessary to check the calibrations, for the galvanometer constant may have changed in the meantime. This may easily be done by using a standard Daniell cell and a 10 ohm coil as in Exp. G — 21. If the standard cell is not available, a low reading (3 volt) voltmeter may be used. Connect the voltmeter terminals to two points on the circuit to be used in the experiment, so as to get a fairly large reading, say about 1.5 volts. Then without removing the voltmeter terminals connect the galvanometer terminals to the same two points, and compare the readings. If the galvanometer does not read correctly, adjustment of the series resistance must be made. In this case the ammeter side must be adjusted by changing the positions of the sliders. It will be more convenient in this experiment to use deflections of the galvanometer in one direction only. This may be done by making use of the reversing switch. Connect the voltmeter terminals of the galvanometer to the terminals of one of the cells on open circuit. Observe and record the position of the reversing switch, the direction in which the galvanometer deflects, and which terminal is on the left of the other. This will be the arrangement for + P.D. or rise in potential when the terminals are connected in the same order. If in making your measurements the galvanometer deflects in the opposite direction, throw the reversing switch. This, then, will be the arrangement for — P.D. By following this definite arrangement you will always be sure whether you are measuring a rise or a fall of potential. Test Questions. 1. Draw conclusions from your diagram regarding (a) general parts of a circuit in which there. is a rise of potential, and in which there is a fall of potential; (b) the fall of potential in the resistances and comparison of rates of fall within and without the cells; (c) the fall or rise of potential through the cells; and (d) the point at which the diagram ends. 2. What is represented by the slope of the lines of the diagram? Explain. Compare the slope of the different parts of the diagram and explain. 3. What are the tests of accuracy? 4. What is the general equation representing the relationship between the four quantities involved in a circuit? Show how all the different cases are involved in this experiment. 5. Why is the E.M.F. divided into two parts in the diagram? 6. When the circuit is open, the difference in potential between the termi- nals of the cell is equal to the E.M.F. of the cell. Why is this not true when the circuit is closed? POTENTIOMETER MEASUREMENTS 277 experiment v-22 Principle of Potentiometer Measurements E.M.F. OF Different Types of Cells We have so far made use of a direct reading voltmeter and a sensitive galvanometer as instruments for the measurement of the E.M.F. of a cell. In each case we measured the P.D. at the terminals of the cell when no current was flowing; that is, we assumed that the voltmeter resistance being so high, the current through the voltmeter was negligible. But when we use a low reading voltmeter the resistance may be only 300-400 ohms, and ■the current flowing through the cell may cause an error in the observed E.M.F. of from 1 p.c. to 2 p.c. To overcome this serious error we must make use of a method that will allow absolutely no current to flow. This is accomplished by the potentiometer method. If we connect a battery, B (Fig. 74), in series with a large resistance, R, we know that the fall of potential through each ohm of the resistance R will be constant as long as the current is constant. Now, if a galvanometer, G, be shunted about a ^H^> j D-] [Hd->| Cywvwwww Fig. 74. Fig. 75. part of this resistance, we shall, in general, have a current through G. But if we place a cell C in the galvanometer circuit, whose E.M.F., Eu opposes the current from the main circuit, the direction and amount of the current in G will depend upon the relative fall of potential through r^ (between a and b), and the E.M.F. of C. In fact, if B is large enough, h may be varied until the fall through ri equals the E.M.F. of C, and then no current flows through G. Under such circumstances, when we 278 MAGNETISM AND ELECTRICITY have no deflection of G, we know Ei = Iri. Then, if we substi- tute another cell whose E.M.F. is £2 for £1 and change /i (keeping R, and consequently /, constant) to ra, so we again have no current in the galvanometer, £2 = /^a or £2 = Eiri/ri. If we know £1 as that of a standard cell, and determine ^2 and ri, we can readily compute £2 of the unknown cell. Procedure. Prepare the standard Daniell cell as directed in Exp. G — 21. Connect in series three gravity cells and two exactly similar resistance boxes of at least 10,000 ohms each, Ri and R2 (Fig. 75). In shunt with i?i connect the two cells to be compared, a galvanometer, and a resistance of 10,000 ohms (two 5000-ohm spools). The two cells, S and X (standard and unknown, respectively), are to be joined to a 3-pole key at K such that either may be put in circuit by pressing the proper contact. Both are to be connected so as to oppose the E.M.F. in the main circuit, i.e., all positive or all negative terminals connected to a. With apparatus thus connected, pull all the plugs from R2. Then vary the values of i?i (transferring plugs from i?ito the corresponding hole in R2, thus keeping the resistance abc con- stant) until on pressing the key for 5 no deflection is obtained. Call'this value of i?i, i?s. Repeat for X. Call this new value J?i. From these two readings and the known E.M.F. of S, determine the value of X. Note: In feeling for a balance the key K should be struck and immediately released, making the instant of contact as short as possible, in order that too great a deflection may be prevented. The 10,000 ohms may be reduced and finally cut out altogether when the proper balance is nearly secured. In this way make at least two determinations each of the E.M.F. of a gravity cell, a Leclanche cell, a potassium bichromate cell, and a dry cell, or a corresponding number of other cells available. Compare with values found in text book. As a further check the E.M.F. of each of the above cells may be determined by a voltmeter. Tabulate all data as in table. Test Questions. 1. Why is a low resistance voltmeter unreliable for accurate determination of the E.M.F.'s of cells? 2. A voltmeter applied to the terminals of a cell which has an internal resistance of 8 ohms reads 1.050 volts. If the resistance of the voltmeter is 300 ohms, what is the true E.M.F. of the cell? POTENTIOMETER MEASUREMENTS 279 TABLE LXI E.M.F. of Standard cell = 1.105 volts (within 0.2 per cent.) Cell Rx R. RJR. E.M.F., By Voltmeter From Reference 3. Outline the principle. of the potentiometer. 4. Describe the standard cell used, also briefly describe the various cells tested. 5. Show how, by using a third resistance box and a suitable number of cells, the apparatus might be made direct reading, for example, so that when Ui = 1085 ohms, the E.M.F. in shunt with it would be 1.085 volts, or in other words, so that the fall of potential through each ohm would be .001 volt. How would you calibrate such an apparatus from the standard cell? 6. Do your results agree with table values for the electromotive forces of cells? If not, how can you account for discrepancies? GROUP VIIl-X. MISCELLANEOUS If one of the following experiments is chosen or assigned, the student is expected to read up the subject, prepare an outline of theory and procedure, and submit it to the Instructor in charge for approval before beginning the experimental work. experiment x-21 Theory of Shunts. Experimental Verification experiment x-22 Verification of Kirchoff's Laws experiment x-23 Joule's Equivalent by Electrical Method experiment x-24 The Standard Electrometer experiment x-25 The Ballistic Galvanometer. Comparison of Condensers TABLE OF CONTENTS PART III — THIRD TERM CHAPTER V. NOTES ON ADVANCED MECHANICS I. Errors SEC. PAGE 301. Kinds of Errors . . .301 302. Errors of Observation . 301 303. Errors of Derived Results 302 304. Constants by Least Squares 303 II. The Beam Balance 305. The Beam Balance . . 305 306. The Balance Equation . 306 307. The Knife Edges . . .306 308. Sensibility Curves . . 309. Inequality of Arms .■ 310. Ratio of Arm Lengths III. Accelerated Motion 311. D'Alembert's Principle . 308 312. Motion of Center of Mass 310 313. Accelerated Rotation IV. Elasticity 314. 315. Introduction Young's Modulus 312 313 ■316. 317. Modulus of Rigidity Moment of Torsion 307 308 308 311 314 316 CHAPTER VI. EXPERIMENTS IN ADVANCED MECHANICS Group I — M. Measurements EXPERIMENT PAGE M-^1. Radius of Curvature of a Lens . 319 The Spherometer. Probable Error. Reliability M — 32. Sensibility of a Spirit Level 321 Radius of Curvature and Tests for Adjustment M-^3. Adjustment and Use of a Dividing Engine 324 Graduation of Scales and Verniers M — 34. Calibration of Micrometer Microscopes 327 Standardization of a Scale M-^S. Erros of a Graduated Scale by Precision Methods . . . 329 Errors of each Tenth of a Meter Scale Group II — W. Weighing W — 31. Study of the Beam Balance, and Balance Tests . Correction for Buoyancy of the Air W— 32. Calibration of a Set of Weights Comparison by Method of Vibrations and Reversals Group III — A. Accelerated Motion A— 31. Speed and Acceleration of a Falling Body . . . Constants by Method of Least Squares A— 32. Relations of Force, Mass, and Acceleration . . Application of D'Alembert's Principle 330 335 341 345 Xll CONTENTS EXPERIMENT PAGE A — 33. Simple Harmonic Motion of Translation 347 Relations of Quantities Derived and Verified Group IV — R. Rotary Motion R— 31. Accelerated Rotation of a Wheel 349 Moment of Inertia, Energy, Friction R — 32. Harmonic Rotary Motion and Moment of Inertia .... 354 Relations of Quantities Derived and Verified R — 33. The Compound Pendulum 356 Period and Equivalent Length Group V — E. Elasticity E— 31. Young's Modulus of Elasticity of a Wire 358 Measurements with an Optical Micrometer E — 32. Laws of Flexure of Rectangular Beams 360 Method of Derivation of Empirical Formulas Group VI — T. Torsion T — 31. Laws of Torsion Verified with the Torsion Lathe . . . 364 Modulus of Rigidity. Moment of Torsion T — 32. Moment of Inertia by the Torsion Pendulum' 365 Period by the Method of Middle Elongations Group VII — I. Impact 1—31. Laws of Impact for Inelastic Bodies 367 Conservation- of Momentum. Loss of Energy J — 32. Laws of Impact for Elastic Bodies 371 "CoefBcient of Restitution Group VI II — D. Density -D — 31. Calibration of an Hydrometer of Variable Immersion . . 374 The Mohr-Westphal Balance Used as a Standard 3>— 32. Densities of Cast and Stamped Sterling Silver 382 Nicholson's Constant Volume Hydrometer D — 33. Relative Densities of Gases 383 Bunsen's Effusiometer Group IX — F. Fluids F— ^1. Study of the Venturi Water Meter 384 Application of Bernoulli's Theorem F— 32. Specific Viscosity of Liquids 388 Damped Vibrations F — 33. Errors of an Aneroid Barometer 388 Variable Pressure and Temperature PART III— THIRD TERM Advanced Course CHAPTER V NOTES ON ADVANCED MECHANICS I. ERRORS. 301. Kinds of Errors. In physical measurements there are four classes of error, as follows: (1) Errors of observation; (2) Instrumental errors; (3) Errors of method; and (4) Mis- takes or blunders. The first may be decreased by taking the mean of many independent observations, the second by using more accurate instruments, the third by using a better method, and the fourth by care and" repetition. Some method of check- ing, or testing, a result is often useful in detecting errors and determining reliability. 302. Errors of Observation. We may reasonably consider the mean of a large number (w) of observations of a given quantity to be reliable within the average deviation from the mean (d^) ; for da is larger than the quantity commonly called the probable error (e). The probable error of the mean (m) is a quantity e such that the probability that the actual error is greater than e is the same as the probability that the actual error is less than e. The term probable error does not mean that the result is probably in error by the amount e, and does not give any information as to whether the result is too large or too small; neither does it mean that the actual value lies between m + e and m-e, as some assume. The term means rather that the true value (x) is just as liable to be outside of the limits w ± ^ as it is to lie between these limits ; and for that reason m ± e is used to express x. 301 302 ADVANCED MECHANICS It is proven in books on the " Theory of Probabilities " that the probable error of the mean, ' = ■<"« ^ f •':('--»'••'•' ) ' -^fl """'• <"»' in which d^, d^, etc., are the deviations of the n observations; and the probable error (e'} of any single observation is, / = .6745 yj Ml^LAr^JLl^^i^Jl\ = .84 d. approx. (102) Note that the average deviation (da) is larger than ^, and that e' is ^n times as large as e; or e' = V»* ^- Hence it is rea- sonable to assume that the actual error lies) Avithin the limits ± rf, which limits we may call the " possible error " to distinguish it from the probable error which is usually much smaller. The main purpose in estimating the possible error is to determine the doubtful figure in the result; therefore the order (power of ten) of the error is generally more important than the actual value, and the possible error may serve this purpose with less labor than would be required to compute the probable error used in more elaborate refinements in determination of reliability by use of calculus in complicated cases. 303. Errors of Derived Results. For the possible error of a sum or difference take the simi of the possible errors of the terms. To find the possible error of a product or quotient it is necessary to find first the percentage of possible error by takti^ the sum of the percentage errors of the factors. The possible error may then be computed. If the derived result is simply related to the observed factors as in Exp. M — 31. the approximate method of finding possible error as given above is easily applied. To illustrate, — ^let it be required to find the effect of a percentage deviation of 2 per cent, in the value of /, and .3 per cent, in the value of d, on the value of the radius of curvature R computed to be 64.5 cm. by /' d the formula jj = — + — -. The uncertainty due to /2 ig .4 per bd 2 cent. This added to .3 per cent, for d gives .7 per cent, for R. .7 per cent, of 64.5 cm. is .45 cm. Therefore R = 64.5 +. 45 cm.; OT R = 64.5 cm. Showing that there are only three significant ERRORS 303 figures in the result. Note that d / 2 is very small in compari- son with l^y6d; consequently the effect of the error of d/^Z on the error of R is negligible, and that term may be ignored in finding the approximate measure of uncertainty in the value of i?. Those who are familiar with calculus will find good explanations of methods of computing the effects of errors of observation on derived re- sults in Duff and Ewell's Physical Measurements, pp. 5-12 ; in Nichols and Blaker's Laboratory Manual, Vol: I, pp. 20-23, and in Miller's Manual, Sec. 9-11. Neither probable error nor average deviations take into ac- count " constant errors " caused by imperfections in instruments or improper methods of measurement. These constant errors may be large when the variable error is small; the latter indi- cating simply the extent to which the accidental errors of meas- urement have been eliminated. 304. Constants by " Least Squares." This expression is used here to designate briefly the determination by the method of " least squares " of the constants in the equation of the straight line that represents most accurately a set of observed data for two variables having a first degree relation. Let the variables be expressed by x and y and represent their first degree relation by the typical straight line equation y = ax+b. In this equation a expresses the slope of the line [since for two points, y^, x^, and y^, x^, a = {y^ - 3/1) -■ (x^ - ^1)] and b is the intercept on the Y axis, or the value of y when x = 0. These constants generally express, or are closely related to, the results sought. By reason of accidental errors and adverse conditions of observations, the observed values of the variables may be quite irregular so that the slope and position of the line that will best fit the data is uncertain. To illustrate : Plot to a large scale the following data for an accelerated motion due to a constant force. Time in seconds 1 2 3 4 5 6 (=x) Speed in feet per sec. 3.5 4.4 6.0 7.2 8.5 9.6 (= y) Let it be required to find the acceleration, the speed at the 304 ADVANCED MECHANICS chosen zero of time, and how long the force had acted before the zero of time. Draw the line that you think will best fit the data and note that there are small vertical distances between the points and the line. Let these distances, or deviations be represented by d^, d^, dg, etc. These will change if the position or slope of the line is changed. The method of " Least squares " is based upon the principle that the best position of the line is that which will make the sum of the squares of these deviations a minimum, or % d^ = a. minimum. The principle applies to lines of any degree but is limited here to the stright line which is the most useful case. The values of a and b in terms of the observed values x and 3; are determined by the methods of maxima and minima. They are a = [SxSy - n^xy] .=- [ (2.r) ^ - n^x''] ( 103) and b = [S.xS.xy - SyS^"] h- [{txY - n%x^] ( 104) in which n is the number] of observations. (Reference: Nichols and Blaker's Laboratory Manual, pp. 22-28.) A convenient tabulation of data and computations is given be- low: TABLE CI X > xy x^ Summations, Etc. 1. 2. 3. 4. 5. 6. 3.5 4.4 6.0 7.2 8,5 9.6 3.5 8.8 18.0 28.8 40.5 57.6 157.2 1 4 9 16 25 36 91 Sx= 21. Sy= 39.2 2'*y = 157.2 «= 6^ a= 1.143 21. 39.2 b= 3.486 b- 21 X 39.2 - 6 X 157.2 - 120. 21 X 21 - 6 X 91 - 105 21 X 157.2 - 39.2 X 91 - 366. 21 X 21 - 6 X 91 - 105 1.143 = 3.486 THE BEAM BALANCE 3O5 The equation which most accurately represents the observed data is 3»= 1.143 ji: + 3.486 or speed s= 1.143 t + 3.486 The constant acceleration is a= 1.14 ft./sec.^ The speed when t = is b = 3.49 ft./sec. The time when J = is t = - h/'a = - 3.05 sec. To locate the line find 3; for two values of x: (say x.^ = 0, and jTj = 6) and draw a straight line through the plotted points. Any serious error in computations may be detected by noting whether the line located by least squares lies close to all of the observed points. The analytical method of obtaining desired results is also use- ful in the case in which the precision of measurements is so great that an inconveniently large and accurate plot would be neces- sary to give an adequate graphical solution of the problem. II. GENERAL NOTES ON THE BEAM BALANCE 305. The Beam Balance. The Balance consists of a double lever or beam, A B, (Fig. 101) supported upon a knife-edge through its center and perpendicular to its length. The knife- edge rests upon the top (T) of a fixed vertical column. Knife- edges near the ends of the beam, equidistant from the center, support the pans and their loads (L). A long slender pointer (P) is attached rigidly to the beam. Its lower end moves over a scale when the beam is deflected by putting a very small excess weight (m) in one of the pans. The deflection per milligram (on centigram) of excess load is called the sensibility {s) of the balance. If m milligrams of excess weight causes a deflection of d divisions, the deflection per milligram is cf/m = s. When the equal arms A C and C B are equally loaded so that the beam is at rest in a horizontal position, the center of mass (G) of the beam and pointer is directly below C by a small dis- tance (r). If the plane of the knife-edges is deflected to the position A' C B', the center of mass swings from G to G' ; so that the weight {w) of the beam is a vertical force having a moment arm (fl = r sin 6) equal to the distance of G' from the vertical plane through the knife-edge. Therefore there is a mo- 3o6 ADVANCED MECHANICS ment (wxr sin 0) tending to bring the beam back to its initial position of equilibrium when deflected in either direction. Fig. 101. 306. The Balance Equation. The equation for the sensi- bility of the balance is derived from the fact that the excess weight deflects the beam until the moment of w is equal to the moment of the excess weight (m) which acts upon an arm a' =: I cos e. Then mxl cos = wxr sin 6; and = !f^xtan(? = !^^ / I p' in which p is the length of the pointer (in scale divisions) and d is the number of divisions over which its point is deflected by m, so that tan ff = d/p and, since s = d/m,, s = -^ (105) wr Therefore the sensibility is directly proportional to the length of beam, and the length of the pointer; and inversely propor- tional to the weight of the beam, and the distance between the center of mass and the knife-edge. 307. Relative Positions of the Knife-edges. The deviation above relates to the case illustrated in Fig. 101, in which the three knife-edges lie in the same plane for all loads ; so that the sensibility is independent of the load. If, however, the center knife-edge is above the plane of the end ones, the moment arms of the loads do not remain equal. The length of the rising arm THE BEAM BALANCE 307 increases at a decreasing rate until it reaches the horizontal po- sition, while the falling arm is shortened more and more per degree, the farther it moves from the horizontal plane. Conse- quently the sensibility is decreased^ and is variable for different parts of the scale, with a maximum at the center which is be- low the sensibility in the preceding case. In the third case, when the center knife-edge is below the plane of the end ones, the rising moment arm shortens at an increasing rate and the falling one lengthens at a decreasing rate until it passes through the horizontal when the reverse occurs. This again gives a variable sensibility for different parts of the scale but the maximum does not occur at the center and may exceed that of the first case; or the equilibrium may pass from stable to neutral or unstable equilibrium for the larger deflections, thus destroying an essential condition. In review, note that in the first case the sensibility is con- stant for all loads and all parts of the scale. In the second case the sensibility decreases towards the end of the scale, and the decrease is greater for the larger loads. Even the maximum is lower than in the first case.. In the third case the sensibility is variable with a tendency toward instability if the increase of load does not cause this case to approach the first. 308. Sensibility Curves are plotted with* loads as abscissas and the corresponding sensibilities as ordinates. If the balance meets the conditions of case one, it will be a straight horizontal line which holds for all parts of the scale. It is possible, how- ever, that slight bending may occur as the loads increase, and the sensibility curves for the ends of the scale may differ from the one for the center. If for the three scale readings, two left and one right, the mean of the lefts is added to the right, the sum is twice the read- ing for the equilibrium position ; and the ratio of the differences of the sums equals the ratio of the differences of the correspond- ing equilibrium readings. Therefore the fraction of a milli- gram (or centigram) may be determined from the differences of sums, as explained in Experiment W — 2, Part I, and the sen- sibility may be expressed by the change of sum. per milligram instead of the change of scale reading per milligram. 308 ADVANCED MECHANICS 309. Inequality of Balance Arms. If the length (/) of the left arm is not exactly equal to the length (/') of the right arm, the weight W^ on / to counterbalance a weight X on /', differs from W on /' to counterbalance X on /. The moment equa- tions are, Wl = Xl' anAXl^WV Multiplying equations gives, X^ll'=WW'll' 3.nAX=^/ WW (106) When W and W differ but little y/ WW' \s very nearly equal to i (W+W) which is easier to compute and sufficiently ac- curate for ordinary uses with good balances. 310. Ratio of Lengths of Arms. Place in the pans weights of the same nominal value. If a small weight a/ must be added to J^ on Z to produce equilibrium with the weight W' of the body on /', and w" must be added to W on l' to produce equilib- rium when the weights are reversed, then l(W + w')=rW', and IW' = 1' (W + w") Multiplying equations and cancelling W' /2 {W + ■!!/) =/'2 {W + w"); and L = , I w + w' = 1 + "' ~ ^' ' I \ w + w" y W+w" .•. r = 1 + — ^r-TTr — approximately. ( 107) / 2 vv * V (l + c) =1 + J c + i c2 + etc. = l + ic approx. when c is small, v/ or w" will be negative in sign if placed in the other pan from the one speci- fied. For further notes on the balance, weighing and calibration of weights see Experiment W — 31 and W — 32. III. NOTES ON ACCELERATED MOTION 311. D'Alembert's Principle. If a resultant force Z^, acts upon a particle w^, and produces an acceleration 0^= (v^- v^) /t, it changes the Kinetic energy of the particle from \m.p.^ to \in^^ ; and this change of energy equals the work ( W) done by /i in the time t and distance j. '■W = f^s = f^{v^^-v^^)/2a^ = \m^{v^^-v^^), from the law for conservation of energy ; and /i = »«i fli ACCELERATED MOTION 309 But /i is the resultant of all of the external forces, S fei plus all of the internal forces 2 f u. Then 2/ei + S/n = m^a^ ; and S/es + 2/13 = Wis Og , etc., would in like manner hold true for other masses m^, m^, etc., and SFe + 2F, = Swa for a system of particles or bodies. By Newton's third law, the internal forces always occur in pairs, equal and opposite. 2Fi=0, and 2Fe = %m a. Here 2wJ a is numerically equal to- the resultant of the ex- ternal forces that produce the acceleration, and has been called the " effective force " ; also - S w a has been called the " reversed effective force," and represents the opposition to acceleration due to inertia. It is not an active force doing work, but is the passive oppo- sition to the acceleration that accompanies the transformation of work into kinetic energy. It exists only during acceleration. It is capable of offering a reaction equal and opposite to the impressed action of the accelerating force in agreement with Newton's third law. This fact has probably led to the use of the term " reversed effective force," and the form of equation, %F--S,ma = 0. -(108) This equation expressed in words is commonly known as D'AIemberfs Principle; and may be stated as follows : — The external forces (S F) acting on a system of particles, together with the reversed effective forces {-%ma) due to the accelera- tion of the system, form a system of forces in equilibrium. It should be noted that the equation %F -'S,ma = is general and far-reaching in its possibilities of application. In the gen- eral case 2 F and "Zma are the vector sums ; but when limited to the components in a given direction, the vector sum is the 310 ADVANCED MECHANICS algebraic sum. If desired the general equation may be re- placed by three for the rectangular components, as follows: — SF^-2wa^ = (109) 5Fy-2way = (110) 2F,-Sma, = (111) If the acceleration is zero, then %F = 0, which is a condition essential for static equilibrium of bodies. 312. Motion of the Center of Mass. In applying D'Alem- bert's principle it may be more convenient to use in place of %m a, the product of the mass (M = 2 w) of the system of par- ticles, and the acceleration (a) of the center of mass of the sys- tem, i. e. ^ma = M a. To prove this relation we may start with the definition of the center of mass ; viz : — The center of mass is a point so located that the product of the total mass (Af = 2 w) by the distance (d) of this point from any plane of reference equals the sum of the products of each mass by its distance from the same plane. Then, m^ d^ +m2d2 + Wg d^ + etc. = [m^ + 'm,2 + m^ + etc.] d. Let the masses be in motion for the increment of time (f), during which the equation above changes to m^ d\ + Wj d\ + Wg d\ + etc. = [m^ + m2 + m^ etc.] d'. If the terms of these equations are subtracted, and each change of distance is divided by t (for example, Wi {d\-d^')/t = m^ v^), the equation of differences becomes, m^v^ + m^ v^ + m^v^ + etc. = [m^ -{-mi + m^ + etc.] v. in which the velocities are the components normal to the plane, and the products are the components, in the same direction, of the momenta of the masses. If the velocities then change dur- ing" a second increment of time f, the velocity equation becomes, OTi v\ +m^7/^ + Wj v'^ + etc. = [w^ + m^-\-m^ + etc.] v'. Subtracting the two velocity equations and dividing the differ- ences by t' gives, m^ a^ + m^a2 + m^ o, + etc. = [w^ + m2 + m^ + etc.] a. for the components of force perpendicular to the plane. Since ACCELERATED MOTION 3II the equation holds true for any plane of reference, the equa- tion must be true in the general case and 5wa = Mo. (112) 313. Accelerated Rotation. All motions may be resolved into component translations and rotations. It is evident that D'AIembert's principle applies to cases of accelerated transla- tion, but how it may be utilized or applied in dealing with ac- celerated rotation may need further explanation. If a particle m^ is at the distance r^ from the axis of rotation, and is acted upon by the force f^ perpendicular to r^ so as to produce the angular acceleration oi, then /i - »i Oi = by D'AIembert's principle. Multiplying by ^i and substituting for Oi its equivalent o^ r^, gives f^r^-mrj^ a = for the particle, and for the whole body. If T is used for torque, or moment of force, instead of / r, and / is used for the moment of inertia in- stead of m r^, the equation becomes %T-%Ia = (113) By analogy, -%I a may be called the reversed effective torque ; which together with the impressed torques gives a system of torques in equilibrium during acceleration. This result could have been predicted from the analogies that exist between all corresponding formulae for translation and rotation. If force, mass, and linear displacement, velocity, and acceleration in formulae for translation are replaced by torque, moment of inertia, and angular displacement, velocity, and ac- celertion respectively, the resulting formulae will hold true for rotation. If the angular acceleration is zero then sr=SFr = 0; which is one of the conditions essential for static equilibrium of bodies. 312 ADVANCED MECHANICS IV. NOTES ON ELASTICITY 314. Introduction. Elasticity is the tendency of a body to regain its original form or voliune after deformation. A perfectly elastic body is one in which a deforming force will develop an equal and opposite force of restitution that will not diminish with lapse of time and that will immediately restore the body to its original shape or size on the removal of the deforming force. A highly elastic body offers a great resistance to distortion and is very nearly perfectly elastic. The elastic forces of restitution act between contiguous parts within the body. Imagine a sectional plane of small area (^), in the interior of the body subjected to external forces. The portions of the body on the two sides of this section act on each other in a certain direction with a force F. If this force be divided by the area the quotient is the force per unit area and is called the stress on the section. When the force is uniformly dis- tributed Stress = total f orce / area = F / .^4 (114) The stress is called a normal stress if F is perpendicular to A ; and a tangential or shearing stress if F is parallel to A. When F is in any other direction the stress on the section can be resolved into normal and tangential components. If the stress is the same in magnitude and direction on all parallel sections of a body, the body is said to be under homogeneous stress. Perfect fluids offer no resistance to shearing stress and pos- sess elasticity of volume only. Bodies that resist a shearing stress are said to possess rigidity and have elasticity of shape as well as of size. Within certain limits Hooke's Law, that the deformation is proportional to the deforming force, is obeyed approximately by nearly all substances; the limits, however, are very often narrow. The elastic limit of any given material is the stress that cannot be exceeded without producing a permanent set and a depar- ture from Hooke's Law. The principal kinds of stress are: longitudinal, shearing and ELASTICITY 313 hydrostatic. For each kind of stress there is a corresponding kind of deformation or strain. 315. Young's Modulus. A longitudinal stress produces a linear extension or compression. The longitudinal strain is the change of length per unit of length, or Strain = elongation / length = E / L (115) If a rod or bar be stretched but does not reach the elastic limit the elongation (£) is directly proportional to the tension (F) and the length (L) ; and inversely proportional to the area of cross section {A^. These relations can be verified experi- mentally. Algebraically expressed, — E oi FL/A, or E = KxFL/A (116) The proportionality factor K is the coefficient of elasticity of tension and may be interpreted physically to be the strain pro- duced by unit stress, since K= (£/£) /(F/^) = strain /stress. (117) The elongation of a cubic centimeter by a tension of one dyne would be an extremely small fraction of a centimeter, especially for metal and many other solids. The reciprocal of the co- efficient of elasticity is a very large number called the Modulus of elasticity which for tension is known as Young's Modulus (M). It may also be defined to be the ratio of stress to strain; or the stress that would produce unit strain (i. e. double the length) if the elastic limit were beyond that stress. M = \/K={F/A)y^{E/L)=FLyAE (118) M is commonly used instead of K in comparing the tensile elasticities of materials. If the values of elongation produced by variable tension be observed the values of stress and the corresponding strains may be computed. If these values of stress and strain are plotted on section paper the line is called a stress-strain diagram. The stress at the upper limit of the straight portion is the elastic limit. The modulus may be computed from the slope of the straight part, the resilience^ from the half product of the 1 Reference: Ferry and Jones, p. 140. 314 ADVANCED MECHANICS coordinates* at the elastic limit, and if the stress is carried to the breaking point, the tenacity (maximum stress) and brittle- ness (ratio of elastic limit to tenacity) may be found. The maximum elongation in per cent, of length is sometimes used as a measure of ductility. 