ALBHRT R. MANN LIBRARY AT ( ORNELL UNIVERSITY Cornell University Library QC 35.B63 Experiments in physics for students of 3 1924 002 964 066 The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002964066 Experiments in Physics FOR STUDENTS OF SCIENCE BY ERNEST BLAKER Assistant Peofessor of Physics IN Cornell University Printed expressly for students in a first year laboratory course in general physics in Cornell University J* S/f CARPENTER & COMPANY ITHACA, NEW YORK 1914 Copyrighted 1905, 1909, 1914 BY ERNEST BLAKER First edition, Sept. 1905, second edition, revised and rewritten July 1909 ^j^/ Reprinted Sept. 1910 ''mM"'*'^'^ /, .V-- Revised Sept. 1914 ..'•^r^ ""''''£J^' nECHRXilk PREFACE TO FIRST EDITION The collection of experiments printed in this book has been arranged and modified to meet the demands of a par- ticular elementary course in laboratory physics at Cornell University and therefore not to meet the requirements elsewhere. The laboratory course for which this manual is intended as a hand-book supplements a lecture course and also a recitation course in general physics in which the text used serves as a general reference for the theory on which the experiments are based. For the most part the experiments, selected from many sources, have been used in the work for several years. The directions being type written or mimeographed have been changed from time to time as was deemed necessary. Graphical as well as numerical methods are insisted on. Occasional notes and explanations are given, and forms are many times suggested for tabulating data and results. In some cases no suggestions for tabulation are given in order that the student may show what he can do for himself. At the end of the book will be found a hst of tables of physical constants, useful numbers and natural trigonomet- ric functions. I take this occasion to thank the many who have aided in one way or another to make it possible to print the book and especially to give thanks to Professor H. J. Rogers of Leland Stanford University who first developed the course, and to Mr. Willard J. Fisher, instructor in Physics at Cor- nell University, for many valuable suggestions and aid in getting out the work. Ernest Blakee. PREFACE TO SECOND EDITION The writers have taken advantage of the printing of the second edition of this manual to correct errors of the first edition and to include a few experiments that have been added to our Hst. The laboratory experience of the time elapsing since the first edition has also resulted in the revi- sion of many experiments, in some cases the revision result- ing in almost a complete rewriting. Mr. Willard J. Fisher, Instructor in Physics in Cornell University, joins in the general revision of the book. To Mr. H. G. Dorsey, Instructor in Physics in Cornell University, the undersigned is very grateful for many helpful suggestions and for rewriting some of the experiments. Ernest Blakek. July 15, 1909. PREFACE TO THIRD EDITION Advantage has been taken of the necessity to print another edition of this book to make some corrections, to revise many of the experiments and to add others. This has made desirable the rearrangement of the order of some of the experiments and to a renumbering of many of them. It is a pleasure to acknowledge the value of suggestions of Mr. E. C. Mayer and Mr. C. E. Power in connection with the revision. Ernest Blaker. Sept. 8, 1914. TABLE OF CONTENTS Introduction 9 Pabt I — Mechanics EXPERIMENT PAGE 1 Comparison of the Inch and Centimeter 17 2 CaUbration of Instruments 18 3 Determination of the Gauge of Wire 20 4 Density Determination from Mass and Volume 22 5 Study of Uniformly Accelerated Motion 24 6 Determination of Spring Constant and "g" Simple Harmonic Motion 28 7 Relations of Force, Mass and Acceleration. Newton's Laws _3g^,- 8 Equilibrium of Concurrent Forces 36 9 Equilibrium of a Rigid Body; Forces and Moments 37 10 Wheel and Axle 40 11 The Pulley and Pulley Systems 42 12 IncUned Plane 45 13 The Balance. Weighing by Vibrations 46 14 Friction, Coefficient of 49 15 Work and Power. Dynamometer 50 16 Determination of "g" by "Free fall" 52 17 The Simple Pendulum 53 18 Uniform Circular Motion and Acceleration Toward the Center 55 • — 20 Boyle's Law 60 21 Density of Liquids. Hare's Method 63 22 Specific Gravity. Hydrometers 64 24 Specific Gravity of Solids not Soluble in Water 65 25 Specific Gravity. Jolly Balance 66 26 Angle of Contact of Mercury and Glass 68 27 Surface Tension 69 28 Capillarity and Surface Tension. Use of Cathetometer . . . 71 29 Young's Modulus. Tension 75 30 Young's Modulus. Flexure 78 31 Torsional Rigidity 80 Part II — Heat 40 Comparison of Thermometers 83 42 Coefficient of Expansion of Air at Constant Pressure 85 43 Linear Expansion of Metals 87 Notes on Calorimetry 89 6 TABLE OF CONTENTS EXPERIMENT PAGE 45 Water Equivalent of a Calorimeter and Specific Heat of Copper 91 46 Specific Heat of Metals 91 47 Heat of Fusion of Ice 93 48 Heat of Vaporization of Water 95 49 Hygrometry and Dew Point Determination 96 Part III — Sound 60 Experiments in Wave Motion 100 61 A Study of Vibrating Strings. Melde's Experiment 102 62 Determination of the Frequency of a Tuning-Fork by Means of a Vibrating String 104 63 Velocity of Sound in Air. Resonance Tubes 106 64 Velocity of Sound in Solids. Kundt's Experiment 109 65 A Study of Beats and Interference of Sound 112 66 Determination of the Frequency of a Tuning Fork 113 Part IV — Light 70 Photometry. The Bunsen Photometer 116 71 The Laws of Reflection 118 72 Determination of the Angles of a Prism. Pin Method. ... 119 73 Determination of the Index of Refraction of a SoUd and a Liquid 121 74 Determination of the Index of Refraction of a Prism 123 75 Measurements of Angles of a Prism using the Spectrometer 126 76 To Determine the Focal Length and Radius of Curvature of a Concave Mirror 128 77 Focal Length of Convex Lenses. Parallel Ray Method. . . 130 78 Focal Length of Convex Lenses . Ob j ect and Image at Finite Distances from the Lens 132 79 Focal Length of a Convex Lens. Changing Position of Lens 134 80 Focal Length of a Concave Lens. Method of Divergent Rays 136 81 Focal Length of a Concave Lens. Auxiliary Lens Method 138 82 Simple Microscope Magnifying Power 139 83 Compound Microscope Magnifying Power 141 84 Astronomical Telescope Magnifying Power 142 85 Terrestrial Telescope Magnifying Power 143 86 Wave Length of Light by Diffraction Grating 144 TABLE OF CONTENTS 7 Part V — Elbctkicity and Magnetism EXPERIMENT PAGE 90 Electrostatic Fields 147 91 The Gold-Leaf Electroscope 148 92 Faraday's Ice Pail Experiment 149 93 Holz Electric Machine 151 95 Magnetic Figures with Iron FUings 152 96 Determination of the Pole Strength of Two Magnets 153 97 Determination of the Moment and Pole Strength of a Magnet 155 98 Determination of the Product of MH 157 99 Determination of the Ratio M/H 159 100 Tangent Galvanometer 1 162 101 Tangent Galvanometer II 166 102 CaUbration of a d'Arsonval Galvanometer 168 108 Ohm's Law for a Simple Circuit 171 104 Determination of tM E. M. F. of a Battery 173 105 Measurement of Resistance by Comparison 174 106 The Slide-Wire Bridge 176 107 The Wheatstone Bridge, Box Pattern 178 108 Determination of the Resistance of a High Resistance Galvanometer 179 109 Resistance of a Cell by the Half-Deflection Method 180 110 Potential Variation between Generator Terminals 181 111 Ohm's Law for a Series Circuit. Potential Difference Measurements 184 112 Comparisons of E. M. F's. of Batteries by Potentiometer Method 191 113 Electrolysis 193 114 Measurement of the Capacity of a Condenser 195 116 Study of a Magneto Machine 198 O TABLE OF CONTENTS Part VI — Tables NO. PAGE 1 Some Useful Numbers 203 2 Work and Power 203 3 Density of Some Substances 204 4 Coefficients of Frictions 204 5 Elastic Constants 205 6 Heat Constants of Solids 206 7 Velocity of Sound 205 8 Vapor Pressure and Hygrometry 206 9 Index of Refraction for Sodium Light 207 10 Electro-Chemical Equivalents 207 11 Specific Resistances and Temperature Coefficients 207 12 Electromotive Forces of Cells 208 13 Wire Gauge -and Copper Resistance Table 208 14 Natural Trigonometric Functions 209 INTRODUCTION The object of the experiments given in this manual is not only to impress more fully upon the mind of the stu- dent the fundamental physical principles involved, but also to teach systematic methods of observation and analysis. The work being of necessity in general quantitative in nature, in order to bring out more completely its qualita- tive aspect requires some elementary mathematics. It is to be borne in mind however, that the mathematics is used only as a means to an end and while its correct use is es- sential to correct analysis, its importance should always be kept secondary to the physical principles involved. In the following paragraphs will be found a few rules of the laboratory and some general hints as to methods of doing the laboratory work and writing reports. It is expected that every student will be prompt and regular in attendance. Excused absences may be made up at periods to be arranged for by the student with his in- structor, and no reduction in grade will be made on ac- count of such excused absence. The notes in the manual should be carefully read and the topics to be investigated should be studied in some standard text book. There will be found in the laboraroty a list of references for the various experiments to one or more standard text books, this method of giving reference being more flexible than printing lists in the manual in connection with each experiment. It is well to be familiar with the notes and references on two or three 'experiments in advance as it frequently is not convenient to assign ex- periments in regular order. The experiments are arranged in groups. In general each student is to perform one experiment from each group. Each student is held responsible for all the apparatus he is using whether working alone or with a partner. Ap- 10 INTRODUCTION paratus taken from the apparatus room will be charged to the student when drawn out and he will be credited with it when returned. In performing experiments and making computations students working together should share alike, but all other work should be strictly individual. OBSERVATIONS. — ^All original observations are to be recorded in the note book at the time of performing the experimental work, in such a manner as to present a neat appearance and to follow some logical order which may be easily followed by one having a knowledge of the require- ments of the experiment. In many cases some such order will be suggested in the manual. Errors of judgment as well as mistakes are always to be carefully guarded against, and previous observations and preconceived notions must not be allowed to bias one's ob- servations. All observations are to be made with care and are to be recorded exactly as made, even when apparently wrong. When an observation is known to be wrong it may be omitted from the computations, reasons being given. In measuring quantities which require scale readings, it generally happens that the readings to be taken will not fall on marked divisions. It is then necessary to estimate tenths of divisions, as in estimating tenths of degrees on a thermometer graduated in degrees only. The smaller the quantity observed the greater should be the care used in making the observations. The percentage error, that is, the ratio of the absolute error to the value of the quantity measured, increases with a decrease of quantity measured. The accuracy of the result depends on the percentage error. As an example, suppose the volume of a long slender wire is -to be measured. Y=nr^l. Since r, the radius, is supposed to be small compared with I, the length of the INTBODUCTION 11 wire, the accuracy of the result depends more on an ac- curate measurement of r than of I. In general one observation of a quantity is not sufficient. COMPUTATIONS.— Simple numerical errors are of fre- quent occurrence and are to be carefully guarded against- Reports containing numerical errors will not be accepted- In making computations the physics of the experiment must not be lost sight of. The slide rule is a very useful instrument for making computations such as multiplication, division, finding powers and extracting square roots, and in most cases the results are sufficiently accurate. It has the great advantage of being a great time and brain saver. A little careful practice will soon make it possible to carry out computa- tions with an error of much less than one per cent- Do not carry out computations to a degree unwarranted by the accuracy of the observations. Observational errors will usually be not less than 1% and may sometimes be as great as 5%, therefore it is generally not necessary to carry computations beyond the third significant figure, irrespec- tive of the position of the decimal point. Zeros must often be used as significant figures. If a weighing be made that is equal to seventeen grams with an error in tenths of milli- grams it should be indicated thus 17.000. If the weighing is correct to hundredths of grams it should be written 17.0. If a certain computation has twelve places to the left of the decimal point only three of which are accurate the result is not to be expressed as 217384950607 nor as 217,000,000,000 but as 217x10' or 21.7x10"'. It is often convenient to ex- press such qualities as .000328 in the following form, 328x10"*. This method of writing results has the merit of brevity, but its greatest utility is in indicating the accuracy of the work. The whole number with the power of ten in- 12 INTRODUCTION dicates the magnitude. The whole number without the power of ten indicates the accuracy. Carry divisions out as far as the corresponding multipli- cations would be carried. Several independent determina- tions of the same quantity will usually differ. If this dif- ference occurs beyond the third significant figure com- putations should be carried out until the difference does appear. The percentage deviation of a few of the inde- pendent determinations from the mean should be found. REPORTS. — Always begin a report at the top of a left- hand page, so that at least two pages of the report may face each other. Head the report with a title and the date. Do not crowd the data or other matter, or mix portions of different reports. Try to give the report as neat an appear- ance as possible. An example of a page heading is given below : Group 8, Expt. .17. The Simple Pendulum. Apr. 20, 1905. Before leaving the laboratory get the instructor in charge of your section to put the laboratory stamp in the blank space at the left of the heading. Leave the top of the right-hand page blank, for the instructor's use in indicating corrections, etc. Also leave page 1 blank for similar use. Do not erase correction marks. Give definitions of physical terms used for the first time, and a full statement of new laws as they come up. Include all equations used and give indicated numerical substitutions in them. Put in the reports explanatory diagrams; and always give complete diagrams of electrical connections exactly as made. Whenever possible graphical representa- tions of results should be made. Finally, state clearly what facts have been verified, and give a clear formal conclusion to the report, showing what principles the experiment INTRODUCTION 13 illustrates. Index all reports on the first page of the note- book. CURVE PLOTTING.*— The following brief rules and accompanying diagram will illustrate the process. 1. Never use less than half of a page. In many cases a full page is advisable. 2. Draw a horizontal base line and erect near the left hand edge of the paper a vertical line intersecting it. These ^782- ■jo 8 3^ ■fSO Me- SJCZ p. a' .rnl — f yt ! !i«, HLft y 5^ ^im: /Aoo /4D0 ISoc 03 o 380 a /4 3 O l> /O Z40 1/ 5 8g« r- P res^ Boy are le's i a w ;o SO 30 :?o 5o So TO So qo Too 775 lio Pressure in Centimeters of Mercury Figure 1 lines, the axes, serve for guides. Choose such scales for plotting the variables that the largest values of them will be near the right hand and upper edges of the plot, using scales that may easily be read, such as one division for one. * Consult standard texts on "Graphical Representation" of various phenomena, e.g., Watson's General Physics §4. 14 INTRODUCTION two, five or ten or like sub-multiple units. More than one curve may be plotted on the same sheet. 3. Put the scales along the axes together with the names of the quantities plotted. The arbitrary variable should generally be plotted along the horizontal or X axis. 4. Points indicating the experimental values to be plotted should be marked plainly by means of crosses, dots or dots surrounded by circles. 5. A smooth curve is then to be drawn so that it may best fit the observed points but not necessarily passing through them. This curve is to represent the continuous changes of the variables plotted. The deviation of points form the curve usually indicates errors of observation. 6. Give a title to each curve which usually will call attention to the variables plotted and the conditions under which the particular values were obtained. Study carefully the preceding example of curve plotting. EXAMINATIONS : A written examination, covering all phases of the work in the laboratory, will be held at the end of each term and questions may be asked about any experiment performed at any time during the course. The grade of each student will depend on the amount of experimental work successfully performed, the neatness, arrangement and completeness of data and reports and final examinations. Students electing more than one hour per term may be required to take an oral examination as well as the pre- scribed written examination. NOTES ON UNITS.— The measurement of physical qualities always involves a unit upon which the measure- ment is based. It is said, for example, that a certain line Note. Consult also some standard text on units. INTRODUCTION 16 is twenty-five centimeters long. This involves the idea of a standard of comparison, and the standard is the centi- meter, a unit of length. The fundamental units of measurement in the system generally employed in physics are called the centimeter, the unit of length; the gram, the unit of mass; and the second, the unit of time. This system is called the C.G.S. system. For some purposes the fundamental units are not well adapted and other units which are multiples or sub- multiples of the fundamental units are used. The second- ary units are called derived units. Another system of absolute units is called the foot- pound-second system, in which the units of length, mass and time are the foot, pound, and second respectively. The unit of force based on this system is called the poundal, and is that force which will produce in a pound mass an acceleration of one foot per second for every second the force acts. Among engineers so called gravitational systems of units are in general use in which a force unit takes the place of the unit of mass. In the English system the weight of a pound mass, about 32 poundals, is taken as the imit of force. The unit of force is commonly called a pound which often leads to confusing it with the unit of a mass of the F.P.S. system on which it is based. In the engineering system the unit of mass, is 32.2 pounds. According to Newton's second law of motion the. ac- celeration produced in a body is directly proportional to the force acting on the body and inversely proportional to the mass of the body. If the mass of a body be m and its acceleration a, then the force producing the change of motion must be / = ma from the above law. Now if this force be that due to gravi- tation then / = mg where "g" is the acceleration due to 16 INTRODUCTION gravity. The force with which a body is attracted by the earth is called the weight of the body, that is the weight of a body is a particular value of a force; W = mg. From the above discussion it is seen that the ratio be- tween the foot-pound-second system and the gravitational system is g, the acceleration due to gravity. The mass of a body is constant everywhere, but its weight varies from place to place. Part I MECHANICS Experiment 1 The Estimation or Tenths A Comparison of the Inch and Centimeter In engineering and other scientific work it is often de- sirable and sometimes necessary for accuracy of measure- ment to estimate scale readings of tenths of the smallest divisions into which the scale is divided. Practice and good judgment are essential to good work. The object of this experiment is to make some trials of estimating tenths and to find the relation of the inch to the centimeter. The distance between two points is to be measured in centimeters and inches. Place a scale graduated in millimeters parallel with a line joining the two points. Do not place the end of the scale at either point. Read the position of the two points on the scale, taking care that the parallel be as small as possible. In general if a division mark of the scale be op- posite one point the other point will be between division marks. Estimate the position of the second point to tenths of the smallest division; that is, to tenths of millimeters. Measure the distance between the points several times using different parts of the scale as starting points. Make one end of the line correspond to division marks in one set of readings, the other end of the line to correspond to divi- sion marks in another set, and neither end correspond to division marks in a third set. In all cases put reading in the note book as soon as made. From these readings the length of the line is to be computed. 