316. Modulus of Rigidity. A body is said to possess rigidity if it resists the action of forces tending to slide adjacent parallel planes within it upon each other. Such forces are said to cause a "shearing" stress which is measured by the force per unit area. This we will denote by S. The relative displacement (parallel to the fordes) of planes unit distance apart is called the "shearing strain." In Fig. 102 the shearing strain is E G/E B or E G when E B- is unity. The arc of with the radius E B '\s very nearly equal to E G; then since the ratio of arc to radius is a measure of the angle in radians <(> = E G/^E B very nearly, and it is customary to use the angle of distortion <^ as a measure of the shearing strain or " shear." The modulus of simple rigidity (R) is the ratio of shearing stress to shearing strain. R-S/4> (119) An important case of shearing stress and strain, in which the relations of quantities may be derived, is the twisted cylinder. Fig. 103 represents the cross section, and Fig 104 a side view. B Fig. 104 E G Fig. 103 of a cylinder in which the element initially at B £ is twisted to the position 5 G by forces acting in the plane of cross section at the upper end and perpendicular to the radial moment arms of the section. Assume the lower end to be fixed. * Reference: Carhart, Vol. I, p. 102. ELASTICITY 315 Let 6" be the stress at the circumference and S' the stress at the distance / from the center. Then since the stress is directly proportional to strain, and therefore to distance r' from the center, r' r The arc E G = r6 = I cj). Therefore, Re/l = R ^/r = S/r. The moment of the force /' on the element of area a' is f r = S'a'r' = S (r'/r ) »' r' = (S/r) a' r^ The total moment of force on the section is L = f'r'= (S/r) S of r'2, in which 2 a' r"^ is the moment of inertia of plane figuffe for the cross section of the wire. 5 o' r'2 = ^ ^ y2 for the circle of radius r and area A, by the same mathematical process that the equation I = 'S,mr^-^M R^ is obtained for moment of inertia of a circular disk. Therefore L=(R e/l) I A r2, or L=i7rRxe r^/l, then ( 120) Loidr^/l (121) Since R is constant. The interpretation of this last equation is, — that for rods of a given material (for which R is constant), the montent of the twisting force is directly proportional to the angle of twist, and to the fourth power of the radius; and inversely proportional to the length of the rod. These statements are called the laws of torsion. If the cross section is rectan^lar with dimensions x, y, the polar m'o- nlent of inertia of plane figure is V^ A (;f2 + y2)=S o'r'^. For a square cross section Vu A (x^+y^) reduces to iAx' = ix*. If this is substi- tuted for itrr* in Eq. 121, L = ^.Vex^;a.ndL a: ^ (122) By experiment it has been found that Z, = .841 Sf^ for a square shaft 6/ S'howing that the results computed from the polar moment of inertia are too large for square or rectangular sections. ■ 3l6 ADVANCED MECHANICS 317. Moment of Torsion. The moment of torsion of a wire is the mcmient of the couple that will twist the wire through an angle of one radian. Let it be denoted by Ci; then L = C,9; and from the notes in the preceding part it follows that C^ = irr*R/2l (123) The moment of inertia of a body suspended by the wire and the period of vibration of the body as a torsion pendulum are also related to the moment of torsion. This relation may be de- rived as follows: Take the case of a torsion pendulum in motion. The kinetic energy of the rotating mass when the angular displacement is zero is all converted into potential energy in the twisted wire when the angular displacement is a maxi- mum; hence, the expressions for the energy in these two cases inay be placed equal to each other and the value of C^ may then be readily determined in terms of / and T, as below : £is = ^Ita^ and w = v/r ( 124) in which v is the maximum linear velocity of a pwint on the body and r is the distance of the point from the axis; v = A Ztt/^T in which A = rff , ff being the maximum angular dis- placement. (0 = 2ire'r/T r = IwO' /T and E^ = I2i^e'^/T' (125) £p = Qe' X 6l'/2 (126) since the potential energy is the product of the average torque and tfie^Sistance through which it acts. Since E^ = E^, I lir'&^/T' = C^ff ^/2 and r2 = 4 ,r2 //C, r = 2,rVVCi (127) From this last equation any one of the three quantities may be obtained if the other two are known, or can be found by ex- periment. The term " Constant of Torsion" (C) is sometimes used in- stead of moment of torsion, but has a different meaning. The constant of torsion is the moment of the couple that will twist wiit length through unit angle. It may be found by multiply- ing the moment of torsion by the length. (C = QL). ELASTICITY 3 17 Equation 127 may be derived in another way, as follows: Since the motion of the torsion pendulum is harmonic r = 2,rVTV^ (128) The moment of inertia (/) is the moment of the couple that will produce unit angular acceleration ; or / = C^B/^a and dy/a = XyC^. Therefore ^ = 2^^/(77c^) (129) CHAPTER VI EXPERIMENTS IN ADVANCED MECHANICS GROUP I — M. MEASUREMENTS experiment ivi-31 Radius of Curvature of a Lens The Spherometer. Probable Error. Reliability The experiment involves the measurement of a very small length with a good degree of accuracy, and the determination of the probable error. The Spherometer. The essential parts of the spherometer are a micrometer screw and a small tripod in which the screw is mounted centrally with its axis perpendicular to the plane through the points of the pointed feet of the tripod. The axis intersects this plane at the center of the circle through the three points, which are equidistant from the point of the micrometer screw and from each other. If the spherometer is placed on a spherical surface, and the micrometer screw is adjusted so that the four points will all touch the surface at the same time, the point (p) oi the microm- eter screw will be a small distance (d) above the plane of the other three points (p^, p^, Pa). The relation of this distance to the radius of curvature (R) may be derived as indicated in the following outline: ■Let / be the length of the sidds of the equilateral triangle. The radius of the circumscribing circle r = | -\/l' -\l^= V J l^ ; and r is a mean proportional between d and 2 R-d. d r , , . , ., 1 ,2 2R- d from which if i l^ is substituted for r. ^- bd^2 Method of Procedure. In using the spherometer to deter- mine the radius of curvature of a spherical surface, the read- 319 320 ADVANCED MECHANICS ing of the micrometer is taken first when adjusted on a plane surface, then when adjusted on the curved surface. In each case all four points must be in contact with the surface when the micrometer is read. The difference between the means of the two sets of readings gives d. The determination of R from measurements of d and / is an illustration of indirect measurement, which in many cases is the only method that can be used. The measurements of d and / should be made within about the same percentage of error (Sec. 39). Since d is in general very small in comparison with /, it must be measured with especial care, and a good mi- crometer, to make the fractional error as small for d as for a direct measurement of I with a good scale. Make at least ten independent settings and readings for the plane surface (more are better), and an equal number for the curved surface. Compute in each case the probable error of the mean of readings, and the probable error of any single observation, as indicated in Sec. 302. Compare the results for e computed by the long and short formulae. The length / may be measured by applying the scale directly to the points, also by measuring the distances between the cen- ters of impressions left on a page of the note book after the instrument is lightly pressed upon it. If the results of the two methods are equally reliable, take the arithmetical mean; but if one is judged to be more reliable than the other, express the relative reliabilities by numbers called " weights," and find the " weighed mean " by multiplying each result by its weight, and dividing the sum of the products by the sum of its weights. The quotient is then called the " weighed mean." If the proba- ble errors are known, the weights are, according to the theory of probabilities, proportional to the reciprocals of the squares of the probable errors. For approximate results the reciprocals of the probable errors may be used for the weights. Remarks: Try to find a good method of making the spherometer settings to insure the simultaneous contact of all four points. Compare the average deviations for the plane and the spher- MEASUREMENTS 321 ical surface, and decide whether the plane is sufficiently reliable to use. Partners should work on different lenses with different spherometers so that it will not be necessary for one to wait for the other. Tabulate the deviations and their squares for use in finding the average deviation and probable error. Preparation for the experiment should include careful study of Sec. 301-303. Test Questions. 1. Define " least count," and explain how to find it. what is the least count of the spherometer used? Tell how to read and set the spherometer. 2. How may the plane surface be tested? Where does the doubtful figure occur in the readings? Is the uncertainty due principally to the unavoidable errors of reading, the uncertainty of making settings, or the inaccuracy of the plane? 3. Be prepared to draw figures and explain why r = | V l^-p/4, and why r is a mean proportional between d and 2R - d. Interpret the spherometer formula. Under what conditions is the last term negligible? 4. Define probable error, and tell how to find it for the mean, and for any single observation. How are they related? Are as many ob- servations needed for the latter as the former? 5. If the mean diameter of a cylindrical rod is .7460 + .0015 cm., and the length is 20.42 + .02 cm.; what is the volume, and its approximate probable error? 6. Estimate the fractional error liable to occur in the measurement of / and d, and find from them the fractional error that may occur in the radius. Write the radius of curvature with the error to which it is liable, annexed by the sign +. experiment m-32 Sensitiveness of a Spirit Level Radius of Curvature and Tests for Adjustment The important part of a level is the upper side of the bubble tube. In a good level the inner surface, against which the bubble rests, is slightly curved. A longitudinal vertical section is the arc of a circle of large radius, so that a small inclination from horizontal will cause a relatively large displacement of the bubble which always comes to rest at the highest point of the arc. If the curvature of the longitudinal section is the samie at all parts, the displacement of the bubble {d) will be propor- tional to the angle (a) through which the level is tilted. The distance (rfj) that the bubble moves when the inclination is 322 ADVANCED MECHANICS changed one minute is a measure of the sensitiveness of the level. Since the distance rfi is proportional to the radius of curvature (:K), the latter may also be used as a measure of sensitiveness. The constants of the level to be determined are d^ and R. The same unit of length should be used throughout the com- putations. If the longitudinal section is not the arc of a circle, d^ and R will have different values for different parts of the curve. For particular uses, such as measurements of small inclina- tions with an engineer's transit or level, the sensitiveness should be the same throughout the useful range of the bubble-tube scale, and should be of the same order as the sensitiveness of the telescope, i. e., a readable angular displacement of the tele- scope should produce an equally readable movement of the bubble in the level. If the level is to be used in the zero posi- tion only, as a carpenter's level, the uniformity of curvature is not important. . The temperature of the level should be uniform; if it is not, the bubble tends to move toward the warmest part of the tube. The level tester, on which the level is placed for the test, is a platform resting on two short pointed legs at one end and on a micrometer screw used as a single pointed leg at the other •end. The end supported by the screw is raised or lowered when the screw is turned. Procedure. Part A. Determine the pitch of the screw and the number of minutes (or seconds) of angle (a') that the table is tilted by one-fourth turn (or other convenient part of a turn) of the micrometer screw. Let n be the number of quarter turns of the screw, then a'n = a is the corresponding angle in minutes. Show that a in radians = d/^R ; a in minutes = 3438 x d/^R = ■a'n; d^ = d/a'n; and R = 3438xd^: One radian = 360°/'25r = 206265 seconds. Adjlist the level on the tester so that the bubble is near the left hand end, and adjust the micrometer head to the zero reading. Call the center line 10 on the scale of the bubble tube in order to avoid negative readings. For convenient reference call the left hand end of the level A, and the right hand end B. MEASUREMENTS 323 Take readings at each end of the bubble ; not at the edge of the liquid since that is less sharply defined than the rounded end of the bubble. Tilt the level by uniform steps by turning the micrometer head one-fourth revolution (or other convenient fraction) at a time. When the bubble gets near the right hand end, reverse the steps to the point of beginning. Data and Results. Tabulate the observed data, venient form is indicated by the following headings: A con- TABLE CII Bubble Readings A to B Bubble Readings B to A From A to Center From Center to B From B to Center From Center to A Left Right Mean Left Right Mean Left Right Mean Left Right Mean The means of the left and right readings give the positions of the center of the bubble. Corresponding values of these should be tabulated side by side, in another table, together with the average of corresponding means. These average values repre- sent the positions of the center of the bubble for each quarter turn. Plot a curve with average of mean readings as ordinates, and quarter-turns as abscissas. If there is no apparent regularity in the deviations from a straight line, draw the straight line that best fits the data. The slope of this line together with the pitch {p) of the screw, and the distance {D) from the point of the micrometer screw to the line joining the points of the two feet at the other end, will give sufficient data for computing the radius of curvature and the sensitiveness of the level. If there is a fairly regular deviation from a straight line, draw the proper curve and find, from the slope at different parts, the curve representing the variation of sensibility. Will the two curves be similar in form, or not? Procedure. Part B. If a level that is permanently at- tached to a telescope is chosen or assigned for this experiment, 324 ADVANCED MECHANICS the preceding instructions and the remarks below are to be carefully studied in order to get a clear understanding of what data is essential; then outline a method of procedure that will give the data desired without the aid of the level tester (See question 4), and present it to the Instructor for approval. Remarks. Note that the displacement of the bubble divided by the corresponding angle of tilt in minutes gives the displace- ment per minute of arc. Express this algebraically, and show that d^ = l/a' X {d/n) in which a' = 3438 p/AD and d/n is the slope of the curve; also show that R = {AD/p) d/n. To test the adjustment of a level used for horizontal surfaces, take a reading on a suitable level surface, then reverse the ends of the level and read again. The deviation of the center of the bubble from the center mark should remain the same in amount and direction from the center. If the deviations for the two readings differ, turn the adjusting screw enough to cor- rect hcdj of difference. Repeat trials until ■ correctly adjusted. Test Questions. 1. What is meant by the limitations of a level? 2. Show that the angle, in radians, through which the level is tilted, is measured by the ratio of the run of the bubble to the radius of curva- ture d/R. 3. What data is essential in order to find di and i?? How is the data obtained and used to find each? 4. If the level to be tested were permanently attached to a telescope, as on a surveyor's transit, how could its sensitiveness be tested by means of a scale at a convenient distance without the aid of the level tester? Outline the method of obtaining the essential data. 5. Describe the process of testing the adjustment of a level used for horizontal or vertical surfaces. experiment m-33 Adjustment and Use of a Dividing Engine Graduation of Scales and Verniers Before beginning this experiment, the construction and mode of operation of the engine should be studied, and the precau- tions necessary to insure good results and prevent injury should be carefully noted. Read the description following this out- line. (Further information and illustrations are given in -Manuals in the reading room if desired.) Study the adjustments and details of manipulation by inspec- MEASUREMENTS 325 tion and use of the engine. Verify approximately the pitch of the screw by observing the number of turns and fraction of a turn to cover a given space (say 2 inches) on a reliable scale. Note how to use the engine to measure an unknown length. For preliminary practice, graduate on waxed glass a 5 or 10 cm. scale in millimeters, and a two inch scale in some convenierit small fraction of) an inch. Etch with hydrofluoric acid and present for inspection. Graduate on metal a scale and vernier (or two verniers) to be specified by the Instructor. The following description and directions are taken from " Laboratory Instructions " by Prof. Wm. J. Hopkins : " The instrument used in this laboratory was made by the Societe Genevoise and its general construction and mode of operation are described below. " Upon a wooden base is fastened a massive casting of iron, to act as a rigid bed for the moving parts. Upon the top of this, prepared for the purpose, slides the carriage, upon which is clamped the object to be measured or examined, or the scale for graduation. The most important part of the engine is the screw. Upon the accuracy with which this is cut and upon the manner in which its mounting guards against trouble due to wear and lost motion, depend the value of the engine as an instrument of precision. The method of mounting the screw will be evident, upon inspection, and no change is to be made in the adjustment of these parts. " A; split nut travels upon the screw and bears in one direction only, against the carriage. ' Backlash,' or looseness of fit of the nut on the screw, is thus avoided, since the turning of the screw can propel the carriage in but one direction, and the threads of the nut always bear against the same side of the threads of the screw. From the under side of the nut projects a rod which bears against a steel template in the bed-plate. This template is adjustable. When adjusted so as not to be parallel with the motion of the carriage, a slightly greater or a slightly less motion than the pitch of the screw results from one revolution of the handle, since the nut turns somewhat, either one way or the other, with or against the screw. By 326 ADVANCED MECHANICS proper shaping of this template, inaccuracies in cutting the screw could be corrected. " The screw is turned by a handle at the left, provided with a milled head divided into 200 parts. There is also, in connection with this head, an adjustable stop, by the use of which it is possible to move the carriage easily and without unduly frequent observations of the head, through successive equal intervals of space of any desired amount. The principle, and the mode of use of the stop, are best learned from an inspection of the in- strument. " The ruling device consists of a tool-holder, pivoted in a rigid frame in such a way as to have motion across the car- riage only. In the tool-holder may be placed a drawing pen, pencil, or steel graver. In making the mark, turn the milled head, winding up the cord, until the tool is lifted clear of the carriage. Then, with the left hand, raise the pivoted tool- holder until it brings up on its stop, turn the milled head slowly in the reverse direction, letting the tool down gently upon the body to be marked, and move the tool-holder back to the other limit. The tool must then be raised clear of the carriage before the screw is turned. These movements must not be violent or hasty, for that would introduce errors into the work. Above all, never let any moving part bring up hard against a stop. The extent of the movement of the tool is limited by adjustable stops, and as it is usually desirable in ruling scales to make lines at certain regular intervals more prominent than the others, provision is made for accomplishing this by the use of toothed wheels, which can be adjusted to each other in pairs. Some of the cuts between the teeth are deeper than others, thus allowing a greater motion of the stops and a consequently longer mark by the graver. " In using the dividing engine, the greatest care must be ex- ercised not to injure any part of it. It is not an instrument that will bear careless handling; but especially is it necessary to preserve the screw from injury. It should be inspected before use, as should all parts of the engine. Remove any dust with a soft brush, or, if necessary, with a soft cloth slightly oiled. Neither dust nor lint must be left upon the thread of the screw MEASUREMENTS 327 and but little oil. Put the nut in place carefully, with the two little studs bearing- against the lug at the end of the carriage, and the bar downward. Adjust the microscope at about the middle of the bed, and having made sure that all the bearing surfaces are clean and properly (not too freely) oiled, become familiar with the operation of the engine." Test Questions. 1. Describe briefly the essential parts of the divid- ing engine. What precautions should be taken in operating it? 2. How should it be adjusted to rule a scale in 20ths of an inch? Specify also the setting for ruling a 25 part negative vernier for use against the scale ruled in twentieths of an inch. 3. WTiat would be the least count of the vernier described in 2? What are the characteristics of positive and negative verniers? experiment m-34 Calibration of Micrometer Microscopes Standardization of a Scale The micrometer microscope is a combination of the microm- eter screw and the ordinary microscope. The objective of the microscope produces an enlarged image of the object. This image can be adjusted, by focusing to lie in the plane of a sys- tem of fixed and movable cross wires. These very fine wires and the image are observed with a magnifying eye piece. The movable cross-wires are mounted on a frame which is moved at right angles, to the line of sight by a fine micrometer screw. The number of complete turns of the screw is marked by a fixed saw-toothed scale in nearly the same plane as the cross- wires. The eye piece should be focused on the cross-wires, before the microscope is focused on the object. The calibration constant for any given system of units is the number of turns required to move the cross-wires over the en- larged image of a standard unit of that system, or of some con- venient fraction of such unit. Procedure. Part A. The first part of the experiment is to determine the number of turns of the micrometer screw that will carry the movable cross-wires over the enlarge(J image of a millimeter. A standard scale with fine accurate rulings should be used and the observations should be repeated several times using different parts of the toothed scale in order to determine 328 ADVANCED MECHANICS if the constant is the same for all parts of the field of view. Find the calibration constants for the millimeter and the inch, and use the instrument on some very small length (such as the diameter of a needle) which may be verified by a micrometer caliper. (Reference, Miller's Manual, pp. 20-23.) Procedure. Part B. The second part of the experiment is to test the length of a given scale by comparison with a standard meter bar. Do not allow anything to touch the graduated part of the standard meter. Two microscopes on tripods are used for that part, and the scales are supported on a frame so ar- ranged that first one, then the other, of the two scales, can be brought into position for reading under the microscope. Such an arrangement for facilitating the comparison of scales is called a comparator. The microscopes are set at a distance apart equal to the lengths to be compared. Their axes should then be rendered parallel and vertical by use of a mercury surface, as follows: Place a fine cross-hair (horizontally and parallel to the micro- meter cross-hairs) over a shallow dish of mercury at such a height that the microscope may be focused on the cross-hair, then on its image in the mercury, without touching the cross- hair. Adjust the levelling screws so that both the cross-hair and its image when separately focused, will give the same read- ing on the micrometer scale. The scales are so adjusted that the graduations limiting the length to be tested (e. g., 1 and 99 cm. marks) will be visible near the center of the field. The microscopes are set first upon the standard scale, then upon the other. The difference between their lengths is read and recorded in turns of the micrometer screws. The operation should be repeated two or more times. If only one micrometer is calibrated, the other may be used as an index to which one of the graduations on each scale is set. Note the temperature {t) of the bars when compared. Obtain from the Instructor the temperature {f) at which the standard meter bar is cor- rect; also its coefficient of expansion {e). Find the length (Lj) between the observed marks on the standard scale at the air temperature {t). If the marks 1 and 99 are chosen, L^ = 9'&{l + et) / {\ + ef). Derive this formula from the gen- MEASUREMENTS 329 eral one L = L„ {l + ct). From L^ of the standard and the ob- served difference (d) the unknown length (X^) at air tempera- ture may be found. Then find the temperature at which the marked value of the scale tested is the true value; or, if that temperature is far from the ordinary air temperature, compute the percentage deviation of the marked value from its true value at 20° C. Remarks. The relative dimensions of the object and the image in the plane of the cross-wires are proportional to their distances from the lens in the lower end. If the pitch of the micrometer screw is known the magnification of the objective may be found. Test Questions. 1. What are the essential parts of a micrometer microscope? How calibrated? How used? 2. What is meant by the calibration constant? How may the calibra- tion be checked? 3. How may the axes of the microscopes be adjusted to be parallel and vertical? 4. What are the essential parts of a comparator? It should include provisions for what adjustments and motions? How used? 5. Interpret the formula it =98 (1 + e t)/'(,l + e t'). How may the dis- tance between two marks on the standard at the temperature t be com- puted? What data must be given? 6. What are the essential steps in the standardization of a scale? 7. What figure in the readings made with the micrometer microscope is doubtful? How determined? experiment m-35 Errors of a Graduated Scale by Precision Methods Errors of Each Tenth of a Meter Scale The experiment is to be performed as directed and illustrated in Miller's Laboratory Physics, pp. 32-36. Test Questions. 1. Describe the method of finding the relative cor- rections for each tenth. What data must be observed? How arranged? 2. What additional test must be made before the true corrections can be determined? 3. In which test must temperature be considered ? In which should change of temperature be avoided if possible? Explain. 4. How may progressive errors be eliminated? 330 ADVANCED MECHANICS GROUP II. WEIGHING experiment w-31 Study of the Beam Balance and Balance Tests Corrections for Buoyancy of the Air Theory. The theory to be studied relates to (a) the derivation of the equation for the sensibility of the balance (see Sec. 305-306) ; (b) the interpretation of the equation for the conditions upon which sensibility depends; (c) the mechanical conditions required, and effects of defective construction (Sec. 308-315) ; (d) the precautions to be observed in using the bal- ance; and (e) how to correct for the buoyancy of the air. The study of the theory found in Sec. 305-310 should give a clear understanding of the reasons for the following: If three knife edges are initially in the same plane, the sensi- bility is greater, (1) the smaller the distance between the center knife edge and the center of mass of the beam; (2) the longer the beam, (3) the lighter the beam, (4) the less it is bent under load; and (5) the smaller the friction of the knife edges. The more important mechanical conditions relate to the knife edges. The center edge should be equidistant from the other two, making the arms equal. The three knife edges should be parallel and lie in the same plane. The center of mass of the beam and the center knife edge should be in the same vertical plane. The pans should have equal masses. Precautions. The precautions to be observed in using the balance relate principally to the care of the knife edges, the weights and the pans. (1) Under no circumstances should weights be put on, or taken from, the pans when the balance is resting on its knife edges. In releasing and arresting the beam carefully avoid any jerk or jar; and arrest the beam as it passes its position of equilibrium. The sharp, steel, or agate, knife edges and the agate plates on which they rest when in use, are easily injured by careless handling. They should be in contact as little as possible. Do not overload the balance. (2) Never touch the marked masses, or the pans, with the WEIGHING 331 fingers or with anything that may injure them. Use forceps. A measurable change in mass may be caused by a single touch. Liquids should be weighed in stoppered bottles. Water vapor will ruin steel knife edges. (3) Care should be taken that the pans do not swing sidewise and that the case is closed and air currents have ceased before the final readings of a weighing are taken. Po not weigh a body when hot; for the air currents will affect the weighings. Keep the case closed except when changing weights. (4) Keep all standard masses in their proper compartments in the box when not in use on the balance pans. The pans should be carefully dusted with a camel's-hair brush before using. It is best to try the marked masses methodically in their proper order. The larger weights should be placed near the center of the pan, and the others so distributed that the pan will not swing to one side when released. Use the rider instead of masses smaller than 5 or 10 mg. (5) To avoid mistakes, the weights should be counted two or three times and be carefully recorded immediately. Count them in order of magnitude while in the pan. Count them by the places in the box from which they were taken; and count them as they are returned, in order of size, to the box. (6) Avoid parallax in reading by having the line of sight perpendicular to the scale. Read the scale to tenths of the small divisions. Call the middle division 10 to avoid negative readings. The release of the arrest will usually cause sufficient vibration of the beam. To increase the swing, gently stir the air above, or below, one of the pans. (7) Before leaving the balance, be sure that the beam is lifted, set the rider at zero, dust off the pans and floor of the case, and close the case; also be sure that all the marked masses are in place in the box. Corrections for Buoyancy. If the density (rf) of a body is less than the density (d') of the weights used to balanced it, the volume (V) of the body will be greater than the volume of the weights, and will consequently be subjected to a greater buoyant force. By Archimedes's Principle the buoyant force is meas- ured by the weight of the air (or fluid) displaced. The ap- 332 ADVANCED MECHANICS proximate density of air is .0012 gm. per c.c. (For more ac- curate values see Miller's Manual, Table 12, p. 382.) There- fore the buoyant force is .0012 gV= .0012 g m/d. The true weight {mg) of the body less the buoyant force of the air on it, balances the true weight (w' g) of the marljed masses less the buoyant force of the air on them; or, mg - .0012 mg/d = m'g - .0012 m'g/d' .". m = m' + .0012 {m/d-m'/d') If m in the perenthesis is called m', and d' for brass weights is 8.4, m = m' + W\2m' {\/d-\/?>A)=m' + c The errors due to inequality of balance arms and masses of pans may be eliminated by weighing the body first in one pan and then in the other; then if m' and m" are the results with the marked masses in the right and left hand pans respectively, the mass m = Vw' x m" ; but the average of m' and m" is an ap- proximation sufficiently close for any ordinary purpose. If / and r are the lengths of the left and right arms respectively, the ratio of arms is given by ly/r= V m'/m". (Sec. 309-310.) Methods of Weighing. Methods have been devised in which the errors due to inequalities of arm lengths and pan ihasses are eliminated, and the zero, or no load, reading is not required. The method of Vibrations and Reversals is given in Part I, Exp. W — 2. It may be modified to advantage by using the rider instead of the milligram weights if the balance has a Tider scale each side of the zero so that the rider may be trans- ferred to the other side when the loads are reversed. See first illustration below. If the balance has only a single rider scale, a method that does not require reversal of loads is more con- venient. Weighing by Vibrations and Substitution is a method in which substitution replaces reversal, and the advantages given by use of vibratiwis and the rider are retained. It is equally accurate and requires no more time than weighing by reversal. A load of known masses (Af), larger than the unknown {X), is placed on the right and the rider is on the right near the mid- dle of its scale. These are balanced approximately by any con- WEIGHING 333 venient " tare " in the left pan. Readings are taken ; then M is replaced by X and known masses are added, on the right, until by aid of the rider a balance within the range of the pointer readings is secured. A second set of readings is then taken. If the sensibility for the given load is not known, it should be found by moving the rider one milligram and taking a third set of readings. The additional details and the form of record are indicated in the second example below. The method is a modi- fication of Borda's method of weighing by tare. If several weighings are to be made this method is much shorter than the one by reversal; for, after the first weighing, each new mass may be found by making only one additional set of readings, as indicated for Y and Z in the illustration. Since the load is constant, the sensibility remains constant and does not need to be redetermined for each new mass as in the method by reversal. The reversal requires at least two additional sets of readings in place of the one. If it is desired to adopt a single method suitable for all pur- poses, the method by Vibrations and Substitution (or weighing by constant load, or by " tare " as it is sometimes called) is recommended for its accuracy, simplicity, and economy of time. WEIGHING BY VIBRATIONS AND REVERSALS. TABLE CHI L.Pan R. Pan Rider Cor. Pointer Readings Sum Diff. Results M M M = 26.47 X X + 5 -5 -7 +.0006 7.6 13.5 7.8 8.2 11.8 8.4 8 7 15.2 8.9 21.2 20.2 24 + 1.1 3.9 26.4756 .00028 -¥=26.4759 gm. WEIGHING BY VIBRATIONS AND SUBSTITUTION. TABLE CIV L.Pan R. Pan Rider Cor. Pointer Readings Sum Diff. Results "Tare" A/= 50.00 + 5 +.0004 7.4 12.0 7.6 19.5 41.3637 it X+ 8.64 + 2 -.0003 8.2 11.9 8.3 20.2 -0.7 - .00017 H •' + 3 5.9 10.2 6.0 16.1 4.1 -y=41.3635gm. K=37.5793gm. fc44 79836 gm. tt + 12.42 + 8 -.0014 6.0 11.4 6.2 17.5 + 2.0 n + 5.20 + 4 +.0027 8.0 12.8 8.2 20.9 -1.4 Procedure. Part A. The experiment consists in determin- 334 ADVANCED MECHANICS ing the true mass of distilled water that completely fills a specific gravity bottle at the temperature observed when filled, and in determining a sensibility curve for. the balance used. First, find the apparent mass of the dry bottle alone; then fill it with dis- tilled water at a known temperature and find the apparent mass of the bottle and water. (It is desirable that the temperature of the water be slightly higher than that in the balance during the weighing. Why?) Use the substitution method of weigh- ing. Make the corrections that are necessary in order to determine the true mass of the water. Explain why you make them as you do. The density of glass may be determined from tables, if needed. Note that the buoyant force on the glass cancels in subtracting weight equations. From the true mass of the water and its known density at the temperature when filled, compute the volume of the water, or the cubical contents of the bottle at that temperature, and at the standard temperature marked on the bottle. Note the principal sources of error and the mode of minimiz- ing them. (If preferred, the determination of the density of a gas may be substituted for the calibration of a specific gravity bottle. Call for oral instructions.) Take the necessary data and plot a sensibility curve. Note the period of the pointer for no load and for full load. Part B. The following may be substituted for the outline given under A, by obtaining special permission to change the ad- justment of the balance for variation of period. [See (c) be- low]. The balance must be lifted from the knife edges before changing adjustment. Make tests to determine the following: (a) Ratio of lengths of arms.- (b) Variation of sensibility with change of load. (c) Variation of sensibility with change of period. (d) Variation of sensibility with position of pointer. (e) Reliability of pointer readings and results. (f) Deviation from equality of masses of pans and supports. Plot curves for (b), (c), and (d), and interpret them. Leave the balance in good order, and adjusted to the same pe- WEIGHING 335 riod it had at the beginning, or to the period specified by the instructor. Remarks. The columns headed Cor. in the tables are for entering the calibration corrections of the set of weights used. (See Table CVIII). If there is no calibration table with the set, this column may be omitted. Test Questions. 1. (a) What is meant by the sensibility of a balance? How measured? (b) On what does sensibility depend? How does it vary with each? 2. If the sensibility of a given balance is 2.4 divisions per milligram, the length of pointer 120 scale divisions, the mass of beam 25 grams, and length between pan knife edges 20 cms,; (a) what is the value of r? (b) How much must the center of mass be lowered to decrease the sensibility 10 per cent.? [Answers. Ca) .02 cm. (b) .0022 cm.] 3. (a) Why should the knife edges all lie in the same plane? (b) Upon what condition does stability depend? (c) How may the inequality of arms be detected, and the effect elimin- ated? 4. (a) Under what conditions is a correction for buoyancy needed? (b) Explain how to make the correction. 5. (a) If the apparent mass of a platinum beaker is 12.486 grams, when weighed with brass weights, what is its true mass? (b) If the water that fills a specific gravity bottle at 4° C. has an ap- parent mass of 25.000 gm., what is the capacity of the bottle? 6. Name the more important precautions to be observed in using the balance. experiment w-32 Calibration of a Set of Weights Comparison by Method of Vibrations and Reversals The errors of adjustment in sets of weights are frequently too great to be neglected in weighings where a good degree of ac- curacy is required. This may be true even of sets presumed to be of fine quality. A convenient method of calibration is there- fore important. Procedure and Computations. In the method by compari- son each weight is compared with another single weight, or with a combination of weights of nearly equal mass so that the num- ber of weighings is one less than the number of dififerent weights or groups used. If the mass of some one of the weights or 336 ADVANCED MECHANICS groups is assumed to be correct as marked, or better, if its true mass can be determined by comparison with a standard mass, the number of unknowns will be the same as the number of weighings; therefore, the weighings will furnish enough equa- tions to make possible the computation of the error of each weight. The computation is facilitated by a systematic arrangement of the equations so that the method adapted for one group of weights may be used in each succeeding group. Such an ar- rangement is given below. Let the masses be represented by capitals and the differences by small letters as in the following outline. The values of these differences may be either positive or negative. OUTLINE FOR THE CALIBRATION OF A SET OF WEIGHTS. TABLE CV Mark Mass 50 20' 20" 10 A B C D 5 2' 2" 1 F G H I .5 .2' .2" .1 K L M N .05 .02' .02" .01 P Q R S .01 Group E J O T Y 1. A = B + C + D + a = 5J+ a+ b + 2c + id = 5J+ a' = 2. B= C +b=2J +b+ c+ d=2J+y = 3. C= D + J + c =2J +c+d=2J+c' = 4. D = J + d = J j^ = J + d = 5. Let £ = 10 J + y =WJ+ a + 2h + ic + 6d = 6. Theny = a + 2b + Ac + bd; and J = 1/10 of (£ - j) = i/5 of (A - a') = etc Since many trustworthy weighings are to be made it is im- portant that the method adopted be short and accurate. Any method which depends upon the determination of the zero point must be untrustworthy since the position of equilibrium of the unloaded balance may not coincide with its position of equilib- rium when loaded. It is practically impossible to obtain and maintain exact equality of balance arms and pan masses, and the sensibility generally changes somewhat with change of load. All of these sources of error should be taken into ac- count and should be eliminated as far as possible. All of these conditions are fulfilled by the method of weighing by vibrations and reversals, as illustrated in the following data sheet, and ex- plained in Exp. W — 31 and W — 2. The signs of the differ- ences a, b, c, etc., may be easily determined by noting that, for the WEIGHING 337 two masses compared, the pointer approaches the lighter mass; i. e., the lighter the mass on the right, the greater is the sum of readings. (To find the sum, add to the sum of the first two, half of the difference between the first and third readings.) The corrections are to be tabulated as indicated in Table CVIII. Especial care should be taken to prefix the proper sign. If the mass is less than the marked value the correction is negative. If any one of the masses A, B, C, D, E, T, is determined by comparison with a standard (Z), the correct value of each may be computed from the equations p. 338; or if any one of these masses is assumed to be what it is marked, the values of the others relative to the one assumed may be computed from the equations given. If the one assumed to be correct as marked is later found to be incorrect by a small amount {z'), when com- pared with a standard (Z') each of the other values will be incorrect, and must be given a correction proportional to its mass. For example, if the true value' of A exceeds the assumed value by the amount ^ , the true value of B will exceed th^ value computed relative to the assumed value of A by the amount 2/5 of z^. If the sum of the computed values A, B, C, D. does not check with the computed, assumed, or known value' of E, the cause should be found if possible. An incorrect sign is a common source of error. To check the signs of the differences note that the pointer swings toward the lighter mass. The data given in the sample sheet (p. 338) illustrates the mode of procedure when the errors are so large that extra masses must be used to keep the pointer on the scale, and when the rider can be used on only one end of the beam. The 20 mg. and 15 mg. used with A should be either reliable weights, or be cali- brated by use of a reliable rider if they belong to the un- calibrated set. Note that 20 mg. on the left pan when the rider on the right arm is at 2 mg. is equivalent to 18 mg. with A, or to 15 mg. with A and the rider at 3 when the masses are reversed. The pointer readings show that A + 18 mg. is larger than B + C + D ; hence the .3 mg. must be subtracted from the 338 ADVANCED MECHANICS O H U «1 o o Pi < o S H £ K H o (d H « < < > < ^ m 00 00 ON o O "O s O rt ..H O 1 o o o o o o o o £ lO « M "-I II II M II < oa u Q .s t>. CO ~ --I ■* t^ o 35 •" "^ 41 CO .-1 o g o o B B o o § o o 9t eS 1 1 1 1 |cS II II II II II 5 « -O V, Ts N "s o CO eO 0^ vO ^ + 11']' + °i ON vO tNI NO ■" NO l-( ICO <; =0 « •S CO • • -1 NO nO NO 2 mIvo , rH M NO Q 1 ' ' J V -1 e + (S ^eomNOinoMoooc^c>.^^ ooo ONi>^«n 00 oeo»o Nvo ^ in 00 a^^^ " + f-H W C« 1-H M .-H i-H CS ^ »-l M ^ ^ 1-H 0 O m ^ f-t 00 o^^^ c rH *.^ ■o 1 lA OOlOOrrO'^OlOMCSOOv lOOO tH OnOO 1-H ooo C^l-H O ^ ON O ffv ^ i-(»-lT-(»-4»-( i-)i-(i-(rHi-l »-( & a 'I ij >o 00 m MIC vo lo eo o in o in end t^ >o t>. oo On »n >o o m -^ 1-) 00 ONt^ »-i 1-1 'S, ■S esco 1-1 o oc^oowoocNj ovi-n (2 Q bo «rf + B y S" ■g,g oS; oeqi -,J: -,|: s< + + ++ + II ffl< QCJ Q N mQ hi] 2l 1+ PS c N-H i-iCN»^ ^ ,1-H 1-H C.^ . OOOOO I 1 II 11 ■SioNooo II II ^'R'^ o OCSIM,-l§ + + + II II II II II -e>-« SN:c.X■^^^., + + + + + + + « 'SN^^'^ o II II II II II II II II II \:b \,.^ is unity ; then -L = C^. For a particle rotating about an axis at a distance r, the linear displacement {x) is r times the angular displacement (<^) ; and the linear acceleration (a) is r times the angular acceleration (o) . Therefore x = r, a = ra, and x/^a = ^/a, but <^ = - Ly/C^, and a = Lyi (from definition of /). Therefore -y'a=tyCi. For linear harmonic motion T = ZttV ( -■«'/^;) but since -x/a = r = 2^V(VCx), (Eq. 1). for harmonic rotation. State the principal fact expressed by the equation T = '2tt V (//Cj) and express how it may be verified experimentally. Compare the equation with the corresponding one in Experiment A — 33. Procedure. Obtain data for corresponding values of T and mr^ when different masses {m) are placed at the same distance (-r) from the center, (use the brass weights near the ends of^he cross bar) ; also observe data for the same masses (the two balls provided) at different distances from the center. Plot a curve T^ and / as coordinates. Mark differently the points representing the two sets of data. Extend the plotted line across the / ^xis and interpret the intercept. Discuss the results and draw conclusions. ROTARY MOTION 355 Obtain a series of deflections (<^) and the corresponding mo- ments (L) of the deflecting forces. Plot a curve with and L as coordinates and find from it the moment (Ci) when <^ is unity. Select from you data some one case, and find for it the maxi- mum value of a from L = I a, and check the result by use of Eq. c' in the Remarks. Find also the kinetic energy of rotation -when t = T/6. Remarks. If the phase is zero when a particle vibrating harmonically is at the center of its path, the equations for x, v, and a for harmonic translation are: x = A sin {2-,rt/T) Eq. a. v^A (2 7r /T) cos (2Tt/T) Eq. b. a = A(2w /T) 2 sin (2 ■rt/T) Eq. c. Let P (Fig. 106) be any position of a particle vibrating har- monically along the arc SOQ, of radius r, about the center C; being the mid-point of the path at which the phase is zero. Let ^' be the varia- ble angular displacement subtended by the linear displacement x; and ^m be the maximum angu- lar displacement. The amplitude (A) of the har- monic motion is ^ = OQ = r<^m, and x = r^'. If F>g- 106 the angular velocity and acceleration about the center C are denoted by «' and a respectively, the equations above become by substitution and cancellation of r. 4>' = m sin i2'7rt/T) Eq. a' o>' = ^{2^ /T) cos {2 7rt/T) Eq. b' a' = ^ (2 ^/Ty sin (2 ^t/T) = (4 ^'/T') 4>' Eq. c' Since these equations do not contain r they apply to all par- ticles of a body rotating harmonically. In harmonic translation a cc x. In harmonic rotation a a ,j,. (See Eq. c'.) The condition essential for harmonic translation is F oc x; for harmonic rotation it is L a . Test Questions. 1. Define harmonic rotation, and show that the mo- tion of the vibrating system of the apparatus used is harmonic if the damping due to friction is neglected. 356 ADVANCED MECHANICS 2. Interpret the equation r = 2TV(//Ci,) and state what relations may be tested experimentally? How? What quantities should be plotted? 3. How could the relation of T to C, be tested experimentally? (The test is not a required part of the experimental work.) 4. State the conclusions drawn from the computed results and the curves. 5. If. two bodies are supported successively by the same wire and vi- brated as torsion pendulums, how are their moments of inertia and pe- riods of vibration related? 6. Describe a general method for finding the moment of inertia of an irregular body about an axis through its center of mass; also about some parallel axis outside of the body. EXPERfMENT R-33 The Compound Pendulum Period and Equivalent Length Formulas. The compound pendulum describes an oscillat- ing motion along a curved path. The acceleration toward the mid-point is proportional to its displacement from the mid- point. Therefore, the motion is harmonic, and but from the torque equation of the pendulum /a = - Mgh sin ^ = - MghO approximately and T= 1: \Mg. a- Mgh ' ' yj Mgh in which h is the distance from the axis to the center of mass. Comparing this equation with the one for period of the simple pendulum having the same period it is evident that the length I of the equivalent simple pendulum is numerically equal to I/Mh, Therefore, / = I/Mh Procedure. Suspend a long bar by a knife edge near one end and attach a bob or disk near the other. Make all of the measurements needed to determine the moment of inertia of the oscillating system about the knife edge. Compute (in metric units) the moment of inertia, the period, and the equivalent length. Adjust a simple pendulum to the computed length. Start the two pendulums together. If they do not have the ROTARY MOTION 357 same period, adjust the simple pendulum until they have the same period, which should then be determined as a check on the computed value. Find the percentage difference between the computed length and period and the observed check values. Convert the metric data into English units and compute I, I and T. Check with the results in metric units. When the quantitative work is completed, note in a general way without measurements how the period of the pendulurri is affected by changing the position of the bob (especially when above the knife edge), and the position of the knife edge on the bar. Remarks. The period of a given pendulum varies directly as ^I and inversely as the ^/h. I may be decreased by putting the bob above the knife edge, but h may be reduced even to zero in which case T and / would be infinite ; so it is possible to make / any value above a certain minimum. If the dimensions common to both numerator and denomi- nator of the fraction under the radical sign are cancelled out, only T^ remains. Hence, T is independent of the system of units used. Test Questions, i. (a) What data are needed to find / about the knife edge? Indicate how the data are used. 2. (a) If the inch is the unit of length in which / and h are expressed, what value of g must be used in computing T? Explain. (b) If the thickness of both the bar and the bob were doubled, thus doubling the mass, what would be the value of T? Explain. 3. Derive the conversion factor for finding I in Gravitational Ens'ish units from / in metric units. Use it to check your two determinations of /. 4. Is it possible to have an equivalent length much longer than the bar? Kxplain. 358 ADVANCED MECHANICS GROUP V—E. ELASTICITY experiment e-31 Young's Modulus of Elasticity of a Wire Measurements with an Optical Micrometer Study the Notes on Elasticity, Sections 314-315 in order to become familiar with the meaning of terms, and relations of quantities. The experiment includes the determination of corresponding values of stress and strain. From these a stress-strain diagram is plotted, and the Modulus of rigmiy yis found from the slope of the straight part of the curve. If preferred, the loads and elongations may be plotted instead of stress and strain, to which they are proportional. The modulus may be computed from the slope of the load-elongation curve if properly interpreted. The stress is conveniently varied by adding weights on a weight-hook supported by the wire to be tested. The initial load should be large enough to straighten the wire. The important part of the experiment is the measurement of the elongation. Any of the more reliable methods of measuring small increments of length may be used; such as the optical micrometer, the micrometer microscope, the electrical microm- eter, etc. If the elongation is to be measured by any other means than the optical micrometer described below, special notes will be given out if needed. The Optical Micrometer. Gaertners's Apparatus for Young's Modulus uses a three pointed optical micrometer. It consists of a small mirror mounted on a horizontal bar which is supported at one end by two pointed legs (p^ p\) and at the other by one (^1). The two points rest on a fixed arm of the frame, and the point p-^ rests on the upper end of a small short cylinder concentric with the wire to which it is clamped by a set screw. As the wire is stretched, />i falls and the mirror {M) is tilted. The amount of tilt is read by a telescope and vertical scale at a distance. To illustrate, let the distance (D) of the scale from p^ p\ be 20 times the distance (rf) of p^ from the ELASTICITY 359 line joining p^ p\. Then if p^ drops 1 millimeter, the mirror will be tilted so that a perpendicular from the mirror will rise 20 mm. along the scale. This, however, corresponds to a change of 40 mm. in the scale reading. This is due to the fact that the line perpendicular to the mirror is midway between the ray from the scale reading to the mirror, and the ray from the mirror to the telescope and eye. Draw a diagram and prove that the elongation e = (S^ - 6"i) dy/2 D approximately in which (S^-S^) is the change of scale reading due to the elongation e. Let k = rf/2£> and note that k is the elongation per unit change of scale reading. It is expedient to set the scale at such a distance that k will be a convenient fraction, as illus- trated above, where k = 1/40. Procedure. Adjust the optical lever, and the telescope and scale so that a clear image of the scale is seen in the telescope by reflection from the mirror. Add each successive weight carefully and take a series of readings for increasing load, also a series for decreasing load. If these differ appreciably, seek to remove the cause. If small differences remain, use the average of the two series for plotting. Note and observe all of the additional data needed for the determination of stress and strain. Data and Results. Tabulate clearly and in good form, the miscellaneous data, the observed data, and the computed re- sults. Express M in dynes per square centimeter, and in pounds weight per square inch. Explain clearly how the modulus is found from the curve. Remarks. Be sure that the collar that supports p^ is free to move vertically with negligible friction on its guide. If the wire is quite stiff and bent, two or three kilograms may be needed for the initial load. If the weights have not been ad- justed to agree with their marked values, the mass of each should be found and a designating number be given. The stress-strain diagram is an exception to the general rule to plot the independent variables as abscissae. Leave room be- '36o ADVANCED MECHANICS low the origin to plot the initial load, also leave some space at the left to extend the line. Test Questions. 1. (a) Define stress (in tension), strain, and Young's Modulus of elasticity, (b) How are they related? (c) Does the modu- lus change with the dimensions of the test piece for a given metal, or with the load? Explain, (d) Show how to find M from the slope of the curve. 2. If an optical lever is S cms. long and the scale is two meters from the mirror, what is the elongation per centimeter change of scale reading? Explain. Why is the value of k the same for inches as for centimeters? 3. (a) Between what points should the length of the wire be meas- ured? Why? (b) Would a long length have any advantage over a short one? 4. How many significant figures in your result? How estimated? . 5. Indicate clearly the derivation of the factor used to convert dynes per square centimeter to pounds weight per square inch. 6. (a) Which way would the curve bend if the elastic limit were passed ? (b) If your curve reaches the elastic limit, find what load will stretch a rod of like material and 8 sq. in. in cross section to its elastic limit. (c) Find the elongation of a ten-foot rod at the elastic limit. If the elastic limit was not reached, answer (b) and (c) for the maximum' load uesd. (d) Should the initial load be used? Show how to use the curve to find data for computing (b) and (c). experiment e-32 Laws of Flexure of Rectangular Beams Method of Deriving Empirical Formulas An empirical formula is one derived from experimental data and usually expresses important relations of quantitie.« or laws of nature. Explanation of Method. If a rod of rectangular cross sec- tion be supported near its ends on knife edges and a load is located at its center, it would be expected from common experi- ence that the deflection (d) would increase if either the load (F) or the length (L) between knife edges were increased, also that the deflection would decrease if either the breadth (B) or the depth (D) were increased. If the length be doubled for a given load and cross section, we know that the deflection would be much more than twice as great; and if the depth were doubled, the deflection would be very much less than half as great. In general, it would seem reasonable to suppose that the deflection ELASTICITY 361 varies as some power or root of each separate variable and may be expressed by the general formula d a: F^L^B°D^ ( 1 ) in which m, n, o, and p are constant exponents having positive signs if they denote powers and negative signs if they denote reciprocals of powers. The purpose of the experiment is to illustrate a method of finding the values and signs of the exponents from experimental data, and to derive the laws of flexure of beams from the re- sulting empirical formula. Arrange the conditions of the experiment so that three of the four quantities F, L, B, and D will remain constant during the taking of any one set of observations, then vary the fourth and note the corresponding values of d. For example, with L, B, and D constant, F may be given values F^, F^, F^, etc., and the corresponding deflections will be d-^, d,, d^, etc., then d^ = kF^'^D'B<'D^ (2) in which k is the arbitrary constant that must be put in to change from a variation to an equation. If this equation is di- vided by a like one containing d„ and F^, the result will be d,/d, = F,^/F-^ (3) The corresponding logarithmic equation is log rfi - log ^2 = m (log Fi - log FJ or (4) m= i\ogd^-\ogd^)/(\ogF^-\ogF,) (5) To determine whether m is constant or not, its value may be found in like manner from other pairs of corresponding values of F and d. (Six pairs may be obtained from four values each of F and d; viz: 1-2, 1-3, 1-4, 2-3, 2-4, and 3-4.) If these values of w do not differ by an amount larger than can be reasonably ascribed to unavoidable experimental errors, m may be considered constant; furthermore, if the mean value is very near an integral number it is reasonable to suppose that the integer is the true value of the exponent. If preferred, m and other exponents may be found graphically from the slope of a curve plotted with differences of logarithms 362 ADVANCED MECHANICS of the two variables as coordinates. Prove from equation (5) that the exponent is the slope of such a curve. When each of the four exponents have been obtained as in- dicated, for the first, solve for k in at least one equation from each of the four sets of data and take the mean as the constant for that material. If a different material were used the values of the exponents would remain the same, but the value of k would be different. It is thus evident that fe is a constant depending- upon the material. Infer from the formula, when found, how k is related to stiffness. Note that the results above are obtained on the assumption that k cancels out in dividing, or is the same for the different bars used for variable breadth and depth. It is difficult to get several bars near enough alike to give closely concordant re- sults. Procedure. Use at least three bars. They should all have th same thickness and different widths. It is convenient, but not necessary, to have the widths simply related, say in the ratio 1:2:3. The work should be carefully planned and systematic- ally done in order to get all that is required and not do a lot of unnecessary work. The deflection may be observed by use of any convenient ap- paratus such as an optical lever, a micrometer screw with an electric detector of contact, or a microscope with the micrometer eye piece. Be sure to keep far within the elastic limit of the bar. Data and Results. The folloviring general form is sug- gested for recording each set of data and computed results. TABLE CXII Constants F Logf R d Logd m L - 1 2 3 4 I _ 2 JB — 1-3 D — 1-4 Fq— 2-3 R„— 2-4 3 J 4 ELASTICITY 3,63 Fi F^, etc., are the loads in excess of the initial load F^, and R the observed reading of the apparatus used to measure the de- flection. If a more condensed tabulation of data and results is desired, the following is suggested: TABLE CXIII Single Rod All Rods Flatwise L = .. . Bet. Supports Flatwise F= ... at Center Flatwise F^... L=... Edgewise F=:. L=..B=.. F d L d B d D d m = . P= • State in words the laws algebraically expressed by the formula derived. Remarks. A comparison of the empirical formula with the formula derived from theory shows that for end supports and a central load k = 1/4 M; M being Young's Modulus. Test Questions. 1. (a) Compare the stiffness of two joists under like central loads if one is 2" x 10" and the other is 3" x 8", if the sp^ns are each 12 ft. (b) How much may the depth of the latter be reduced to give the same deflection on a 10 ft. span? (c) What reduction of width would give the same deflection on a 10 ft. span? 2. Find Young's Modulus from k = l/AM and check by reference to tables. 3. (a) State the general plan of procedure in taking the data. (b) Which is the least reliable part of the data? What are the chief sources of error and unreliability? (c) What precautions must be observed with respect to choice of load? 4. If the depths of two beams of the same length and width are in the ratio 1 : 3, what is the ratio of their deflections for a ^iven load? 5. What curves may be plotted as a means for finding the exponents from their slopes? 364 ADVANCED MECHANICS GROUP VI— T. TORSION experiment t-31 ■Laws of Torsion Verified with the Torsion Lathe Modulus of Rigidity. Moment of Torsion Preparation. Study the Notes on Elasticity, Sections 314, 316, 317. The experiment may be divided into four parts, as follows: 1. To determine if L oc ^ 2. To determine if L oc r* 3. To determine if L oc 1// 4. To determine values of R for steel drill rod and brass. Procedure. 1. Use the smallest round rod (A). Vary the torque and read the angular deflections. Plot L and 6, and draw conclusions. Plot the independent variable as abscissae. 2. Use three or more sizes of steel drill rod. Find the torques that will produce the same angular deflection for each of the three rods of equal lengths. Plot L and r* and draw conclusions. 3. Find the torques that will produce the same deflections for the whole length of the smallest steel rod (A) and for part of its length. Clamp it at the marked end and at the middle. Test also the square iron rod (D) in a similar manner with four or more lengths from about one-third to full length. Place brass pieces between the square rods and the screws to keep the screws irom cutting the rod. Plot the results and interpret the curve. 4. Compute the values of R for half of the rod A and the •whole of aftother, and compare them. Determine the modulus of rigidity of brass using the round brass rod. Find experimentally the moments of torsion of a square iron rod and a square brass rod of the same dimensions. Find also the ratio of their moduli of rigidity. Test Questions. 1. Define Torque, Modulus of rigidity, and Moment of torsion. How are they related? 2. State the laws of torsion and be able to explain any of the steps in their derivation. 3. Show how the laws are verified by use of curves plotted from the data. 4. (a) How may yielding at the set screws be detected and eliminated? (b) Are there any evidences of a third variable in part of the experi- ment? TORSION 365 experiment t-32 Moment of Inertia by the Torsion Pendulum Period by the Method of Middle Elongations Study the Notes on Elasticity, Sec. 314, 316, 317, relating to Stress and Strain, Modulus of Rigidity, and Moment of Torsion. The experiment may be divided into two parts, as follows: Procedure, Part A. Determine the moment of torsion of a wire by using it to suspend a cylinder (or other regular body) the moment of inertia of which may be readily computed from measurements. The suspended cylinder, when started, vibrates as a torsion pendulum, the period of which is observed by the method of middle elongations described in Miller's Manual, sec- tion 83; or in Nichols and Blaker's Manual, Experiment Aj, Vol. I. When the moment of torsion is found, determine from it the modulus of rigidity (R) of the wire. Procedure, Part B. Determine the moment of inertia of a large wheel (or other body the moment of inertia of which cannot be obtained from measurements) by suspending it as a torsion pendulum in place of the cylinder. Its period is observed as before and the moment of inertia of the wheel can be found from the moment of inertia of the cylinder and the periods of the cylinder and the wheel. Prove that if the moment of torsion is the same for both bodies. Remarks. By the method of middle elongations it is pos- sible to find the period to five significant figures if tenths of seconds are estimated, and three sets of ten readings are taken at proper intervals. Note that the unavoidable error of observa- tion is a very small fractional part of many periods. Test Questions. 1, (a) Define Modulus of Rigidity, and Moment of Torsion. (b) How are the two quantities related? 2. How is the moment of torsion of the wire related to the period and moment of inertia of the vibrating mass suspended by the wire? 3. (a) Point out the distinctive features of the method of middle elongation, and its advantages. 366 ADVANCED MECHANICS (b) The high degree of accuracy depends upon the application of what general principles? (See Remarks and Sections 35, 38.) 4. (a) What should determine the length of interval between sets of readings ? (b) Why may the second and following intervals be much longer than the first? 5. (a) Why is it important to note carefully the direction of the first transit of each set? (b) Is the number of periods in an interval necessarily integral? 6. (a) Why does it make no difference if the marker or telescope is moved between sets of observations when the time for each transit, or each third or fifth transit, (successive ones in opposite directions), is recorded, but does destroy the results when each second or fourth transit (all in same direction) is recorded? 7. How may the moment of inertia of an irregular body be determined? IMPACT 367 GROUP VII — I. IMPACT experiment 1-31 Laws of Impact for Inelastic Bodies Conservation of Momentum. Loss of Energy The laws considered in this experiment may be stated as fol- lows: 1. The vector sum of the momenta of colliding bodies is not changed by impact. 2. The sum of the kinetic energies of colliding inelastic bodies is less after impact than before. 3. When a moving mass (w^) strikes an inelastic mass (m^) at rest, the fractional loss of kinetic energy (I) is equal to the ratio of the mass at rest to the sum of the masses [Wj/Cwj + Wg)] and is independent of the velocity of impact. The third statement relates to only one simple case ; however, the experience in dealing with this will indicate the way to solve more complicated cases. The object of the experiment is to deduce these laws from fundamental principles and then test them experimentally in the case of the collision of two pendulums. Derivation of Formulas. During the collision of two masses Wi and m^, the force /a acting on Wj equals (by Newton's third law) the force f-^ reacting on m^. .'