18 MECHANICS Then using the inch scale find the length of the line in the same manner, estimating tenths of the smallest sub- division of the inch. Find the mean of the length of the line in centimeters and in inches and compute the number of inches to the centimeter, and also the number of centimeters per inch. Referring to the text, find the errors of the results. What are the sources of error in the determination and how may they be reduced to a minimum ? Tabulate results as follows : Using Metric Scale Readings on Readings on Distance Point A. Point B. m cm. 27.50 43.78 16.28 32.23 48.49 16.26 38.81 55.10 16.29 41.17 57.41 16.24 53.49 69.76 16.27 Mean length in cm. = Using Foot Scale Readings on Point A. Readings on Point B. Distances in inches Mean length in inches = Experiment 2 Calibbation of Instruments In scientific work where accuracy is to be attained the MECHANICS 19 measuring instruments used must be studied and their errors investigated in order that the readings may be re- duced to as nearly the true value as possible. These errors may be due to periodic or constant errors in graduation or to both, or to the error of the zero point if that be always a point of reference. For ease and rapidity of reduction of the readings to the approximately true values of the quantity measured a calibration curve is plotted. The object of this experiment is to calibrate a spring balance, graduated in pounds, in terms of kilograms. The spring balance gives a measure of the force with which gravity pulls on a given mass. The reading will vary from place to place on the earth's surface owing to the variation in the acceleration of gravity. Suspend a spring balance at such a height that the pointer may be on a level with the eyes of the observer for any reading. First, read carefully the position of the indicator on the graduated scale when there is no load. Take readings of the indicator when 1 Kg. is added, and for every additional kilogram until the limit of the scale is reached, if it be a twelve pound scale. If the scale be a twenty-five pound scale add two kilograms each time. Reduce errors due to parallax to a minimum by keeping the eye on a level with the indicator when making readings. For each mass added make at least two readings. First lift the mass so that the pointer is raised two or three divi- sions, then carefully decrease the upward pressure to zero and take a reading. Next, exert a pressure from above and depress the pointer two or three divisions, then slowly reduce the pressure to zero and read. Correct the mean readings for the constant zero error. 20 MECHANICS if there be one, and compute the value of the kilogram in pounds. Find the error of the mean value of the kilogram. Tabulate observations as follows : Reading of indicator for no load, zero error = Load in Kgs. Scale Mean Readings Readings Mean reading corrected for zero error Pounds per Kg. Mean value of pounds per Kg. = Plot a calibration curve using values of load for ab- scissas and mean scale readings (column 3) for correspond- ing ordinates. (See paragraph on Curve Plotting, p. 13.) Explain what the curve indicates. If the curve does not pass through the origin what does that fact indicate ? Experiment 3 Determination of the Gauge of Wire Micrometer Caliper Sizes of wire, twist drills and the thickness of sheet metal are often specified in commerce by gauge numbers. There being several wire gauges it is necessary to specify V J (1 V J PP" J J Fig. 2 — Micrometer Caliper MECHANICS 21 the one used; as the B. & S. gauge, the Birmingham gauge, or music wire gauge. An instrument called the micrometer caliper is com- monly used to determine the thickness of sheet metal and the diameter of wire. From these dimensions the gauge of the material is determined. The micrometer caliper consists of a frame containing a nut fitted with a screw whose "head" or barrel is grad- uated around its circumference. For accurate work the screw must have the threads parallel and the same distance apart. One revolution of the head advances the screw one division. The graduation of the barrel admits of an accurate reading of a small por- tion of a turn. In making a reading always take care not to jam the nut, and to use as nearly as possible the same pressure. Determine the pitch of the screw and test for the zero error when the jaws are closed on each other, making five readings. Measure carefully the diameter of several wires, making at least five readings on each wire moving the caliper to different points on the wire. 1 Devia- Zero Reading ou Deviation Reading on tion readings Wire No. 1 from mean Wire No. 2 from mean 1 Mean= Mean Deviation = Mean zero = Material of wire. Diameter = Gauge = 22 MECHANICS From the mean zero error and the mean of the reading on each wire determine the diameter of the wire, and the corresponding B. & S. gauge from tables. Find also the Circular mils in their cross-sections. The electrical engineer uses units called the mil and the circular mil. One mil is a thousandth of an inch. One circular mil is the area of a circle one mil in diameter. Find the average error in the readings on each wire. Experiment 4 Density Determination Fkom Mass and Volume Vernier Caliper The mass of a body is generally defined as the amount of matter of which the body is composed. The density of a substance is the mass per unit voliune. The object of this experiment is to determine the den- sity of several substances from their respective masses and volumes, and to get experience in using the vernier caliper. Fig. 3 — Vernier Caliper The masses are to be determined by means of the bal- ance, and the volumes are to be found from the linear measurements. MECHANICS 23 (1) Find the mass of each body in grams; first put- ting the body on one of the scale pans then on the other pan, taking the mean as the final value of the mass. (2) Find the dimensions of the body, using the vernier caliper, in centimeters and also in inches. Measure each dimension several times along various lines. If it be a cylinder find its diameter at six or more places along three diameters at each end say, and its length along several elements of its surface. In making readings with the caliper do not exert too much pressure, which may spring the jaws. If the jaws are not parallel always make measurements by bringing the points on the jaws which bear on each other when they are closed, to bear on the object measured. Determine the zero error of the instrument when the jaws are in contact, making a number of readings and taking the mean for the zero correction. Correct the mean reading of any dimension for the zero error. Reduce the mass to pounds (see table 1) and find the mass in grams per cubic centimeter, and also in pounds per cubic inch. From the mean linear dimension determined find the relation between the centimeter and the inch, the cubic centimeter and cubic inch. Compare the densities and relations found with those given in tables. Note. If the jaws of the caliper will not take in a di- mension of an object use centimeter and inch scales to determine the dimension, reading the scales to tenths of the smallest divisions. Tabulate your data carefully. 24 MECHANICS Experiment 5 Gallileo's Experiment A Study op Uniformly Accelerated Motion An inclined plane down which a steel ball is allowed to roll is used to study uniformly accelerated motion. Adjust the inclination of the plane so that the sphere, starting from rest, rolls about the length of the plane, 300 centimeters in 4 seconds. This may be done by placing a marker near the lower end of the plane and changing the inclination until the ball passes the marker at the fifth click of an eleetro-magnet connected to a seconds pen- dulum, the sphere being automatically released on the first click. A metal block may be substituted for the marker and the inclination of the plane changed until impact takes place at the fifth click. Take observations only when the ball starts rolling on the first click, and continues to roll without wobbling. Adjust the groove for straightness, and remove dust. To eliminate the error of the ear the settings ii^^be made by using a movable metal block and adjustin^^K)osition so that the sphere may strike it at the click of ^m sounder,- starting with the block too high up the plane^nd moving it down gradually until the sounds coincide, fljeasure the distance passed through by the ball. Make ai^pieF setting starting with the block too far down the plarw moving it up gradually until the ball strikes the block with the click of the sounder. Measure the distance as before. Do this for 1, 2, 3 and 4 seconds. Weigh the ball. Measure the inclination of the groove, using a straight edge, a spirit level and a meter stick. (These last two measurements will be used, in Experi- ment 15.) MECHANICS Tabulate data and results as fdllows 25 Data Results Time in Sees. Total Distance 19.8 21.2 82.2 82.7 184.6 185.4 326.7 328.2 Distance Mean , Pa^^ed _. . ! over Distance i during sec. Velocity at end of sec. Acceler- ation 41.0 42.0 39.0 40.8 1 2 3 4 20.5 82.5 185.0 327.4 20.5 62,0 102.$ 142.4 4^.0 83.0 122.0 162.8 Average acceleration 40 . 7 In accelerated motion the velocity is constantly chang- ing. If the motion be uniformly accelerated the changes of velocity in equal increments of time are equal. The velocity at anj' given instant may be taken as the average velocity over a very short interval of time. If ds is the space passed over in the time dt the average velocity for the time dt is i' = ds^dt, of which the particular instant is the mid-time. (For the significance of dt see note in Wat- son's' A Text Book of Physics, at bottom of page 27. See also Crew's Physics, §32.) Give indicated numerical examples of computations. *This column also gives the average velocity during the second, and, since the velocity is supposed to be uniformly accelerated, the velocity at the middle of the second. 26 MECHANICS Read carefully the directions regarding curve drawing in the introduction to the manual. Plot the following curves on the same sheet and same axes, using carefully chosen scales : (1) Time in seconds as abcissas, total distances passed over as ordinates. (2) Square of time as abcissas, total distances passed over as ordinates. (3) Time in seconds as abcissas, velocities acquired as ordinates. (4) Time in seconds as abcissas, and accelerations as ordinates. Explain the method of determining the velocity at the middle of a second and also at the end of a second. Define speed, velocity, acceleration, and uniform ac- celeration. \ ~ .In the expression for an acceleration, 50 cm/se&, why is the exponent ^ used ? From data and results as illustrated by curves draw conclusions with regard to motion down an inclined plane. These conclusions should state tlja relaton between : • 1. Velocity acquired aiffllstime. 2. Total distance travefeed and time. 3. Acceleration and time. 4. Distances traversed in successive equal intervals of time. Each of these conclusions should be stated in words, in symbols, and finally in symbols with numerical coefficients. The results of this experiment will probably differ from ideal results which the student may be looking for. This difference is due to several causes : Errors of observation, MECHANICS 27 Je» «ro r V it s •J /5b /e* ^ Fig. 4 (an error of .05 sec. in setting the marker at the end of the plane will correspond to a distance of about 8 cm), friction, air resistance, hopping and wobbhng. 28 MECHANICS Is rolling a simple or complex motion ? Explain why the ball rolls, using force diagram. Experimenft 6 Determination op Spring Constant and g by Elonga- tions AND Vibrations. 1. To find the Spring Constant by Elongations. If a mass m is suspended from a spring and is allowed to I Fig. .5 come to rest it is in equilibrium under the action of two forces, the upward pull of the spring and the downward pull of gravity. Here the resultant force acting upon it is zero, (or it would not remain at rest). To specify its po- sition we can fix a vertical meter stick beside it and by means of a pointer attached to m find the scale reading corresponding. If a greater mass is attached the spring will be stretched more, and the rest position will be lower. The reverse is true for a less mass. The relation between MECHANICS 29 the stretching and the force acting on the bob m is that an increase in force produces a proportioQal elongation or increase in length. Let I represent the scale reading specify- ing any position of the bob, w the corresponding earth-pull, or weight, acting on the bob-mass m. To indicate that a certain scale-reading, weight, and mass correspond to one another we may use subscripts; thus li wi and mi mean the scale-reading, weight, and bob-mass which correspond to one another. Then l^ — k, means the change in length, or elongation, of the spring when the weight is changed from wi to W2 by changing the suspended mass from mi to mi. The elongations and weight-changes being propor- tional, an equation may represent the relation thus : k (h — li) =W2 — wi. The letter k stands for a constant multiplier, whose meaning is clear if the elongation k — k happens to have the value one centimeter; then the equation stands fc = W2 — Wi- In words, k is the change in grams weight which produces Table or Data Obsn. Gram's Scale Diff. Diff. No. wt. Readings Grams Cms. 1 12th-lst 150 39.85 2 10 2.64 llth-2nd 90 23.72 3 20 5.23 10th-3d 70 18.49 4 30 7.86 9th-4th 50 13.23 5 40 10.48 8th-5th 30 ~ 7.97 6 50 13.22 7th-6th 10 2.65 ! 7 8 9 60 70 80 15.87 18.45 21.09 Sums 400 105.91 400 10 90 23 72 11 100 26.36 105.91 12 150 39.85 Spring number 34 Mass of spring 63 . 6 g. 30 MECHANICS an elongation of one centimeter. This may be called the spring constant and may be different for different springs. The equation k{k—li)=Wi—wi, is an algebraic statement of Hooke's Law for the spring. To determine the value of k for a particular spring, suspend it from a bar with a mass on it suflBlcient to elongate it 2 or 3 cm. Clamp a meter stick beside it as in the figure" Bring the bob to rest and take the reading of the pointer. Add successively 10 or 20 grams at a time, and take the readings each time. As soon as taken, enter the date in your notebook and compute as indicated on page 29. Plot a curve with gram-weights as ordinates and scale readings as abscissas. It should be a straight line, with its slope nearly equal to the value of k computed from the data. Why is it straight ? Does it pass through the origin ? Why? 2. To find the Value of g by Vibrations of the Spring. Suspend a known mass m from the same spring. The mass will rest by the scale at a position which may be called k- If now it be pulled straight down and let go it will vibrate up and down for a while, gradually travelling shorter distances until it comes to rest again at lo. Call its position at any instant while in motion I. Its displacement from the rest position is then l — k- The force acting on the mass m is the resultant of the spring-pull upward, and the earth-pull downward; but this resultant is zero only when the mass m is passing through its position of equilib- rium, since the mass moves with changing speed. (New- ton's Law II.) At k the spring-pull balances the earth- pull w. Below lo the spring-pull is in excess by —k{l—lo)- Above k the earth-pull is in excess by a like amount. The resulting force acting at I is therefore in grams-weight — k (l—k) and in dynes — gffc (l — k)-' This force is propor- MECHANICS 31 tional to the displacement from k and directed opposite to it. When a body has simple harmonic motion its accelera- tion is proportional and opposite to the displacement. a= — cerx= X By Newton's Second Law, the resultant force acting on a body is equal to its mass multiplied by its acceleration. In 4ir^m , . X, measured m the case of S. H. M. force = mx=- dynes If we put {l—k) instead of a;, ioTce= —gk{l—lo)'= =^ ('~'-o)) or = gk. Here m and T are found by experiment, and k is known from Part I ; hence g is determined. To find the period,* set the bob in vibration with an amplitude of about 5 cm.; note the time of starting, and the time of the 10th, 20th, 90th returns to the start- ing point. Do this with four widely different bobs; tabulate Table of Data No. Oban. No. Return Time of Return Vibns. Time Seca. h. m. s. 1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 2 24 14 21 30 38 46 53 25 1 8 16 23 lOth-lst 9th-2nd 8th-3d 7th-4th 6th-5th 90 70 50 30 10 69 55 38 23 7 Sums 250 192 ^= 250 =-^^ *Students working alone ask for special directions. 32 MECHANICS and compute the periods as shown above. Accuracy in results depends upon great pains in taking the periodic time. Weigh the spring and the bobs used. Plot a curve with bob-masses m, as ordinates and squares of periods T^, as abscissas. It should be straight, with a slope kg/^^. Find g from its slope. Explain the meaning of its m-intercept. Write in your note-book the following definitions, etc.: Simple Harmonic Motion, using the circle diagram; dis- placement,; phase, period, frequency, linear and angular velocity, acceleration, dyne, gram's weight; give the relsr tion between these two. What is Hooke's Law ? If a body is subject to forces that act according to Hooke's Law, what must be the nature of its motion ? Why ? What is assumed when it is stated that mass (grams) X acceleration (cm./sec^) equals dynes ? What is the meaning of ^ ? When the bob is vibrating, where are its velocity and acceleration greatest and least ? Why ? How does change in amplitude affect the period, according to your observa- tions ? Experiment 7 Relation of Force, Mass and Acceleration. Newton's Laws. If equal masses be attached to the two ends of a cord passing over a pulley as nearly frictionless as possible, it will be found that unless started the masses will not move; and that if set in motion the motion will become gradually slower owing to the friction of the pulley. If just enough excess mass m be added to one of the large masses so that the excess force acting on one side is just equal to the slight retarding force of friction, and the masses be giv^i a small MECHANICS 33 motion, the motion will be found to be uniform, since all the forces are balanced. Now, if a still greater mass be added to the one side and the system be released there will be an unbalanced system of forces, and a uniformly accelerated motion will result. The force producing the motion is that unbalanced force due to the action of gravity on the second added mass which, from Newton's second law of motion, is equal to the added mass times the acceleration of gravity. It may be expressed symbolically as fi = nhg. Let mi, be the mass added to overcome frictional forces. Since no acceleration is produced the force mig does not produce motion and is therefore not considered. Let M and M be the masses on the two ends of the cord and Mo be the equivalent mass of the pulley. Again from Newton's second law the total mass moved {M+M+Mo+mi+nh) times the acceleration produced (o) must be equal to the accelerating force which we know to be Si = mtg. . ■ . mig = {M+M +Mo+Tni+m2) a If all the masses be known and the acceleration be ob- served, g the acceleration of gravity may be found. The relation connecting the mass, acceleration and force is f=ma f or a= — m. which indicates that the acceleration produced in a system varies directly with the force applied and inversely as the mass acted on. The object of this experiment is to determine whether 34 MECHANICS the above relation holds and to find g the acceleration of gravity. Experiment ■■" The Atwood's Machine used for this experiment consists essentially of the nearly frictionless pulley, cord and masses of the above discussion, with additions to assist in measur- the distances and the times. The mass which is to fall i^ released by a hinged shelf which is dropped by an electro- magnet operated by a push button, clock and battery. The release occurs as the clock pendulum passes the center of its swing, and as long as the push-button is held down the magnet gives a click at each second, thus measuring the elapsed times. The large masses are composed of two equal hangers, one for each side of the wheel, and equal cylinders which slip over these. On these are to be placed smaller masses, known as riders, because they ride down as the system moves, until they meet a movable ring, through which the hangers and. cylinders pass freely, but the riders are removed, so that the distance through which the riders move is regulated by changing the height of the ring. The ring should be so placed that the elapsed time from start- ing the fall until the removal of the riders shall be an exact number of seconds or half-seconds if a half-seconds clock be used. Weigh both hanger and cord, one cylinder and each of the riders. Using 1 cyhnder on each side, and three riders, find the distance fallen by the riders in 1 sec, 2 sec, and 3 sec, (or 2 sec, 3 sec, and 4 sec.) Using 2 cylinders and 3 riders, same. it *i it *' 1 " ** t{ ^ *' "2 " ^' MECHANICS 35 Longer times of fall may be used, if desirable. Obtain from the apparatus room the equivalent mass of the wheel. Before beginning the runs, in each case place on top of the cylinders on the falling side enough tin foil or paper discs to cause the system to move with approx- imately uniform motion when once gently started by the hand. Note that, for computing, the total mass set in motion consists of the wheel, the chord, two hangers, two, four or six cylinders, and, one, two or three riders, with the small mass used to overcome friction and get a uniform motion. Table of Data ilfo=96.1g. ; cord+two hanger8 = 91.1.; each cylinder =83. 5ff | No. of Cylds. Eiders Mass Total Fall Accel. a 1 a 9 No. Mass TO2 Sec. Cms. 2 2 12 12 7 4.85 4.85 3.35. 4.85 .8.20 359 362 5 4 164 172 13.1 21.5 .0764 .0465 971 949 Mean g = = 960 For the total masswith three cylinders on each side find how the accelerations and the rider-masses vary in relation to each other. For the three riders find how the accelera- tions and the total masses vary. Plot a curve, using total masses as abscissas and the reciprocals of the correspomd- ing accelerations for three riders as ordinates. This curve should be nearly straight. Its slope is equal to the reciprocal of 7n2g. Using this fact find g from the curve. Plot another curve showing the relation between rider 36 MECHANICS masses and the corresponding accelerations of the three large masses on each side. What does the curve show ? How does this experiment illustrate Newton's Law of Motion ? Experiment 8 Equilibrium of Concurhent Forces Three concurrent forces are in equilibrium when they are parallel and proportional to the sides of a triangle and in the directions in which these sides are successively passed over in going round the triangle. 