.f2 = -fia.ndmr,a^ = -m^a^ (1) Let Ml, u^, be the respective velocities before impact and v^j v^, the velocities after impact; then the changes in velocity are (Mj- z/j) and {u^-v^, and these divided by the duration of the collision give the mean rates of change of velocity which may be substituted for Oj and fflj in (1), giving Wj (Mj-w,) =mi (z/i-Mi) (2) Derive from this equation a result that expresses the first law, and note that it is general, being without limitations as to elas- ticity or direction or magnitude of velocity. Let £' and E" denote the sums of the kinetic energies before 368 ADVANCED MECHANICS and after impact respectively; then the loss of kinetic energy is E'-E" and the fractional loss (/') is (E'-E")/E'; or /' = (£' - E")/E' = 1 - E"/E' /'=!-( m^v^^ + WjZ'/ ) / ( m^u^^ + m^u^^ ) ( 3 ) If Wj is at rest before impact, M2 = 0; and if the colliding sur- faces are inelastic the bodies will have the same velocity after impact, or v^ = v.^. Why ? With these limitations Eq. 3 reduces to I' = \- {m^ + m^) v^/m^u^ (4) and Eq. 2 reduces to (Wi + mj) ^2 = OTiMi (5) Substituting in Eq. 4 the value of v^ from Eq. 5 gives V = \-\m^/{m^\m^\=m^/{m.i^-\-'m.^ (6) Show that the second and third laws are expressed in full for the given case by Eq. 6. Be able to supply any steps omitted above in the derivation of Eq. 6. The first law may be verified for the case stated in the third by the use of two colliding pendulums suspended by pairs of cords from a common horizontal axis. They are shown in Fig. 107 in which the starting point for w^ is at s, the center of mass of m^ at impact is at />, of m^ at 0, and of (Wi + Wa) at q. Fig. 107 After impact the center of mass of the moving system rises to ^. To find the momenta immediately before and after impact it is necessary to get data from which Mj and v^ (in Eq. 5) may IMPACT 369 be computed. The velocity at any point on the arc below the starting point is numerically the same as the speed acquired by a free fall from rest the same vertical distance ; since the loss of potential energy is the same along the arc that it is along the vertical path and equals the gain in kinetic energy when gravita- tion is the only force that affects the speed. This last condition is fulfilled by the pendulum; for the pull of the cords is per- pendicular to the path and can neither hasten nor retard the motion. If the vertical distance (h^^) through which m^ falls before im- pact, and the vertical distance (h^) through which (m^ + m^) rises after impact can be found, Mj and v^ may be determined from the fact that for a free fall from rest v'' = 2gh, or z; = V {2gh) In Fig. 107 h^ = en- cm = R cos p -R cos .r (7) and h^^R cos q-R cos s' (8) The increase in momentum of the moving system (nii + m^) dur- ing its fall from 5 to is neutralized by its decrease during an equal rise to q' on the other side of o; consequently, the momen- tum at q is lost during the rise hr. from q' to /. The points s, p, q, O, /, may be located by reading on the cir- cular scale placed just below the swinging masses. Determine whether the scale is divided in degrees or not. If it is, the angles s, p, q, s', may be obtained directly from the scale readings and the cosines be found in tables. A light marker placed near s' is pushed along by m^ and serves to locate the limit of its swing. The impact is rendered inelastic by placing a piece of wax at the point of impact. Determine the momenta immediately before and after impact; and compare their values for three or more starting points with >% falling and one or more with m^ falling. Find the loss of kinetic energy during impact; also the frac- tional loss and the percentage of loss. Compare heat produced by impact when Wj falls with that produced when m^ falls from the same starting point. Draw conclusions. (Reference: Millikan, Experiment 6.) Tabulate the data and results as indicated. 370 ADVANCED MECHANICS R = P = q-- S'°: = Observed Data and Formulas : (to center of mass) fe=? (In terms of i? and cosines) (between centers') ^2= ? (in terms of i? and cosines) qt= ° Mi2=? (In terms of gji?, and cosines) s, = r° - ? trjS = ? ( In terms of g, R, and cosines) mz = mi + W2 TABLE CXIV ■ Center of m\ Falls Marker Case Reading from s" r° 1 10° 2 IS" 3 20» OT2 from 4 15° Center of Rises to h-L »i M omentum Before Impact Momen- tum After Impact TABLE CXV Case Energy Just Before Impact Joules Energy Just After Impact Joules Loss of Energy Joules Frac- tional Loss m (at rest) Heat Produced Calories Ha m\ + ma Ht 1 2 3 4- H2 = Remarks. In question 3 note that the kinetic energy at impact equals the work done by the force. Test Questions. 1. If a S pound mass moving 12 ft. per second strikes a IS pound mass at rest with inelastic impact, (a) what will be the velocity after the impact? (b) How much is the momentum of each mass changed? (c) Is the change of momentum contrary to the first law; or may the first law be used as the basis for solution? Explain. 2. Compute the kinetic energy of each mass in the first question be- fore and after impact. (a) Does the fractional loss of kinetic energy agree with the third law? (b) What becomes of the lost kinetic energy? (c) Under what conditions would practically all of the energy be transformed ? 3. One forrri of Impact machine for testing cast iron uses a heavy pen- dulum bob to deliver blows of known variation of energy. The test piece receives the blow when the velocity of swing is a maximum. If the length of the pendulum^ is 6 ft., and the mass of the bob is 20 pounds, what is the average force of the blow if the center of mass of the bob swings 20° before impact and .2° during impact? Under what conditions B'ould the maximum force be twice the average? IMPACT 371 experiment 1-32 Laws of Impact for Elastic Bodies Coefficient of Restitution The Laws to be considered may be stated as follows: 1. The vector sum of the momenta of colliding bodies is not changed by impact. 2. When two elastic masses (m^, m^) collide, the velocities after impact {v^, z\) are related to the velocities before impact {«!, Ms) as follows: V, - V, (1) ^ _ (mi - m^e) «i + (1 + e) m^u^ ,^. rriy + m^ (3) _ (1 + ^) niiUi + (m^ — ftiie) u, trii + m^ in which e is the ratio of the velocity of separation just after impact to the velocity of approach just before impact; and is called the coefficient of restitution. This coefficient is constant for two given bodies if the impact is not violent enough to pro- duce permanent deformation. 3. The sum of the kinetic energies of colliding bodies is less just after than just before impact, except when they are per- fectly elastic. (Deduced from 4.) 4. When elastic masses collide, the loss (L) of kinetic energy is x = (i-/).l.^jL-(„,_„,r (4) I nil + m, 5. When an elastic moving mass, m^, strikes an elastic mass, JWj, at rest, the fractional loss (l) of kinetic energy is, / = (1 _ /) _??i^?L (5) wzi + m^ and is independent of the velocities at impact if there is no permanent deformation. At impact elastic bodies are deformed so suddenly that there is internal friction even in bodies which show practically perfect elasticity by static tests. If perfectly elastic bodies could be found, the coefficient of restitution would be unity and the frac- 372 ADVANCED MECHANICS tional loss in Nos. 3 and 4 would reduce to zero. This is sup- posed to hold true for gas molecules which, according to the kinetic theory of gases, are constantly colliding with each other and bombarding the sides of the containing vessel. Derivation of Formulae and Procedure. The first law may be proved analytically as in Experiment I — 31. Verify it by experiment for the case when one of the masses is at rest. At first let the smaller mass be at rest, as indicated in Fig. 108. Read the explanation given in Experiment I — 31 and be able to prove that TOi V (cos p - cos j) = Wi V (cos p - cos /)+w8jV(1- cos s") (6) •v:?^. ^^v.l. Fig. 108 If the larger mass is at rest, the direction of the smaller body will be reversed at impact. Draw a figure for this case similar to Fig. 108, and write the corresponding momentum equation. Compare the momenta before and after impact, obtained from observed data, for both cases. The second law is based upon the definition of coefficient of restitution, from which the first equation is obtained directly. The second and third equations are obtained from the first and the equation expressing the first law. Express e in terms of cosines of the angles for the case shown in Fig. 108, and also when the masses are interchanged. Solve for e in each case, using data obtained for the momentum equa- tion. Draw conclusions. Compute Wj and v^ as indicated in equations 2 and 3, and check with the values computed from cosines, etc. Show that e = Vj^/'ui when a small ball is dropped on a large IMPACT 373 fixed plane at rest ; and test the constancy of e when u^ is varied, and the compression at impact is within the elastic limit. The third law is derived from the fourth, which must be con- sidered first. Let L denote the loss of kinetic energy at impact. Then I ni^u^^ + 1 m^Mj^ = i m^Vi^ + ^ m^v^ + L (7) Transposing, \ m^ (u^^ - v^^) =^m^ {v^^ - «/) +L (8) Factoring, i m^ (u^ - wj (u^ + v^) =i m^ {v^ - u^) (j/. + mJ+L (9) From 1st law, i »% ("i - v^) = ^m^ (v^ - u^) (10) Transposing, i m^ {u^ - v^) (m^ -u^ + v^-v^) =L (U) or iwi (mj-z/,) (1-e) (Mi-mJ =L (12) tut i 171. («. - t..) = i (1 + .) -^ («. - «,) (13) when the value of v^ in equation 2 is substituted. Therefore i = (l--Oi^f^(«.-«.) (14) To derive the fifth law, let / denote the fractional loss. Then I = L/\ {m^u^ + m^u^) (15) for the general case. For the special case when u^ = Q /=(1-^^)-^ (16) »2i + rm When m^ is negligible in comparison with m^ l={\-e^) (17) Tabulate the observed data required to determine the various quantities involved, also the computed results. Verify the laws, check results, and draw conclusions. Test Questions. 1. (a) Define the coeMcient of restitution, (b) What are its limiting values, and under what conditions do the limits oc- cur? Illustrate, (c) Under what conditions is the coefficient a constant? 2. If a small bicycle ball drops 50 cms. on a case hardened anvil, and rises 40 cms., what is the coefficient of restitution for the ball and the anvil ? 3. If a 5 pound mass moving 12 ft. per second strikes a IS pound mass at rest with elastic impact and a coeffiicient of restitution of .9, (a) What will be the velocity of each after im^iact? (b) How much is the mo- mentum of each changed? How much is the sum_ changed? (c) Com- pare these results with those for inelastic impact in I — 31, Q. 1 ; also compare the losses of kinetic energy. 4. Be able to explain each step in the derivations. 374 ADVANCED MECHANICS GROUP VIII — D. DENSITY experiment d-31 Calibr^vtion of an Hydrometer of Variable Immersion The Mohr-Westphal Balance Used as a Standard The Object of the experiment is to study hydrometer scales and methods of testing the reliability of hydrometers. The problem is either to test the reliability of one that is already calibrated, or to calibrate one having only an uncalibrated scale of equal parts, and to plot a calibration curve for each of two or more hydrometer scales in common use. Scales. Hydrometer scales are calibrated to read (a) density directly; (b) specific gravity relative to water at 4° C. or at 15° C. ; (c) specific gravity in degrees, the number of degrees being 1000 times the value of the specific gravity; and (d) arbitrary degrees corresponding to certain specific gravities. Hydrometers are also called areometers if calibrated for specific gravity, and densimeters if calibrated to read density directly. Some hydrometers are named from the liquid for which they are especially graduated, as the lactometer, alcoholom- eter, salinometer, etc. The common scales for specific gravity read as specified in (b) or (c) above. Specific gravity relative to water at 4° C. is numerically the same as density in metric units, but not in English units. The principal arbitrary scales are the Baume, Twaddell, and Beck. The specific gravities corresponding to given scale read- ings, or degrees, are shown in Table CXIV. Baume used for one of his standard solutions a 15 per cent. solution of salt of density 1.11383t at 17.8° C. The mark to which the hydrometer sunk in this solution he called 15 on his scale, the zero of which was the mark to which it sunk in pure water. fThis value, given in Carharfs University Physics, page 119, is prac- tically the same as the 1.1160 given by other authors for specific gravity relative to water at 15° C. in connection with the relation between Baume readings and specific gravity; but these do not agree with table values for density of a IS per cent, salt solution at 15° C. The latter range from 1.1100 to 1.1115. DENSITY 375 For light liquids the scale reading in water is 10, the zero being the reading in a 10 per cent, solution of salt of density 1.0736. The noteworthy characteristic of the Twaddell scale is that the specific gravity is one plus one two-hundredth of the scale read- ing. TABLE cxiv LIGHT LIQUIDS HEAVY LIQUIDS Scale Reading .Beck Baume Baume Beck Twaddell Sp. Gr. Sp. Gr.* Sp. Gr.* Sp. Gr. Sp. Gr. 1.000 1.000 1.000 1.000 0.971 5 1.035 1.030 1.025 0.949 1.000 10 1.073 1.062 1.050 0.919 0.967 15 1.114 1.097 1.075 0.895 0.936 20 1.158 1.133 1.100 0.872 0.907 25 1.205 1.172 1.125 0.850 0.880 30 1.257 1.214 1.150 0.829 0.854 35 1.313 1.259 1.175 0.810 0.830 40 1.375 1.308 1.200 0.791 0.807 45 1.442 1.360 1.225 0.773 0.785 50 1.517 1.417 i:250 0.756 0.764 55 1.599 1.478 1.275 0.739 0.745 60 1.691 1.S4S 1.300 0.723 65 1.795 1.619 1.325 0.708 70 1.912 1.700 1.350 Formulas. The fundamental equations are the defining equa- tions for density, heaviness, and specific gravity. Density (D) is the mass per unit volume (absolute units), heaviness (H) the weight per unit volume (gravitational units), and specific gravity (S) is the ratio of the density (or heaviness) of the substance to the density (or heaviness) of the standard. D = M/V; H = W/V; S = D/D^ = H/H^ ( 1, 2, 3) in which D^ is the density of the standard. * Relative to water at 4° C 376 ADVANCED MECHANICS Let i/= the average volume per unit length of stem F = volume immersed in water = Lv' V = diif erence of volumes immersed in water and liquid = It/ in which L is the length of a cylinder of uniform cross section equal to that of the stem and containing the volume F, and I is the length of stem between the two readings. V may be found by dividing the mass (M) of the hydrometer by the density of water at the temperature of the test. If the cross section (A) of the stem is uniform, L = V/A. For liquids heavier than water, the Baume scale readings (S) are proportional to the corresponding values of v,or v t^ B. Since the specific gravity is expressed by the ratio of two volumes, any unit of volume may be used to express V and v. If the unit of volume is the change of immersed volume corresponding to each unit of B, then v = B, and, solving S= V/{V -v) in Eq. 4 for V in terms of 5" and v, and substituting B for v, Sv _ SB 1.11383x15 ,.. ^-T^l-J^- 1.11383 - 1 ~ ^^^-^^ ^^^ A c ^ 146.78 ,.. for heavy liquids relative to water at maximum density, or 4° C. For liquids lighter than water, the Baume scale reads 10 when V is zero, and v = B -\0 for all values of v and B. Since the volume immersed in a light liquid is greater than V, S= F/(F + i;)inEq. 4, and „_ Sv S(B- 10) 1.0736(0-10) ^~T^S~^r^ 1 -- 1.0736 ^ ^^^-^^ ^^^ 9 = ^ = 145-88 ,„. V + B-10 135.88 + B ^^ relative to water at 4° C. Formulae 6 and 8 give the values tabulated above. Note that the units of volume are slightly different for heavy and light liquids. For heavy liquids the unit volume is one- fifteenth of V between readings in water and in the 15 per cent. DENSITY 377 salt solution, and for light liquids it is one-tenth of v between readings in water and in the 10 per cent, salt solution. The first unit is F/'146.78, and the second is F/14S.88. The error introduced would be small if a single unit V /XAd were used for both scales. The Beck scales are like the Baume in having equal volume intervals, but are unlike the Baume in having readings (i?) pro- portional to V for both light and heavy liquids, and v = R when the unit of volume is V /\7Q. Then .S"= 170/(170 ±i?) (9) in which the plus sign is used for light liquids and the minus sign for heavy liquids. In the Twaddell scale the intervals are unequal but indicate equal changes in specific gravity. So that 5"=l+i?/200=l + .005i? (10) In all cases the buoyant force is in equilibrium with the weight (j^) of the hydrometer; and W = H^V = H(V±v) (11) for water and another liquid respectively. Methods of Test or Calibration. The first three methods given below are less reliable than the last two. The objects in describing the first three are, first, to use the relations of the various quantities involved in this subject; and second, to make it possible to make a test that will be sufficiently accurate for ordinary purposes by means of apparatus that is apt to be readily available. 1. Stem Uniform. If M and A are determined, L becomes known from L = M/D^ A ; then from Eq. 4 the I corresponding to any given density (or specific gravity) may be computed and used to locate the mark for that value on the stem. 2. Stem of Variable Cros^ Section. The variable cross section may be determined at regular intervals by use of a micrometer,, caliper, and plotted against stem divisions. On this curve find the average cross section between the reading in water and suc- cessive points along the stem. Plot another curve with these 378 ADVANCED MECHANICS average cross sections to successive points as ordinates. From this last curve the value of v to any given point may be com- puted, and the corresponding density or specific gravity deter- mined from Eq. 4. 3. Hydrometer for Light Liquids. If an hydrometer of say 20 gm. wt. sinks to a given mark in a liquid having a heaviness of .8 gm. wt. per cc, it will displace 20 gm. wt. of liquid (by Archi- medes's principle), or 25 cc. To sink the hydrometer in water, of unit heaviness, to the same mark would require a downward force of 25 gm. wt., a force greater than the 20 gm. wt. in the same ratio as the heaviness of the two liquids; or S = H/H^=W/{W + w) (9) in which w is the added weight to sink the hydrometer to the mark corresponding to S relative- to water at the temperature of the test. This suggests a convenient simple method of testing" by use of added weight to sink the hydrometer to varying depths in pure water. If the scale is specific gravity relative to water at 15° C, the water should be cooled to that temperature for the test. Equation 9 may be derived from the fact that in water the ratio of the buoyant forces W/{W + w) equals the ratio of the volumes V/{V + v') =S. 4. General Method of Test by Use of the Mohr-Westphal Bal- ance. The hydrometer to be tested is read in several liquids of varying specific gravity, such that the readings will be suita- bly spaced throughout its range. The specific gravity of each is then determined by use of the Mohr-Westphal balance. The precision and reliability of the balance is much greater tharr that of the ordinary hydrometer. The important parts of the balance are, a small cylindrical glass sinker suspended by a very fine platinum wire on one end of a horizontal counter- weighted beam supported on a knife edge. In air the system- is in equilibrium ; but when the sinker is buoyed up by complete immersion in a liquid, equilibrium is restored by placing riders on the sinker end of the beam. The largest rider is adjusted to balance the buoyant force of a liquid of unit specific gravity when placed at the end of the beam over the sinker. The beam )c DENSITY i(79 is decimally divided to give tenths with a second large rider, and smaller riders having masses .1, .01, and .001 as large as the unit riders are provided. If when the sinker is immersed in a given liquid, the instrument is balanced with the unit rider ift the eighth notch, and the other three small ones in the third, fifth, and eighth notches' respectively, the specific gravity is .8358. If a notch is occupied by one rider, a smaller one may be hung on the hook at the lower end of the larger. The riders are usually adjusted to read specific gravities relative to water at 15° C. (Reference: A good description of the balance and mode of use is given in Practical Physics, by Ferry and Jones, Expt. 31, to middle of page 101.) If used at other tempera- tures and for specific gravities relative to water at 4° C, it is necessary to apply a correction as described below in the direc- tions for calibration. To calibrate the balance, find its reading for distilled water at the observed temperature. Find from tables of densities of pure water at diflferent temperatures data from which the specific gravity of the water used may be computed relative to water at 15° C. ; for example, if the temperature of the water used is 25° C, the required data would be the densities at 15° and 25°. Then the balance correction {k) for water is the difference between the specific gravity {S) and the balance reading (6') ; or S=h' + k and for substances heavier or lighter S=b + bk = b{\ + k), if the error per unit heaviness is k. 5. Method by Use of the Specific Gravity Bottle. This method is like the preceding one except that the specific gravi- ties of the several liquids are determined by weighing in a specific gravity bottle, or pyknometer, as described in Experi- ment D — 2 in Part I. The pyknometer method of finding density and specific gravity of liquids is probably the most ac- curate method of all, if a good chemical balance and carefully calibrated weights are used. Procedure. Read the hydrometer to be tested in pure dis- 380 ADVANCED MECHANICS tilled water as free from air as possible. For the first reading be sure that the stem is not wet more than a few millimeters above the reading. Then read very soon after submerging about half of the stem, and again soon after nearly all of the stem is wet. Note approximately the amount the reading is changed, and how soon it returns nearly to the dry stem reading. Take several readings, avoiding parallax as much as possible. Lift or otherwise move the hydrometer between readings. De- termine the doubtful figure in the readings and estimate the range of uncertainty or deviation that is quite liable to occur. By use of a micrometer caliper (used very lightly against the very thin glass stem) determine roughly whether the variation of cross section of the stem will greatly exceed the unavoidable errors of reading. Note that an error of .1 per cent, in the diameter causes an error of .2 per cent, in the cross section. Report to the Instructor in charge the result of this prelimi- nary test, and if the method of test is not assigned, select one of the less accurate methods of test and obtain the necessary data. If the second method is used, determine the mass of the hydrometer, and measure the diameter of the stem at each of the principal division marks. Measure two diameters 90° apart or three 60° apart, and use the average diameter for plotting a curve of variation of diameter. Place a scale against the stem and take readings opposite each principal division line. Use the distances of these division lines from the zero line as abscissas of the variation curve. Determine as precisely as possible the hydrometer reading in pure water, and record the temperature of the water. If the instrument has not been calibrated, plot a calibration curve for specific gravities relative to water at 4° C. and scale readings, and another curve for Baume readings and scale read- ings. If the instrument is calibrated, plot a correction curve coordinating the readings and the corrections to be applied, also plot a calibration curve for corrected readings and actual read- ings. Check two or more points on the stem scale or scales by read- ings in liquids, the specific gravities of which are determined by DENSITY 381 use of the Mohr-Westphal balance after its correction has been found. For one of the heavy liquids use a 15 per cent, solution of salt. Remarks. The hydrometer may be kept in place near the center of the jar by placing a card, having a triangular hole in its center, over the stem and resting on the top of the jar. The hole should be larger than the stem. To avoid parallax in reading the scale, place the eye below the level of the liquid surface to get an approximate reading; then raise the eye until it is in the plane of the surface, and make the final estimate of the number of tenths of the small division to complete the reading. In the 3d method small copper burrs or washers are con- venient weights to use. These are hung on a fine wire hook bent to fit the lower end of the hydrometer. The ratio of their weight in water to their weight in air is (S' -Si)yS', in which S' is the specific gravity of the copper and 5"i is the specific gravity of the water. The weight of the hook must also be taken into account. A good reference on a graphical method of constructing a density scale for any given hydrometer may be found in Nichols' Laboratory Manual, Vol. I, pp. 89-90 (Old edition). Test Questions. 1. Define density, heaviness, and specific gravity and express definitions by formulae. 2. What is the weight in tons of 550 cu. ft. of brine having a specific gravity of 1.125, if the heaviness of water is 62.4 lbs. per cu. ft.? 3. What arbitrary specifications fix the Baume scale and the relation between scale readings and specific gravity? 4. Account for the absence of readings between zero and ten on the Baume scale for light liquids. 5. In what respects do the Beck scales differ from the Baume? 6. How are Twaddell scale readings related to specific gravity? 7. How many figures in ordinary hydrometer readings are significant? Is the number the same for all scales? 8. Describe the method ofl calibrating the Mohr-Westphal balance. How is the correction factor determined and used? 9. Which hydrometer scales have divisions of unequal length and in which are the divisions approximately equal? 10. If a very accurate determination of density of a liquid were re- quired, what iTiethod would you use? Why? 11. Find the value of V in Eq. 5 if 5" is referred to water at 15° C. instead of water at 4° C. 382 advanced mechanics experiment d-32 Density of Stamped and Cast Sterling Silver Nicholson's Constant* Volume Hydrometer The Nicholson Hydrometer used in this experiment is of the constant volume type with a pan at the top above the water, and a basket at the bottom. The problem is: to find the density of a silver coin, to com- pute the density of cast sterling silver assuming that there is no change of density due to alloying, and to find the increase of density due to the combined effects of alloying, rolling and stamp- ing. Procedure. By the use of known weights on the pan, with and without the coin, find the weight of the coin in air; then place the coin in the basket and find its loss of weight in water. If the zero mark is above the water surface before adding the smallest weight used, and below when the weight is added, the fractional part of the small weight required to bring the mark to the surface may be estimated by interpolation between the positions above and below. All air bubbles must be carefully removed before each adjustment of weights. Derive the necessary formulae and compute the density of cast sterling silver, assuming the densities of cast silver and cast copper to be 10.45 and 8.85 respectively, and the composition of sterling silver to be 9 grams of silver to each gram of copper. Find the increase in density due to stamping, etc. Test Questions. 1. Write an equation expressing' the fact that the total volume equals the volume of the silver plus the volume of the copper. Express volumes in terms of mass and density. Solve for density of the mixture in terms of densities of the parts. 2. Formulate in words a general statement expressing the relation of the density of a mixture to the density of its parts, and relative masses of the parts. 3. Derive the relation' of the density of a mixture of three or more parts to the densities of the parts and the relative volumes of parts. 4. If the density of cast aluminum is 2.57 and of cast zinc is 7.10, what is the density of an alloy containing 91 gm. of Al to each 9 gm. of Zn, if the density is not affected by alloying? 5. What is the ratio of the volumes of Al and Zn in No. 4? density 38j experiment d-33 Relative Densities of Gases Bunsen's Effusiometer For the theory of the experiment, description of the apparatus, and mode of procedure, see Ferry and Jones, Experiment 23, pp. 107-109. Study carefully the proof that the densities of two gases are directly proportional to the squares of the times required for the effusion of equal volumes under the same pres- sures. Find the density of illuminating gas relative to air, and rela- tive to hydrogen. Test Questions. 1. How are speed, pressure, and density of gases re- lated during effusion through a very small hole? What is the method of obtaining the relation? 2. How are the densities of two gases related to their speeds of ef- fusion under equal pressures,! and to their times of effusion for equal vol- umes through the same orifice? 384 ADVANCED MECHANICS GROUP IX — F. FLUIDS experiment f-31 Study of the Venturi Water Meter Application of Bernoulli's Theorem . The Objects of the experiment are : ( 1 ) To study the steady flow of a liquid through a pipe of varying cross section with respect to velocity, pressure, and energy as an illustration of Bernoulli's fundamental principle of hydraulics; (2) To deter- mine experimentally the relative pressures at three points, A, B, C, (by use of manometer tubes as shown in Fig. 110) for varying rates of flow; (3) To find the relation between manom- eter readings and rates of flow; (4) To find to what extent the pressure differences are due to velocity differences and to fj-iction respectively; (5) To compare the theoretical pressure required to produce a given rate of flow of a frictionless incom- pressible fluid free from eddy currents, with the observed pres- sure less that lost in friction for the same rate of flow; and (6) to draw conclusions relating to disturbing factors not pro- vided for in the fundamental formula. Fluids in Motion. In general the velocity of flow of a fluid varies from point to point along its path, and is not constant at any given point. This general case, which is beyond our reach, will be omitted; and the following discussion will relate to the case in which the velocity at any given point is constant although velocity may vary from point to point. Steady Flow. A motion which may vary with position but does not vary with time is said to be steady. The path described by a particle of the fluid during steady flow is called a stream line. Stream lines never intersect. A portion of the moving fluid bounded by stream lines is called a tube of flow. These tubes of flow may have cross sections, that differ widely at dif- ferent parts of their length. The velocity of flow varies in- versely as the cross section, since the same volume must pass through each cross section in the §ame time. Bernoulli's Theorem. Let a pipe of varying cross section, as FLUIDS 385 shown in Fig. 109, be filled with a liquid of density D flowing steadily. Then the total energy per unit volume at any given level will consist of three parts due respectively to velocity, to Fig. 109 Fig. 110 height, and to pressure. If the pipe is large so that each tube of flow will have a practically frictionless boundary, and, for steady flow of an incompressible fluid, will be free from internal friction, then no portion of the fluid within the tube can receive or give out energy; and by the law of conservation of energy, the total energy per unit volume will be the same in one part of the tube of flow that it is in another. The energy may change in form, but not in amount. To express these statements in algebraic form, let v^ and p^ denote the velocity and pressure at /»i, and v^ and p2 the velocity and pressure at h^. Then ^Dv^^ + Dg\ + Pi = \T^v^ + Dgh^ + p2 = constant ; therefore, at any point in a tube of flow ^Dv^ + Dgh + p = constant. ( 1 ) This expresses Bernoulli's theorem, and is the fundamental equa- tion of hydraulics. It expresses the fact that : The total energy per unit volume of an incompressible fluid free from friction and eddy currents is constant during steady flow. If the pipe is horizontal, the second term of Eq. 1 becomes constant, and i Dt/' + p = constant (2) in which it is evident that if v increases, p must decrease to 386 ADVANCED MECHANICS k^ep the sum constant. Hence, according to Bernoulli's prin- ciple : // a fluid Hows from a point of lower speed to a point of higher speed, it must flow from a point of higher pressure to a point of lower pressure, and vice versa, if the pipe is horizontal, or if the change of pressure due to change of head is less than the change of pressure due to the change of velocity. Application of Bernoulli's Theorem to the Venturi Meter. To apply Eq. 2 to the Venturi meter shown in Fig. 110, let the pressure, velocity, and cross section at A be denoted by piV^a^; and at B, by p^v^a^ ; also let Q express the rate of flow in giriime C.c . per second. Then Q = "iVi = a^Vi 2 /-12 / 2 2 /^2 / 2 Substituting in Eq. 2 and subtracting to eliminate the constant, p, + i D Q^ja^ = constant pi + \ D Q^joi = constant Q'^\D dla,' - 1/V) =P.-P. Let K=2 a,WlJ^W - «^) Then Q'=K (p, - p,) (4) in which the pressure is in dynes per square centimeter. One •centimeter difference of level of mercury of density D^ and •covered by water equals (D^-D)g dynes per sq. cm., or 12,300 "dynes/sq. cm. at 20° C. Therefore, if manometer readings i?i, i?2 at 20° C. are substituted for p^ a;nd p^, (^ = K(R, - RM2i00 (5) O' -- K'{R, - R,) iff^'=i^iooK{6) Q = K" V R,-R, ) f K "= Vk' (7) For kilograms per minute (ikf) the constant (C) will be C = 60 i<::^1000. Therefore ^=i!r?x M-cVr,-R, (9) For litei-s per minute (L), L=(C/D) V(^i--R.) (10) FLUIDS 387 ■procedure. In setting up and using the apparatus the manom- «eter tubes require especial care. The tubes should be filled twith water above the mercury and the flow should be turned on slowly to avoid throwing- the mercury out of the tubes and -down the waste pipe. Do not let the mercury surface in the Mgh pressure tube get too close to the bottom of the tube. Place the scale to read from the top downward. Why? Manometer readings (R^, R^, R^) and flow per minute {M) are to be observed for varying rates of flow within the range «of the apparatus. When the flow is steady, set the markers on the manometer tubes at the average readings; then catch the flow for one minute in a suitable vessel, and during the minute mote carefully on each tube the amount by which the average reading deviates from the marker settings and record the cor- rected average reading for each tube. If there are no markers, -take as many readings as possible during the minute and record the estimated time average if the readings are variable. Tabu- late the data in suitable form to use in studying the objects speci- fied at the beginning of the outline. Plot y/XR-i- R2) a^d kilograms per minute, also V(i^3 -^2) and kilograms per minute from the data. If it is assumed that the loss of head due to friction (/) is the same in the two parts, friction is eliminated by plotting VT^T^^i^aT-^a) and M. Compare this curve with the curve plotted for Eq. 9, using any convenient value of square root of pressure difference; since only one point besides the origin is necessary to locate the line. If this line does not coincide with the curve based on the as- sumption that friction is eliminated, try to account for the dif- ference by considering the objects of the experiment. Test Questions. 1. Define steady flow, stream lines, and] tubes of flow. 2. Upon what dftes the energy of any portion of a flowing fluid depend? Write andi explain the algebraic expression for the three parts of the total energy per unit volume. 3. State Bernoulli's Theorem. Note especially the conditions under which it isl applicable. .. . . „ 4 Does the experiment indicate whether the conditions specihed in Ber- noulli's theorem are fulfilled in the Venturi meter used? Is there any probability that the case would be any different for a large commercial Meter? Give reasons for the conclusions drawn? S. Compare the meter constants for the manometers AB, and BC, and draw conclusions. What is indicated by the difference of level in tubes A and C? 388 ADVANCED MECHANICS 6. Are there any indications that eddy currents have any part in the case? 7. Commercial meters have a short taper for entrance to the neck of the meter and a very long taper for the flow away from the neck, and they use only a single manometer. What advantages are gained by this arrangement of parts? Is there any advantage in having the sam'e taper on each end of the experimental meter? What! disadvantages as a meter? experiment f-32 Specific Viscosity of Liquids Damped Vibrations Follow the Method described in Ferry's Practical Physics, Experiment XXXIX. Combine Experiments XXXVII and XXXIX. Call for assignment of liquids to be tested. experiment f-33 Errors of an Aneroid Barometer Variable Pressure and Temperature The method of testing for variation of pressure at a given temperature is described in Miller's Laboratory Physics, Experi- ment XXXIV. Devise a method of finding the effect of tem- perature changes and report it to the Instructor in charge for approval before performing. TABLE OF CONTENTS PART IV— FOURTH TERM ADVANCED COURSE CHAPTER VII. EXPERIMENTS IN HEAT Group I — T. Thermometry and Hygrometry EXPERIMENT PAGE T — 41. Calibration of a Mercury-in-glass Thermometer 401 Thermometer Corrections T — 12. Calibration and Study of Thermo-Electric Couples 408 Thermo-Electric Power T — 13. \'ariation of Vapor Tension with Temperature 419 Static Method T — 44. Variation of Boiling Point with Pressure 419 Dynamic Method T — 15. Hygrometry: — Dew Point, Vapor Pressure 421 Absolute and Relative Humidity Group II — H. Heat Constants H — 41. Calorimetry : Heat of Vaporization of Water 425 Method by Superheating H — 42. Calorimetry: Heat of Vaporization of Alcohol 425 Ferry Calorimeter H — 43. Specific Heat of a Liquid by Cooling 426 Laws of Cooling H — 44. Specific Heat of a Liquid by Heating 429 Magie's Method H — 45. Specific Heat of Materials Used for Heat Insulators 429 Method of Mixtures H — 46. Ratio of the Specific Heats of Air 430 Adiabatic Expansion H — 47. Radiation and Absorption Constants of Different Surfaces. . 430 Graphical Solution and Check H — 48. The Mechanical Equivalent of Heat 433 Callender Apparatus H— 49. The Electrical Equivalent of Heat 433 The Constant Flow Current Calorimeter CHAPTER VIII. EXPERIMENTS IN WAVE MOTIONS AND SOUND Group III — W. Wave Motions and Sound W — 41. Study of Wave Motions 434 Longitudinal and Transverse Waves W — 42. Velocity of Sound by Resonance 440 Measurement of Wave Length 6 xiii XIV CONTENTS EXPERIMENT PAGE W^3. Velocity of Sound in Metals 443 Young's Modulus — Kundt's Method CHAPTER IX. EXPERIMENTS IN LIGHT Group III — L. Light L — 41. Fundamental Principles Relating to Photometry, : 445 Measurements with Simple Photometers , L — 42. Laws of Reflection and Refraction 450 Tracing Reflected and Refracted Rays L — 43. Study of Spherical Mirrors 454 Reflection at Curved Surfaces L — 44. Laws of Lenses and Lens Constants 460 Experiments with Converging and Diverging Lenses L — 45. Adjustment and Use of a Spectrometer 470 Index of Refraction and Angles of a Prism L — 46. Study of the Essential Parts of Optical Instruments 475 The Telescope, Field Glass, and Microscope L — 47. Study of the Compound Microscope 476 Optical Principles, and Adjustments of Accessories L — 48. Study of the Projecting Lantern 477 Adjustments of Arc, Condenser, Slide, and Lens L — 49. Calibration of a Diffraction Grating 478 Measurement of Wave-length L — SO. Calibration and Use of a Spectroscope 478 Bright Line Spectra of Metals PART IV— FOURTH TERM Advanced Course CHAPTER VII EXPERIMENTS IN HEAT GROUP I— T. THERMOMETRY AND HYGROMETRY experiment t-41 Calibration of a Mercury-in-glass Thermometer Thermometer Corrections The calibration includes the determination of: I (a) The limits of the fundamental interval ; (b) The average value of the scale unit (0° to 100° C); (c) The correction for stem exposure; (d) The lag ice reading (after the steam reading) ; II (e) The corrections for variable capillary; (/) The calibration curve for temperatures and readings. Part I. The ice-reading (i?o) and the steam reading (Rs) may be obtained as directed in Expt. H — 1, pp. 144-147, with the following modifications. If pure shaved or crushed ice, or snow, is available, it may be packed around the thermometer bulb in a funnel to determine Ro if preferred to the methods given. The steam reading (i?«') with the stem exposed to room temperature should be taken before the reading (Rs) with the whole thread of mercury at the temperature of the steam. The top of the hygrom- eter may be removed while R/ is being obtained. The bulb must be above the water. Use the special tops provided. The correction for stem exposure depends upon the length of thread having a lower temperature than the bulb and upon their difference of, temperature. Let c be the correction factor, or depression per cm. and per degree, k the length exposed, 5 the temperature of the stem, and t the temperature of the bulb. Then the correction is c(t — s)le, in which c is the apparent expansion of mercury in glass which is about .000156. 401 402 • HEAT If the glass bulb, after being exposed in the steam, is air cooled to about 40° C. and is then put into ice again, the contraction of the glass lags somewhat behind the change of temperature. Since the bulb is slightly larger than before, the reading is lower due to the lag, and is called the lag ice-reading (i?o') or depressed zero. The lag reading is lower after a decrease of temperature and higher after an increase. The lag is greater the greater the temperature interval through which the thermometer is heated or cooled, and the more rapid the heating or cooling. If cooled from 100° C. to 0° C. in one or two minutes, the lag will probably be as large as .1° and may reach .5°, depending upon the kind of glass. The lag disappears very slowly at temperatures below 50° C. If rapidly cooled 100° C, the rise of reading might continue several months at 0° C, a few days at 100° C, and a few hours at 200^ C. The rate of change decreases as time increases. From this it is evident that readings may be affected by residual lag and the previous history of the thermorneter. A large part of the lag due to cooling from the high temperature at which the bulb is blown may be removed by proper annealing. The data required and the results to be obtained are indicated in the following illustration. TABLE CL. PART I Thermometer No. 8677. Made by H. J. Green. Divided in cm. and mm. Ro =- 0.055 cm.; Barom. Reading = 76.845; Room Temp. (Tr) = 27.0° C. Cor. Barom., B = 76.52; Rs = 27.96 with /. = 0; Manom. Reading, = O.S cm. Steam Temp., Ts = 100.20° C; R,' = 27.76 with /e = 20 cm. and 5 = 30° C. (?) Ro' = — 0.09S, S min. after removal from steam. (Air cooled to 12 cm.) c = {Rs - Rs')lh(T, - S) = .000143 cm. per cm. and degree difference. Av. degrees per cm. (i?o to Rs), D = 3.577° C. or 6.438° F- Part II. The cross section of the capillary is inversely pro- portional to the length {]) of the short thread of mercury used for the calibration. For convenience consider the stem divided into suitable equal parts, each of length p, between principal scale lines. If the cross section of n parts were uniform, the degrees per part {D) would be one nth. of the temperature interval for n parts. If some principal line A is arbitrarily chosen as the starting point and ta is the corresponding temperature, then, in the ideal case with a uniform cross section, the temperature for a THERMOMETRY AND HYGROMETRY 403 reading, B, n parts above ^ , is /„ + wD = 4. If the cross section of the «th part is increased, the reading for fc will fall below B. Let the deviation be denoted hy d. If the cross section of the other parts is also increased, the reading will fall still lower by an amount equal to the sum of the deviations due to each part (2d) ; therefore the reading for 4 when the cross section is in- creased becomes B — Xd, in which I,d is the calibration correction desired. The calibration correction for any given temperature is the deviation of the actual reading from the ideal reading that would be obtained if the cross section were uniform. If it were possible to have the length of thread of such value {I') that the average of n' lengths equals p, then lid for the whole range would be zero and the deviations for the separate parts would he p — v. Of course it is impossible to get just the right length; but if I differs but little from p, the reduced lengths, I', may be obtained by multiplying each of the n' observed lengths by the ratio of p to the average of the n' observed lengths. The de\'iations may be determined beyond the n' lengths in either direction. The readings from which the values of I are obtained should be taken with special care. The illustrative data given in Table CLI was read to estimated twentieths of a millimeter by the aid of a reading glass so used that parallax was avoided. The ther- mometer was laid before a window on a sheet of white paper so that its scale read from left to right. A straight line was drawn across the paper at right angles to the thermometer, and the reading glass was supported at such a height above the ther- mometer that its scale was clearly focused. If the line, the optical axis of the lens, and the eye are in the same plane normal to the paper, parallax will be avoided. If the eye is moved to right or left, the part of the line seen through the lens becomes curved, due to the ends having a greater apparent lateral dis- placement than the center, and the alignment of parts of the line is destroyed. If desired, the initial adjustment of the lens may be tested by use of a small piece of mirror laid on part of the line to give an image of the eye directly below the line when the eye is properly placed above the line to give perfect alignment of parts and freedom from parallax. Call for oral instructions by the instructor in charge about 404 HEAT how to detach the proper length of thread and how to move it along the capillary; also what precautions are necessary. Set the lower end of the thread successively very near to each of the lines limiting the parts, and read at each end of the thread. After one has made a setting and recorded his readings, the other partner should read the same setting and record his results with- TABLE CLI. DATA FOR CAPILLARY Thermometer No. 8677. Made by M. J. Green Divided in cm. and mm. Scale Readings Average Length Partner's Readings Lines Top Bottom Length Length Top Bottom -2 -0.995 -2.030 1.035 1.033 1.030 -0.995 -2.025 -1 +0.025 -1.000 1.025 1.025 1.025 +0.025 -1.000 +0.990 -0.035 1.025 1.025 1.025 +0.995 -0.035 1 2.010 +0.995 1.015 1.015 ' 1.015 2.015 + 1.000 2 2.975 1.950 1.025 1.023 1.020 2.975 1.955 3 4.040 3.020 1.020 1.020 1.020 4.040 3.020 4 5.005 3.985 . 1.020 1.020 1.020 5.005 3.985 5 6.020 5.000 1.020 1.020 1.020 6.020 5.000 6 7.005 5.985 1.020 1.017 1.015 7.005 5.990 7 8.030 6.985 1.045 1.043 1.040 8.030 6.990 8 9.040 8.000 1.040 1.035 1.030 9.035 8.005 9 10.030 8.995 1.035 1.030 1.025 10.030 9.005 10 11.025 9.985 1.040 1.035 1.030 11.020 9.990 11 12.015 10.970 1.045 1.040 1.035 12.015 10.980 12 13.040 11.990 1.050 1.047 1.045 13.035 11.990 13 14.050 13.000 1.050 1.047 1.045 14.045 13.000 14 15.020 13.980 1.040 1.037 1.035 15.015 13.980 IS 16.020 14.975 1.045 1.040 1.035 16.015 14.980 16 17.050 16.000 1.050 1.053 1.055 17.060 16.005 17 18.040 16.980 1.060 1.057 1.055 18.035 16.980 18 19.050 17.990 1.060 1.060 1.060 19.050 17.990 19 20.040 18.975 1.065 1.067 1.070 20.040 18.970 16 16.955 16.015 0.940 0.943 0.945 16.955 16.010 17 17.965 17.015 0.950 0.947 0.945 17.965 17.020 18 18.950 18.000 0.950 0.950 0.950 18.950 18.000 19 19.970 19.010 0.960 0.960 0.960 19.975 19.015 20 20.965 20.005 0.960 0.957 0.955 20.960 20.005 21 21.995 21.035 0.960 0.960 0.960 21.995 21.035 22 22.985 22.025 0.960 0.957 0.955 22.985 22.030 23 23.970 23.010 0.960 0.957 0.955 23.965 23.010 24 24.980 24.025 0.955 0.953 0.950 24.975 24.025 25 25.990 25.035 0.955 0.957 0.960 25.990 25.030 26 26.965 26.005 0.960 0.960 0.960 26.970 26.010 27 27.970 27.015 0.955 0.955 0.950 27.970 27.015 28 28.950 28.000 0.950 0.950 0.950 28.950 28.000 Average of 20 lengths of first thread (lines 0-19), 1.0366 cm. = h Av. of lines 16-19, 1.0592 and 0.9500. Ratio of lengths, h/h = 1.115 Length factor first thread, ©.9647 ; second thread, 1.0756 THERMOMETRY AND HYGROMETRY 405 out knowing what the first reader obtained. The second reader may then reset the thread and take the first reading for the new position. Continue to alternate to the end. If the thermometer has no safety bulb at the top it may be necessary to use one thread for one part of the length and another thread for the remainder, as shown in Table CLI. Tables CL, CLI, and CLI I, and Fig. 150 illustrate the method of using the observed data to obtain the desired calibration curve. Fig. 151. TABLE CLII. DERIVED RESULTS Scale Reduced Eq. Vol. Deviations Corrections Stem Cor. Cor. for Lines Lengths Points above 8 cm. H. Scale — 2 0.996 -1.985 +0.015 +0.030 -I 0.989 -0.9S9 +0.011 +0.018 All Neg. t All Neg. 0.989 0.000 0.000 0.001 0°.000 1 0.978 0.989 -0.011 0.021 0''.023 2 0.987 1.967 -0.033 0.051 0°.042 3 0.984 2.954 -0.046 0.073 0°.0S8 4 0.984 3.938 -0.062 0.097 0°.073 5 0.984 4.922 -0.078 0.121 0°.085 6 0.981 5.906 -0.094 0.146 0°.095 7 1.006 6.887 -0.113 0.172 0M02 8 0.998 7.893 -0.107 0.175 0M07 9 0.994 8.891 -0.109 0.186 * OMU 10 0.998 9.885 -0.115 0.200 "0.002" 0M14 11 1.003 10.883 -0.117 0.210 0.005 0M15 12 1.010 11.886 -0.114 0.216 0.008 OMU 13 1.010 12.896 -0.104 0.214 0.013 0M12 14 1.000 13.906 -0.094 0.212 0.018 0M09 15 1.003 14.906 -0.094 0.220 0.025 0M05 16 1.016 "15.909 -0.091 0.226 0.032 OMOO 17 1.029 16.925 -0.075 0.219 0.040 0°.094 18 1.023 17.945 -0.055 0.208 0.050 0°.087 19 1.032 18.968 -0.032 0.192 0.060 0°.080 20 1.030 20.000 0:000 0.168 0.072 0°.072 21 1.032 21.030 +0.030 0.146 0.085 0°.064 22 1.030 22.062 0.062 0.122 0.098 0°.055 23 1.030 23.092 0.092 0.112 0.112 0°.046 24 1.025 24.122 0.122 0.080 0.128 0°.038 25 1.030 25.147 0.147 0.064 0.144 0°.028 26 1.032 26.177 0.177 0.042 0.162 0°.019 27 1.027 27.209 0.209 [0.019 0.180 0°.009 28 1.022 28.236 0.236 0.000 0.200 0°.000 29 29.258 0.2S8 *lfeS?Cor. = 3.5/ X .000143 X {R - sy, assuming i = 30° C. t From the following data, from Miller's Manual, p. 166, bulb of Jena glass 16: Temp. 0° 10° 20° 30° f40° 50° 60° Cor. 0°.000 -O°.0S5 -0°.090 -0°.109 -0°.115 -0M09 -0°.096 Temp. 70° 80° 90° 100° Cor. -0°.076 -0°.053 -0°.027 -0°.000 4o6 HEAT r ~ 1 1 1 "^ 1 r ■ "', ^^ _ - f "APILL/ RY onp R RCT >1S - ~" ,, ^ k / ^ H / N / - -X / \ / , 1 / \ / / ' Y \. / / z o 1 J ^ \ / 1 ' 1- / ,■-' / 1 / / / nL X < ; ^ \ / 13 f" / \ / Z /i ?'- / \ z h ^^ k/ N 1/ •^^ ^r z r , ^'' / / -< 1 r ^'- S / \ 1 E / r / ^ s 1 •- / / / 1 \ / \ / 1 \ 0. 1 1 f y .' \^ \, j.ni-:- / V V \ :^=7 / / \ / \ n[no l.OOS // \ / \ ^ \ 2L^ K 2 * e ? * i » z I * 1 s 1 B 2 z z 4 ^ e I k 3 D nin? _. T HE RM ow E E Sc AL E _ Qv ly _ Z \ ^ -1- n 714 o.L(« \ ^ -x/ «7 il n!nR Q^7m \ 'a. S / o nlfift \ 's z \ / nlio , / — > 1 \ , y _flll2 s ^ Fig. ISO. Fig. 151* illustrates the use of multiple scale plotting, the object of which is to use a large scale for plotting on a small curve sheet. The plotting space is chosen with some convenient integral number of squares each way. If the light dotted line, for the ideal calibration, were plotted to the same scale on a large sheet having a plotting area twenty-five times as large, it would be a straight line between the ice and steam points (i and s). The deviations are laid off horizontally from the ideal line, and through the points thus located the full line is drawn, which is the real calibration curve. * The original drawing was made on section paper ruled in twentieths of an inch, two twentieths being used for each millimeter. When the five small divisions per square are left out, the scale of abscissas appears to violate the general rule for selecting proper scales. THERMOMETRY AND HYGROMETRY 407 Two points {i and x) at the ends of the first run (0° - 20°) are carefully determined and plotted. A straight line between thes e gives the slope to which all of the other lines are drawn. Whenever the line meets the boundary of the plotting area, an offset is made to the opposite side from which point the line is continued. The line starts at i and ends at 5. If the ideal line should not pass through 5, it indicates some error in slope or drawing, or both. To illustrate the use of the tables and curves, let it be required to find the temperature {T) corresponding to a reading, R = 24.35 cm., taken with the stem exposed above 8 cm., 5 being 30° C, and R' = - 0.090. The stem correction is 0.133 cm., and the temperature corre- sponding to 24.48 cm. is 88.°15 C. from the calibration curve, if the lag is neglected, the curve being started from the average ice reading §(-^o + Ro) for the fixed zero method of reading. The result may be obtained from the tables alone, as follows: r =fR- Ra' + Stem. Cor. + Capillary Cor.j D - H. Scale Cor. =(24.35 + 0.090 + 0.133 + 0.072) 3.°577 - 0.°03 1 1 1 1 1 1 1 1 1 1 T h ■IRM'OrtETEK CALIBRj ■iTION Sf oV '' ^y / / •^z ' 'y iir f. */ << / '/ / ^ / '/ ^ •* »> '/ , / ' y 'y / V / '/ / '/ / /- V / ^ y / Id 1+ K ^ ^> < / / /y /-^ ■"y / ■' V* rf / ^ ^ ^ — 1? / / '< A /' /' , ■J / / r V // /■ / to / ^ /^ ' >" /y /y / 'y /, '/ ^» /■ y v « ' ^ ■>;j r^ / — tr / / /" < ^^ ( / / / C3 UJ. Q / / / ^' i '', y y / / / / ^' ?\ rO' / / /. ' / /. / V h^- V / /, / /- / / > r . / /^ ' / / /. / ?A '/' an fiO fln ?n n *nt / /y / / ?y (." I 1 2 3 \ 9 6 6 7 8 » X 1 1 1 2 1 ? 1 3 1 4- 1 5 1 > i 7 1 8 1 9 z ° 2 2 1 2 Z 2 3 2 1 z 5 2 B 2 7 2 7 2 5 2 9 3 3. A -E H th Dl NO S ' N c :nti ^1E TE R 1 ... _, 1 Fig. 1,51. 408 HEAT = 88.°15 - 0.°03 = 88.°12 C. on the hydrogen scale Remarks. The method of calibration described may be ap- plied with slight modifications to thermometers graduated for direct, or approximately direct, reading in degrees, either Centi- grade or Fahrenheit. The divisions may be used as scale units regardless of the original plan of graduation. The method may be used even with "pointed" thermometers having nonuniform scale divisions. If the internal or external pressure is changed, it may affect the reading, especially if the bulb is thin and large. The reading in a horizontal position may be greater than in a vertical position. If further explanation is desired, a good discussion of "Accurate Mercurial Thermometry" is given in Theory of Heat, by Thomas Preston (revised by Cotter), pp. 118-134. A few shorter references are as follows : (1) Laboratory Manual of Physics, by Nichols and Blaker, pp. 32-35. (2) Practical Physics, by Ferry and Jones, pp. 155-166. (3) Laboratory Physics, by Miller, pp. 159-166. (4) Heat for Advanced Students, by Edser, pp. 23-30. (5) Practical Physics, by Franklin, pp. 140-143 and 148-151. (6) Physical Laboratory Notes, by Holman, pp. 49-57. experiment t-42 Calibration and Study of Thermo-electric Couples Thermo-electric Power Study the following Notes on Thermo-electricity, especially the thermo-electric diagrams, to answer the following questions : (A) What will be the general form of the curve for temperature and electromotive force (a) for metals whose lines intersect within the range of temperature used; (b) for metals having parallel lines, if such can be found? (B) How may the E.M.F. for a given temperature difference be computed from data obtained from the thermo-electric diagram ? (C) If the junctions of an iron-zinc couple are joined directly by zinc, (a) will the iron wire from the hot junction carry current to or from the galvanometer? (b) If one junction is at zero THERMOMETRY AND HYGROMETRY 4O9 degrees, the other must be heated above what temperature to cause the current to reverse? Notes on Thermo-electricity. Introduction. If two wires of different metals are joined at both ends, as shown in Figs. 152 and 152', no current will fidw if the junctions (a, b) are at the same temperature; but if the temperature of one junction {b) is higher than that of the other (a), a current will flow as indicated by the arrows. The hot junction is like a generator in this respect, that current flows through it from the metal of lower potential to the metal of higher potential. The electromotive force (£) depends upon the metals used and the temperatures of the junctions. «-/coid "' tfotVi Fig. 152. Fig. 152'. Thermo-Electric Power and E.M.F. The change of E.M.F. per degree change of temperature of the junction is called the "thermo-electric power" of the junction. Let p be used for thermo-electric power, and let Ae and Ai be corresponding increments of electromotive force and temperature; then The E.M.F. of the couple is the summation of the increments for each degree between the temperatures of the junctions. It is equal to the product of the average thermo-electric power and the difference of temperature of the junctions. If £ is the E.M.F. of a couple when the temperatures of the junctions are ti and t2, then £ = SAe = l^p^t (2) and E = pavit2 - ti) (3) Thermo-Electric Diagrams. Thermo-electric diagrams repre- sent graphically the variations of thermo-electric power with temperature for the various metals when used with lead 410 HEAT Thermo-electric powers are plotted as ordinates and tempera- tures as abscissas, as shown in Fig. 153. The thermo-electric power of a junction of any two metals at any given temperature is represented by the vertical distance between the lines for those metals, measured along the ordinate for that temperature. i 1 : ; i 1 1 M i 1 i 1 i ^ - t 1 1 : ■ i i| ■ :: i II 1 1 ! 1 n i 1 =s 1 J S " : : i i i H 1 ; 1 * 1 1 H 1 ; ■4 : :: :; :: ;; :: 1 \ \ 4+ 1 I 11 in ■.tk : tt : H; i i 1 1 X 1 1 1 tls= Pi 1 TiH- Fig. 153. To illustrate the interpretation and use of the diagrams and equations, consider the lines for iron and cadmium as shown in Fig. 154. If the limits of t are 0° C, and t-t C, the E.M.F. of an iron-lead couple (£') is (from Eq. 3) the product of the average ordinate of the part ab of the iron line, by the difference of temperature, or the corresponding abscissa. But this product expresses the area of abfe in Fig. 154. In like manner the E.M.F. of a cadmium-lead couple (£") between the same temperature limits is expressed by the area cdfe, and the differ- ence between these two areas (that is, the area abdc) expresses THERMOMETRY AND HYGROMETRY 411 the E.M.F. of an iron-cadmium couple (E) between the same temperature Hmits. Hence, from Eq. 3, these electromotive forces are: For Fe - Pb, E' = pj{h - h) = 14.84 X 90 (4) For Cd - Pb, E" = pav"{h - to) = 4.80 X 90 (5) ForFe - Cd, £ = {pav' - pj'){k - h) = 10.04 X 90 = 904 mv. (6) in which the numerical values are taken from the thermo-electric diagram for the junction temperatures 0° C. and 90° C. For the general case, when the lower temperature is t\ instead of io, Eq. 6 becomes E = ipav' - Pav")it2 - ti) (7) This equation expresses the general rule for finding the E.M.F. in a thermo-electric circuit; which is, — Multiply the difference of thermo-electric powers at the mean temperature of the junctions by the difference of the junction temperatures . The E.M.F. curve in Fig. 154 is plotted for values of E com- _ "G .. n ___ y S t' ^ ■y H Y' / \ c a <> / \ J in A \ -z OS / N p i ifi "^ ^ kj / \ \ \ '^ 1 s ' <^ -^ 5 ^ nn p 14 \ \ \ k ^ > \ r' j -a ? \ \ \ ^ , ^ '^ 1 :* a d 1? \ \ \ \ \ \ K ^ V5 \ b \ \ w \ N^ \ C; V j T . !d \ \\ X \ \ \, 1 \ \ \ > -^ " 1 CJ H \ \ /. \ \ \ \ ^ \ c \ \ \ \ ?■ \ }■ •\ OP u. fi \ \ \^ >< V^ \ K. a \ y X ^ fe tv \ 'b ^ \ y x" ^ !^ r V X ^ f- y >^ X f Y ^ X '^ p p w- X x; Y. f L ~ « lU ?°'' ( n 2 g^ 3 XT 1 |t b: l£. EB k •u Rfl [c Ep [T t^ ^D? _ L. 1— 1_ Fig. 154. puted as indicated in Eq. 6. The 904 microvolts is plotted at the point m. The relation of the ordinates of this curve to the corresponding areas between the thermo-electric power (T.E.P.) 412 HEAT lines may be illustrated as follows: Since the vertical side of each small square is 1 microvolt, and the horizontal side is 10°, its area represents 10 microvolts, and ten times the number of small squares between abh and cdg up to any given temperature is the ordinate of the E.M.F. curve for that temperature. The increments of E for each 10° grow smaller and smaller up to /„ where the lines intersect. This is called the neutral tempera- ture. It is the temperature of one junction for which p = 0, and E = a. maximum, if the temperature of the other junction is fixed. Above i„ the E.M.F. decreases due to reversal of the resultant thermo-electric power, and becomes zero if the junction temperatures are equidistant from <„, as are 0° and ti°. The area gnh is equal to the area anc, so that the E.M.F represented by one is neutralized by the opposing E.M.F. represented by the other. The equations for the T.E.P. lines may be used if desired to find the average values of p, and the corresponding value of E. The T.E.P. lines are straight lines, and therefore may be expressed by the typical equation y = mx + h. The variables y and x axe. p and / respectively and m is the slope. Let pa express the intercept on the F-axis at 0° C. ; then p = Po +mt , (8) For Fe - Pb, p' = 17 - 0.048 / (9) For Cd - Pb, p" -= 3 + 0.04 1 (10) The average value of p between the temperatures to and 1% is the initial value (^o) plus half of the increase {^m{t2 — to)); or Pav' = po + Wih -to) (11) Pav" = po" + Wih - to) (12) ■ £m = (po - po")(t2 - to) + Urn' - m"){t2 - toY (13) = (po' - Po"){t2 - to) - him"- - m'){t2 - toY (13') £„ = 14 X 90 - .044 X 90^ = 904 mv. (Plotted at m) (14) If the lower temperature is t\ instead of 0° C, E = (Pi' - Pi")(<2 - vCu -/Heated CootedV Fe\ /F e. •< Fig. 155. If H calories of heat are absorbed or emitted in r seconds when the current flowing is Ic C.G.S. units, H X 4.18 X 10' = P X /. X T = £ X /c X T ergs (19) in which P is the coefficient of the Peltier effect. It is the number of ergs transformed per second by a current of one C.G.S. unit. From Eq. 19 it follows that P may be regarded as an E.M.F. It is measured by the product of the difference of the thermo-electric powers (in C.G.S. units) and the absolute temperature {T) of the junction. One microvolt equals 100 C.G.S. units of E. :. P = \OQ{p' - p")T C.G.S. units (20) Ei= iPi' -pnTimv. (21) E, = {P2' - p2")T2 mv. (22) Sir Wm. Thomson (Lord Kelvin) predicted in 1857, and later demonstrated experimentally, the existence of reversible heat effects in other parts of the circuit besides the junctions in which the Peltier effects are located. A slope of potential exists between the hot and cold parts of the same metal. For example, hot copper is positive to cold copper, and hot iron is negative to cold iron. In general, a potential slope exists where there is a tempera- ture gradient, indicating an E.M.F. in that part of the cricuit. Heat is absorbed if the current flows up a slope of potential, and vice versa; therefore the change of temperature due to flow of current is reversed if the direction of current is reversed. This is the reversible heating effect predicted, and now known as the Thomson effect. The heating or cooling of parts of the circuit when current from an outside source is sent through the circuit indicates the existence of electromotive forces in the parts affected, since THERMOMETRY AND HYGROMETRY 415 energy is transformed wherever current flows through a source of E.M.F. The peltier effect indicates an E.M.F. at each junc- tion (£1, £2). The Thomson effect indicates an E.M.F. in each metal where the temperature gradients exist (Ec, -Ei)- Metals may be classed, with respect to these electromotive forces, into two classes. Copper is the type of one class, and iron of the other. Ec is used to denote the E.M.F. of the copper class, and Ei the E.M.F. of the iron class. The T.E.P. lines of the copper class all have a positive slope (upward toward the right), and those of the iron class a negative slope in a diagram plotted relative to lead. Lead is chosen for the axis, or line of zero slope, because the Thomson effect is zero (or so small that it is negligible) in lead. The electromotive forces, Ec and Ei, are each measured by the product of the difference of the thermo-electric powers at two points and the mean absolute temperature. Taking the illustrative case represented in Fig. 154 for cadmium and iron, Eo = {p2" - P.") X \{T, -f T,) = m"iT2 - T{) X h{T, + T2) (23) Ei = (p^' - P2') X 1(^1 + T2) = - m'iT2 - Ti) X KTi + T2) (24) From what is given above it follows that Ec is from cold to hot, and Ei from hot to cold. £1 and £2 are both from cadmium to iron for temperatures below /„. These directions are indi- cated in Fig. 156. Note that the resultant E.M.F. is Ec ► ^°" :. y ^i Ei Fig. 156. £ = £„ + £2 + £< - £1 (25) From equations 7, 15, 21-25, £, = W'(T2' - Ti^) (26) Ei= - \m'(Ti - Tx') (27) £2 = (/,/ - p,")T2 = ipo' - Po")T2 - im" - m')Ti' (28) - £1 = - (Pi' - pi")Ti = - iPo' - po")Ti + (m" - m')T,' (29) 7 4i6 HEAT Adding, E =.iPo' - P,")iT, - TO - Um" - m'){Ti' - T,^) (30) E = (/,/ - pn{T^ - TO - §(w" - m'){T^ - ri)2 (150 E = {pj - pav"){Ti - TO (T) In Fig. 157 a part of Fig. 154 is replotted to absolute scale of temperature, and each of the five electromotive forces is repre- ented by an area as follows : Area rcdq = (p," - p,") X i(T2 + TO = E. (31) = 3.60 X'318 = + 1145 mv. Area qdbp = {p^' - p^") X T^ = E^ (32) = 60.8 X 363 = + 2207 mv. Area pbao = {p^' - Po') X HT^ + TO = Ei (33) = 4.32 X 318 = + 1374 mv. Area oacr = - (Po' - po") X To = -Eg (34) = - 14.00 X 273 = - 3822 mv. Area cdba = {pj - pa.") X {T^ - To) = E^ (35) = 10.04 X 90 = + 904 mv. ?(\ ^ n — — — I a V, n ■^i to \ ^ ■^ il ^ ;^ "^ 1 — — — — — — — J — — — - =>* ■^ \ T _. _. ._. ._. ._ f^ ^ ^ ) 1 r 2 r 1 pf ?» T; 4( M 5 )0 Tl ^MPE 1 — 1 — 1 — 1 J_. 1 ^BS. sc^ U.I y Fig. 157. From equations 23-35 inclusive it follows that Seebeck's resultant electromotive force is the algebraic sum of the four com- ponent electromotive forces . References: Mag. and El., by Brooks and Poyser, pp. 461-479 EI. and Mag., by Starling, pp. 203-223 Mag. and EI., by Hadley, pp. 359-381 THERMOMETRY AND HYGROMETRY 417 Procedure. Preliminary Test. Make an iron-copper couple by connecting a piece of iron wire between two pieces of copper wire. The twisted part of the closely twisted joints should be one-half inch or more in length. Scrape the wires before twisting. Connect the free ends of the copper wires to the galvanometer, through about 300 ohms, or enough to keep the galvanometer reading on the scale when the deflection due to heating one junction is a maximum (15 to 20 cms.) in the direction of the first movement. Support the junctions in air above the table. Bring a Bunsen burner flame, turned low, (or a candle flame) to the tip of one junction. Note carefully the movement of the scale reading as the junction is slowly heated until about half of the twisted part is a dull red; then slowly remove the source of heat and continue to trace the change of deflection as the junction cools to air temperature. The deflections are proportional to the thermal E.M.F. in the circuit. Describe the variation of E.M.F. with temperature by reference to Fig. 154. Draw a diagram of connections and note whether the readings are on the black or red part of the scale. Determine the effect when the other junction is heated. Plot an E.M.F. -temperature curve for an iron-zinc pair having the following thermo-electric powers at the temperatures given : Temperatures 0° 50° 100° 150° 200° 250° 300° Th. El. Powers.. -M6 -M2 -|-8 -f-4 -4 -8 Assume one of the junctions to be at 0° C. and plot to a scale of one-half inch per 100 microvolts. Test of Pt-Cu Couple. Each junction is bound to the bulb of a thermometer and immersed in thin clear oil in a test tube. The tubes are supported on arms projecting from a standard. A small tubular electric heater, made to fit the test tubes, is adjusted to cover all but the top of the tube containing the junction to be heated. The current in the heater (1 to 2 amperes) is controlled by incandescent lamps in the circuit. The other tube is sus- pended in air at room temperature; or, if preferred, may be sur- rounded by an ice bath. Assemble the apparatus, connect the galvanometer and heater, and draw a diagram of connections, which should be shown to 4l8 HEAT the Instructor before proceeding to take data. Observe gal- vanometer readings and temperatures for the Pt-Cu pair unless another pair is assigned. (If the making of couples is assigned, call for oral instructions and materials.) Do not go above 200° C. Determine the direction of current in each metal and indicate it on the diagram of connections. No added resistance is needed to prevent large deflections. Plot the galvanometer deflections as ordinates and hot junction temperatures as abscissas. Find the E.M.F. per centimeter deflection; the temperature of the neutral point ; and the temperature at which the current will reverse. Check with the diagram (Fig. 153) and draw con- clusions. Determine the thermo-electric power (from the slope of the E.M.F. curve) and plot a curve of thermo-electric power of platinum relative to copper. Compare it with the diagram given in Fig. 153. Test of Fe-Ni Couple. In order to avoid having galvanometer deflections that are either too large or all too small, compute approximately the resistance to use in the galvanometer circuit to keep deflections within 20 cm. for a temperature difference of say 150° C. Use the thermo-electric diagram, and the constant and resistance of the galvanometer, which may be obtained from the Instructor. Observe the precautions and directions for procedure given for the Pt-Cu couple, and plot the data as there directed. Find the E.M.F. per centimeter deflection and draw an E.M.F.