1. Cause three forces to act at a common point by means of weights attached to cords which pass over pulleys Fig. 6 to the common point. (See diagram.) Move the inter- section of the cords up and down, to the right and to the left, and finally leave it at the point where you think it would come to rest if there were no friction at the pulleys. (To get the best results, the three angles should not be greatly different.) Measure and record the three forces; measure and re- cord the three angles. Repeat all observations for two MECHANICS 37 different combinations of forces, using a different ratio of loads in each case. Draw force diagrams using a whole page in your note- book for each diagram. From a common point draw heavy lines representing to scale the three forces in direction and magnitude. Put an arrow head at the end of each of these lines showing the direction in which the force acts. On any two of these lines complete a parallelogram by drawing hght lines. Draw the diagonal of this parallelogram and compare its magnitude and direction with the remaining line representing a force. From the results of this experiment, draw a general conclusion. 2. Cause four forces to act at a common point, by means of weights, cords and pulleys, or by means of cords and spring balances. Measure angles, and observe forces recording them at once in your note book as in part 1. As before, draw heavy lines representing in direction and mag- nitude the four forces. Divide these lines into pairs, and complete a parallelogram on each pair. Compare the diag- onals of these two parallelograms. Find percentage devia- tion in each case. Draw conclusions. What is the meaning of the resultant of two or more forces ? Experiment 9 Equilibrium of a Rigid Body; Forces and Moments When a body is acted on by several forces, two condi- tions are necessary in order that no motion of translation or rotation shall take place, namely : The resultant of all the forces in any direction must be zero; otherwise there is a translatory motion in the direc- tion of the resultant force. The sum of the torques (moments, of all the forces) 38 MECHANICS about any point arbitrarily chosen as an origin or center of moments must be zero, or a rotation will take place. 1. Balance a straight bar on a knife edge to find its center of mass, as at B. Determine its mass B. Then suspend the bar horizontally by means of spring balances as shown in figure 7. From any point along the bar as at A, a known mass is suspended. Assume any point as a center of moments. Then for equilibrium the following expressions must hold, assuming forces acting downward a,s positive and those acting upward as negative, clockwise moments as negative and contra-clockwise moments as positive; C+B+A-X-Y=o Cc+Aa+Bb-Xx-Yy = o (1) (2) h <- sMC •91 -> Fig. 7 s Measure the distances a, b, and so on. Observe Cy B, A, X, Y, etc., and reduce them to the same units. Verify the above relations of forces and moments of forces in equi- librium. Note that in equations (1) and (2), which are only sample equations, there is for each external force a term in each MECHANICS 39 equation. If a different number of forces from that in the figure be used, change the equations to correspond. 2. Mount the gate (Fig. 8) or derrick (Fig. 9) in a ver- FiQ. 8 "S ^ -c * Fig. 9 tical plane as shown in the figure. With a given mass Fi, adjust the apparatus until the bar supporting, the mass Fi, is as near horizontal as possible. Find the mass and the center of mass of the frame. Observe the value of the 40 MECHANICS force Fi. Express all forces in the same units. Having the three known forces find the magnitude and direction of the fourth force, by means of a large force polygon accurately drawn to scale. Draw accurately to scale a large plot of the apparatus, showing the four forces drawn to the same scale as used in the force polygon. Find the sum of the moments of all the forces first about the point and then about any point P outside the figure. Compute the percentage error of the results and state the principles on which the computations are based. Does any moment appear in the last computation that did not appear in the first ? Why ? Tabulate your readings and draw a diagram to scale representing the forces and lever arms. Experiment 10 Wheel and Axle Relation Between Applied Force and Force Overcome A cord to which a weight is attached passes over the axle of a wheel and axle figure 10 and is fastened to it. This weight is the force overcome, call it W. Over the wheel passes a cord to which is attached a smaller weight. Determine the least value of this smaller weight necessary to main- tain the larger weight in uniform upward motion when started; call this force /i. In like manner de- termine the force which will just allow the weight to fall with uni- form motion when started; call this force /j. The mean of these two forces gives the applied force Fig. 10 MECHANICS 41 / which would be required if there were no friction. Ex- plain why. Measure the radius of the wheel and of the axle, also the corresponding distances moved by the two forces. Represent in direction and magnitude all the forces acting on the wheel and axle by a carefully drawn diagram. Make observations using the following values of W, .1, .2, .3, 5., .7, 1.0, 1.5, and 2.5 kgs. In the experiments on simple machines they are to be studied in two ways; (a) to find the law of ideal machines, (b) to find the law of actual machines. (a) Draw a curve with mean f as ordinate and W as abscissas; interpret its slope and intercepts. (b) Draw a curve with W as abscissa and % efficiency as ordinate; interpret. Is the origin a point on the curve ? (For definition of efficiency see Experiment 11.) In com- puting efficiency use /i. Diameter of wheel = 2R = " axle =2r = Distance moved by W = L = " " " f =1 =- Table OF Data Wt. lifted = W Force applied w// I/L % effy. h h mean/ Draw conclusions and show that the same conclusions might be deduced from the principle of torque and equi- librium. 43 MECHANICS Note. The word "power" in common use must not be confused with the term meaning the rate of doing work. In this case it means the working force. Experiment 11 The Pulley and Pulley Systems Efficiency of a Pulley System Two pulley blocks each containing one or more wheels called sheafs, are often used as a mechanical device in which vy Q) fti Fig. 11 a small force acting over a comparatively large distance overcomes a large force acting over a small distance. The cord or rope to which the working force is attached passes in turn over the sheafs of the two pulleys and has one end fixed to a pulley block. If the sheafs were frictionless the tension in all parts of the cord would be the same and MECHANICS 43 the weight lifted would be equal to the number of parts of cord connecting the two blocks multiplied by the tension in the cord. In practice, these conditions are not fulfilled on account of the weight of the block to which the weight to be lifted is attached and to the friction of the sheafs on their axles. The efficiency of a system is the ratio of the useful work done to the amount of work done on the machine. Use loads W of .1, .2, .3, .5, .7, 1.0, 1.5, and 2.5 kgs. Place enough mass in the pan at the free end of the cord so that it will move downward with a uniform motion; call this force /i. The force /i, must then be sufficient not only to put the system in equilibrium but also to overcome the friction of the machine. To eliminate the amount of force necessary to overcome friction, decrease the mass on the free end until that end will move upward with a uniform speed; call this force fi. Then fi is equal to the force necessary to produce equi- ibrium minus the force necessary to overcome friction. Therefore the force necessary to produce equilibrium is 2 ■'■ The weight of the pan is included in /. Find the force / necessary to balance the different weights at W. Find the relative distance passed through by the work- ing force /i, and the load W. Remembering that the several values of /i are those necessary to operate the machine as a machine for doing work, find the efficiency of the machine for the different loads. Draw a diagram showing the forces acting. 44 MECHANICS Arrange data and results as follows : fi+h ' Distance passed over Efficiency LoadT^. 100 h- h- 2=^- 16.8 100 % 29.0 58 200 82 16.6 40.5 400 123 7 65 16.7 54.3 600 158 46 102 16.9 64.2 1000 243 83 163 16.7 68.8 1500 338 162 ■ 250 16.7 74.1 2000 442 232 337 16.6 75.1 Draw the following curves : (1) With W as abscissa and corresponding mean f as ordinate. (2) With W as abscissa and corresponding /i as ordinate. W tn qrtiyna. Fio. 12 Meee ^ MECHANICS 45 (3) With W as abscissa and corresponding % efficiency as ordinate. Discuss curves. Is the origin a point on the second ? Why? Experiment 12 The Inclined Plane Relation Between Applied Force and Weight Lifted A weight W rests on an inclined plane whose height is h and whose length is I, Figure 13. Determine the force parallel to the plane by means of a weight attached to a cord passing over a pulley, which will be just sufficient to -maintain uniform motion up the plane when started by the hand; call this force /i. Now determine the force which t±]j: Fig. 13 will allow uniform motion down the plane; call this force /a' The mean of these two forces is the applied force which would be required if there were no friction; call this force/. Measure the height and length of the plane. If the apparatus given you is fitted with a heavy roller, make seven observations, starting with as small a slope as will cause the roller to move. Plot two curves; (a) with sin

W Construct a diagram illustrating all forces acting on the weight in direction and magnitude. Draw conclusion. Show that the same conclusion can be drawn from the principles of force and equilibrium. Experiment 13 The Balance The general theory of the balance as given in Watson's Text Book of Physics or any other standard text should be thoroughly understood before performing the experiment. The experiment consists in weighing some masses, and determining the ratio of the lever arms of the balance. MECHANICS 47 The balance to be used is not considered a sensitive one but it will fill the wants of the experiment. In making weighings do not allow the arms to swing violently. Always put masses on the scale pans when they are resting on the table on which the balance is supported. To Make a Weighing. With no mass on either scale pan, raise the pans off the support and allow the balance to vibrate so that the pointer does not pass beyond either end of the graduated scale near the end of the pointer. Air currents affect the vibration of the balance and therefore should be eliminated. Determine the zero point by the method of vibrations. Allowing the pointer to vibrate over the scale take readings on the turning points, estimating tenths of divisions and assuming the zero of the scale at the long mark to the left. Take one more turning point on one side than on the other. Find the average reading on each side. Find the mean of these two averages, which will give the zero reading of the balance. The following is an example of a zero determination : Scale Readings. 2.8 17.0 16.6 3.2 16.6 3.0 2)6^* 16.3 3)49.9 2)19.6 3.0 9.8— Zero 16.6 Make two more determinations of the zero of th^ bal- 48 MECHANICS ance. A good agreement among the zero determinations indicates that the balance is in good order. To determine the mass of a body place the body on one scale pan and some weights on the opposite pan. Raise the central knife edge by means of the key to see if the weights added are about right. If they are not, lower the knife edge, change the weights and try again. Continue the op- eration until a combination of weights is found that will not carry the pointer off the scale. Then make readings of the turning points as before, to determine where the pointer would come to rest. In general this rest point will not coin- cide with the zero previously determined. To find what weights would bring the pointer to the proper zero add 0.1 g to the weights and find the rest point by the method of vibrations. The difference in the two rest points will give the change in reading for a change of 0.1 gram. If the balance assigned be very sensitive use 0.001 g in place of 0.1 g. Knowing the change of rest point for 0.1 g and the de- sired change to bring the rest point to the zero, the amount of weight that must be added to or taken from the weights to make the proper balance is easily computed. This method assumes that there is no shift of the zero when the balance is loaded and that the balance arms aie equal. No correction is here made for the buoyancy of the air. Double Weighing and Ratio of Balance Arms To eliminate errors due to unequal arms the mass may be weighed first on one pan, then on the other. Let li and Zj be the lengths of the two balance arms. Let wi and w^ be the weighings when x the unknown is at the end of the arms U and li respectively. MECHANICS 49 For the first weighing, xh = Wili (1). When X is placed in the other pan, xk = Wil2 (2). Multiplying these two equations together and solving for X we have X = s/wiWi If the arms are very nearly the same length the arith- metic mean of the two weighings is accurate enough for most purposes. If equation (1) be divided by equation (2) the following expression showing the ratio of the balance arms may be obtained : h _ | wi ll '\ W2 Find the ratio of the balance arms using the method above described, making two determinations using different masses. Experiment 14 Coefficient of Friction The object of this experiment is to find the coefficient of kinetic friction between two surfaces and show that it is independent of pressure. The coefficient of friction is defined as the ratio of the tangential force required to maintain uniform motion, to the normal pressure. The coefficient depends on the nature and condition of the surfaces. On that account do not touch the surfaces with the hands, because it will change the condition of the surfaces and consequently the coeffi- cient will be changed. Put a mass of 2 Kg. on the block, then add enough mass to the hanger to produce uniform motion when the block is 50 MECHANICS started byhaird. Take readings for masses of 2, 4, 6, 8 and 10 Kg. Taking account of mass of block compute the sev- eral values of the coefficient. Plot a curve using tensions in the cord as ordinates and corresponding masses for abscissas. Consider the pulley as frictionless. Draw conclusions and discuss the curve. Mass of Block = Mass of block plus added mass Force producing mo- tion in grams wt. Coefficient of fritcion, fJL Experiment 15 Work and Power (A). Work against gravity Weigh yourself, obtain from apparatus room the dis- tance from first to third floor, climb the stairs and observe the time i-equired. State whether the stairs were climbed very quickly, quickly, moderately, slowly, or very slowly. Tabulate data and results as follows : English Engineering Units French Engineering Units C. G. S. Units Meter-Kilo- gram Sec. Units Weight Vertical Dis- tance Time Work Done Power ex- pended lbs. ft. sec. ft-lb. ft-lb. sec. < H. P. Kg. M. sec. Kg.m. Kg-m. sec. F.H.P. dynes. cm. sec. ergs. ergs, sec. dynes. meters. sec. Joules Watts MECHANICS 51 Is the French horse-power greater or smaller than the English ? (B) Work against friction. The Friction Dynamometer. A strong cord passes around the groove of a wheel. Its two ends are fastened to two spring balances, the latter be- ing fastened to a solid support. One end of the cord may have a known weight suspended from it in place of a spring balance. If the wheel is turned rapidly the tension Fi of one branch of the cord will, on account of friction, be greater than that- of Fs of the other. The amount of work done will be the same as in the case of a fixed wheel and a moving cord. If I is the distance moved by the cord (or by the rim of the wheel if the cord is fixed) then the posi- tive work is F2 I and the negative work is Fi I. The net work on the wheel is (F1—F2) I. If the wheel makes n revolutions we have 1 = 2 n r n and the work will be W={Fy-F2) 2nrn= L «P. in which L is the resultant torque and $ is the angle turned through expressed in radians. Observe the tensions as indicated by the two balances both when the wheel is at rest and when it is in uniform motion. Count the number of revolutions in 1 min., ob- serving F\ and Fi at intervals of 10 seconds. Reduce data to consistent units. Tabulate data and results as follows : 52 MECHANICS 2r = diameter of wheel = cm. = in. Rev. in 1 min. Fi Fi Work Done Power H. P. Watts ft.-lbs sec. r i (C). Refer to data taken in Experiment 5. In the actual experiment, what energy changes occur aS the ball starts from rest and moves down the groove ? Com- pute in ergs the gain in kinetic energy due to the change in linear velocity, and the loss in potential energy, for the four second run. Are they equal ? Explain. Experiment 16 Acceleration op a Fbeely Falling Body In the apparatus used for this experiment a steel sphere is automatically dropped from the pointed end of an electro- magnet, while at the same instant a pendulum is released and swings across the path of the falling ball. The bob of this pendulum has an adjustable slit at right angles to the plane of vibration. If the slit is immediately below the ball's starting-point at the instant when the ball reaches the level of the slit, then the ball can pass through; other- wise the ball will strike one or the other jaw of the slit; which one may be known from the sound, as one jaw is faced with steel, the other with lead, and so it is possible to tell whether the ball arrives at the slit too soon or too late to get through. The period of the pendulum is ad- justable by altering its length or by sliding an iron weight on the pendulum rods. To adjust the apparatus : (1) With the pendulum at rest in the center, turn on the current by a small rotating switch mounted on the MECHANICS 53 same board, place the steel ball on the pointer of the magnet armature above, break the circuit and note if the ball passes through the Slit, or on which side it strikes. Adjust the jaws of the sUt so that the ball 'will pass through with very little clearance between it and the jaws. (2) Turn on the current and pull the pendulum into contact with the large electromagnet below at the left. The face of the magnet-core should be covered with paper, or other non-magnetic substance, to insure prompt release. Drop the ball as before, and note by the sound on the re- bound on which jaw of the slit it strikes, if it does not pass through. If the sound indicates too short a time of fall, lengthen it by raising the upper electromagnet a little, and test again; if too short, lower the magnet; test until you find that the ball passes through several times. Then the time of fall of the ball is the same as the time taken by the pendulum to swing from its extreme position to the center, which is V4 or ^U or ^A the period of the pendulum; note which. Measure the distance of fall, find the period of the pendulum as in Experiment 6 or 17, and compute the value of g. SUde the adjusting weight on the pendulum, above the knife-edges, either up or down a little, which will alter the period a little, and repeat the test. Make tests with three different periods in all, and average the values of g. What are the possible sources of error in this experiment ? Experiment 17 The Simple Pendulum The object of this experiment is to verify the relation existing between the periodic time and length of the simple pendulum and to find the acceleration of gravity g. The relation is T — 2ti\/ — . Suspend a metal sphere by a long fine cord from a solid support. Set it to vibrating through 64 MECHANICS a small angle. Determine the periodic time for four differ- ent lengths, varying from about 30 cm. to the greatest length convenient using nearly equal intervals. Accurate results depend upon great pains in finding the periodic time. To determine the periodic time observe the time in hour, minute, and second at which the pendulum passes through its mid position in a given direction; then, counting the transits through the mid position in the same direction take the time of transit as follows; with lengths of pendulum, measured from the underside of the support to the center of the metal sphere, of about 30, 70, 100, and 150 centi- meters observe every tenth transit for 270, 190, 150, and 130 transits respectively. Compute the periodic time for each set. Measure each length for which observations of the periodic time was taken, at least three times. Tabulate data and results as follows : Table of Data Transits to Right Differences Times 1 No. h. m. sec. Numbers Min. Sec. 10 10 13 300th-0th 300 5 55 20 10 37 280th-20th 260 5 09 40 11 00 260th-40th 220 4 22 60 11 25 240th-60th 180 3 34 80 11 48 220th-80th 140 2 46 100 12 12 200th-100th 100 1 58 120 12 35 180th-120th 60 1 12 140 12 59 160th-140th 20 23 160 180 200 13 13 14 22 46 10 Sums 1280 25 19 220 14 34 Length of P Bndulum = 35.10 cm. 240 14 59 I 260 15 22 7-= 1.187 =25.39 280 15 46 r2 300 16 08 ? = 985 MECHANICS 55 Plot two curves on same axes, using I as abscissa; for one curve use T as ordinate, for the other T^. Discuss the forms of the curves, and the meaning of the intercepts and slopes. Show by diagram all the forces acting on the pen- dulum when displaced from its mid-position, and discuss the force-system. (Note that resolving a force, as the weight of a pendulum-bob, does not give any new forces.) Experiment 18 Uniform Circular Motion and Accblbeation Toward THE Center When a particle moves in a circle there is a tendency for it to move off in a straight line. There must be a force acting constantly to cause the particle to move in a circular path. Suppose the angular velocity go be uniform. Then the force causing the particle to move in a circular path must be either toward or from the center. If the force acted in any other direction there would be a component of the force acting at right angles to the radius joining the particle to the center of the circle and its speed would either increase or decrease. €> < --- r» «-r,-> Fig. 14 The particle does not move away from the center, but constantly changes its direction of motion in such a manner 56 MECHANICS as to remain at a constant distance from it. Therefore the force acting to produce this change of motion must be toward the center. The expression for the acceleration of a particle moving at a uniform angular velocity in a circular path is a = — r where v is the speed or tangential velocity and r is the radius. Since f=r'fna the force acting toward the center is r But v — r 00 therefore /=m r oo'-. Since co = 2 n n where n is the number of revolu- per second the following expression also holds, /=4 7T^ n^ r m The experiment is divided into two parts either of which or both may be assigned. I. To show that f varies with r. Let two masses mi and m2 be mounted on a bar, Fig. 14, which revolves in a horizontal plane. Consider that the masses may slide on the bar without friction. If the bar be rotated there will be forces acting to cause the masses to move frorn the axis with forces equal and opposite to those necessary to hold them, at a fixed distance from it. These forces will be /i = wii n toi^ fi = nwr2 00^ Since ihe bar is rigid the angular velocity &> will be the same for both masses. If a light cord fasten the two masses together, in general MECHANICS 57 they will move in the direction of the larger force. If the values of n and r^ be properly chosen the two forces will be equal and since they are in opposite directions no slip- ping along the bar will take place when the bar is rotated about a vertical axis. Then/,=/2, or mi n GOi' = nhr2 a^^, and TOi ri = »i2 ra, from which — = — , ri mi that is, the radii for no sliding motion are to each other inversely as the. masses. Experiment : — Fix the relative positions of the two masses a few centimeters apart by means of a small wire. Revolve the bar about a vertical axis, slowly at first and adjust the positions of the masses until no radial motion takes place. Gradually increase the angular velocity, shift- ing the position of the masses when necessary until a very angular velocity may be attained with no radial motion. This is necessary to reduce to a minimum errors due to friction of the masses on the rod. Stop the machine and measure ri and r?. Repeat the experiment with the masses at different distances apart, making four sets of observations. Find the masses of toi and m^ and show the numerical relations among the observations. Care must be taken that the bar rotates in a horizontal plane. II. *To show that f varies with n^. *Henry Crew, School Science and Mathematios, May, 1905, Vol. v., No. 5, p. 331. 58 MECHANICS :i^_t. j, r — «l ■T5SP ► = S J^ A n j i . J ■ I r n J. cr Fig 15 Suppose the mass m be rotated about the vertical axis C E while a spring S pulls m towards the center of rotation. When the speed of rotation reaches a certain amount the outward force mr a? will just balance the inward force / produced by the spring S. If we now measure n, the num- ber of revolutions per second we can determine /. Also, since /, the force in dynes = fc d g, where k is the spring constant in grams per cm. (See Expt. 6), d the amount the spring is stretched and g the acceleration of gravity, if we know k and d we may calculate g. f=k d g — 4 ^^ mr n^ J^n^ r^ mr ^= kd Experiment : — The essential parts of the apparatus are shown in 6 of Fig. 15. The mass m which can slide along the rod A is held against the stop B by the spring S. The entire part shown is rotated about a vertical axis by means of a small electric motor. The electric curcuit is so ar- ranged that when the speed is sufficient to throw m away from the stop B power to the motor is cut off and the speed decreases until m is pulled back to B by the spring when the power is again applied. Thus the rotational speed will oscillate between limits a little too high and a little too low.; but if the average is taken for 30 sec. it will be found to be about the correct value. For each of the two different masses on the horizontal rod make the following observations : MECHANICS 59 With just sufficient tension in the spring to pull the mass TO against the contact point B when displaced not more than a millimeter, find the number of revolutions in 30 seconds when equilibrium speed has been reached, as noted above. Repeat the observations twice and use the average in making computations. Make three other sets of observations with varying tensions, one of which should be such as to give equilibrium for approximately the maximum speed of the motor. The other two should be for well chosen intermediate tensions. The adjustments for each set of observations may be made as follows : Having to a millimeter or less to the left of B move the spring fastener F to the right until the slack is out of the spring and note position of F. Then move F to the Spring stretch for friction d No. of rev. in 30 sec. Rev. in 1 sec, n. n^ Calculated 9 .8 cm. 3.1 31 32 31 .7 cm. 4.2 Av. = 31.3 43 42 42 1.042 1.085 1010 Av.=42.4 1.41 1.99 959 etc. etc. etc. Av. fric- tion = Av.g = r= cm. Spring constant k= . Spring stretch for friction from curve = cm. Slope of curve = . g from curve = 60 MECHANICS right until m just slides. Measure this distance each time the tension is changed. It will be the stretch of the spring necessary to overcome friction. Also measure from the po- sition of the fastener when there is no tension in the spring when m is touching B out to the fastener's position for the tension to be used. The distance will be d, the stretch of the spring. Find the distance r of the center of mass of m from the axis of rotation. Make two tables. Plot two curves on the same axis, using values of d for abscissae and w^ for ordinates. Determine the slopes of these lines. Since the slope of these curves = 4 TT^ m r k g values of g may be found from them. Calculate g from each curve. The values should be nearly equal to those obtained by direct calculation. Experiment 20 Boyle's Law The object of this experiment is to investigate the rela- tion between the pressure and volume of a constant mass of a perfect gas, the temperature remaining constant. ; Obtain the atmospheric pressure from the barometer, which may give readings in inches. Reduce the reading to centimeters of mercury. (a) Boyle's Law Apparatus, Type A. This consists of a mercury reservoir, of glass, which can pe set at any height on a standard by means of a wedge; fi connecting hose; and two glass tubes, one long and open at top, the other short and closed at top. These two tubes communicate at their lower ends through iron tubes with each other and with the hose. Between the two glass tubes a meter-stick is fastened. Set the mercury reservoir at such a height that the MECHANICS 61 mercury shows at the bottom of the open tube The short closed tube should then be half or two-thirds full of air; (if it is not, ask an instructor to adjust it). Wait two minutes, and read the heights of the mercury in the two tubes. Raise the mercury reservoir so that the mercury goes up about one-fifteenth of the possible range in the open tube, wait two minutes and read again. Re- peat these observations, (waiting always two minutes aftef changing the mercury levels), until the mercury can be raised no higher in the open tube. Read the height of the top of the closed tube on the scale. Adjust the barometer and read its height. The time between observations may be profitably spent in computing. (6) Boyle's Law Apparatus, Type B. It consists of a vertical wooden standard supporting two glass tubes and a vertical centimeter scale between them. On the left is a gas tube, closed at its upper end, for holding the air or other gas to be examined; on the right an open tvbe, whose position determines the pressure on the gas; and a hanging rubber hose containing mercury, connecting the two tubes. The open tube slides on the standard, and is supported by a cord which runs over a pulley above, then down, and is fastened by a clutch behind. Make sure that you under- stand the action of this clutch. Observations with Air. Lower the open tube as far as it will go and yet show mercury in tube. Wait at least two minvies, then read on the scale the level of mercury in both air and open 62 MECHANICS tube. Raise the level of the mercury in the open tube about one-fifteenth of the possible range of the apparatus by pulling the open tube up and fastening the clutch; wait two minutes, and read the two levels again; repeat this until the open tube can be raised no farther, waiting al- ways two minutes at least between observations. Read the height of the top of the closed tube, and the height of the barometer. Lower the open tube to a level with the air tube, and leave it so, for safety's sake, and the con- venience of the next student. Scale reading with mercury at the same level in both tubes = Barometer reading, inches = cm. = Closed tube readings Length pro- portional to volume of air column in closed tube Open tube readings Diff. of Hg. levels Pres. in cm. of Hg. P. V. * * Or quantities proportional to the product P. V. Plot curves for pressures both above and below the atmospheric pressure as follows : Use pressures in centi- meters of mercury for ordinates, and lengths proportional to volumes of column of air in closed tube, as abscissas for one curve; and for another use the reciprocal of pressures and volumes, or quantities proportional to them, as or- dinates and abscissas respectively. The origin should be on the page. Discuss curves. (See curves in the intro- duction of the Manual.) MECHANICS 63 Experiment 21 Density of Liquids Hare's Method In finding the relative density of two liquids that mix, a method devised by Hare is easily applied. The principle of /> T. i>anp. the experiment is that the pressure exerted by •tt a liquid is proportional to the height of the /^^^ liquid, to its density, and to the acceleration of gravity, p = hdg where p is the pressure, h the height of the liquid column of density d, and g is the acceleration of gravity. t Hare's method makes use of the fact that t the ratio of the densities of two liquids which 5- exert the same pressure is equal to the inverse ratio of the heights. An inverted U tube, the bend of which is connected to a suction pump, is mounted ver- tically. A scale is fastened to the frame be- tween the two tubes which measures distances ¥ from the lower ends of two wires A A. Before "- performing the experiment clean out the tubes by drawing distilled water into them, Fig. 16 then force air through them until dry. Place two beakers containing the liquids whose densities are to be compared at the lower ends of the tubes and draw the liquids up the tubes until one of them is near the top of the scale. Close the connection to the pump, and by means of a pipette regulate the height of the liquids in the two beakers until their surfaces are at the lower ends of the rods A A. Read the respective heights of the hquids in the two tubes. Allow enough air to enter the upper end of the U tube to lower the surfaces of the liquids by about ten centimeters. 64 MECHANICS Adjust the level of the liquids in the beakers and make a second reading. Make at least six readings like those described above, comparing a number of substances with distilled water, or with a substance of known density. Note the temperature of the liquids and find their specific gravities. Prove that -^ = -r-- ok hi Experiment 22 Specific Gravity Hydrometers Study carefully the theory of the hydrometer as given in some standard text. The experiment consists in finding the specific gravity of several liquids by means of the Nicholson hydronieter. and either checking them by using a calibrated variable immersion hydrometer, or using the results in calibrating such an hydrometer. Find the specific gravities of four salt solutions, pre- pared as follows : Solution Method of preparation (a) Dissolve 200 gm. of salt in 800 cc. of water (b) To '/4, by volume, of (a) add 250 cc. of water. (c) To Va, by volume, of (b) add 333 cc. of water. (d) To V2, by volume, of (c) add 500 cc. of water. (e) To Vs, by volume, of (d) add 666 cc. of water. In the preparation it is assumed that the number of cubic centimeters of water used is numerically equal to the mass. MECHANICS 65 Water in co. Solution Added mass to sink hydrometer to fixed point Spe- cific grav- ity From old solution Added Volume in cc. Mass in grams Grams of salt Concen- tration Mass of hydrometer g. Temperature of solutions Plot a curve with specific gravities as ordinates and concentrations as abscissas. Find the specific gravity of a solid by means of the Nicholson hydrometer. Experiment 24 Specific Gravity of Solids not Soluble in Water (1) Solid; specific gravity* greater than unity. Weigh the sohd in air, then weigh it suspended in water, taking care to dislodge all air bubbles from it before weigh- ing. State Archimedes' Principle and derive a formula for specific gravity. Why does the body lose weight when weighed in the Uquid ? Tabulate data as follows : Substance Wt. in air Wt. in water Loss in water Sp. Or. Measure, weigh and compute the density in grams per *NoTB — "Weight in air divided by loss of weight in water" is not a satisfactory definition of specific gravity. 66 MECHANICS cc. of a geometrical solid (for directions, see Expt. 4,) whose specific gravity is also to be found by the above method. What relation is observable between the volume by meas- urement and the loss of weight in water ? Explain. (2) Solid; specific gravity less than unity. Weigh the solid in air. Weigh a "sinker" in water, and find the combined weight of sinker and solid in water, taking precaution to get rid of air bubbles. Derive a formula for specific gravity for this case. Substance Wt. in air Wt. of sinker in water Wt. of solid and sinker in water Loss of wt. in water Sp. Gr. , Experiment 25 Specify Gravity of Solids and Liquids by Means of Jolly Balance I. The Specific Gravity Bottle. If a bottle be carefully cleaned, then filled with distilled water of known temperature and weighed, the volume of the bottle may be determined; the weight of the bottle being known. If the bottle now be filled with some other liquid, at the same temperature, and weighed, the weight of the liquid may be determined. Since the volume of the liquid and that of the water are the same the specific gravity may be found very easily. Bottles made for this particular purpose are called pyknometers or specific gravity bottles. The specific gravity of a solid which is not acted on by water is easily determined. MECHANICS 67 Weigh the empty bottle. Place the solid in the bottle and weigh again. Fill the bottle with distilled water being careful to get all air bubbles out and find the weight of the combination. Finally^ find the weight of the bottle filled with distilled water. From these data the specific gravity of the metal may be determined. II . Sinker Method for Liquids . Weigh an insoluble solid in air and pure water; also weigh it in the liquid to be tested (in which too it must be insoluble). Find the loss of weight in w:ater and also in the liquid. The latter loss divided by the former gives the spe- cific gravity of the liquid. Show how computations are made in each case, and explain the principles of the methods. III. The Jolly Balance. The weighings in this experiment are to be made by means of a Jolly balance. The Jolly balance is fundamentally a coiled spring fastened at one end and free at the other. To the free end is attached a scale pan in which the sub- stances to be weighed are placed. The spiral spring follows Hooke's Law, that is, equal increments of force acting on the spring produce equal increments of length. (See Expt. 6.) The Jolly balance used in this experi- ment consists of a spring attached at its upper end to a rod which may be raised or lowered so that an indicator may be brought to a given position of rest. The elongation is read by means of a scale and vernier. The whole system is supported on a base Fig. 17 68 MECHANICS having three legs which are supplied with leveling screws. Adjust the apparatus by means of the leveling screws so that the aluminum indicator is central in the glass tube. Raise or lower the support carrying the upper end of the spring until the mark around the lauminum indicator is in the plane with the etched circle on the glass cylinder. Read the vernier. Place the article to be weighed in the scale pan or sus- pend it from the hook. Raise the support until the line on the aluminum indicator coincides with the etched line on the glass as before. See that the indicator hangs free of the tube. Read the vernier. The difference in vernier readings will give the elonga- tion of the spring. If the elongation per gram is known the weight may be easily computed. If the elongation per gram is not known adjust the balance, read the vernier, add weights five grams at a time until twenty grams have been added, taking vernier readings, for each added weight. From these readings the spring constant may be computed. In the determination of specific gravities the constant of the spring need not be known. The reason why it is not necessary is to be explained by the student. Tabulate the readings carefully and show how compu- tations were made. Indicate the numerical operations. Experiment 26 Determination of the Angle of Contact of Mercury AND Glass In the case of a tube of small diameter the meniscus, or surface of hquid enclosed, is nearly spherical in form. That it cannot be exactly spherical follows from the fact that a liquid surface at rest is at every point normal to the resultant force, which in this case is composed of surface MECHANICS ■ 69 tension and gravity. For tubes of small diameter the error is negligible. Hence a thread of liquid enclosed in such a tube may be considered as consisting of a right cylinder capped at each end with a spherical segment of one base. If the base radius r of this segment and its height h are known, the angle of contact a is known from the relation /i/r = tan \ (« — 90°). (To be proved by the student.) Use three tubes and find the meniscus height of mercury and the radius in each as follows : Introduce clean mercury into a dean glass tube, forming a short thread about one centimeter long near the middle of the tube. Fasten the tube horizontally with the clips or with wax, on the stage of a micrometer microscope. Focus the cross-hairs on the edge and then on the vertex of the meniscus, reading both scales of the instrument for each position. The difference of readings gives the meniscus height. Then fasten the tube in a vertical position, and measure its longest and shortest internal diameters at each end. From the mean of these four measurements find the radius. Calculate the angle a. in degrees and minutes. Experiment 27 Surface Tension Surface tension may be measured in either of two ways: first by means of measuring directly the sustaining power of the film with some form of dynamometer; second by capillarity. In this experiment it is to be measured by means of the Jolly balance, a spring dynamometer. (For a description of the Jolly balance see Experiment No. 25.) A clean wire having a shape as indicated in figure 18 is suspended so that the longer part remains horizontal and a reading of the vernier of the balance is made. 70 MECHANICS A beaker containing the liquid whose surface tension is Fig. 18 desired is placed on the adjustable support of the balance and is raised until the horizontal portion of the wire is just submerged. Then no force due to surface tension is acting on the wire. Now increase the tension on the spring very slowly, raising the wire out of the liquid until the film breaks. Read the vernier. The difference between the two readings multiplied by the constant of the balance will give the. pull due to the surface tension in grams weight. If the constant of the spring is not given it may be determined by the method explained in Experiment 6. The maximum surface tension is then kx m (grams wt.) n djmes T = 2a 2a 2a where k is the constant of the spring, x is the elongation of the spring, and a is the length of the wire. Make four trials on each liquid used. Find the result in grams weight and in dynes. Explain fully how the formula is obtained. The aluminum wire and the liquid surface must be clean. Dip the wire in dilute sulphuric acid and agitate it a few moments, after which rinse first in distilled water, then in alcohol, and then dry it. Clean the wire after test- ing each liquid. MECHANICS 71 Experiment 28 Capillary Determination and Surface Tension Uses op the Cathetqmeter I. Adjustment of the Cathetometer. In the instrument furnished the level is fastened to the telescope and it is assumed that the line of collimation of the latter is parallel to the former. Under one end of the telescope there will be found a screw by which the inclina- tion of the telescope can be changed. CaUing the foot- screws of the tripod P, Q, R, and the center of the same B as in figure 19 make adjustments as follows : (1) To adjust the cross- hairs to the focal plane, pull out or push in the eye-piece until the cross-hairs are clearly seen. Place the telescope at the cor- rect height on the pillar and point it toward the object de- sired; pull out or push in the mounting carrying the eye-piece and cross-hairs until the object is clearly seen and the cross- hairs and object apparently have no relative motion when the eye is moved sidewise. The object should be not over 60 cm. from the telescope; it is an advantage to make this distance small, to diminish er- FiG. 19. rors due to poor levelling, crookedness of pillar, width of cross-hairs, etc. 72 MECHANICS (2) To make the pillar vertical and the telescope hori- zontal, turn the telescope parallel to the foot-screws P, Q, level the telescope by means of the telescope-screw beneath it, then turn the telescope 180°. If the bubble changes position, bring it back half with the telescope-screw, half with the foot-screws P. Q. Turn the telescope through a right angle, make it level with the foot-screw R. Repeat these steps, then test adjustment (1); if they have to be changed repeat both (2) and then (1) until all are perfect. //. To determine the Surface Tension of a liquid. In this experiment the liquid rises in a capillary tube the lower end of which is immersed beneath the surface. Since the liquid used wets the surface of the tube the angle of contact a in figure 20 is zero and the expression for the surface tension is T = y^hrpg in which T is the surface tension, h the height the liquid rises in the tube, r the radius of the tube, p the density of the liquid, and g the pull of gravity on a unit mass. I .