- temperature curve. Determine and plot the thermo-electric power of iron relative to nickel. Check the results with the diagram. Remarks. If the wires of which the couples are made are not homogeneous, especially where the temperature gradient is large, the E.M.F. may vary slightly with change of depth of immersion. (Reference: The Thermo-Element as a Precision Thermometer, by Walter White, in the Physical Review, August, 1910.) If it is desired to make qualitative tests for the Peltier and Thomson effects, call for oral instructions. Test Questions. 1. Define thermo-electric power. Describe a thermo- electric diagram. THERMOMETRY AND HYGROMETRY 4I9 2, 3, 4. Answer Questions A, B, C in the Introduction. 5. Which metal is at the higher potential at the hot junction? Which way does the current flow? Does the hot junction behave like a generator in causing current to flow from lower to higher potential within the generator? 6. If the thermo-electric power of a nickel-iron couple is 40 microvolts at 0° C. and the nickel line in the thermo-electric diagram is parallel to the iron line (and below it) : (a) What E.M.F. will be produced when one junction is ISO" C. above that of the other? (b) If the iron wires are joined by nickel, which junction should be con- nected to the -t- terminal of the galvanometer? How determined? (c) If the current per centimeter of galvanometer deflection is 2.2 X 10~' amperes, and the galvanometer resistance is 130 ohms, what external resistance (including the resistance of the wires) will limit the deflection to 15 cm. for 150°? (d) What would be the deflection per degree if the external resistance were only 2 ohms? 7. Compare the results obtained by experiment with Cu-Pt junctions with those given by Ni-Fe junctions, and compare the results of each with those given in diagrams or tables. experiment t-43 Variation of Vapor Tension with Temperature Static Method See Milliken's Molecular Physics and Heat, pp. 152-159, and Problems, pp. 162-163. experiment t-44 Variation of Boiling Point with Pressure Dynamic Method Regnault's method of measuring the pressure of saturated water vapor at different temperatures is, to observe the temper- ature of the steam above water boiling under different measured pressures. When a liquid boils, bubbles of vapor are formed in the interior of the liquid and are maintained by the vapor pres- sure within, balancing the external pressure on the surface of the liquid. Hence, when the surface pressure is reduced, bubbles will form at lower temperatures, and the "boiling point" is lowered. 420 HEAT Procedure. A flask, or boiler (B), is partly filled with water and is closed with an airtight stopper in which a delivery tube and a thermometer tube are inserted. The bottom of the ther- mometer bulb should be at least one or two centimeters above the water. The delivery tube is connected to a filter pump (P) which draws out air and the vapor formed by boiling when heat is applied to B. The delivery tube is also connected by a side branch to the mercury open tube manometer (M) through another flask ( C) which serves as an air reservoir to steady the action of the apparatus, and to act as a trap to protect the manometer. The experiment may begin with the greatest reduction of pressure, in which case boiling will occur at the lowest temperature. If possible, maintain the boiling and manometer reading steady long enough for the thermometer reading to reach the true temperature of the steam. Apply the heat very carefully and avoid rapid boiling. When a pair of readings for pressure and temperature have been observed, increase the pressure by reduc- ing the action of the filter pump, or by increasing the rate of boiling, or by increasing air leakage by means of a needle valve or stopcock near the pump. Continue to observe pairs of readings for increasing pressures and temperatures up to atmos- pheric pressure. A set of readings should also be taken with decreasing values of pressure, in which case the thermometer lag may affect the results in the opposite direction. Observe the barometer reading and correct it for temperature. Also take data for testing the thermometer used. Carefully observe the steam reading of the thermometer for atmospheric pressure as indicated by the barometer and zero reading of the manometer. Observe also the lag ice reading, and, if the ther- mometer scale is incorrect, plot a correction curve assuming the capillary to be uniform. For example, if the ice reading is —0.6, and the steam reading is 102.5 when the corrected barometer reading is 77.35 cm., the correction is —2.0 at the steam reading, and -|-0.6 at the ice reading. If readings are plotted as abscissas, and corrections as ordinates, and a straight line is drawn joining these two points, the temperature corresponding to any given reading is the algebraic sum of the reading and the correction THERMOMETRY AND HYGROMETRY 42 1 indicated by the corresponding ordinates, if the cross section of the capillary is uniform. Correct the observed thermometer readings. Plot a pressure-temperature curve using dots for one set of data, and small circles for the other set. Also plot, by use of small crosses, the corresponding values obtained from Table CLIII for comparison and check. Discuss the results and draw conclusions. See Milliken's Molecular Physics and Heat, pp. 160-161. Test Questions. 1. Specify the apparatus and observations required to determine the pressure above the boiling liquid. 2. Upon what does vapor pressure depend? What is its value during ebullition relative to the pressure above the liquid? 3. What must be done in order to have ebullition and constant pressure exist simultaneously in the flask for even a short time? Why? 4. Is the change of boiling point proportional to the change of pressure? 5. What change of pressure will change the boiling point one degree at 100° C? 80° C? 60° C? 6. How may the boiling point and pressure curve be checked for reli- ability? May Table CLIII be used? Discuss sources of error. experiment t-45 Hygrometry: — Dew Point, Vapor Pressure, Absolute and Relative Humidity Hygrometry relates to the study of the water vapor in the earth's atmosphere. The four quantities involved are: (1) Dew point; (2) Vapor pressure; (3) Absolute humidity; and (4) Relative humidity. The dew point (Td) is the temperature at which the con- densation will begin if the temperature is slowly decreasing. The vapor pressure {p) is the pressure of the water vapor at the existing temperature of the air. The absolute humidity (D) is the mass of water vapor contained in unit volume. The relative humidity, or degree of saturation {r), is the ratio of the density {D) of the water vapor existing in the atmosphere at any given time to the maximum density (£>„) that it could possibly have at the existing temperature. Plot a pressure temperature curve for saturated water vapor between — 10° and 30° C. from the table given in the remarks. 422 HEAT If one of the four quantities given above is found experi- mentally, each of the others may be obtained from the known one and the curve. Procedure. Find the dew point experimentally by use of the dew point hygrometer, which is a small polished cup copied by ice water or by evaporation of a volatile liquid. Find the temperature of the cup when a film of dew first appears as the temperature slowly falls; and the temperature at which the film disappears when the temperature slowly rises. Take the mean of these as the dew point. The vapor pressure of the saturated vapor at the cold surface of the cup is found on the curve at the point corresponding to the dew point temperature, and is the same as the vapor pressure for the warmer parts of the air away from the cup. Hence, p may be taken directly from the curve. The density (D) of the vapor at some point X where the temperature is T^ differs from that at the cold surface where the temperature is Td. The vapor density at the temperature T^ and pressure p may be computed from the known density of dry air at 0° C. and a pressure of 760 mm. of mercury, by use of the gas law, which may be expressed algebraically as follows: di di P2S1 Vi D2 61 and 62 are absolute temperatures. The density of water vapor is .624 times the density of air at the same temperature and pressure. At 0° C. and 760 mm. the density of dry air is .001293 gm. per cc. ; then if the density of air at the temperature Tx be denoted by Da, Da ^ ^X273 ^ D .001293 760 X {T^ + 273) .624 X .001293 ^ ' from which ^ .000290^ (n + 273) ^^> From the definition of relative humidity it follows that r = D/Dm, and since densities are directly proportional to pressures if the temperature is constant, D p ' = D-=P (^) THERMOMETRY AND HYGROMETRY 423 in which P is the largest vapor pressure that could exist at the temperature Tj,. Compute r and express the relative humidity in per cent. , Check the values computed above by use of the wet and dry bulb hygrometer at the same place and under the same con- ditions as nearly as possible. The values of relative and absolute humidity, dew point, and vapor pressure may be obtained from tables given out with the apparatus. After obtaining the data at the place where the dew point is observed, select another place where the temperature is quite different and take wet and dry bulb readings in order to compare the results, especially with respect to the relative and absolute humidity. One set of data with the wet and dry bulb hygrom- eter may be taken indoors and the other outside. Remarks. The state of any given mass of gas is specified by the corresponding values of pressure (P), volume (F), and abso- lute temperature (5) ; and Pi Vijdi for one state of a given mass of gas equals P2V2I62 for another state, which is one way of expressing the "Gas Law" obtained by combining the laws of Boyle and Charles. The table given below gives the pressure P (in mm. of mercury) of saturated water vapor at the given temperature Ta- TABLE CLIII Ti p Ti P Ta P Ti P -10 2.2 20 17.4 70 233.1 99 733.2 - 5 3.2 25 23.5 80 354.9 100 760.0 4.6 30 31.5 85 433.2 101 787.7 5 6.5 40 54.9 90 525.4 102 816.0 10 9.1 50 92.0 95 633.6 105 906.4 IS 12.7 60 148.8 97 681.0 110 1075.4 If the dew point is below zero, frost will form on the cup instead of dew if the temperature is sufficiently reduced by evaporation or a freezing mixture. A preliminary test with ice water will determine whether the dew point is above or below zero. Test Questions. 1. Define each of the four quantities involved in hygrom- etry. 424 HEAT 2. What are some of the sources of error and uncertainty in the determi- nation of the dew point? What precautions will help to reduce errors? 3. Under what conditions will it be possible for D to be greater and r less outdoors than inside? Under what conditions will the converse be true? 4. State the "Gas law," and Dalton's law relating to the pressure of mixed gases. 5. Which is heavier, water vapor or air? Does the vapor affect the barom- eter? Which way? Explain. HEAT CONSTANTS 425 GROUP II— H. HEAT CONSTANTS experiment h-41 Calorimetry: Heat of Vaporization of Water Method by Superheating Use the outline given in Nichols and Blaker's Laboratory Manual, Vol. I, Experiment Ij, including the General State- ments, and Parts I, II, and III of the introduction, pp. 133-144, except that the water equivalent of the calorimeter may be computed from its mass and specific heat, omitting the experi- mental determination if there is lack of time. The temperature of part of the calorimeter may not change as much as that of the contents. To compensate, use a smaller mass in computing the water equivalent. Correct results may often be more closely approximated by recourse to a judicious guess. The apparatus includes a superheater in which the temperature of the steam is raised 5 or 10 degrees. Account of this should be taken in the calorimetric equation which should be written out and used in connection with the sixth question below. Test Questions. 1. Define the terms water equivalent and radiation constant. 2. Specify what data should be obtained, and explain how the data is used to determine each. 3. What difficulties are met and what precautions takeli? 4. Upon what does radiation per unit time depend? Upon what does the rate of cooling depend? 5. Define the heat of vaporization, and state the laws of ebullition. 6. How will the result be affected (a) by wet steam; (b) by ignoring radi- ation or absorption; (c) by disregarding air currents; (d) by too high final temperature; (e) by having the thermometer bulb too near the top and the water not properly stirred ; (f ) by not taking into account that the temperature of part of the calorimeter and stirrer is not changed as much as that of the water; (g) by assuming the pressure to be normal when it is above normal; and (h) by superheating without including it in the computations? 7. How may wet steam be avoided? How much error in reading the temperature of superheated steam will be equivalent to the error due to 1 per cent, of moisture in the steam? experiment h-42 Calorimetry: Heat of Vaporization of Alcohol BY THE Ferry Calorimeter In this calorimeter the vapor is condensed in a copper worm 426 HEAT and collects in a copper bulb ; the worm and bulb being immersed in the known mass of water in the calorimeter, to which the heat emitted during condensation of the steam and cooling is imparted. The liquid in the boiler is heated by an electric current through a coil of wire covered by the liquid. Follow the directions given in Practical Physics by Ferry and Jones, Experiment 53, or Ferry, Experiment LX. Note especially the graphical method of finding radiation corrections. Ferry and Jones, pp. 209-215, or Ferry, pp. 197-204. Remarks. The thermometer in the vapor shows the boiling point of alcohol unless the upper end of the heating coil projects above the liquid in which case the vapor becomes superheated. The thermometer should be watched and if more than a degree or two of superheat occurs, the experiment should be stopped. The specific heat of alcohol may be taken as ^cal. per degree C. Put pieces of broken glass in the boiler to avoid bumping. Test Questions. 1 . Write and interpret the calorimeter equation. 2. What masses and temperatures must be observed? Is stirring im- portant? Why? 3. How do you determine the radiation or absorption represented by each square of the area between the two temperature-time curves on your curve sheet? experiment h-43 Specific Heat of a Liquid by Cooling Laws of Cooling The determination of specific heat by this method depends upon the fact that when a body cools in an enclosure the heat emitted per unit time depends only upon the difference of temperature between the body and the enclosure, and the nature of the radiating surface. If a small calorimeter filled with a hot liquid is enclosed in a water jacketed enclosure, it will emit a certain number of calories per minute when its temperature exceeds that of its surroundings by a given amount regardless of whether the liquid is water or something else. If m grams of liquid of specific heat 5 are cooled from t2° to t°, in t minutes in a calorimeter of water equivalent e with the surroundings at the temperature t^, the heat emitted per minute is {ms + e){t2 — and ms -\- e T m's' + e (2) Notice that e does not cancel out, and is not negligible unless it is small and the numerator is nearly equal to the denominator. If the masses, times, and water equivalent are determined, and the specific heat of one of the liquids is known, the specific heat of the other may be computed. Procedure. Put equal volumes of water and of liquid to be tested into calorimeters or containing vessels that are exactly alike, especially as to nature of surface and radiating area. Heat them to about 50° or 60° C. if the liquid is not volatile and does not have a low boiling point. A safe way is to place them in a sand-bath • over a burner. The polished surfaces might be blackened if the flame came into contact with them. When they are removed, wipe the outer surfaces clean and place them in cooling chambers that are alike and at the same temperature. , Use for cooling chambers calorimeters in larger ones with the space between them filled with water. The inside walls of the cooling chamber should be kept dry. If a suitable sand-bath is not available, the liquids to be cooled may be carefully heated in small glass flasks, with a thermometer in each, to a temperature a little higher than that chosen for starting the data; after which the liquids are poured into their respective calorimeters and the data for the cooling curves observed. Read the thermometer in the cooling liquids at convenient equal time intervals, say one minute if the cooling is not too slow. If the cooling "becomes quite slow at say 30° C. it may be discontinued. Read the temperature of the jacket water occasionally. 428 HEAT Plot a temperature-time curve for each liquid and the jacket water on the same sheet. Take from the curves times for 5° temperature intervals (or other convenient intervals) for the computation of specific heat from different parts of the curves. Laws of Cooling. Newton's law of cooling states that " The rate of cooling of a body is proportional to the difference between its own temperature and that of its surroundings." This law is not even approximately true for large temperature differences, but is more convenierit to use for small differences than the more exact law based on Stefan's law for radiation, which states that "The rate of radiation from a body is proportional to the fourth power of its absolute temperature." If a body at the absolute temperature d emits the energy E in the short time t, and received in the same time the energy Eg from its surroundings at the absolute temperature 6^, then the rate of cooling is - - - = c(e^ - &/) (3) T r by Stefan's law, c being the proportionality factor. This may be written ^^-^" = c{e - e,){e^ + e^e, + eSs^ + e,^) (4) r If S is nearly equal to ds ^~^' = AcG'id - e,) = C{B - dg) (5) T (approximately), which expresses Newton's law of cooling. The range through which Newton's law may be used may be judged by testing as illustrated below. Let t and t' be the times required for one of the liquids to cool two degrees, from 51 to 49 and from 31 to 29 respectively, in an enclosure at t°; then, since the rates of cooling are directly proportional to the differences of temperature, the times will be inversely propor- tional, and T30 _ 50° — te . , T50 30° - t. ^^> if Newton's law holds true. If the terms are not equal, find the percentage of deviation. By the more accurate law T30 ^ 323^ - (f. + 273)^ T60 303^ - {t, + 273)* ^'^ HEAT CONSTANTS 429 Find the percentage of deviation, if any, and draw conclusions. Test Questions. 1. Define specific heat. 2. Derive and interpret the formula for computing specific heat from the experimental data. 3. What data must be obtained? How used? Explain. 4. Should the masses or the volumes of water and liquid tested be the same? Give reasons for answer. 5. Show that the thermal capacity of the containing vessel, etc., is not eliminated from the computation, even by cooling both liquids in the same vessel. 6. Show that the thermal capacities of equal volumes of mercury and glass are nearly the same; and that the thermal capacity of the thermometer is a little less than .5 calorie per c.c. immersed. 7. State Newton's law of cooling, and its limitations. Also state Stefan's law of radiation. 8. How may the agreement of your data with the laws of cooling be tested? Through about what range of temperature do yoU' think Newton's law may be applied? References: Milliken, p. 209; Miller, p. 186; Ferry and Jones, p. 219. experiment h-44 Specific Heat of a Liquid by Heating Magie's Method See Miller's Manual, Experiment LXXXH. Find the specific heat of brine containing a given percentage of salt, or of some other liquid specified by the Instructor. experiment h-45 Specific Heat of Materials Used for Heat Insulators Method of Mixtures The ordinary methods of calorimetry are not suitable for finding the specific heat of substances that are very poor con- ductors of heat unless the methods are somewhat modified. The method of mixtures may be used if the solid is finely powdered and mixed with a liquid that will not dissolve it; as directed in Miller's Manual, p. 190; or the process can be made slow enough to use larger pieces if Dewar bulbs are used as calorimeters, and proper precautions and checks are used. The details are left as a problem for the student. The outline proposed should be submitted to the Instructor before beginning work. 430 HEAT experiment h-46 Ratio of the Specific Heats of Air Adiabatic Expansion See Physical Measurements, by Duff and Ewell, Experiment XX; or Franklin, Experiment 58. experiment h-47 Radiation and Absorption Constants of Different Surfaces Graphical Solution and Check Radiation and Absorption. Prevosfs theory of heat exchanges leads to the conclusion that the rate at which radiant energy is emitted from a surface depends only upon its nature and absolute temperature, and does not depend upon the temperature of surrounding bodies. Stefan's Law states that this rate of radiation is proportional to the fourth power of the absolute temperature. The rate at which energy is received by a surface does depend upon the absolute temperature of its surroundings; and the portions of this energy that are absorbed or reflected depend upon the nature of the surface. The rate of cooling of a body depends upon the difference between the rates of emission and absorption through its surface. For black body, or non- selective, radiation this rate of cooling is proportional to the- difference of the fourth powers of the absolute temperatures of the body and surroundings; or if the temperature difference is small it is approximately proportional to the first power of the difference of the temperatures; as explained in Experiment H— 43. The ratio of the radiation | , , , j- by a surface to that 1 h h H I ^^ ^ lamp black surface of the same area and temperature in the same time is called the \ , , . ]■ power of L absorbing. J ^ the surface. Tu u c ^u 1 • r radiated from ") The number of the calories | ^^sorbed by / """^ ^'^"^''^ * Read the sentence first using the upper words, then read again using the lower words. HEAT CONSTANTS 43 1 centimeter of surface in one minute, for a difference of one degree between the temperature of the surface and its sur- !• • II , , r radiation i roundmgs is called the -^ , ^ . }■ constant of the surface. L absorption J The objects of this experiment are: to determine the radiation and absorption constants {K or Kr and K^) of different surfaces and to illustrate the use of graphical methods in the solution and checking of results. The constants may be determined by dividing the heat {H) lost (or gained) by a vessel in a given time, by the time (t), the area {A) of the vessel, and the average difference between the temperature of the surface (i) and that of its surroundings (<,). ^ ^ rXAX{t-t,) ^^) If the walls of the cooling vessel are thin and of some highly conducting metal, the temperature of the contents next to the inner surface will be nearly the same as that of the outer surface, so that the reading of a thermometer in contact with the inner surface and contents of the vessel may be taken for the tempera- ture of the outer surface without any great error. Procedure. Use for the radiating vessels two small calorim- eters A and B that are exactly alike except that the surface of one (A) is nickel plated and the other (B) is a dull black. Each of these is to be enclosed in two larger vessels, one within the other. The space between the outer two is filled with water at a temperature slightly higher than that of the air, to keep the walls surrounding the radiating surface at a constant (or nearly constant) known temperature. If the water jacketed enclosures are not available, the experiment may be performed in the open air where it is fairly free from currents or changes of temperature. Fill A, then B, with equal amounts of water at the same temperature, 15° or 20° C. warmer than the air. Place them within the water jacketed enclosures with a good thermometer in each of the radiating vessels close to one side. The thermom- eter should be read at intervals of one or two minutes depending on the rate of cooling. The temperature of the jacket should be taken at intervals of about five minutes. Continue the observations for half an hour or more. 432 HEAT To obtain data for absorption proceed in like manner with the exception that the temperature of the water in the absorbing vessel is 15° or 20° below that of the water in the jacket, which is slightly below air temperature. Plot curves with temperature (i) as ordinates and times (r) as abscissas. From Newton's law of cooling and the definition of radiation constant in which C is the heat capacity of the calorimeter and contents, A the radiating area, ts the temperature of the surrounding jacket, and Kr the radiation constant, which may be computed from the observed data and the value obtained by finding the slope of tangents at different parts of the plotted curve. The mean value of K is the one sought, and the agreement of the values indicates the degree of reliability. Another method of using the data, which is perhaps more instructive than the one given above, is indicated in the following explanation : Equation (1') may be written dt KA ,, , ,^. By integration KA Log« (< - i.) = -^ X r + c (3) in which c is the constant of integration. If log {t — ts) and T are plotted as coordinates, the result should be a straight line, the slope of which is KA/C, from which K may be computed and checked against the first determination by Equation (1'). Equation (3) is correct when the first term is expressed in Naperian or Natural logarithms. If common logarithms are used for plotting Logio jt - t,) _KA 1343 --C^^ + ' ^*) and ^ = fx^^x5^"^ (5) A .4343 312 — :ki in which .4343 is the modulus of the common system and Xiyi and X2y2 are coordinates of points on the curve. HEAT CONSTANTS 433 Test Questions. 1. State Prevost's theory of exchanges, and Stefan's law for radiation. 2. Define emissive and absorbing power, and radiation and absorption constants. 3. What data must be observed and how used to find Kr and KJ 4. What indicates the reliability of the results in each method of finding K} Which is the better check? Do the results of the two methods agree? 5. Modify equation (3) so that it will be correct when common logarithms are used instead of natural. experiment h-48 The Mechanical Equivalent of Heat Callender Apparatus Perform the experiment according to the outline given out with the apparatus. experiment h-49 The Electrical Equivalent of Heat The Constant Flow Current Calorimeter See Experiment 57 in Practical Physics, by Ferry and Jones. CHAPTER VIII EXPERIMENTS IN WAVE MOTIONS AND SOUND GROUP III— W. WAVE MOTIONS AND SOUND experiment w-41 Study of Wave Motions Longitudinal and Transverse Waves The object of the experiment is an experimental.study of wave motions with respect to (1) reflection of waves at fixed and free boundaries; (2) the formation of standing waves, and the phase relations of particles in different parts of a standing wave; and (3) the relations of velocity, frequency, and wave-length. Definitions of longitudinal and transverse waves, wave- length, wave front, wave train, stationary wave trains, segments, nodes, antinodes, change of phase, change of sign, etc., may be found in the notes preceding the Test Questions. When a wave reaches a boundary where there is a change in speed or in the freedom of motion, part of the energy will be reflected in a returning wave. A wave is reflected with change of phase if the displacements of the particles in the reflected wave are opposite in direction to those in the direct wave. A wave is reflected with change of sign if a "compression" is changed to "expansion" (or vice versa) by reflection; or if a "crest" is changed to a "hollow" (or vice versa) by reflection. Part I. Longitudinal Waves. A long spiral spring of large cross section is supported in a horizontal position by light cords so as to be free to move endwise. When a compression traverses the spiral spring, the kinetic energy of a given turn of the wire is received from the turn behind it and is expended in accelerating the one ahead of it. When the end turn at a free boundary is reached, its energy must be expended in accelerating the one behind by pulling it forward since there is none ahead to be pushed. This pull produces an expansion which traverses the coil in the reverse direction and is 434 WAVE MOTIONS 435 called the reflected wave. Reflection is said to occur with change of sign if an incident compression is reflected as an expansion or vice versa; or if a crest of a transverse wave is reflected as a hollow or vice versa. Study the action at free and fixed boundaries when compressions and expansions are reflected. Procedure: 1. Reflection. Strike the right hand end of the spring a quick blow (not too hard) tending to compress the spring, and observe the advancing and the reflected wave. Determine whether the reflected wave is a compression or an expansion by noting whether the first effect of the returning wave is to push the end turn outward (toward the right) or draw it inward (toward the left). From this observation determine whether the reflected wave is a compression or an expansion. Make observations with the left-hand end free and with it fixed. Give the spring a quick short pull (at the right-hand end) tending to stretch it, and observe the result when the expansion thus produced is reflected at the left end. Note the sign of the reflected wave when the left end is free and when it is fixed. Determine whether reflection is with or without change of phase, and with or without change of sign, when : (a) A compression is reflected at a free boundary. (b) A compression is reflected at a fixed boundary. (c) An expansion is reflected at a free boundary. (d) An expansion is reflected at a fixed boundary. 2. Standing {or Stationary) Waves. The two conditions for standing waves are, like wave trains in opposite directions. If one of these trains tends to displace a particle in one direction at the same instant that the other train tends to displace it in the other direction, the particle will remain at rest. If the action of one train is neutralized at a given point by the action of the other at a given instant, the actions will continue to neutralize each other as successive parts of the trains reach the point, since the trains have the same speed, wave length, etc. Draw the curves and verify the statement. The point (or plane) where the medium is thus kept at rest is called a node. If the trains are continuously opposite in phase at a given point, they will also be opposite in phase at other points one-half wave 436 WAVE MOTIONS AND SOUND length apart; but will be in the same phase at points (called antinodes) midway between the nodes or points at rest. The vibratory displaceinent is a maximum at the antinodes where the effects of the trains are additive. From a node to an antinode is one-fourth wave length. The portion between consecutive nodes is called a segment. There are two segments in a wave length. By means of properly timed impulses imparted by the hand (acting through a very short distance in opposition to the pull or push of the wire), set the whole spring into vibration with one or more nodes between the ends. If the motion of the end held by the hand is slight, that end may be called a node. Show by diagram where the nodes are, and describe the motions, relative to each other, of the particles in a single seg- ment, between two nodes; also of the particles' in adjacent seg- ments. Note where the greatest variations of pressure occur, and where the variation is a minimum. 