-ir-- ^ Fig. 20 Use a dean'* glass tube, supported vertically over a glass cell with plane sides; pour in clean water to fill the * Success in this experiment depends on cleanliness. MECHANICS 73 cell 2 or 3 cm. above the bottom of the tube, and wait 2 min. or more, until the water has risen as high as it will go; raise the tube about 1 em.; this insures that the inner surface of the tube be wet above the level of the water column in it. The tubes should not be handled with the fingers. After removing them from the cleaning solution they may be washed with tap water and then with distilled water. Air bubbles collecting within the tubes may be removed by using an air blast. Filter paper may be used to handle the tubes. The heighth h may be found in either of two ways (a) by direct measurement with a scale or (b) by using a cath- etometer. If a scale isused hold it vertically with the lower end touching the liquid surface in the cell and read the height of the capillary column of liquid. Repeat the meas- urement twice. If the cathetometer be used set its cross-hairs on the lower edge of the mensicus, and read the venier; then on the level of the water in the cell, and read the venier; do this 3 times. The mean difference in the readings is h. Measure diameter of tube with the micrometer micro- scope. This is done by bringing the cross-hair of the mi- croscope tangent first to one side of the image of the hole in the tube, then tangent to the other, and taking the readings of the micrometer screw in the two positions. Repeat the observations as a check. The bores of the tubes are usually not truly circular, therefore measure along four diameters at approximately 45° intervals. The bore varies along the tube and as the bore is wanted at the top of the meniscus, measure both ends and take averages of all readings. 74 MECHANICS Use three tubes of different diameters and find the surface tension of three liquids for each tube. Find the temperatures of the liquids and get their den- sities from tables. As a correction to the height of the column, due to the fact that the liquid in the meniscus has not been taken account of, ^/s the radius of the tube may be added to the measured value of h. In place of the instrument described in part I a uni- versal reading microscope may be furnished to measure the heights of the liquid columns, as with the cathetometer, and also the internal diameters of the tubes. It consists of a rigid frame to which is attached a movable carriage which supports the reading telescope either horizontally or vertically. The carriage to which a venier is attached moves along a rod graduated in millimeters, so that a length may be easily found by setting the cross-wires of the tele- scope on the extremes of the length to be measured, read- ing the scale and vernier in each position. To make accu- rate settings of the carriage there is a clamp nut which fixes the position of a block against which the end of a micrometer screw working in a nut in the carriage rests making very small changes in the position of the telescope possible. The micrometer screw is to be used in determin- ing the internal diameters of the tubes which are to be so placed as to make possible reading end on. The micrometer screw carries a head divided into 100 parts. The pitch of the screw is to be determined by the student. In using this apparatus the same readings are to be made as with the ordinary form of cathetometer as indicated above. MECHANICS .75 Experiment 29 Elasticity Young's Modulus Tension Young's Modulus is defined as the force per unit cross- section necessary to increase unit length of a substance by unit amount, provided the elastic limit of the substance be not reached. In most cases such increase is absolutely im- possible, only a very small increase in length being possible within the elastic limit. Within the limits of this experi- ment we may assume that the increase of length is propor- tional to the force applied. From the above definition it follows that q I qe where F is the force; q is the cross-section; e the elonga- tion; and I the total length of the wire from the support to the mark where the elongation is measured, expressed in proper units. One end of the wire is fixed. A mark near the other end is to be used as a reference point. The stretching force is to be applied to the free end of the wire. Focus the cross-wire of a micrometer microscope on the mark near the free end of the wire and read its position when enough force is applied to take the kinks out of the wire. Add a force equal to a kilogram's weight to the force acting and read again. Proceed in like manner until at least five elongations have been determined. Reduce the force acting stepwise to the original value taking readings as a check on the original observations. If 76 MECHANICS there be much discrepancy in the elongations repeat all the readings. Determine the pitch of the micrometer screw. Compute the mean elongation per gram by the method of Expt. 6, part I. Measure the diameter of the wire in at least five places, using a micrometer gauge. (See Expt. 3.) Define elastic limit, permanent set. Arrange data as follows : Laboratory station. Material of wire. Length of wire used. Diameter of wire. Cross-sectional area of wire. Zero reading of micrometer. Pitch of micrometer screw. Force producing elongation in kg. wt. Reading Elongation e. Modulus in gram's wts. Modulus in dynes Plot a curve using total elongation as y and total F as x. What does the curve indicate ? If the wire is 4 meters or more long, a vernier caliper may be used instead of the micrometer microscope. It must be firmly clamped to the bench, with the movable jaw directed at the wire and near enough to be almost in contact with a marker fastened to the wire. Get directions from your instructor as to the proper loads to use. MECHANICS 77 Fig. 21 A third apparatus for this experiment is Searle's level apparatus. In the figure, A' is the wire to be tested, A a comparison wire whose length is kept unchanged throughout the test. From A' hangs a brass frame C D', carrying a micrometer screw with large divided head S, and a miUi- meter scale R, for measuring whole turns of the screw, also a pan P below, on which the varied load is to be placed. A similar frame CD hangs from A, but without the screw and scale. From it there is suspended a load M. The two frames are connected by two links K and K', which prevent them from becoming non-parallel, but allow freedom of vertical play. A level L turns on pivots H in the frame CD, and rests on the top of the screw. With a suflScient load on M and P to straighten the two wires, turn the screw until the level-bubble is centered, and read the screw-head and scale; add to the load on P, which will throw the bubble out of center; raise the screw S, until the bubble is again centered, and read the head and scale. 78 MECHANICS The difference in readings gives the elongation correspond- ing to the additional load. This operation is to be repeated up to a given maximum load on P, and the load diminished to its original value, changing by the same steps as were used in the increase. Get directions from the instructor as to the proper loads to use. Experiment 30 Yottng's Modulus Flexure The bars to be tested are rectangular in cross-section. They are to be supported on two knife edges a a with a knife edge c supporting the weights used to deflect them. The experiment is to be performed as follows. Shift the knife edges used as rests to points near the ends of the base and equal distances from the center of the base. Pass the bar to be tested through the U carrying the central knife edge. Shift the bar until the rod connected to the U passes centrally through the hole in the base. The axis of the bar should be at right angles to the knife edges. Adjust the micrometer screw so that its axis is central over the knife edge supporting the weight. Suspend 1 Kg. from the central knife edge and make a reading on the micrometer. Add a known mass to increase the deflection and read the micrometer. If the deflection is ^ Fig. 22 MECHANICS 79 less than a complete turn of the micrometer screw add a greater mass. In like manner make five readings on the deflection of the bar. Do not take any weights off until the micrometer has been raised two or three centimeters. Make two sets of readings on the bar using different values of distances between knife- edges. Make two sets of readings, using bars of the same thick- ness but of different widths. Use the same distance between knife edges as in the first series of readings on the first bar. Make a series of readings on bars having the same length and width but with various depths. In making the adjustments the contact of the micrometer screw with the central knife edge will be indicated by a de- flection of a galvanometer in circuit with a cell, the mi- crometer and the bar. The circuit will be open except when the bar and micrometer are in electrical contact. Determine the width and thickness of the bars used with a micrometer caliper. For a bar supported at two points with the deflecting force applied midway between them PP where e is the deflection due to a force P, I is the length between knife edges, b is the width of the bar, d its depth, and M is Young's Modulus. ,^ PI' iebd^ Arrange data as follows : Material of bars. 1= cm., 6= cm., d= cm. Micrometer zero = Pitch of micrometer screw = turns per cm. 80 MECHANICS P in kg. wt. Micrometer Readings e in cm. Plot one curve using values of P as abscissas and cor- responding values of e as ordinates. Plot one curve using values of P as abscissas and corres- ponding values of e as ordinates. Plot one curve using values of — as abscissas and cor- b responding values of e as ordinates. Plot one curve using values of —- as abscissas and cor- cP responding values of e as ordinates. What do the curves and data show ? Experiment 31 ToBsioNAL Rigidity If a torque acts on one end of a wire the other end re- maining fixed, an element originally parallel with the axis of the wire will be twisted into the form of a spiral, any radius being turned through an angle the value of which depends on the material of the wire or rod, its radius r, the distance I from the fixed end to the radius in question, and the torque acting L, such that ^ = where is the MECHANICS 81 Fig. 23 angle turned through in radians and n is a constant de- manding on the material of which the rod is composed. If rods are square in cross-section use the following for- aula : o V (1 + 0.00367 t). WAVE MOTION AND SOUND 109 Show by diagrams how the air column breaks up. Ar- range data as follows : Frequency of fork= Radius of tube = Temperature = Correction added = 4 Cl 3, i2+.6r=— \ 4 Vi X h-h= - 2 vs Mean velocity = Velocity at 0°C.= Experiment 64 Velocity of Sound in Solids. Kundt's Experiment* Rods made of elastic substances are capable of vibrating both transversely and longitudinally. This experiment is intended as a means of determining the velocity of sound in metal or glass rods along their length. The experiment depends on the resonance of a column of air or some other gas or gases in which the velocity of sound is known. It may also be used to find the velocity of sound in a gas by comparison. I. A rod of the substance to be studied is clamped at its middle (Fig. 30). Longitudinal vibrations are excited in the rod by stroking it with a piece of leather covered with rosin if metal, or with 4 wet cloth if the tube is glass, in the direction of the axis at the end B. A piston on the other end of the rod fits into the tube T at A snugly without binding. The tube T, which must be dry, contains some lycopodium powder or cork dust along its lower side. *Read the theory in connection with Experiment 63. 110 WAVE MOTION AND SOUND J3ig ^ nA I Fig. 30 When the rod is in vibration the vibrations are trans- mitted to the air in the tube by the piston A. The piston D is adjusted until the tube is of proper length for reso- nance. When resonance is obtained the lycopodiiun powder or cork dust will be collected at the nodes. There will be a node at the movable piston D and one near the movable piston A attached to the rod. The piston A will be at some point in the vibrating segment. Excite the rod and adjust the piston D until resonance is obtained. Measure the position of the middle of each dust-heap from some fixed point, and calculate the average interval between them by the same plan as is used for ob- taining periods in Experiments 6, 17, etc. Compute the wave length of the sound in air, remember- ing that the distance from one node to the next is half a wave length. Note the temperature of the air and assuming the velocity of sound in air at 0°C. is 332 meters per second 1 t find the velocity of sound in the tube. v = Vo I 1+ "otq"" From the results obtained compute the frequency of the sound. Since the rod is free at both ends and fixed at its center it vibrates like an open organ pipe sounding its fimdamen- tal. From this fact and the computed frequency find the velocity of the sound in the rod. Disturb the powder in the tube, change the position of the piston A with respect to the tube and make two or three more trials. WAVE MOTION AND SOUND 111 Arrange data systematically and show how computa- tions were made. II. This experiment has important applications beside the one given above being used to find the velocity of sound in other gases than air, and in liquids and solids, to find elas- ticity coefficient, and the ratio of the specific heats of gases. The following will illustrate some of these, and extend. the method by varying the points of clamping. Clamp the bar in the center and make the observations called for in the Manual three times, noting the tempera- ture in the tube; do the same with the bar clamped at points one-fourth the length of the bar from each end, and again clamped at the center and one-sixth the length of the bar from one end. These will give the fundamental and the first and second overtones, if care is taken to stroke the bar near the nodes of undesirable tones; otherwise the tone pro- duced will not be pure, and the dust-heaps formed will be of complicated forms. Measure the positions of the nodes and compute the wave-lengths as described in part I. Find the velocity of sound in the bar for each arrange- ment of the clamps. Measure and weigh the bar, and from the average velocity obtained and the density calculate the value of Young's Modulus, using the formula in Watson, § 281, in which, for this experiment, E is the modulus. With the bar clamped as in the observations on air, make a set of observations on some other gas, and calculate the velocity of sound in it at 0°C. Assuming Newton's formula for velocity, compare the results for air and the other gas, if its density is known, and also see if they agree. If any discrepancy is found, give the explanation. Also 112 WAVE MOTION AND SOUND find the ratio of the two specific heats for the other gas, assuming that for air the ratio is 1.4. Throughout the experiment care must be taken to keep the bar from warming under friction, as this alters its elasticity constants. Experiment 6S A Study of Beats and Intebfekence of Sound Two tuning forks of very nearly the same frequency, mounted on their resonance boxes are used in the following experiments. The frequency of one fork is given. Remove one of the forks from its box and cause it to vibrate. Hold it by the stem in front of its resonance box, or a tube adjusted to resonance (See Experiment 63) and rotate the fork slowly about its own axis. It will be found that for certain positions of the fork the resonance is a minimum. Explain why this is so. The same phenomenon may be observed if the fOrk is caused to vibrate and is held by the stem, prongs down- ward, and slowly revolved about its axis, near the ear. II. Mount the forks on their respective resonating boxes, and set both in vibration. Determine the number of beats per second; and having given the number of single vibra- tions of the known fork per second, compute the correspond- ing number for X from the frequency of beats. III. (1) Stick a small lump of wax on each prong of one of the forks and note the effect on the beats. (2) Adjust the wax weights so as to make the forks in unison. On WAVE MOTION AND SOUND 113 which fork must the wax be placed ? Why ? (3) Observe the effect when the wax is removed from the one weighted to give unison, and large lumps are placed on each prong of the other fork. If the experiment is carefully worked through and thought out in detail, the written report may be made very brief as suggested in the following outline. Outline of Results of Experiment I. Minimum sound in ...positions. Diagrams.* " due to.. Waves reach ear in phase. Why ? II. Frequency of beats C marked single vibrations per sec. X makes.. " " " III. (1) Period of beats ...(increased or decreased). (2) Wax on fork Why ? Pitch of .....(raised or lowered). (3) Tones differ by half steps (approximate number). Effects on ear, how changed ? Experiment 66 Determination of the Frequency of a Tuning Fork In this experiment the periodic time is compared with the periodic time of a short pendulum or a fork of known frequency by causing both to make tracings simultaneously on a moving glass plate coated with "Bon Ami," or whiting and alcohol. The fork is clamped in a holder so that it may vibrate in a horizontal plane with the ends of its prongs very near the short pendulum which is so mounted as to vibrate in a *Relative positions may be represented by points, two for prongs, and one for ear. 114 WAVE MOTION AND SOUND vertical plane. The glass plate is mounted on a carriage which enables the plate to be moved in a horizontal plane parallel with the plane of its face. A stylus attached to the pendulum and another attached to one prong of the fork are adjusted to bear Ughtly on the uncoated glass. Coat the glass with a smooth even layer of the Bon Ami, not too thick. Adjust the plate on the carriage so that when the pendulum and fork are vibrating and the carriage is moved, two sinuous lines will be traced on the surface of the glass. Set the fork and pendulum vibrating and obtain curves' Get three sets of curves by repeating the process shifting the plate on the carriage sufficiently so that there is no overlapping. The curves traced on the glass will be sine curves pro- vided the motion of the plate is uniform. The curves enable one to count the number of vibrations which the fork makes during one vibration of the pendu- lum. This can be done even if the plate were not moved with uniform speed although the curves will not then be true sine curves. Determine the periodic time of the pendulum in the same manner as in Experiment 17, taking the time of every tenth transit to the right from 1 to 91. This gives the equivalent of 250 vibrations. In getting the corresponding number of vibrations of ihe two styli account must be taken of their distance apart in the direction of motion of the plate. Tabulate data for the periodic time of the pendulum as in Experiment 17. WAVE MOTION AND SOUND 115 Number of vibrations of pendulum in a given length Number of vibrations of the fork in the same length Ratio of vibration of fork to pendulum Period of pendulum = Period of fork, computed = Frequency of pendulum = Frequency of fork, computed = Marked frequency of fork= Part IV, LIGHT Ejcperiment 70 Photometry The Bunsen Photometer The Bunsen photometer is to be used in the following described experiment to compare the brightness of different light sources and to study the distribution of the horizontal intensity from one or more light sources. The fundamental principle of photometry (sometimes called Bouger's Principle) is, that if two sources produce equal intensities of illumination at a given place their candle powers are directly proportional to the squares of their respective distances from the place, the areas of the sources being small compared with their distances from the photometer. The apparatus to be used in this experiment comprises a photometer, a graduated photometer bar and two Ught sources, one of which may be assumed as a standard. The two standards in most common use are the Hefner lamp, burning amyl acetate, and the standard sperm candle weigh- ing six to the pound and burning 120 grains per hour. The ordinary parafine candle burns about 123^/2 grains or 8 grams per hour. Directions for performing the experiment. If a candle is used as a standard light source, it should be burned for a little