3. Speed, Wave Length, and Frequency. If the end is struck a light blow, the disturbance will travel to the further end and back many times before it becomes too small to be detected when it returns. Find the number of round trips in an observed time interval (say 20 or 30 seconds), and compute v. Compute, from the length of the coil, the wave lengths (X) when the number of segments changes, by steps of one-half segment, from one to four or five segments. A half segment is formed at a free end. Since the wave length is the distance traversed by the dis- turbance during one period of vibration, the velocity is equal to the number of periods {n) per second, times the wave length; or z; == «X. n is called the frequency. Compute the values of n corresponding to the wave lengths found above. Set the coil vibrating in stationary waves of 1|, 2|, 31, seg- ments successively when one end is free ; then fix one end and set it vibrating in 2, 3, and 4 segments successively. Note the number of vibrations per minute, and find the frequency for each wave length. Compare the observed frequency with the computed. How are frequency and wave length related? Part II. Transverse'jWaves. A long light chain is used to illustrate transverse vibrations. Fasten one end three or four WAVE MOTIONS 437 feet above the floor. Hold the other end in the hand. Pull hard enough to make it clear the floor. Strike the chain a quick sharp blow with a small rod 15 or 20 cm. from the hand. Ob- serve, and describe by use of a diagram, the advancing wave and the reflected wave. Determine whether reflection occurs with or without change or phase, and with or without change of sign at the fixed boundary ; then make the end where reflection occurs comparatively free laterally by inserting 9 or 10 feet of light cord between the chain and the fixed point of attachment. Take observations for relative phase and sign of the direct and reflected waves when the end where reflection occurs is free. Try to produce a wave in which the front and rear parts are not alike ; then note whether the front part of the advancing wave is ahead or behind in the reflected wave. If a second advancing wave (like the first) were to meet the first after reflection at a fixed end, what would be the result at the point where they meet? Explain by diagram assuming the Fig. 159. advancing wave to have the form, shown in Fig. 159. Would the result be any different if the wave were sinusoidal? What would be the result if a continuous train of advancing waves were sent out? Try to produce and maintain stationary waves. Describe the relative motions of particles in the same segment and in adjacent segments. How may the number of segments be changed? What is the effect when the tension is changed? Notes on Waves and Wave Motions. If one particle of an elastic medium is displaced, the equilibrium of its neighbors is disturbed, and these in turn transmit their motion to particles beyond. The progressive motion of the disturbance from particle to particle is accompanied by a transfer of energy from the initial point to all points affected in succession. In a homo- geneous elastic medium the disturbance progresses with a definite uniform speed away from the initial point of disturbance, but each particle of the transmitting medium in time vibrates through a very limited range and pa^-ses its kinetic energy on to 438 WAVE MOTIONS AND SOUND its neighbor. This progressive disturbance, or transmitted energy, is called a wave. A periodic succession of waves is called a wave train. Note that there are two distinct kinds of motion : the uniform motion of transmission, and harmonic vibratory motion of the particles. The kinds of waves are named from the relative directions of these two kinds of motion. When the vibratory motion of the particles is parallel to the motion of transmission of the disturbance, it is called a longitudinal wave. If the particles vibrate at right angles to the direction of trans- mission, the wave is called transverse. Longitudinal waves may be transmitted by either fluids or solids, but transverse waves can be transmitted only by media possessing rigidity, or elasticity of form ; which is a characteristic of solids. Fluids can transmit longitudinal waves only. Sound is transmitted through air by longitudinal waves. If a periodic succession of alternating compressions and rarefactions follow each other to the ear, they produce the sensation of sound. The speed of transmission through the air at 20° C. is about 344 meters per second. The maximum speed of the vibrating particles depends on the amplitude and frequency of vibration. Loudness of sound depends on the amplitude, and pitch depends upon frequency. [Problems. (1) Find the maximum speed of air particles, assuming their amplitude to be 2 mm. and the frequency to be 150 per second. Ans. 1.885 meters per second. (2) How many seconds are required for a bugle note to travel one mile if the temperature is 20° C? Ans. 4.68 sec] The distance that a wave travels during the time of one complete vibration of a particle is called a wave length. If the velocity is denoted by v, the frequency by n, and the wave length by X, then v = n\; for the distance that sound travels in one second equals the product of the wave length and the number of wave lengths that pass a given poipt in one second. The continuous locus of particles that have the same phase is called a wave front. In a homogeneous medium it is a surface perpendicular to the direction of progression of the disturbance. If sound radiates from a point, the wave fronts are spheres. For a single wave train, wave fronts one wave length apart are WAVE MOTIONS 439 in the same phase or part of their cycle of motion, and those one-half wa\e length apart are in opposite phase. The displace- ments of particles in the same phase are in the same direction. The displacements of particles in opposite phase are in opposite directions. The displacements for any given phase are pro- portional to the amplitudes. If the motion of particles is harmonic, a sinusoidal curve may be used to represent the relative phases and displacements of either successive particles at a gi^'en instant, or a single particle at successive instants. More frequently the former, in which case it is sometimes called a wave form. If two or more wave trains traverse the same portion of the medium simultaneously in the same direction, the resultant displacement at any point may be determined by taking the algebraic sum of the displacements that would be produced by the trains separately. Like wave trains in opposite directions produce stationary or standing waves, which are characterized by vibrating portions or segments between fixed points called nodes, one-half wave length apart. The component waves are in opposite phase at the nodes and in the same phase at the antinodes, which are midway between the nodes. Particles in a given segment have the same phase, but particles in adjacent segments are in oppo- site phase so that particles on each side of a node swing simul- taneously toward the node, then away from it. Consequently, the maximum variations of pressure occur at the nodes and the maximum motion of particles occurs at the antinodes. Like wave trains in opposite directions may be obtained by reflection of a wave train at either a fixed or a free boundary. A fixed boundary corresponds to a node and a free boundary to an antinode, consequently reflection occurs with change of phase at a fixed boundary and without change of phase at a free boundary. Test Questions. 1. Define longitudinal and transverse waves. 2. State the conditions essential for stationary waves. 3. Describe the phase relations of the component waves at the nodes and at the antinodes of the resultant stationary wave. 4. Explain how nodes and antinodes are formed, and why an antinode and a node are 1/4 wave length apart. 5. Describe the phase relation of particles hi a single segment, and of particles in adjacent segments. 440 WAVE MOTIONS AND SOUND 6. Locate the points of maximum and minimum changes of stress or strain of medium in a stationary wave train. 7. Is reflection with or without change of "phase" at (1) A fixed boundary? ] f (A) For longitudinal waves. (2) A free boundary? J L (B) For transverse waves. 8. Is reflection with or without change of "sign" at ' (1) A fixed boundary? "1 J" (A) For longitudinal waves. (2) A free boundary? J \ (B) For transverse waves. 9. How is the speed of propagation related to the wave length and frequency of vibration? 10. Upon what does the maximum speed of the particles depend? Solve Prob. (1) in the Notes on Wave Motions and compare the result with the speed of propagation. 11. Would the speed of transmission of waves be affected if the diameter of the spiral coil were decreased? Explain. How is the speed of a transverse wave in a cord affected by ten.sion? experiment w-42 Velocity of Sound by Resonance Measurement of Wave Length The Frequency (n) of a given pitch is the number of complete vibrations per second: The wave length (X) is the distance any wave front travels during one complete vibration ; hence V = n\ = \/T V being the velocity of sound and T the period of vibration. The distance between adjacent nodes of a standing' wave is one-half wave-length (^X). A wave front is the locus of adjacent particles in the same phase of vibration. It is a surface normal to the direction of propagation. Resonance is the reenforcement of sound by means of sym- pathetic vibrations. It relates to the vibratory motion induced and increased by periodic impulses of the same frequency as the natural frequency of the resonating body. The waves of energy from one vibrating body are absorbed when received by another body having the same frequency. The condition essential for resonance is : The natural frequency of the resonator must be the same as the frequency of the exciting waves. The resonator used in this experiment is the air column in a tube of variable length. The length may be varied by means of VELOCITY AND WAVE LENGTH 44I a piston in a horizontal tube, or water may be raised and lowered in a long vertical glass tube by raising and lowering a bottle of water connected by a siphon and flexible tube to the bottom of the large glass tube. The top of the water in the tube may be regarded as a piston or fixed boundary from which waves running down the tube are reflected. An incident wave is reflected without change of sign at the piston, and with change of sign at the mouth, or free boundary. Imagine a condensation to enter the tube from a tuning fork \ibrating near the mouth. It is reflected as a condensation at the bottom, and when it returns to the mouth, this condensation is reflected back into the tube as a rarefaction. If the round trip down the tube and back has occupied half a period (or an odd number of half periods) a rarefaction from the fork will enter the tube and unite with the reflected rarefaction to give a wave of increased energy. When this rarefaction is reflected from the bot- tom and reaches the mouth, it is reflected as a compression, and its energy is further increased by an added compression from the fork. The energy of the vibrating air column in the tube is thus increased until the rate of dissipation of energy in friction, etc., equals the rate of supply from the fork. The like wave trains in opposite directions give a standing wave in the tube with a node at the piston and an antinode at the mouth. Three nodes are indicated at 0, 0', 0" in Fig. I60. The upper row of arrows indicates the directions of motions of the segments at a given instant, and the lower row the directions half p'tea Fig. 160. a period later. Place the vibrating prongs of the fork as indicated at pp'. The antinode at the mouth is a little beyond the end of the tube. For this reason it is better to determine X from the distances between nodes, omitting the half segment at the end. Procedure. If the water tube is used, place a support for the bottle at such a height that the water in the tube rises nearly to 442 WAVE MOTIONS AND SOUND the top. When the fork is in position and vibrating, set the bottle on the floor and note the positions of the water surface when maximum resonance occurs as the water runs down the tube. Mark these places by the small rubber bands on the tube. Place the bottle on the upper support and repeat the observations with the water rising. The flow of water may be controlled or stopped at any time by pinching the rubber tubing. Repeat the observations up and down until the node markers are adjusted as precisely as possible. Then take readings for the position of each node with the zero of the scale placed at the top of the tube. Partners should exchange parts and take another set of observa- tions with a fork of different pitch. Tabulate the data and compute the velocity. Observe the room temperature and compute the velocity at 0° C. by the formula Vt = FoVl + .003665/. Note the deviation from 331.6 m./sec. (which is the mean of the values found by the experimenters) and draw conclusions. From the velocity ob- tained with one of the forks compute the frequency of the other fork and check with the marked value. Remarks. Koenig's forks are marked in single vibrations per second. The frequency is half of the number on the fork. The "perturbation" at the mouth of the tube may be found by subtracting the first node reading from half of the average distance between nodes. It may be as large as three-tenths of the diameter. The effect of temperature on the velocity of sound is derived in terms of the notation used in the text, as follows: F = Fo X V 1 + .003665/ F = Fo + iFo X .00366/ approximately F = Fo + -607 / m./sec. approximately Test Questions. 1. Define frequency, wave-length, wave front, and resonance. 2. State the condition essential for resonance. 3. State the law for reflection of longitudinal waves at fixed and free bound- VELOCITY AND WAVE LENGTH 443 4. Describe the steps by which sound is reenforced, starting with a ra- refaction reflected at the second or third node. 5. State the conditions essential for standing waves. 6. Describe the phase relations of particles in adjacent segments of the standing wave. 7. Why is the maximum loudness much lower for nodes farther from the mouth? 8. Can you give any measure of the probable reliability of your deter- mination of velocity? 9. Explain why the velocity is not affected by change of pressure; and why it is increased by a rise of temperature. 10. Describe a method of finding the frequency of a fork. experiment w— 43 Velocity of Sound in Metals Young's Modulus — Kundt's Method The experiment illustrates a method of finding Young's Modulus of metal rods by use of the formula Vr = ylM/D (1) in which Vr is the velocity of sound in the rod, M is Young's Modulus, and D is the density of the rod. Since V = n\ for longitudinal waves in both air and metals, the ratio of the velocities in rod and air ( Vr/ Va) is the same as the ratio of the wave-lengths (X,/Xa), if n, the frequency, is the same for both. Therefore, Vr = Va\rl\a (2) and since X, = 21, I being the length of the rod, M = DV,^ 4.PI\a^ (3) in which Va is known and \a may be found by use of the appa- ratus, the construction and operation of which is briefly described as follows : The air in a glass tube is made to vibrate in standing waves by the longitudinal vibration of the rod clamped at its center with its length along the extended axis of the tube. To one end of the rod is fastened a light disk which fits loosely in one end of the glass tube. The other end of the tube is closed by a piston which may be adjusted so that the enclosed air column will be of the right length to give maximum resonance when the free end of the rod is stroked with a damp cloth or resined leather. 444 WAVE MOTIONS AND SOUND Before the tube is put in place a little lycopodium powder is dropped (from the end of a small knife blade) into one end of the tube held obliquely so that it will run in a straight narrow line to the other end of the tube. Carefully adjust the tube with the line of powder at the bottom. The positions of nodes will be indicated by the little heaps of powder formed when maximum resonance occurs. From the average distance between nodes, the average value of Xa may be found. Test Questions. 1. State the conditions that are essential for resonance. 2. Is the end of the air column next to the disk a node, an antinode, or neither? 3. Locate the antinodes and node of the rod. Where do reflections occur? Are they accompanied by change of sign or phase? Are the phases of particles at the two ends of the rod the same or different at a given instant? Explain. 4. Why is a very small amount of powder better than much? What is the principal source of unreliability of the result? Mark the figure that you consider doubtful in your result. CHAPTER IX EXPERIMENTS IN LIGHT GROUP III— L. LIGHT experiment l— 41 Fundamental Principles Relating to Photometry Measurements with! Simple Photometers The intensity (7) of light at any point may be defined to be the rate of flow of light energy, per unit sectional area, past the point. This also expresses the intensity of illumination of a surface if the beam is perpendicular to the surface. If energy is radiated uniformly in all directions from a source as a center, the intensity at the distance d from the source is measured by the total light energy per second (P) divided by the area {A:Td?) of the sphere of which d is the radius. Then ^ and for conical beams (i. e., spherical wave-fronts) the intensities at different points are inversely proportional to the squares of their distances from the source at the vertex of the cone. This is called the Law of Inverse Squares. Since P is the rate at which light energy is radiated, it may be called the radiating power of the source. This rate of flow of light energy is sometimes called the "flux" of light from the source. Since there are 4x units of solid angle about the center of a sphere, and P is the total flux, it follows that Pjiir expresses the light flux per unit of solid angle, which is taken as a measure of the brightness, or luminous intensity, of the source. For a given constant source, Pj^ir is a constant. Let it be denoted by C; then I = C/d^ (2) The standard light recently adopted by the standards laboratories of the United States, Great Britain, and France, is called the 445 446 LIGHT "International Candle"; and C is measured in candle power. (C.P.). (See remarks for other units.) The fundamental principle of photometry (sometimes called Bouguer's Principle) is, that if two sources produce equal intensities of illumination at a given point, their candle powers are directly proportional to the squares of their respective distances from the point. To derive this principle from the law of inverse squares, let /i = Cifdi^ for one and /a = CaM^ for the other source. Then since /i = I2 at the given point, The objects of the experiments are (1) to show experimentally that the intensities of illumination at different distances from the one source vary inversely as the squares of the distances ; and (2) to derive and apply the principle that the candle powers of two or more sources are directly proportional to the squares of their distances from the point where they produce equal in- tensities. Part I may be performed with Rumford's Shadow Photometer as indicated in the following outline and in Fig. 161. The appa- ratus needed is, a long graduated bar, a photometer and the two sources, of which one is constant and the other \'ariable in brightness, with a known ratio of values. Fig. 161. The intensities (7) at the screen due to a constant source (Li) at different distances are matched with the different intensities of a variable source (Z2) at a fixed distance (^2) , the brightness of which can be varied in a known ratio. The distance (di) of Li is observed for each value of I. If two quantities \'ary inversely, their product is a constartt. Hence, if the product of the corre- sponding values of / and di^ is constant, the intensities due to the constant source \'ary inversely as the squares of the distances from the source. Plot 1/7 and d"^, and interpret the results. PHOTOMETRY 447 The brightness of the variable source is varied in a known ratio b\- turning on successively like incandescent lamps, placed in a vertical plane and adjusted, by means of sliding screens, to give separately equal intensities on the screen when compared with Li at a fixed distance (say 160 cm.). Be careful not to move the screens after the initial adjustment. For twice the initial inten- sity turn on the lamps in pairs. If four lamps are used, make four settings (^i) of Li for the pairs 1-2, 2-3, 3-4, 4-1, and use the average value of di. This procedure will help to eliminate errors in the initial adjustment of the separate lamps. Proceed in like manner with intensities three and four times the initial intensity. Each step in the experimental verification of the Inverse Square Law should be clearly understood before taking the observed data and should be included in the report. Note that the shadow a from ii is illuminated by ia, and the shadow h from Li is illuminated by Lx. [An interesting illustration of complementary colors and effects of difference of color may be shown by placing a plate of colored glass before Lx. If the light from Li comes through blue glass, yellow is absorbed so that shadow h receives less yellow than a which receives white light from Li, but appears yellow by contrast with b. Use red glass before Zi and determine the complementary color by inspection of the shadow a.\ Part II includes the measurement of candle power, current, and impressed P.D.; when the last is varied, and the variable candle power of the lamps tested (L^, Ly) is compared with that of a constant standard lamp (L,) by means of an enclosed Bunsen or Jolly photometer. Ask the Instructor in charge about the standard lamps, the range of P.D. to use on the lamps tested, and the precautions that are necessary. Test one lamp {Lx) having a carbon filament, and another (L„) having a metal filament if no directions to the contrary are given. Tabulate the observed data and the derived data required to plot five curves using candle power for ordinates of each, and volts, amperes, watts, watts per candle, and resistance, respec- tively for the abscissas of the curves. All may be put on the same curve sheet. Interpret, compare, and discuss the results. Make ten or more independent photometer settings with constant conditions, and compute the percentage deviation and 448 LIGHT the probable error for the set of readings. Partners may read alternately in order to determine whether one sets uniformly above or below the other, as may be the case with even expert observers. Each should record his own readings and not announce them to the other until the observations are completed. While one is setting the photometer, the other should watch the constancy of P.D., etc. Draw a diagram of the apparatus and circuits. Remarks. Light Units. The British standard candle was made of pure spermaceti with wick and dimensions such that the rate of burning was 120 grains per hour as the standard. The Hefner standard lamp, used in Germany, burns amyl acetate, and is constructed to standard dimensions. For ex- ample, the wick tube is 8 mm. in diameter, and the flame must be very carefully adjusted to a height of 40 mm. Its reliability is far beyond that of the standard candle. The Carcel standard lamp, used in France, is an oil lamp with circular wick. It burns Colza oil, which must be kept at a con- stant level. 1 International candle = 1 American candle = 1.111 Hefner units = 0.104 Carcel unit Some primary standard, such as the Hefner lamp, is used to calibrate incandescent lamps as secondary standards for ordinary work. The rate of flow of light energy through unit sectional area may be called the "sectional intensity" {!,) of a beam; and the rate of flow per unit solid angle of the source, the conical intensity (Ic). The former varies inversely as the square of the distance (d) from the source. The latter is independent of distance, since the light in a given cone remains in that cone if it proceeds from a point source. Is = IJd' (1) The conical intensity of a horizontal beam from a standard Hefner lamp is called a hefner-unit, or a hefner. The candle-unit equals 1.111 hefner-units. The sectional intensity at a distance of one foot from a standard candle is called a "foot-candle" (f.c); and at a distance of one meter, a "meter-candle" (m.c). From Eq. 1 PHOTOMETRY 449 Foot-candles = candle-units/sq. of dist. in feet (2) Meter-candles = candle-units/sq. of dist. in meters (3) From Eq. 2 the intensity of illumination at a distance of four feet from a 16 C.P. lamp is one foot-candle (1 f.c); and at ten feet, it is .16 f.c. Another unit of sectional intensity is the lux. It is the sec- tional intensity of a horizontal beam from a Hefner lamp at a distance of one meter from the lamp. From this definition and Eq. 1, Luxes = hefners/sq. of dist. in meters (4) The intensity of illumination of a surface upon which a beam falls perpendicularly is measured by the sectional intensity of the beam and may be expressed in luxes. If a lamp gives one hefner-unit in every direction, the rate at which light is emitted by the lamp, or the total light flux, is one spherical hefnerr and the flux per unit solid angle is the lumen. Since there are 4x units of solid angle about a point. One sph. hefner = 47r lumens (= 0.90 sph. C.P.) (5) The area of a spherical surface having a radius of 100 centi- meters is 40,000 -K sq. cm. If 47r lumens radiate from the center of the sphere, the flux per sq. cm. at the surface is one ten- thousandth of a lumen, and is also one lux by definition. From this it follows that the flux in lumens (Z) through a given section area A (in sq. cm.) where the sectional intensity is Is luxes is l = AX IJW (6> Test Questions. 1. Define intensity of light and state how it varies with distance from a point source. How may the law be verified experimentally? Explain each step. 2. What is meant by the brightness of a source? Name and specify twO' units of brightness (or conical intensity). 3. Derive and state Bouguere's fundamental principle of photometry. 4. Describe the essential parts of two or more photometers. 5. Specify some measure of the reliability of photometer settings as de- termined from your set of readings made for that purpose. 6. How is the per cent, of probable error of the candle power related to the per cent, of probable error of the readings? Why do many independent readings give more reliable results? The probable error of 25 independent readings is what part of the probable error of 9 readings? 450 LIGHT 7. In what respects do the results of the tests of the carbon and metal filament lamps differ? What conclusions may be drawn from the curves? 8. Find the intensity 10 ft. from a 50 c.p. lamp in foot-candles and luxes. 1 f.c. = ? luxes? experiment l-42 Laws of Reflection and Refraction Tracing Reflected and Refracted Rays Part A — Reflection at a Plane Surface The General Law for reflection is : The angle of incidence equals the angle of reflection; and. the incident ray, the reflected ray, and a line perpendicular to the surface at the point of incidence are all in the same plane. The image of a luminous point in front of a plane mirror is located by the following law: The point (P) and its image (P') are on the same normal to the reflecting surface, and equidistant from the surface. Procedure. Place a leaf of the report paper on a small drawing board. Draw a straight line across the sheet near the middle and place the mirror on the paper so that the plane of the mirror coating coincides with the line and is perpendicular to the board. Fig. 162. To trace the path of a ray of light before and after reflection, set two pins, P, ^, in front of the mirror so that a line through them will meet the mirror obliquely (as shown in Fig. 162), then REFLECTION AND REFRACTION 451 set two more, B, C, in line with the images of the first two. Place another pin, P', behind the mirror so that the part of it seen above or below the mirror will appear to be in the same vertical line as the image of P viewed from different points in different parts of the mirror. Use the pin holders to support the pins in a vertical position. When adjusted press the pin points into the paper and note carefully whether the pins are still properly adjusted. Lines drawn through PA and BC should intersect at the re- flecting surface and give the point at which reflection occurs. Draw a normal to the mirror at this point and determine whether the angles of incidence and reflection are equal. Compare the distances of P and P' from the reflecting surface and note whether a line joining them is perpendicular to the surface or not. To check the location of the image of P, trace two or more additional rays from P which meet the mirror at different angles. Produce each of the reflected rays backward and note whether they intersect at P' or not. Draw conclusions. Considering P to be the source of light, draw a wave front (before reflection) through A , also one (after reflection) through C. Part B — Refraction at Plane Surfaces Obliquely incident rays are traced through the parallel faces of a thick plate or block of glass by the use of pins (A, B, C, D) set as shown in Fig. 163. The General Law for Refraction is : The ratio of the sine of the Fig. 163. angle of incidence to the sine of the angle of refraction is a constant for two given media; and the normal at the point of incidence, 452 LIGHT the incident ray, and the refracted ray, all lie in the same plane. If the ray passes from a rarer to a denser medium, it is bent toward the normal, and the constant ratio is called the index of refraction (w). It is greater than one, and equals the ratio of the velocities in the two media. Procedure. Place the glass on a sheet of report paper and draw fine lines along its edges. Draw lines Im and no with especial care, so that they are in the same vertical planes as the refracting surfaces. Set two pins {A , B) on one side of the plate so that two others (C, D) may be placed on the other side of the plate in line with the images of the first two when viewed through the plate by the eye placed in line with the second two. Use the pin holders to support the pins perpendicular to the paper. When C and D are precisely adjusted in line with the images of A and B, look along the line AB to see if A and B are in line with the images of C and D. Carefully press the pin points into the paper, and make the final inspection of adjustment. When the plate and pins are moved, draw lines through AB and CD intersecting Im and no at p and p' and join pp'. The line App'D traces the path of a ray from A to D, or from Z> to .4. Draw perpendiculars to Im and no through p and p' , and measure the angles between the perpendiculars and the rays with a protractor. Compute the index of refraction («) from the sines of the angles and check graphically by drawing S' and S" parallel to Im from points equidistant from p on pA and pp'. Then n = S'jS". Why? Determine whether CD is par- allel to AB, or not. Compare n with values found in tables, and draw conclusions. Part C — Index of Refraction Micrometer Microscope Method The index of refraction of the glass plate used in Part B is measured by use of the micrometer microscope as a check on the value found in Part B. When a wave front bac, from the point p, emerges from the glass into the air, it travels faster than it did in glass. This causes a change in the direction of the rays, which, viewed from above, appear to come from p' instead of p. REFLECTION AND REFRACTION 453 Let Vi be the velocity in glass, v^ the velocity in air, and n the index of refraction ; then = ^ = oa^ _ ab 'llp 'V _ pb'_ vi aa" ~ ab'lipV ~ J'h' If b' is the point of emergence of a ray very near to the ray pae, as is the case for all rays entering the eye through the micrometer n = Fig. 164. microscope, pb' becomes the thickness of the glass, and p'b' the distance from p' to a at the surface, and pb' _ Thic kness of glass p'b' Depth of p' below surface Procedure. Place the plate of glass to be tested on a surface plate under the micrometer microscope to which is attached a scale and vernier for readings to give the change of height of the barrel. Place a very small amount of lycopodium powder on the upper face of the glass, and gently blow off the coarser par- ticles that do not adhere to the glass. Focus sharply on the remaining powder and observe the scale reading which should be read to tenths of the least count by estimated interpolation on the vernier. (See page 42, Exp. 1-2.) Turn the microscope out of focus, then reset and read again. Repeat readings for at least five independent consecutive settings. Turn the glass over and focus on the image of the powder viewed through the glass. Take readings as before for at least five settings. Re- move the glass and focus on the powder that remained on the surface plate. Again take readings for at least five settings. Tabulate the readings and their mean values, and determine from them the index of refraction of the glass plate. Test Questions. 1. State the laws for reflection; and the laws for location of images with plane mirrors. 454 LIGHT 2. Are there any apparent deviations from the laws in the experiment? Can you explain them? Within what percentage are the laws verified? 3. State the laws of refraction and define index of refraction. 4. Describe two methods of finding the index of refraction of a plate of glass with parallel faces. 5. Prove that the real depth of water is n times the apparent depth when viewed normally. 6. How does the apparent depth change as the angle between the line of view and the normal increases? Explain. 7. Compare the values of n and discuss sources of error. 8. How may a check on the reliability of the measurements in Part C be obtained? 9. What would be the percentage of error in Part C, (o) If each length were .2 mm. too long? (6) If each reading were .2 mm. too high? (c) If each length exceeded its correct value by 2 per cent.? 10 Is there any advantage in using the same method to measure both lengths in Part C? Explain. experiment l— 43 Study of Spherical Mirrors Reflection at Curved Surfaces Introduction. The curvature at any point of a curve is the change of direction per unit length. If d is the angle subtended by a very short arc o,th& curvature is 0/a; but S = ajr; therefore, the curvature is l/r, the reciprocal of the radius of the circle which most nearly coincides with the curve at the point. From this it follows that the curvature of a spherical wave front is measured by the reciprocal of its radius. The object of the experiment is to study experimentally the effect of a spherical reflecting surface in changing the curvature of a wave front incident upon it. Consider the following cases: (fl) The simplest case is that in which a plane wave front strikes upon a concave reflecting surface of radius R. The curvature of the incident wave front is zero (1/°°), and the distance from the reflecting surface to the image is, by definition, the focal length (F) of the mirror. The curvature is changed in this case from zero to IjF. The point where parallel incident rays intersect after reflection is called the principal focus. In the converse case wave fronts proceeding from the principal focus will become plane after reflection as indicated in the figure. SPHERICAL MIRRORS 455 Note that curvature decreases as the wave front moves away from the source, and becomes zero after reflection, showing that the concave mirror decreases the curvature. Any further de- crease of curvature would change the wave front to concave, which may be called negative curvature in contrast with that of a convex front. (6) If the center of curvature of the incident wave front coincides with the center of curvature of the mirror, each ray (perpendicular to the wave front) will strike the mirror normally and be reflected to the point from which it came ; and the curva- ture 1/i? of the convex incident wave front will be changed to — 1/i? for the concave reflected wave front. The source and image will lie in the same plane at the same distance R from the mirror if the aperture is small. (c) If the distance of the source from the mirror is denoted by u, and the distance of the corresponding image by v, the curva- tures of the incident and reflected wave fronts are 1/m and l/v respectively. Let the curvature of the wave front be positive when the rays are divergent and negative when the rays are convergent; then if divergent rays from a real object are con- verged after reflection to a real image, the curvature is decreased from + 1/m to — 1/v, and the decrease of curvature is the curva- ture of incidence less that at reflection; or, since 1/m — (— l/v) = 1/m + l/v, the change of curvature is the sum of the reciprocals of the conjugate focal distances, as the corresponding distances M and V are called. (d) In (a) the change of curvature is l/<» — (— 1/F) = + l/F = 1/F. In (&) the change of curvature is 1/i? - (- 1/i?) = 1/i? + 1/i? = 2/i?. In (c) the change of curvature is 456 LIGHT 1/w — (— l/n) = 1/m + Ijv on the assumption that object and image are real, giving divergent incident rays and convergent reflected rays. Let the image be located by the point of inter- section of the reflected rays; and the object by the point of intersection of the incident rays. If the reflected rays are divergent, they intersect behind the mirror, the image is virtual instead of real, and its distance from the mirror is — t* instead of the + V used when the image was real. If the incident rays converge, their intersection lies behind the mirror, the object is virtual (since there is no real intersection), and its distance from the mirror is — u. Procedure. The questions to be solved experimentally are, 1st, Does any given mirror change the curvature of all wave fronts incident upon it by the same amount? 