time, until the wick is in good condi- tion, then the flame extinguished and the candle weighed. Place the two light sources on the photometer bar as far apart as possible, at the height of the photometer disk. LIGHT 117 Take frequent photometer readings, for a half hour, being careful to displace the photometer after each reading, and then readjusting it until the disk is equally illuminated over its entire surface. On account of a difference in the color of the light sources this may be somewhat difficult. Do not throw out any readings because they seem wrong. If the light source tested has flat surfaces, as in the case of a common gas burner, place the plane of the flame at right angles to the axis of the bar and take four to six read- ings; then bum the flame through an angle of 15°, take four to six more readings and so on until the flame has been turned through 180°. Do not let the gas flame blow. Sketch the form of the flame before extinguishing it, giving side and top views. At the end of the run extinguish the candle, noting the exact length of time it has been burning, and find its loss in weight. From this loss compute its candle power in terms of the standard sperm candle. From this candle power, and the average distances of the sources from the photometer disk for any position of the secondary source, compute the intensity of the second source for the different orientations. Using polar coordinates, plot a curve showing the dis- tribution of the light about the secondary source. State why the curve is not circular. Mark the position of the flame on the diagram. Original mass of candle = g. Final mass of candle = g. Time at which candle was lighted = h. min. sec. Time at which candle was extinguished = h. min. sec. Whole time candle was burned = h. min. sec. Computed candle power = 118 LIGHT Distance between sources = Kind of secondary source, cm. Position of source (an- gle of plane of flame with bar) Distance of standard from photo- meter, di Dist. of sec- ond source from photo- meter, ck •hyd,' I Average Average Experiment 71 The Laws of Reflection This experiment is given to verify the laws of reflection. The apparatus necessary consists of the note book, a mirror mounted vertically and some pins. Mount a plane mirror vertically in wooden clips. Place it so that the face of the mirror lies along a line on a page of your note book. Stick a pin at ten or fifteen centi- meters from the mirror at b, see figure 31. If there be no vertical markings on the mirror stick pins at points along its face about two or three centimeters apart as at c, d, e Mark the points c, d, e on the note book page. Place the eye successively at such points that looking along the surface of the paper at the reflection of the pin at in the mirror, it will appear to be behind the points c, d, e — Mark points on the lines of sight with a pin point, as at m, n, p, r. Remove the mirror, carefully marking the position of the mirror surface. Connect the points ma, nc, etc., by lines, producing them -IS. LIGHT 119 i \ A g/ b Fig. 31 backward until they intersect. These points of intersection should be coincident if the conditions of the experiment be perfect. Suppose 0' be the point of intersection. Draw the Une 00' cutting the mirror line at h. Draw some line es, perpendicular to AB. Show that the angle of reflection is equal to the angle of incidence, using protractor. Remember that all the images of appear to be at 0' but as 0' is behind the mirror no real images exist. The images are also erect which is in accordance with the general law that all virtual images are erect. Compare the distance from to the mirror at the points a, h, c, d with the distances from 0' to the same points. Explain any peculiarities in the differences. Distance from to mirror at Distance from 0' to mirror at Differences Angle of incidence Angle of reflection u 6 c d e a b c d e 120 LIGHT Experiment 72 Determination of the Angles of a Peism by Reflection* The polished faces of a prism may be used as mirrors and by means of reflection from the surfaces the angles of the prism may be found as in the following experiment. Place a prism with the intersecting edges of its rectangu- lar faces vertical, on a blank page of the note book. Carefully trace the outline of the prism on the paper. Place a pin at 0, Figure 32, in front of the angle A to be measured. Observe the reflection of the pin from each reflecting face, getting the reflected ray to come from as near the apex A as possible. Then set pins along the. re- flected rays, ten or fifteen centimeters from A at e and /, sighting along the surface of the paper. Drop perpendiculars from the points e and / to the sides of the prism produced as at the points m and n. Lj An— An Ad, and LeAm^ LmAd. (Prove.) The required angle, is equal to Z?ivld+ ^wiAd there- fore it is also equal to Z/Aw+ Z eAm. Fio.32 *Read Experiment 71. LIGHT 121 Measure very carefully the lengths fA and /n, and also the lengths eA and em. fn em Then—— = sin ZjAn, and—- = sin ZeAm. fA eA From a table of natural sines find the angles fAn and eAm from which compute the required angle. Find each angle of the prism. Angle fn. fA fn/fA /.fAn em eA. em/eA ZeAm. ZBAC 1 2 3 The sum of the three angles = Difference of sum from 180° = Experiment 73 Determination of the Index of Refraction of a Solid AND A Liquid The theory of the experiment in outline is as follows : A ray of light passing from a rarer to a denser medium is bent toward the normal to the surfaces of contact of the two media at the point of incidence. A ray passing from a denser to a rarer medium is bent from the normal. If i be the angle of incidence and r be the angle of re- fraction for any two media the following relation is found to hold sin z sin r = M (1) In the above expression /i is called the index of refraction for the two media used and is numerically equal to the ratio of the velocity of light in the two media. 122 LIGHT Place a transparent plate with parallel sides on a page of the note book, about the center of the page. Stick two pins vertically into the sheet at a distance of ten or fifteen centimeters apart so that the line joining them makes some angle between 30° and 60° with one edge of the plate. On the opposite side of the plate adjust the line of sight until the two pins are in line looking along the surface of the paper. Then mark out the line of sight with two other pins. Draw the outline of the plate on the page and remove it Draw the lines abo, and edc, through the location of the pins to the points of intersection with the outline of the plate at o and c. Connect o and c. Erect a perpendicular to the edge of the plate at o producing it to intersect the opposite edge at k. Drop a perpendicular from o to the line edc produced Fig. 33 LIGHT 123 at h. Drop a perpendicular from a to some point / on the line k f. af Then — = sin Z a of = sin i ao ck and — = sm Z c o & = sin r, CO from which find ju using equation (l). Repeat the above experiment for two other different angles of incidence. The index of refraction of the liquid is found by putting the liquid in a thin walled glass ceU and proceeding in the manner explained above. The walls of the cell being thin, the error introduced by them may be neglected. Set af. ao. sin. i. ck. CO. sinr. M 1 2 3 The emergent ray is parallel to the incident ray but is displaced a distance oh depending on the angle of incidence, the index of refraction and the thickness of the plate. Displacement of ray 1. 2. 3. Experiment 74 Determination op the Index of Refraction of a Prism* It has been shown in the previous experiment that the direction of a ray of light passing from one substance to another of a different optical density is deviated from its original path so that the following relation holds true : *Read the theory in experiments 72 and 73. 124 LIGHT Sin I -. = M sin r It has also been shown that if the ray passes through a transparent plate with parallel faces that the emergent ray is parallel to the incident ray. If the faces of the plate are not parallel the emergent ray will not in general be parallel to the incident ray, but if produced backward will make an angle with the incident ray produced forward. This angle is called the angle of deviation. It may be proved that if the angles of incidence and emergencies are equal, the angle of deviation is a miminum. The amount of the deviation depends on the angle of the prism and the index of refraction of the material from which the prism is made. When the deviation is a minimum the following relation holds : sin i (a + S) M = sin i a Fig. 34 LIGHT 125 in which d is the angle of minimum deviation, and a is the angle of the prism. Describe a circle on a page of your note book of which the radius is about 10 cm. Stick a pin at some point A on the circumference and at a second point K on a, radius OA, 5 or 6 cm. from A. Place the prism approximately in the center of the circle. Place the eye so that, looking through the prism, the pins A and K appear to be along the line of sight. Rotate the prism about a vertical axis with the fingers, following the images of the pins with the eye until on turning the prism in either direction the line of sight comprising the images of the two pins A and K will make a greater angle with the real line connecting the pins. The prism will then be placed so that the angle of devia- tion is a minimum. Place two pins along the line of sight as at ilf and D. Draw an outline of the prism on the paper and remove the prism. Draw the lines DM and A K intersecting the traces of the prism faces at C and B. Produce the lines until they intersect at some point 0. The angle NOA is the angle of minimum deviation. Draw perpendiculars to the face traces at C and B. The angle at the intersection H will be equal to the angle of the prism. Connect the points B and C and drop a per- pendicular from H to the line thus obtained. Drop per- pendiculars from the points A and D to the lines BE and CF. The angles ABE and DCF are the angles of incidence and emergence. Show that they are equal. Show also that the angles HBC and HCB are equal. Determine the angles a and S and compute the index of refraction of the prism. 126 LIGHT Tabulate data and results as follows AE = cm. AB = cm. sin i = i FD = cm. CD = cm. sin i' = i' HP = cm. BH^ cm. sin r = r HP = cm. CH = cm. sin r' = r' KN = cm. K0 = cm. sin d= d sin u = i {a+S) a = sin I a Experiment 75 Measurement of the Angles of a Prism Using a Spectrometer Index of Refraction of a Prism* This experiment may be divided into two parts, the first part being to determine the angles of a prism and the second to find the index of refraction of the prism. The methods involve the principles already described. s "-Zot.-- Fig. 35 *Read the theory of experiments 72 and 74. LIGHT 127 I. In order to find the angles of the prism it is first necessary to adjust the spectrometer. Adjust the telescope T by focusing for parallel rays using some distant object, at the same time adjusting the position of the cross-hairs by means of the adjustable eyepiece until they are distinct when the inage of the distamt object is distinct and there is no shifting of the cross-hairs over the image when the eye is shifted about in front of the eye piece. Next swing the telescope about its axis until it is in fine with the collimator C and the slit in the end of the colli- mator tube is visible. Then adjust the length of the colli- mator tube until the slit is sharply focused and no parallex is observable. If the table P is not adjustable the instru- ment is ready for use. Place the prism well back on the table as shown in the figure, with the refracting angle to be measured toward the colUmator. The edge should be verti-cal. Move the telescope until the image of the slit is seen re- flected from one face of the prism. Set the cross-hairs on the image and read the vernier of the instrument. Revolve the telescope about its axis until the cross-hairs are set on the image of the slit reflected from the opposite face of the prism. Read the vernier. The angle through which the telescope has been moved is twice the angle of the prism. Shift the position of the prism so that the venier will not read the same as before and make a second determination of the angle. Make three determinations of each angle of the prism. II. Mount the prism approximately on the center of the table with the refracting edge vertical, one face being in the path of the parallel rays from the collimator. Revolve the telescope about the axis of the instrument until its axis lies within the refracted beam. If white light be used the re- 128 LIGHT fracted light will be a spectrum. If monochromatic light be used a monochromatic image of the slit will be seen. For this experiment sodium light is to be used. A Bunsen burner tipped with asbestos soaked in a common salt solu- tion will answer. HaAdng found the refracted image of the slit rotate the table carrying the prism, at the same time following the image by rotating the telescope, until a position is found such that if the table carrying the prism be rotated in either direction the deviation of the ray will be increased. The deviation is then a minimum. Set the cross-hairs on the image and read the vernier. Remove the prism from the table, rotate the telescope and view the slit direct. Read the vernier again. The angle turned through by the telescope gives the required angle. Make two independent observations of the angle of mini- mum deviation first with the light incident on one face adjacent to the refracting edge, then on the other face. From the mean of these angles and the refracting angle of the prism compute the index of refraction for the prism Tabulate all observations carefully and show how computa- tions were made. , Experiment 76 To Detebmine the Focal Length and Radius of Curva- ture OF A Concave Mirror The principal focus of a mirror is that point at which parallel rays incident on the mirror parallel with the' axis will interest. The focal length of the mirror is the distance from the vertex of the mirror to the principal focus. The focal length is equal to one-half the radius of curvature. The above applies to mirrors of small aperature, that is the radius of the reflecting area is small compared to the radius LIGHT 129 of curvature. When this iS not true the rays do not cross at a point but along a surface, called a caustic. Place the mirror and light source with its adjustable support on the optical bench, with a ground glass screen between them. So adjust the position of the screen as to receive an image of the source. Measure carefully the distance from the image and from the luminous source to the vertex of the mirror. Make six such observations for different distances from the source of light to the mirror, and compute / from each pair of ob- servations, and also r. If u and V be the distances of the light source and screen respectively from the mirror, then J_ J L- 1. u' V f r If an object be placed at the center of curvature the rays striking the mirror face will be reflected directly back on themselves, since they will strike the surface normally, and the image will also be at the center of curvature. To test this place the back of the mirror before a window, and a wire or pencil in front of the mirror in such a position that the image of the tip corresponds with the actual tip. When this position is obtained the object and image will hold their relative positions no matter how much the ob- server may move his eyes, yet looking toward the mirror. Measure the distance from the point to the vertex of the mirror. This will be the required radius. Make four or five independent determinations of r in this manner. For a mirror of a given radius of curvature, assuming an object, show how to construct the image graphically. 130 LIGHT Prove that 1 1 1 u V f u. V. /• T. Direct measurements ofr. Mean computed r = Mean observed r = Experiment 77 Focal Length op Convex Lenses Parallel Ray Method The focal length of a thin convex lens is defined as the distance from the optical center to the point on the axis of the lens at which rays incident on the opposite face of the lens parallel to that axis are brought to a focus. It must also be remembered that rays passing through a thin par- allel plate are not deviated and that the displacement may be neglected if the plate is very thin. Close to the optical center of the lens the faces may be considered parallel. From the above statements the law of the lens may be obtained for a thin lens of small aperture. Let be the optical center of a thin lens of which the principal focus is at F, a distance of / from o. is- ^^^ rS^^ F- «'.. Fig. 36 LIGHT 131 An object ab is situated at a distance om from o. Let this distance be called u. The image of ab will be at a'h' a distance on from o. Let the distance on be called v. Assmne directions opposite to that of the incident light as positive and those in the direction of the incident light as negative From the similar triangles aob and a' oh'. ab u a'V .^ -V and from the similar triangles cdF and o' V F cd -/ u . — T— = — - = — — smce ab = cd. a'b' -v+f -V This expression may be changed into typical form J 1 ]_ f V u If the object be at infinity the incident rays will be paral- lel and — will be zero, therefore— r = — from which f = v. u J V Point the optical bench, carrying the lens and ground glass screen, at an open window in such a manner that some distant object may be focussed on the screen. Get the best possible focus and measure the distance from the lens to the screen. This gives the focal length di- rect. Make several independent settings of the screen, noting in each case the distance measured. Turn the lens so that the faces be reversed and make three or four more determinations. Find the mean of all the readings for the mean focal length. The experiment is also to be performed with an auxiliary plane mirror. Place the mirror on a table, with the lens lying on it. Place an object above them, in such a position that the reflected image shall be situated in the air immedi- ately beside the object. Test the adjustment by theparal- 132 Fig. 37 lax method i. e., by moving the eye sidewise and observing if there is an apparent motion of the image with respect to the object. If there is no apparent motion the object lies at the principal focus of the lens. Measure the distance from the mirror to the object, and from this subtract one- half the thickness of the lens.* Make this test three times and find the mean. Explain the principle of the method. Experiment 78 Focal Length of Convex Lenses Object and Image at Finite Distances from the Lens Relation of Size of Object and Image f In this experiment the apparatus required consists of the lenses to be tested, a luminous source, a screen and an optical bench. If u and V be the respective distances of the object and image from the lens, of which the focal length be /then Jl 1_ J^ f V u *This correction is not exact, but is near enough, considering the errors of observation. fRead experiment 77. LIGHT 133 Place the lens to be tested between the ground glass screen and the luminous source. Directly in front of the luminous source place an opaque screen with a small sharp opening. Consider the opening as the object. Adjust the positions of the lens and ground glass screen until a distinct image of the edges of the object is obtained on the ground glass screen. Measure the distances from the object to the lens and from the image to the lens. Measure also one dimension of the object and the cor- responding dimension on the image. Change the position of the lens by a few centimeters and again set the screen to receive a distinct image. Make meas- urements as before. Get eight sets of measurements as directed above; four sets with one face of the lens toward the object and four sets with the opposite face toward the object. Note the effect on the position of distinct focus in one set when red or blue light instead of white light is used. Show, by diagram, how to find the position and size of the image, when the object, its size and distance from the lens, and the focal length of the lens are given. Show from the diagram that 1 _ 1 J_ f V u and also that where i is the size of the image and o is the size of the ob- ject, while u and v are respectively the distances of the object and image from the lens. 134 LIGHT Arrange data as follows . u u. V. /« 0. I. V i Experiment 79 Focal Length of a Convex Lens Changing Position op Lens* The apparatus needed is the same as in experiment 78. The general equation for the convex lens is 111 -T = 1) f V u where u and v are the distances of the object and image from the center of the thin lens. If Z be the distance from the screen to the object then —u+v=l OTU=—{l — v) (2) If this value of u be substituted in equation (1) we have 111 (3) / V l—v The solution of (3) for v gives 2 2>| P- 4fl (4) which shows that for a given lens and a given fixed distance between the object and screen, there are two positions of the lens, symmetrically situated with respect to the mid- point between the object and screen, which will give an image on the screen, provided I is greater than 4/. •Read experiment 77. LIGHT 135 For the first position of the lens producing an image on the screen —r = and for the second position of / «! Wl the lens producing an image, I remaining the same \ 1 1_ j 11 u From symmetry— Ml = y and V\=—u. Let the distance through which the lens must be moved from the first to the second position be denoted by a, then a=U\—u=v—V\=—u—v (5) From equations (1), (2) and (5) /= 4Z DiEECTIONS FOR PeBFOBMING EXPERIMENT With the lens between the object and the screen, find some position of the lens giving a good image. Note the position of the lens. Without moving the object or screen, find another posi- tion of the lens producing a clear image on the screen. Note the second position of the lens. The distance from the ob- ject to the screen is called I and the distance the lens was moved is called o. From observed values of I and a com- pute /. I. u. /■ Lens. Mean value of /. cm. 136 LIGHT For a second set change the distance I between the ob- ject and the screen, and find a new value of a. Get sbc sets of readings, three with the lens with one face toward the screen and threee with the lens reversed. Check your results by the mirror method given in Ex- periment 77. Experiment 80 Focal Length of a Concave Lens Method of Divergent Rays* A concave lens is a divergent lens. That is, if a parallel beam of light pass through a thin concave lens of small aperture, parallel with the axis, it will diverge from the opposite face of the lens in such a way as to seem to come from a point on the first side of the lens. This point is called the principal focus of the lens and its distance from the lens will be the focal distance / of the lens. In a manner similar to that used in Experiment 77 it may be shown geometrically that foi" concave lenses the fol- lowing relation holds between the conjugate focal distances u and V, and the principal focal distance /; J_ 1 _ J_ V u f Let a concave lens be mounted in front of a small bright source of light o situated on the axis of the lens. If an opaque screen through which two small holes a and b, 2 or 3 cm. apart be mounted immediately back of the lens two bright spots, c and d, will be found on a screen placed some distance back of the lens. The rays of light oa and ob after passing through the lens and holes in the screen take the directions ac and bd as if coming from some point o' on the axis. The point o' is therefore the virtual image of the source o. *Read experiment 77. LIGHT 137 Fig. 38 The distance oe = u from the object to the lens may be measured directly, but the distance o'e cannot.be so meas- ured. The distance o'e = v may be computed from the similar triangles o'ab and o'cd. Place a screen directly behind the lens being careful to get the holes a and b placed as nearly symmetrical with regard to the axis of the lens as possible. Place the screen S at such a distance from the lens that as clear spots of light be formed as possible and yet make the distance be- tween them quite a good deal larger than between the holes in the screen. Measure carefully the distance between the centers of the holes a and h and between the centers of the spots c and d, also the distance I between the two screens and the distance from the object to the lens. From the distances ah, cd, and I compute v. Having obtained v compute the focal length of the lens, being care- ful to regard signs. Make several determinations of / changing the dis- tances u and /. 138 LIGHT Prove the formula 1 ]_ V u Tabulate data carefully. Find the mean / of the lens. geometrically. Experiment 81 Focal Length of a Concave Lens AuxiLiAKT Lens Method* The method here used for finding the focal length of a concave lens requires the use of properly chosen auxiliary convex or converging lens. The operation is as follows. Find the position of image produced by the converging lens, and measure its distance V from the lens. For practical reasons so adjust the posi- tion of the object that its distance from the lens is con- siderably greater than the image's distance. Place the concave or diverging lens between the con ■ vex lens and the screen and readjust the position of the screen until a new image is obtained. fr^Sj V^s^rsT Pig. 39 ^J/^rrr^ZTit^k^ Measure the distance Vi of the screen from the concave lens. Measure the distance I between the two lenses. As shown in the diagram the image produced by the convex lens may be considered as a virtual object for the *Read experiments 77 and 80. LIGHT 139 concave lens. Its distance from the concave lens will be V—l = Ui. 1 1 1 For the concave lens ■ = — • Vl Ml /i Make four determinations of the focal length /i, chang- ing the position of both lenses, and also the distances be- tween the lenses. Note that when the converging lens is moved a new image due to it must be found in order to get a new value of V. Care must be taken regarding signs. Arrange data thus : V. I. ui = V — L »i. /i- Average value of /i = Experiment 82 The Simple Mickoscope. Its Magnifying Power The eye is an optical instrument of considerable adapt- ability. Objects at a great distance can be seen but not in the detail often desirable. The objects may be brought nearer but there is a limiting positioft for best definition which varies with different eyes but may be taken as about 25 centimeters. This distance is called the distance of distinct vision. Aids to vision are necessary where objects cannot be brought up close or where at the distance of distinct vision the objects or parts to be studied are too minute to be seen. Such aids are the simple and com- 140 LIGHT pound microscopes and terrestrial and astronomioal tele- scopes. The magnifying power of the optical instrument may be defined as the ratio of the angles subtended at the eye by the image and the object provided the object is not nearer than the distance of distinct vision. The SIMPLE MICROSCOPE in its simplest form consists of a convex lens placed at such a position within its focal length of the object that the virtual image will be at' the distance of distinct vision from the eye. In this case the magnifying power will 'be the ratio of the angles subtended by the virtual image and the object at the distance of distinct vision. Set up a scale at the distance of distinct vision from the lens. Between the lens and scale place the object to be magnified. Adjust the positions of the object until no parallax is observed between the scale and the virtual image, using both eyes, one to observe the scale and the' other to observe the image. Read off the scale the appar- ent size of the virtual image. The eye being placed near the lens it is assumed that distances may be measured from the lens to the object and scale. Measure the corresponding dimensionis of the object. Repeat the observations three times and compute the magnifying power in each case. Find the focal length of the lens used by the method of experiment 77, 78 or 79 and construct to scale a diagram to locate the position of the virtual image and to find its size. Find the magnifying power from the diagram. Show that the magnifying power M is approximately M = — LIGHT 141 where 25 cm. is the distance of distinct vision and u is the distance from the object to the lens. Experiment 83 The Compound Microscope. Its Magnifying Power* When a magnifying power greater than about one hundred is wanted the simple . microscope will no longer suffice owing to spherical aberration. The compound microscope, consisting in its simplest form of two convex lenses, is used for high magnifying powers. A lens of short focal length is used to form a much enlarged real image of an object placed at a point a little more than the focal distance from the lens. The second lens is placed so as to form a virtual image of the real image at the distance of distinct vision from the eyes as indicated in experi- ment 82. Find the focal length of two lenses by one of the methods already given in previous experiments. Use the proper lens to form a real image on a screen, of an object placed near its principal focus. Measure the distances of the object and image from the objective. Set up the second lens to obtain a virtual image of the real image as in experiment 82, the screen having been removed. The scale for measuring the apparent size of the virtual image and the eye lens must be moved together their distance apart being fixed until no parallax is ob- served between the scale and virtual image both eyes being used simultaneously. Observe the apparent size of the image and measure the size of the object, and the distance of the eye lens from the objective. Find the magnifying power of the com- pound microscope. *Read the remarks on magnifying power and method of preceedure in experiment 82. 142 LIGHT Make two other independent sets of observations and compute the magnifying power. Make a diagram to scale based on one set of observa- tions and find the magnifying power from the diagram. It is to be noted that in using a compound microscope the objective is generally moved until clear vision is ob- tained rather than the method here used. The method given above proceeds in the direction of the path of the light from the object to the eye. Since the magnification produced by the first lens is vlu show that the total magnification is M= —J— u u where u and v refer to the distances of the object and real image from the objective respectively, u' refers to the distance of the real image from the eye piece, and 25 is the distance of distinct vision. Experiment 84 Astronomical Telescope. Its Magnifying Poweh* A telescope is an instrument for obtaining a magnified image of a distant object. It consists of an object glass of long focal length and an eye-piece. The image formed by the objective is formed a short distance within the focal length of the eye-piece of short focal length. The rays therefore from any point of the image enter the eye as ap- proximately parallel. The angle subtended by the object at the object glass and at the eye are practically the same. The angle subtended by the rays entering the eye from the real image is nearly equal to that subtended at the center of the eye-piece for the reason given above. *Read the remarks on magnifying power and the method of pro- ceedure in experiment 82. LIGHT 143 The ratio of the angles subtended by the image and ob- ject at the eye are therefore approximately equal to the focal lengths of the object glass and eye-piece. F Find the focal length of two lenses. Use the appropriate one as an object glass to focus some distant object on a screen. With a scale 25 centimeters from the second lens ad- just the position of the lens until the scale and virtual image of the real image show no parallax, the screen hav- ing been removed. Read the apparent size of the virtual image on the scale, and measure the distance from the object glass to the po- sition of the screen and to the eye-piece. With the lenses removed but Tidth the scale and eye in place as before ob- serve the distance on the scale subtended by the object. Compare the ratio of these scale distances with the ratio of the focal lengths. Choose other objects and make two more direct deter- minations of the magnifying power. Construct to scale a diagram showing magnification produced by the telescope. Note that with this instrument the images, both real and virtual are inverted. Experiment 85 Teebesteial Telescope. Its Magnifying Powee* In the Astronomical telescope the eye-piece is used to magnify a real inverted image produced by the objective. It is desirable in many cases to have the image erect. This is accomplished in the terrestrial telescope by placing be- tween the object glass and the eye-piece two lenses sepa- *Read the remarks on magnifying power and method of proceedure in ejcperiment 82. 144 LIGHT rated by a constant distance equal to the sum of their focal lengths. This inverting pair is placed so that the image produced by the objective is at the focal distance of the first lens from it. The light from any point of the image is therefore transmitted from one lens of the pair to the other as a parallel beam and it brought to a focus forming an erect image at the principal focus of the second lens where it is acted on by the eye-piece as in the astron- omical telescope. Find the focal lengths of four lenses. Use an appro- priate one as an object glass. Find the position of the image produced. Place the inverting pair in position and find the new image, then place the eye-piece in position as in exper- iment 84 and note the apparent size of the virtual image. Proceed as in experiment- 84 to find the magnifying power and compare it with the computed magnifying power. Choose another object and make a second determina- tion of the magnifying power. Make a diagram to scale for one of the above cases showing the magnification produced. Experiment 86 The Diffraction Grating The diffraction grating to be used in this experiment consists of a plate of plane glass upon which are ruled many equally spaced parallel lines. For light transmitted through the grating the spaces between the ruled act lines as parallel linear sources of light. The grating is mounted with its lines vertical in a holder which may be moved along a track. Near one end of the track is mounted a screen with a nar- row vertical slit. On the same standard with the slit is mounted a scale with its axis at right angles to the track on which distances from the slit may be read off both to the right and the left. If a source of white light be placed LIGHT 145 behind the screen so that the light may shine through the slit along the axis of the track and be transmitted by the grating then an eye placed behind the grating will see on either side of a central white image, spectra with their violet edges toward the central image. The apparent positions of these spectra as measured on the scale will depend on the distance of the grating from the slit and upon the close- ness of the ruled lines on the grating. The spectra will diminish in intensity as their distances from the centra image increase. The first spectra on either side of the slit are called spectra of the first order, the second spectra, spectra of the second order, and so on. If monochromatic light be substituted for white light then instead of contin- uous spectra will appear images of the slit of the same color and at positions corresponding to the same color in the continuous spectra. If the distance from the slit to the grating be great enough to consider the light incident on the grating as normal then the following relation will hold (Crew, Eq. 190) for the bright images (o -|- &) sin = nX in which a+b is the "grating space" or distance from the edge of one line to the corresponding edge of the next, n is the order of the spectrum, A. the wave length of the inci- dent light and the angle between the normal to the grat- ing surface and the path of the light corresponding to the spectrum of the n"* order. The direction in which the rays for a spectrum of any order enter the eye is indicated by the position of the corresponding apparent image on the scale. The tangent of the angle is easily found having measured the distance of the image from the slit, and the distance from the slit to the grating. The angle and its sine may be obtained from tables. If the angle is small the tangent may be substituted for the sine. This will 146 LIGHT make an error of less than 1% for angles up to 6 degreea The experiment consists in making observations from which to find the wave length of sodium light, the number of lines per cm. on the grating being given. Moimt the grating in its holder and set it about 60 cm. from the slit, behind which is a sodium flame. If possible to get so many readings, the positions of four images on each side of the slit should be read, being careful to note their orders. Measure the distance from the sUt to the grating. This constitutes one set of readings. Make two other sets of readings for different distances from the sht to the grating. In making computations readings may be taken in pairs, the two readings being of the same order. A darkened room is necessary but it will be found con- venient to have enough light for the experimenter to read the position of the images on the scale, the apparatus being so placed that his back is to the light. The light may be properly regulated in a room with a well and adjustably screened window. A sample set of data and computations is given below. Grating No. 12. 2540 lines per in. Grating Space = .001 cm. Distance of slit from grating, 100 cm. =L. Spec- trum order n Scale Read- Diff. A tan sin 6 .^ {a-\-b) sin 6 ings. Central image at 50 n .0000494 .0000498 .0000493 .0000491 .0000494 1 2 3 4 45.1 39.9 35.2 30.1 55.0 59.9 64.9 70.2 9.9 20.0 29.7 40.1 .0495 .1000 .1485 .2005 2° 50' 5° 43' 8° 27' 11° 20' .0494 .0996 .1470 .1965 Part V. ELECTRICITY AND MAGNETISM Experiment 90 Electrostatic Fields When a rubber or ebonite rod is rubbed with a woolen cloth or cat's fur it is found that both the rod and the cloth will attract pieces of paper or other light substances. The rod and cloth are said to be electrified. In a following experiment it will be shown that the bodies are electrified in a different manner. The rod is said to be negatively electrified and the cloth or fur positively electrified. We will presently see that in the case of magnetic poles there are fields of magnetic force surrounding them. In an analogous manner there are electrical fields surrounding bodies charged with static electricity. In these fields there are electrostatic lines of force. It is the object of this ex- periment to study those lines of force. The experimental study of electrical fields of force may be pursued by using an indicator consisting of a short rod of some non-conductor terminated at each end by a gilt pith-ball and mounted to swing freely about an axis. For some purposes an aluminum wire terminated by pith-balls may be used. If the non-conducting indicator is used the two pith- balls are to be charged oppositely by contact with the two terminals of an electrical machine. The pith-ball which has been charged positively can be determined by bringing the indicator near a hard rubber rod which has been elec- trified by friction with fur or almost any other non-conductor. If such an indicator is brought into an electrical field the positively charged ball will be acted upon by a force in the direction of the lines of force of the field and the negative ball by a force in the opposite direction, hence if 148 ELECTRICITY AND MAGNETISM free to revolve the indicator will come to rest parallel to the lines of force with the positively charged pith-ball pointing in the positive direction. If the conducting indicator of aluminum is used it need not be given a charge, but when put into an electrical field it will be charged by induction. It will point along the lines of force the same as the non-conducting indicator except that the positive direction of lines of force cannot be determined by it. By means of one of these indicators determine -the di- rection of the lines of force for several different fields in the immediate neighborhood of the following : 1. A charged sphere. 2. A long charged cylinder with hemispherical ends. 3. Two spheres or cylinders charged alike and placed a few centimeters apart. 4. Two spheres or cylinders charged oppositely and placed near each other. 5. A charged sphere or cylinder near a "grounded" metal plate. 6. An ebonite rod which has been charged in spots by being held in contact with the poles of an electrical machine Illustrate the results of these experiments by diagrams Draw two or more equipotential surfaces on each diagram. Experiment 91 The Gold-Leaf Electroscope Read carfully some standard text on the method of electrification by induction before trying the following experiment. I. Charge a gold-leaf electroscope by induction using a rubber rod which has been rubbed with wool or fur. After the electroscope has been charged note the effect ELECTRICITY AND MAGNETISM 149 on the leaves, of bringing near the knob or plate of the electroscope, either the rubber rod or the wool. Do not bring the rod too near the electroscope. II. Discharge the electroscope. Charge the electro- scope again by induction this time using the wool or fur as the carrier of the inducing charge. Prove that the electro- scope is charged oppositely to what it was when the rubber rod carried the inducing charge. Give all the steps in charging the electroscope together with explanatory diagrams of each step showing distribu- tion of charges and lines of force. Give also the sign of the potential of the electroscope at the various stages of charging. Experiment 92 Fakaday's Ice Pail Expebiment* The object of this experiment is to show that when one kind of electricity is induced an equal amount of the oppo- site kind is also induced. Set the apparatus at least two meters away from static electric machines, and within a meter or so of a gas pipe. Connect a hollow sphere or a hollow cylinder, supported on an insulating stand with the knob of a gold-leaf electro- scope by means of a copper wire. See that the apparatus is completely discharged. Charges on parts of the apparatus where they are not known and on clothing will injure your results. Hence it is necessary to discharge all handles, etc., which cannot be done by contact if they are of insulating material. A certain way is to move the thing to be discharged back and forth over a flame, not so low down as to scorch it; the hot gases over the flame being good conductors, carry off the charge. I. Obtain a charge on a proof plane or carrier sphere *Read experiment 90 and 91 before performing this experiment. 