2d, How are the conjugate focal distances, the principal focal distance, and the radius of curvature of the rhirror related? They involve separate determinations of R, F, and conjugate values of u and v. I. Concave Mirror, (a') To find F, obtain on a suitable screen a sharply focused image of some well-defined object so far away that the incident rays are practically parallel, and the wave front a plane. If out-door objects are not available, an incandescent lamp filament at a distance of 10 meters or more will give results within two or three per cent. Is the result too large or too small? Explain. {a") As a check, find the value of F by the following method. Place a scale Si (of cross section paper) on a support in front of the mirror, and a second one ^'2 (like Si) in contact with the face of the mirror. View 52 and the image of ^i through a small aperture in Si. Adjust the apparatus until the lines. of the image of ^i appear to coincide with those of S^; then the small hole in Si will be at the principal focus of the mirror. Show that the rays reflected into the eye at the small hole are parallel from Si to ^2 before reflection. (&') To find R, place an object (or source) and screen side by side on the axis of the mirror so that they are equidistant from the mirror and move together. Adjust them so that the image is focused on the screen; the distance from mirror to screen and object is the radius R of the mirror. A convenient object is a fine wire across a hole in a screen through which light passes to SPHERICAL MIRRORS 457 the mirror from the source, and is then reflected and focused on the screen. If a lamp filament is used as the object, adjust the screen in the vertical plane of the filament perpendicular to the length of the bar. Another good method of locating the center of curvature is to support a pin point in a vertical position in front of the mirror and adjust it so that the real inverted image of the point coincides with the pin point, and the points remain together when viewed from different directions by moving the head sidewise. (c') To find conjugate values of u and v adjust an incandescent lamp (having a single loop filament in a vertical plane) , a concave spherical mirror, and a suitable screen on one of the optical bars, and take bar readings for several pairs of conjugate focal dis- tances, having u> v for some pairs and u < .v for others. Tabu- late the data, and compute results indicated in the following Table. A slide rule or table of reciprocals is useful for finding reciprocals. TABLE CLIV Bar Readings Focal Dist. Reciprocals Results Mir. Obj. Im. u V l/u l/v l/u + llv F=.... R=.... \IF=.... 2IR=.... Average llu+lln=.... Compare the results and draw conclusions. Do the results indicate that l/u + l/v = 2/R = 1/F? What is the approximate fractional deviation from equality? Can you account for the deviations? Does the mirror change the curvature of all wave fronts by the same amount? State in words the relations of focal distances and the radius of curvature of the mirror. {d') If the object is nearer to the mirror than the principal focus, the reflected rays are divergent and appear to come from the virtual image behind the mirror. To find the distance - v from the mirror to the virtual image, locate the image by the method of parallax, as follows: Support a vertical thread as a plumb line in front of the mirror as an object. Place a longer plumb line as a marker behind the mirror where the image appears 458 LIGHT to be. Adjust it so that the parts of the line seen above and below the mirror remain in the same ^-ertical plane with the image when the eye is moved sidewise in front of the mirror so as to view the image from different directions. If the marker and image are at the same distance from the mirror, one will not appear to move relative to the other when viewed from different directions. If they are not at the same distance, there will be an apparent relative motion, the direction of which ■n'ill indicate which way to move the marker if the relative motion is properly interpreted. Measure v and u, and compute R and F by use of the formula in (c')- {d") Another method of locating a virtual image is as follows: Use a lamp filament as object making u < F. Place a converging lens so that the reflected divergent rays will pass through the lens and be focused on a screen. Note the position of the mirror and the lamp. Leave the lens and screen in position, but remove the mirror and move the lamp to the place where the virtual image appeared to be, and adjust the lamp to give a sharp image on the screen ; then the lamp will send out rays that diverge the same as the rays reflected from the mirror in the first case, and the distance from the mirror to the last position of the lamp is — V. From the values of u and v compute i? as a check on the value found in ( F. Convex incident wave fronts are so retarded in passing through the lens that the emerging wave fronts are concave and the rays converge to form the real image at their intersection. The change of curvature is 1/m — (— l/») = 1/m + 1/z;. For plane incident wave fronts (parallel rays) 1/m = 0, and v = F. In this case the change of curvature is — (— 1/F) = 1/F. If a lens changes the curvature of all wave fronts by the same amount, it follows that the change is 1/F in all cases, and 1/m -f 1/v = l/F. If 1/m + l/v is found by experiment to be a constant for a given lens, and if this constant equals l/F determined experimentally for parallel incident rays (and also for the converse case with parallel refracted rays), then the first part of equation (1) is verified and the conclusion may be drawn that the lens changes the curvature LENSES 461 of all wave fronts by the same amount. By the second part of Eq. (1) this amount is (w — 1) times the sum of the curvatures of the lens surfaces. This may be tested experimentally by using the surfaces as mirrors to obtain data for computing n and ^2 by one or more of the methods given in the experiment on mirrors. If the value of n, computed by use of the equation for both converging and diverging lenses, checks with the probable value found in tables, it serves to verify the equa- tion. R = Procedure. Convex Lenses, (a) Focal Length. -,-y^ Find the focal length of lens A as follows: Place if; the lens (Z,) on a mirror {M) under a needle point u» (P) as shown in Fig. 166. The needle projects from I; \ an arm attached to a sleeve {S) sliding on a vertical rod. Another arm from the upper end of the sleeve carries a- small reading glass (i?) so placed that it is '^^. — 777 — '" focused on P. If P is placed at the principal focus of the lens, the rays from P will pass through the lens and become parallel between the lens and the mirror both before reflection and after. The reflected rays again pass through the lens and are focused at P'. When P is adjusted so that P and P' lie precisely in the same horizontal plane and in apparent contact, there is no parallax when either the eye or R is moved sidewise. P is then at the principal focus of the lens. Turn R out of the way, set the zero end of a scale on the top of the lens frame, and read opposite P. Read again with the lens removed and the scale on the mirror. Turn the lens over and repeat the setting and measurements. From these readings find the focal length (P) of the lens, and whether the optical center of the lens is midway between the faces of the lens frame. Determine F from three or more inde- pendent sets of readings, each taken as directed above, and express the precision of F; also .locate the optical center if it is not in the center plane of the frame. {b) Conjugate Foci. Place Lens A in a holder on one of the horizontal light bars, and use as object a single loop lamp filament, the plane of the lamp being perpendicular to the length of the bar, and through the scale index or pointer. Observe and record 462 LIGHT scale readings for the positions of object, lens, and image. The values of u and v may then be found from the readings. Observe several pairs of object and image distances, beginning with u much larger than v. Decrease the distance u by suitable steps to u less than F, and locate the image for each position of the object. The data may be tabulated as follows: TABLE CLV Bar Readings Focal Dist. Reciprocala Remarks Obj. Lens Image u V l/« \lv llu + l/v Averages llu + l/v=.... l/F=:... When u is less than F the rays will still diverge after passing through the lens A , as though they came from some point I back of the lamp as shown in Fig. 167. To locate the virtual image Fig. 167. at I use an auxiliary lens B to focus the divergent rays from A on the screen S. The image formed by lens A at I then becomes the object of lens B, the position of which may be found by re- moving A (after reading Li and 0), and moving the lamp back to 0' so that a sharp image is again formed on S without moving either B or S. To make u negative, change Fig. 167 as follows: Interchange and S; then A receives converging incident rays, and bends them so that they form an image on 5' at O. The position of the virtual object (where the rays incident on A intersect) is located by removing A and moving 5 back until a sharp image is formed by B at 0'. These values of u and v should be included in Table- (c) Lens Graphics. The curve representing corresponding object and image distances, we will for convenience call the "lens characteristic." The pairs of distances are called the conjugate focal distances, and are expressed by u and v. The LENSES 463 focal distances found in (b) give the data for such a curve for lens^; but new data obtained with the non-achromatic con- verging lens C will be of more value in this and other parts of the experiment. For convenience in plotting make one of the dis- tances integral. For example, place the screen so that v is integral and less than u when u has a value about four or five times F; then focus sharply and read. Proceed in this manner increasing v by small amounts (say 3 to 5 cm.) until v equals or Y A M r ^~ — ~~' »n \ 70 1 JEhlS Graphics __ 1 — _ \ 1 fid 1 c H*RACT,ER JSTl.C u \ 1 \ oi^ «: m \ i \ CONVEHGINQ I^NS b \ 1 \ ^n \ 1 \ u _Y V a 1 fln \ \ T / ^ \\ , j / -- .^ ?n ~^ \ \j ' — , iL. -t£ — - — - -- % ■ — . — — — — — - - — -- -- -- — ■7- — — _ ^Ti --' -^ /' '1 ^ — . ^ ^ >- <. / / 1 \ \ \ ^ ___ ^ -- '' ,- ' ~> / / 1 1 \ \ ■^ ^ ■~- K 4 2 \ ■ V 4 6 9 1 ^0 If) A / I^ ^A iE D 31 Mi CE ^ 1 ?n 1 \ 1 1 ' 1 aa 1 1 N' ± Fig. 168. exceeds u; then set the lamp to give decreasing integral values of u that are the same as the values of v already used. The form of the characteristic when plotted is shown in Fig, 168 by the curve MQN, the ordinates of which are u, and the abscissas, v. The scales are alike, and a 45° line from the origin intersects the curve at the point Q, the coordinates of which are equal, and the distance between object and image is a minimum and equals 4F. To prove this, Let u' = v'; then 1/m' -f- \lv' = 1/F = 2/m' and u' = IF and u' -\- v' = ^F — min. dist. 464 LIGHT If lines are drawn joining each pair of values of u and v, they all intersect at the point P at the distance F from each axis. The characteristic is an hyperbola symmetrical about the line OPQ, and is asymptotic to PA and PA' . Note in the data for the characteristic that values of u and v are interchangeable in any pair, or positions of object and image are interchangeable. It then follows that F may be determined graphically by the intersection of the line for u and v and the line for u and v interchanged ; or if F and one of the distances are known, the other may be found graphically from the inter- cept on one axis made by a line through P from the known point on the other axis. When u = F, V = oc , and the line through P is parallel to the X axis. If u is less than F, the line through P intersects the axis to the left of the origin 0, and the — v decreases as the + u decreases, and both the real object and the virtual image approach the lens. If u is negative, v is positive for a converging lens, as shown by the left hand branch of the characteristic in Fig. 168. Proposition. If 1/x + l/y = Ijk = a constant, then a line from any point y on one axis to the corresponding point x on the other axis will pass through the point p at the distance k from each axis. (See Fig. 169.) This proposition may be proven as follows: From Fig. 169, it is evident that Area A Oyp + area A Oxp = Area A Oxy .-. iky + ^kx - iyx where each area is expressed by the product of half the altitude by the base. Dividing by ^kxy gives 1/x + l/y = l/k LENSES 465 In like manner it may be shown that for the line x', y' , p, \kx' - \ky' = \x'y', and l/y' + 1/ - x' = \jk. li X = y, each is equal to 2k, and p is located at the point having equal coordinates \x and \y. This point lies on the 45° line op at its intersection with the line between the' points X and y on the axes (not shown in the figure). The relation of the radii of lens surfaces to F and n is given by the equation l/n + l/>'2 = \IF{n - 1) = a constant for lenses of a given focal length and index. Therefore, in Fig. 170 k' = F{n - 1); and n' = 2jfe' when r^' = 'r4. ■>'. ?< \ LCN b ^ DII 1.1 N \ ri \ / 1 II > / / \. rt y \ \ ? ^ 1 |5 ■ 2 iz Fig. 170. li n = 1.52, the distance k' from C to each axis is 18 X .52 = 9.36, and each radius is 18.7 cm. if they are equal; but if one radius is 25 cm., the other is 15.0 cm. by the construction shown. Plot the data taken with lens C and find both graphically and algebraically the value of F, and the minimum distance between object and image for a sharp image. Measure the radii of curvature of the lens surfaces used as mirrors by the last of the three methods named in Exp. 1,-43, Part II, for Convex Mirrors. Using these radii, find graphically F{n — 1) and check it alge- braically. Using F already obtained, compute n and check its value from tables. Concave Lenses. If parallel rays pass to the right through a lens having a thin center and thick edges, the emerging rays diverge on the right as though they came from a point on the 466 LIGHT left of the lens. From the lens to this point is the principal focal distance (F). Since the rays do not actually intersect at this point, the focus is virtual. All images of a real object are virtual for diverging lenses ; but it is possible to have a real image when the object is virtual due to convergent incident rays from some other lens or mirror provided u is less than F. This fact is used to obtain pairs of conjugate foci by use of an aux- iliary converging lens as follows ; Use the apparatus shown in Fig. 166; and use lens A as the auxiliary lens to find pairs of con- jugate foci for the single diverging lens D. The data gives the position of P at the principal focus of A. Raise the pointer a known distance (say 10 cm.) at first to P". Support D on an arm above A , aiid raise or lower D until a sharply focused real image P'" appears in the plane of P". Adjust to bring the points together so that there is no displacement due to parallax when the line of sight is changed. Measure from the top of the mirror to the optical center, 0, of the lens D; and find the dis- tance from to P"and to P. For the rays downward from P", OP" = u' and — OP = v'\ and for the reflected rays upward - OP' = u" and OP'" = v" . Note that divergent downward rays from P" (real object for £>) are more divergent after passing through D as though they came from P, the virtual image for Z>, and the principal focus for A. The downward rays strike the mirror normally and after reflection return along the same paths to P' and P'" (coincident with P"). A real image is formed at P'" . This image is the conjugate of the virtual object at P' . Change the distance between P and P" to a larger value (say 20 cm.) and repeat measurements for pairs of conjugate focal distances. Use the observed data for u and v to locate p' (in the third quadrant), and determine F from the graphical construction. Check by algebraic computation. Plot the four points on the characteristic given by the observed data. Choose other values of M, both positive and negative, and find graphically the corre- sponding values of v and plot points to locate both branches of the characteristic of D. This characteristic will resemble Fig. 168 if the main part of Fig. 168 is rotated 180° about without rotating the main axes through 0. LENSES 467 Note that no part of the characteristic for D lies in the first quadrant, and no part of the characteristic of A Hes in the third quadrant. The characteristics are in each case asymptotic to lines through the focal points {p' and p) and parallel to the axes. The focal distance F is negative for D, and the focal point p' lies in the third quadrant. Measure the radii of D, using the fages as concave mirrors, and compute n by use of the lens equation. Compare the values of n obtained for lenses C and D with each other, and with table values. Lens Combinations. If light is transmitted successively through several lenses, the image due to one lens becomes the object for the next lens, so that the final image for the combina- tion may be located by tracing stepwise the image position for each successive lens. The direct application of this method to optical instruments is left for a separate experiment. The illustration given in Fig. 171 is for the purpose of showing the steps in the graphical method of dealing with lens combina- n r 1 sa — G \/\ PH cs F Ie NS COMBIN /^T IONS •03 n E- O" w ■J «<: \ (I'uusiration)' 1 w \ V, J)a to. K=20.'f,= -I8., F3y3< ?-,F4=r2 0. \ L.atO.j L^atzo., ILj. ^,/l \ S y > • / ?.n "1 \ i L OC ate J + / / / U4 \ / / in \ \ / / / / / \ J / / f 4 h % \ ^ A / L-t S ■►» n >. f / \ r^ / J3- , ''VI4 "7 -in 1 V 2 V 'd 1 0- -i 1 6 .'7 , fl y 9 ^ I — "" J / / / / -?n J" POSITIONS OF L fd s^ r'AND 1 rK aE< ' ■*2 ^ 1 / . \ ■3(1 I / y Fie 171. 468 LIGHT tions by extending the graphics for single lenses (shown in Fig. 168) to combinations. The first lens (a converging lens) at produces a real image at /i, when the object is 50 cm. to the left of 0. The second is a diverging, lens at the 20 cm. point so that 7i is a virtual object for this second lens ; and Mz is plotted down- ward to the point where a 45° line from I\ meets it. A line from F2 through this point gives the second image position at /a where the line intersects the image axis. Note that F2 is plotted in the third quadrant relative to the lens point at L^ because this is a diverging lens. The real image h in turn becomes the virtual object for L3 at the 40 cm. point. This is a converging lens; hence 7^3 is plotted in the first quadrant on a 45° line from L3. A line from F^ to the lower end of M3 (constructed the same as M2) locates /j another real image which becomes the real object of i4 at 90 cm . The obj ect being real M4 is plotted upward . A line to Fi locates the final image of the combination at I^. Its posi- tion at the left of L^ sho^s that the image is virtual, and that the emerging rays are divergent toward the right as though they came from I^. Use the lenses provided for the experiment to give a combina- tion transmitting light from left to right, ending with a real image, and including at least one virtual image. Choose the object and lens positions and construct the diagram from the known values of F before the image positions are tested experi- mentally. Then determine by trial the position of each image before the next lens is put in place, in order to test and verify the method of construction by which their locations were predicted. Note that positive values of u (distances to left of lens) are plotted upward, and negative values (distances to right of lens) are plotted downward from their respective lens points. Focal lengths are plotted in the first quadrant, relative to their respec- tive lens points, for converging lenses; and in the third quadrant for diverging lenses. The point where a line joining the end of a given u and the corresponding F intersects the horizontal axis is the image position for that lens. The image is real when it falls to the right of the lens, and negative when it falls to the left. The length of the vertical line u is limited by a 45° line through the object point on the horizontal axis. The magnification of li relative to the object Oi is. LENSES 469 lilOx = {vilui){v-,lui){vilu^{vi!u^ = .286 The data and results may be tabulated as shown in Table CLVI. TABLE CLVI Lens No. Lens at Kind of Lens Focal Length Object Dist. (u) Image Dist. (v) 33.3 +54.0 + 16.0 -12.6 Location ol Image Magnif'n v/u i. L-2 U cm. 20 cm. 40 cm. 90 cm. ' D.Cx. D.Cv. Pl.Cx. Pl.Cv. +20.0 -18.0 +30.0 -20.0 50.0 -13.3 -34.0 +34.0 /i at 33.3 h at 74.0 Is at 56.0 I, at 77.4 .667 2.46 .470 .370 Principal Focal Length and Optical Center of a Combination of Lenses. If several lenses are combined, as in a camera or lantern objective, the focal length of the equivalent single lens may be found as follows: Place a lamp as object at a convenient known distance I from a screen. Adjust the lantern objective between the lamp and screen so as to focus sharply with u > v. Then move the objective to the second position for a sharp focus with u ') from alignment with the collimator. (d) Parallel Rays. The rays from the collimator through the prism to the telescope should be parallel rays. These rays will be parallel if the slit is at the principal focus of the collimator lens ; and the telescope will give a sharp image of the slit if focused for parallel rays. A convenient method of adjustment is based upon the fact that the monochromatic image viewed in the telescope, when focused for minimum deviation, will remain focused for greater deviations if the rays are parallel through the prism, but will be blurred -if the rays are not parallel. Turn the telescope so as to increase the angle D' by about three degrees. It may then be necessary to turn the prism table to bring the image into the field of view. Let i be the angle of incidence when the prism is set for minimum deviation. If the prism is turned so that i is increased to i +, focus the blurred SPECTROMETER 473 mage by adjusting the telescope. If i is decreased toi -, focus by adjusting the collimator. Alternate these adjustments until the image remains sharply focused in both positiohs. The telescope and collimator are then both adjusted for parallel rays. Keep this adjustment of the telescope throughout the experiment. If the order of adjustments for focusing is wrong, the blurring due to the change from i -\- to i — and vice versa, will be in- creased instead of decreased. (e) The Prism. Final adjustment. The faces of the prism should be parallel to the vertical axis. Three small levelling screws are provided under the prism table to make this adjustment. Level the table approximately by use of the small level. Center the prism on the table with an angle over each screw. Turn the telescope around near to the collimator (say 30° or 40° from it). Turn the collimator slit to the horizontal position and note the position of its image when rays from the collimator are reflected from one face {AB) of the prism into the telescope. Then turn the prism 180°, in which case the light is refracted on entering {BC) and is then reflected from the surface AB, and passes out through AC to the telescope. (Draw figures.) If the images are not at the same level, correct half of the deviation by turning the screw under the angle C. Proceed in like manner with each of the other faces. (f ) Data for Index of Refraction. The final data for minimum deviation should be very carefully observed for deviation first in one direction, then in the other, for the same refracting angle of the prism. Specify the faces used. Read the settings to minutes, or less, by means of the micrometer microscopes (or verniers) provided. The value of D is half the angular distance the telescope is turned from D in one direction to D in the other direction. Check by taking a reading with the prism removed and the telescope opposite the collimator. Measure very carefully the angle of the prism between the faces used. Check by measuring all of the angles to find the deviation of their sum from 180°. Two methods of measuring the angles are given in Fig. 172 (a) and {b). Rays from the colli- mator are indicated by C and to the telescope by T. In Fig. 172 (a), T and C remain fixed and the prism is turned. The line N 474 LIGHT normal to face AB is midway between T and C, when the setting for reflection from AB is read. Readings are then taken when N' and N" are successively brought to the position of N. End by repeating readings for iV as a check. The angles may then be determined from the angles between the normals. In Fig. 172 (b) the prism edge is set at the center, and readings Fig. 172. are taken for the telescope settings T and T' with the prism and collimator fixed. Repeat for each of the other angles, and check the results. The index of refraction (») is found from the equation, n = sin ^{A + Z')/sin § A in which A is the angle of the prism between the faces used, and D is the minimum deviation. Test Questions. 1. What is the mode of procedure in focusing the eye- piece on the cross-wires to insure the best position of the virtual image at the distance of distinct vision? If the eye lens is moved toward the cross-wires, which way will the image move? 2. Name the essential adjustments of axes. How are they made? 3. What procedure was adopted in using the micrometer microscopes? Is there any advantage in starting with the reading when set on the zero line of the pointer? If the zero reading is 10 and the reading on a degree line is 50, is that sufficient data to find the number of minutes between the points read? Explain. 4. Is the image seen in the micrometer microscope erect or inverted? How determined? What precautions must be used in reading the scale? 5. Show how the readings for each part of Fig. 172 are related to A. 6. If the monochromatic light used were blue instead of yellow, would the index be larger or smaller? Explain. 7. The value of the index indicates what kind of glass? Compare the refractive and dispersive powers of crown and flint glass as given in tables. optical instruments 475 experiment l-46 Study of the Essential Parts of Optical Instruments The Telescope, Field Glass, and Microscope The common optical instruments have two essential parts, viz., the lens toward the object, called the objective, and the lens next to the eye, called the eyepiece. The objective is a converging lens used to form a real image of the object near the eyepiece. The eyepiece in turn gives a magnified virtual image at the dis- tance of most distinct vision when properly focused. The Telescope. In the telescope the real image formed by the objective is quite small ; but the large virtual image is near enough to be clearly seen, and subtends a larger angle at the eye than the distant object. The ratio of the image angle to the smaller object angle measures the apparent magnification. If a distant scale, or uniformly divided object, is viewed directly with one eye and through the telescope with the other, the angles may be compared by noting the number of divisions on the object that are covered by each division of the image. Select suitable lenses from those provided and construct a telescope. Test its magnification and draw a diagram showing the relative positions of object and image for each lens from observed data. The Field Glass, or Galilean Telescope. The eye piece is a diverging lens placed so as to receive the convergent rays from the objective and make them divergent as they emerge and pass on into the eye as though they came from the enlarged virtual image at the distance of most distinct vision (about 10 or 12 inches) . Construct and test a telescope of this form and draw a diagram from measurements observed. The Compound Microscope. Select for an objective a small lens having a short focal length. Adjust to give a magnified real image of some suitable small object. Select a larger lens (or combination) of short focal length to use as an eyepiece, and adjust it to give a magnified virtual image at the distance of most distinct vision. Find the approximate magnification due 476 LIGHT to each lens, and also the ratio of corresponding dimensions of the second image and the original object. Draw a diagram and explain. See question 4. Test Questions. 1. If the focal length of the objective of a telescope is 10 inches, and of the eyepiece 1 inch, how far apart must they be placed to give a virtual image 12 inches from the eye lens, the object being 100 feet away? Show that the di- mensions of the virtual image are approximately one-tenth of those of the object, and that the angle subtended by the image is approximately ten times the angle subtended by the object. 2. Does the apparent magnification depend upon the distance of the object? Does the ratio of the image angle to the object angle increase or decrease as the object distance is decreased? Explain. 3. Answer question 1 for the field glass changing the focal lengths of the lenses to 8 inches and —2 inches respectively. Note that the focal length of the eye lens is negative and the image formed by the objective becomes the virtual object for the eye lens. In this case u, v, and / are all negative in the lens equation 4. Describe and show by diagram the divergence and con- vergence of rays from a single luminous point on the object, through the compound microscope, to the eye. Also describe and show by diagram the relative positions and intersections of secondary axes from two points, widely separated, on the object to the corresponding points on the image. Note carefully the differences between these two cases. experiment l-47 Study of the Compound Microscope Optical Principles, and Adjustments of Accessories The printed matter and references provided with the apparatus are to be studied together with the microscope and accessories. The principal topics to be considered are the following: 1. Precautions, — about handling, care of parts, and adjust- ments. OPTICAL INSTRUMENTS 477 2. Objectives, — focal length, relative resolving power, dry systems or immersion lenses. 3. Eyepieces, — relative focal lengths, positive or negative, effect on magnification. 4. Illumination, — reflection from upper surface of object, or transmission from below; axial and oblique transmission; when to use plane mirror, concave mirror, condenser, and diaphragms. 5. Magnifying power, — use of the camera lucida and microm- eter to test the magnifications • of various combinations of objectives and eyepieces. experiment l-48 Study of the Projecting Lantern Adjustjients of Arc, Condenser, Slides, and Lens Consider carefully the topics indicated in the outline below, using your knowledge of the thelory involved as a basis for drawing conclusions. Where possible test your applied theiory by experi- ment, and note your conclusions. Proceed with the adjustments in the order given below. 1. Adjustment of arc for maximum light toward slide, (a) Crater where? What relative positions of points? Why? {b) What length of arc? Why? 2. Adjustment of condensers for maximum light through objec- tive, (fl) Why two lenses in condenser instead of one? (b) Why use plano-convex lenses? Why others? (c) Why place convex surfaces toward each other? Is the distance between lenses important? (d) What considerations would have a bearing on the desirable focal length of the condenser lenses? (e) What are the chief conditions that determine the best position of the condenser? 3. Adjustment of slide for uniform illumination, (a) How may the defect be remedied when the corners are not properly illuminated? (&) Why should the slide be placed so that the printing is bottom side up, and reversed sidewise when you look through it toward the screen? 4. Adjustment of the objective for a sharp image on the screen. (a) Why is the objective usually constructed with a pair of lenses 478 LIGHT some distance apart, each of which consists of two lenses in contact, one thicker at the edges, and the other thinner? (&) Why is it desirable to have as much light as possible go through the center portion of each lens, and to avoid using the edges? (c) What should be the relative position of the lens and the small- est section of the "neck" between the cones of light having their bases on the condenser and screen respectively? Why? 5. Image and illumination on the screen, (a) Indications of causes of defective illumination and how to remedy defects. (&) Trace the divergence and convergence of rays, from a point on the arc to the image of the arc when the objective is removed. Where is the image of the arc? Why is it not on the screen? experiment l-49 Calibration of a Diffraction Grating Measurement of Wave-length Determine the number of lines per centimeter (also per inch) for the grating assigned, by the use of sodium light, the wave length of which is known. Then use the calibrated grating to determine the wave length of some line (or lines) in the bright line spectrum of some unknown substance to be identified. See Experiment 127 in Practical Physics, by Franklin, Crawford, and MacNutt. experiment l-50 Calibration and Use of a Spectroscope Bright Line Spectra of Metals Follow the outline and directions given in Laboratory Physics, by Dayton C. Miller, Experiment CXVIII, pp. 271-275. /f Z ^>L )^<^''-y'^ ic'l-bJ^