150 ELECTRICITY AND MAGNETISM (a metal sphere with an insulating handle) from an electrical machine and note carefully the action of the electroscope during the following operations : (1) Bring the charged body near the cylinder. (2) Hold it well within the cylinder but not making contact. Move charged sphere about within the cylinder taking care not to make contact. (3) Finally touch the cylinder with the charged sphere. II. Ground the cylinder and electroscope so that they are in a neutral state. (In cold very dry weather touching with the hand may not be sufficient to ground the electro- scope, etc. Always make connection with a wire to a gas pipe.) Also discharge the case of the electroscope in the same way. Obtain another charge on the insulated carrier sphere and proceed as follows, noting the indications of the electroscope. : (1) Introduce the charged sphere within the cylinder, taking care not to make contact, as before. (2) Ground the cylinder or electroscope. (3) Remove ground and make contact between the carrier sphere and the insulated cyUnder. Fig. 40 ELECTRICITY AND MAGNETISM 151 Make careful notes of each step in both sets of observa- tions outlined above and illustrate the distribution of charges and lines of force by diagram. III. Remove the connecting wire from the cylinder and electroscope. Charge the gold-leaf electroscope in the usual manner. Get a charge on the carrier sphere and determine its sign by means of the electroscope. Suspend the charge within the insulated cylinder and test the signs of the induced charges on the inside and out- side of the insulated cylinder by means of a second insulated test sphere and the electroscope. Make contact between the charged carrier sphere and the cyUnder Then test for the signs of the charges on the carrier sphere, the inside and outside of the cylinder Give a general summary in a few words of the conclu- sions to be drawn from the above experiments Experiment 93 HoLz Electkic Machine Make a diagram representing the Holz machine with all parts marked. Run the machine and note the character and maximum lengths of sparks produced without condensers, with small condensers and with larger condensers. With the terminals too wide a part to permit sparking run the machine a few moments, the larger condenser be- ing attached. Stop the machine and test by means of a proof-plane and gold-leaf electroscope the character of the charges on different parts of the machine. Bring the terminals together until a spark passes and test again. Explain the effect of the condensers. Explain how the machine "builds up" when a small charge is ^ven to either inductor the remaining parts being uncharged. 152 ELECTRICITY AND MAGNETISM Experiment 95 Magnetic Figures with Iron Filings The experiment shows the direction, distribution, and characteristic tendencies of magnetic lines of force by means of iron filings. 1. Place a bar magnet in a vertical position in the frame beneath the horizontal glass plate. Sift iron fiUngs evenly over the plate and tap it. Do not use too many filings. Note the arrangement of the filings in radical fila- ments. Are there any indications whether the lines of force are horizontal or otherwise ? Obtain filing figures due to two vertical magnets a few centimeters apart, (a) with unlike poles up; (b) with hke poles up; (c) with a magnet and a soft iron bar. Describe (or draw) and interpret the arrangements of filings. 2. Proceed in like manner with, (a) a two pole magnet placed horizontally against the under side of the glass; (b) a consequent pole magnet; (c) two magnets with like poles first oppositely directed, then in the same direction; (d) a magnet and a soft iron ring in any position desired; (e) a horse shoe magnet. Locate straight lines and neutral points. How is variation of field strength indicated ? Explain. 3. Place a small compass near the magnet pole, then move it so that the motion at any point of the path traced is in the same direction as the needle's length at that point. Draw the line, thus traced, upon the sketch already made. Show by an arrow head the direction in which the positive pole of the needle pointed. In like manner draw a fine in Note. If experiments in magnetism or ciirrent electricity in which tangent galvanometer are employed are to be performed near windows, see that they are closed. The iron window weights are to be in a standard position, in order that the magnetic conditions may be definite. ELECTEICITY AND MAGNETISM 153 some other direction from the magnet. Note whether the lines of force enter or leave the positive end of the magnet. 4. State wherein the figures indicate the two charac- teristic tendencies of magnetic lines of force, viz., they tend to shorten longitudinally and to repel each other laterally. Why do the iron filings map out the lines of force ? Experiment 96 Detekmination of the Pole Strength of Two Magnets The experiment consists in finding the pole strength of two magnets by determining their product mmi and their ratio m/rrii. It depends on the law of force action between poles. Determine approximately the position of each pole. Mark these points with chalk using+and— for the posi- tive and negative poles respectively. 1. Product of pole strengths. Balance A, the lighter of the two magnets with its axis horizontal on one pan of a platform balance. Adjust the magnet B upon outside supports so that it is immediately above magnet A and parallel with it, but separated from it by about 3 or 4 cm. This distance should not be greater than one-fourth the distance between poles of the shortest magnet. Determine the force in grams weight necessary to overcome the at- B_ •' t* -I A Fig. 41 traction between the two magnets and also the force nec- essary to overcome the repulsion when they are placed with like poles opposite. Determine the distance between the two attracting (or repelling) poles. (This is the dis- tance between centers.) 154 ELECTRICITY AND MAGNETISM Tabulate data and results as follows : Vertical distance between axes of magnets with unlike poles near each other, di = cm. Force necessary to balance the attraction of the two pairs of poles in grams wt. = g. /i = dynes. ! cm. = 9- h^ dynes, Product of poles strengths mmi Vertical distance between the axes of mag- nets with like poles near each other, i Force necessary to balance the repulsion of the two pairs of poles in grams wt. Product of poles strength mmi = Average product of pole strengths = Draw a diagram showing graphically all the gravita- tional and magnetic forces for equilibrium. What approx- imations are made ? Discuss fully their influence on the results. 2. Ratio of -pole strengths. Place the magnet A in a vertical position with one of its poles about 8 cm. to the east, magnetically, of the middle point of a good compass. Place the magnet B also in a vertical position with one of its poles to the west of the middle point of the compass. Move B towards or from the compass until the latter points in the same direction as it did when both magnets were ab- sent. Measure the distance of the poles of both magnets from the middle of the compass. Repeat these observa- tions with the magnet pole of A about 10 and 12 cm. dis- tant from the compass. ELECTRICITY AND MAGNETISM Tabulate data and results as follows : 155 1 2 3 Distance of pole A from compass tl tt D CI tt Ratio of pole strength Draw diagram graphically illustrating all the magnetic forces acting on the compass needle. 3. Pole Strength. From the results of the preceding experiment compute the pole strength of each of the two magnets experimented upon. Pole strength of A = Pole strength of B = Experiment 97 To Determine the Moment and the Pole Strength or A Magnet If a two-pole bar magnet is placed in air where it is free from disturbing influence the field at all points equidistant from its poles is parallel to the axis of the magnet and «i oL — .'I »2a. r* --ii^ H ri--^ f«H Fig. 42 156 ELECTRICITY AND MAGNETISM directed from the positive toward the negative pole. The field strengths at these points are inversely proportioned to the cubes of their distances from the poles. (See con- clusion derived below.) If the bar magnet is placed horizontally with its posi- tive pole pointing northward there will be a neutral point on either side where the horizontal component H of the earth's field northward is equal and opposite to the south- ward field / of the magnet. This neutral point may be found by noting where a compass needle or a more deU- cately suspended magnet shows, by pointiug indifferently in any direction, that the resultant field at the point is zero. The distance r of this point from the poles may then be measured directly or may be computed from its distance d, from the center of the magnet and the half length be- tween the poles. In the figure let p be the neutral point where H is equal and opposite to /, the latter being the resultant of the two fields /i and fi due respectively to the north and south poles of the magnet. Let the distance between the poles be 21. From similar triangles in the figure, 21 S H — = —r = — 7^', or 2ml = Hr' r fi mjr M Then since 2mZ=M, M = Hr^ and rn = — ^l The value of H at the station where the experiment was performed may be obtained from the known value of H at a given place by means of a magnetic pendulum vi- brated at the two places. The field intensities are inversely proportional to the squares of the periods. Find the neutral point on each side of the given magnet, measure the distances r and I and compute m and M. ELECTEICITY AND MAGNETISM 157 Experiment 98 To Determine the Product MH.* A magnet suspended by a torsionless fibre will assume a position tangential to the horizontal component of the earth's field, if the axis of the bar be horizontal. If the magnet be displaced through a small angle (so that it may have S.H.M.) about the axis of suspension it will vibrate. The square of its periodic time will be directly proportional to its moment of inertia K about the axis of suspension, and inversely proportional to the strength of the earth's horizontal component H, and the magnetic mo- ment M of the bar. The form of the equation is similar to that in the case of the pendulum. The gravitation pendulum equation is T^ = The magnetic pendulum equation is T^ = 4 Tr'l 9 4: Tt'' K MH The moment of inertia K in the above equation may be F ,-2 shown to be = mass X ( 1 ); where I and r are re- 12 4 spectively the length and radius of the bar. From the above equation it may be seen that if H is known at one station it may be found at a second station by finding the periodic time at both stations, the quantities K and M remaining constant. iTT^ K ^„ 4 7r^ K M Hi MHi *NoTE ON Experiments 98 and 99. If these experiments are to be combined, it is advisable to do them both on the sanie day; combining them amounts to assuming that the magnetic cenditions both of the magnet used and of the locality remain the same, which may not be true if a long time intervenes between the two parts. 158 ELECTBICITY AND MAGNETISM by division The object of this experiment is to find the magnetic moment of the magnet used, the distance between its poles, its pole strength, and the strength of the earth's horizontal component H at one station, H being given at another. If Experiment 99 is also performed H may be found by eliminating between two simultaneous equations. Make a note of the stations used and also the magnet number. Find the periodic time by the method outlined in the table of data below, which is a contraction of that used in Experiments 6, 17, etc. Make two tables of data as outlined below, one for each station. Data No. of transit to right Time of Transit h. min. sec. Sums of numbers Slims of times min. sec. 1 11 21 31 41 51 11 12 5 11 13 21 11 14 38 11 15 53 11 17 9 11 18 25 33 123 90 40 4 51 27 Differences 11 23 Station Data for magnet. Mass = Length = Diameter = Distance between poles = Moment of inertia, K = r= 7.59 sees. Value of H a,t= Y&lneoiM H = Value oiM= Value of m = H 7\ ELECTRICITY AND MAGNETISM 159 Experiment 99 Determination of the Ratio M/H A small compass needle suspended horizontally will assume a position such that its axis will be parallel with the lines of force in the magnetic field in which it is placed. If the field used is that of the earth, the magnet will assume a north and south position. Let a bar magnet be placed in the same horizontal plane with the small mag- netometer needle above referred to, so that its axis is in the horizontal plane through the poles perpendicular to the horizontal component of the earth's field H, through the center of the magnetometer needle. Then at the magnetometer needle there will be two fields at right angles to each other, one due to the earth and the other due to the bar magnet. The field due to the magnet will depend on the strength of the magnet poles and their distances from the needle. The needle will set itself so that its axis will be in the direction of the resultant field. Therefore, if H, the strength of the horizontal component of the earth's field be known the value of the field at right angles to it may be found and also the magnetic moment M of the magnet. The equation giving the relation between the field H due to the earth and that / due to the magnet in the above position is -— = tan q> H If L is the distance from the needle to the center of the bar and I is the distance from the center of the bar to either f 160 ELECTRICITY AND MAGNETISM pole, then form the law of force action, the poles being +m and —m m —m 4: m I L f {L+iy ' (L-iy L'-2L'P^l* iml L AmlL f {V- Py = (I^r^ f™"^ ^hich - = — ^— = tan cp But 2ml = M, the magnetic moment, M {u- py •■• -H = -^iT *"" *^- Place the magnetometer exactly over the station where observations are to be made. Let down the needle. Arith- metical work will be a little simpler if the zero of the scale, which is graduated to 360°, is placed about at right angles to the pointers. Remove all knives, keys, etc., from the neighborhood; stand the magnet with which you are to make observations several feet away, on end. Read both pointers, and take the average of the two readings. Note where this average reading comes on the circle. Clamp the magnetometer-bar nearly east and west lay the magnet upon it with its magnetic center a definite distance east of the needle, read both pointers; take the average and note its position on the circle. Note how many degrees the needle is deflected by the magnet. Transfer the magnet to the same distance west of the needle, with its poles reversed; take the average reading, and find the deflection. If the two deflections are equal, the bar is in the proper position; if not, leaving the magnet on the bar in the position last used, turn the bar so that the needle travels over a number of degrees equal to half the differ- ence of the two deflections and in such a direction as to diminish the larger deflection. Repeat the three sets of readings and the turning of the bar until the two deflections ELECTRICITY AND MAGNETISM 161 are equal. Then go on with the observations called for in the experiment. Place the magnet to be tested on the bar with its axis parallel with the bar, the center of the magnet being far encJugh from the needle to produce a deflection of at least 25°. Read the deflection and the distance from the needle to the center of the magnet. Turn the magnet end for end and read again. Repeat these readings. Then place the center of the magnet at such a distance from the needle as to increase the deflection by 15° or 20° and make another set of readings as above. Then make two sets of readings with the magnet on the opposite side of the magnetometer using the same distances from the needle to the magnet as above. If H be given, from the data taken the magnetic moment may be computed. In performing the preliminary adjustments why stand the magnet on end ? Give diagram with your explanation. If the two pointers do not read 180° apart, what may be the cause ? How does the method above eliminate this error ? Why is the bar moved so as to turn the needle a distance equal to half, instead of the whole, difference of the deflections ? If Experiment 98 has been performed with the same mag- net at the same station, both M and H may be determined. Find distance between poles of the magnet and compute m, the pole strength of the magnet. Locate the positions of the poles in the magnet by laying it in an east and west position with a pocket compass be- side it. Find two points at which the compass needle points at right angles to the magnet, and call these the poles. The data may be tabulated as follows : Number of magnet . Total length of magnet = cm. Distance between poles 21 = 162 ELECTRICITY AND MAGNETISM H = M= m = Center of magnet . . cm. east of magnetometer Magnetometer reading in degrees E W Mean deflection tan 6 AT pole £ N pole W N pole E N pole W 2L Repeat table for the magnet west of the magnetometer. Experiment 100 The Tangent Galvanometeh. I. A. The Formulae and theory of adjustment. The deflection of the needle of a tangent galvanometer depends upon the relative values of the directing force due to the earth's field H acting upon the needle, and the deflecting force due to the field Hi • K i Hi, Fig. 44 produced by the current. When cur- rent flows in the coil the needle turns until its length is parallel to the resultant of the two com- GI POiisnt fields. It is essential that the galvanome- ter be set up so that these components of the re- sultant field will be at right angles to each other. Hi/H = ta.n 6, (1) ^«- ^ e being the angular deflection. The field H^ at the center of a coil carrying a current of / c.g.s. units is proportional to current; '67\ / Hi = ELECTRICITY AND MAGNETISM 163 .■.Hi=GI; (2) G is the proportionality factor called the "Constant of the Coil." It may be proved that gJ-^ (3) r n being the number of turns and r the mean radius of the coil. Substituting GI for Hi in equation (1) gives 7= (ff/G) tan 6 (4) for current in c.g.s. units. For any system of units, Current = Constant X tangent of deflection. (5) H/G is a constant for a given coil in a constant field, and is numerically equal to the current that will produce a deflection of 45°; as tan 45° is unity. The constant by which tan 6 must be multiplied to give the value of current is called the "Constant of the Galvanometer," or the "Re- duction factor." It may be expressed in any of the following forms •.=I'o=H/G = Hr/2 n n for c.g.s. units (6) or h=—p^ = - 7) G 2nn if the current is to be expressed in amperes, since it requires ten amperes to make one unit of current in c.g.s. units. If the value of the constant Id in equation (7) be substituted in equation (4) it is seen that IQH ^ 10 rH tan ^ = tan 0. (8) G 27tn A study of equation (8) shows that the sensibility of a tangent galvanometer depends dixectly on the number of turns in the coil, and inversely on the mean radius of the coil and the strength of the directing field H. Define the c.g.s. electro-magnetic unit of current, and the practical unit. Why should the galvanometer be set 164 ELECTRICITY AND MAGNETISM up SO that the plane of the coil is parallel to the lines of force of the earth's field ? How determine when it is so set? B. Procedure for adjustment. In adjusting the tangent galvanometer for use the two important things to be accomplished are : (1) to get the point of suspension vertically over the center of the gradu- © A XK«j. II ^ • " • ^AA/^^ — Fig. 45 ated circle, and (2) to set the coil with its plane parallel to the lines of the earth's field H. The first adjustment may be made approximately by leveling the needle box by means of the adjusting screws in the base. Then, by means of current, obtain a large deflection and note whether the ends of the long pointers describe arcs of circles concentric with the graduated circle. If they do not, adjust with the level- ing screws until they do. For the first approximation to the second adjustment turn the base of the galvanometer until the coil is approxi- mately in the magnetic meridian. Obtain deflections (somewhere between 30° and 60°) with the current first in one direction then reversed. If the current is constant and the deflections are the same for direct and reversed current, no further adjustment is necessary. If the deflec- tions are not equal the plane of the coil should be turned (by turning the base) until the arcs of the deflections are equal ELECTRICITY AND MAGNETISM 165 for direct and reversed current; then, with no current note carefully the true zero, for each end of the pointer. When the galvanometer is properly adjusted, connect a desingated coil in series with a resistance and a source of FiQ. 46 current. Place in the circuit an ammeter or another gal- vanometer that has been calibrated so that the strength of the current may be obtained. Adjust the resistance so that the deflection is between 30° and 60°. Take readings for each end of the pointer for both direct and reverse currents, and use the mean of the four readings for the deflection. If the currents are obtained by means of another tangent galvanometer make simultaneous read- ings on it and treat the readings in the same manner. If an ammeter be used the current is to be read directly or from a calibration curve for each set of readings on the instrument to be tested. The values of the currents having been obtained from the standard measuring instrument, and the tangents of the angles of deflection of the galvanometer to be tested having been found from a table, the constant Jo of the assigned galvanometer coil may be computed for the par- ticular place in which it is located. C. Calibration of galvanometer. Measure the current through, one lamp alone, two in series, three in series, two in parallel, three in parallel, and a combination of one lamp in series with a group of two in parallel. See Figure 46 for a plan of a variable lamp re- sistance. Compare values of current and explain. 166 ELECTRICITY AND MAGNETISM Compute Jo for each of the above sets of readings. Plot a curve using values of the current as obtained from the standard instrument as abscissas and correspond- ing values of the tangents of the angles of deflection of the tested instrument as ordinates. Discuss the curve. Obtain the value of the mean radius of the coil used and compute the value of H for the given station. Then compute the value of Zo for some other coil of the galvanometer from its dimensions and the value of H as obtained above. Tabulate data as follows : Galvanometer to be tested No Coil.. Ammeter No .-Station- Station... Test Galvanometer 1 Headings of Indicator. Ammeter, readmgs tor current in galvanometer. Direct Rev. Aver. fan e h Direct Reverse Average Current Value of G Value of H Computed Constant for Coil No Number of turns = Mean radius = (? = H = . Jo = Experiment 101 Tangent Galvanometer.* II. Verification of Law and Determination of Constants. I. Connect the galvanometer in series with an am- *Read theory of experiment 100. ELECTKICITY AND MAGNETISM 167 , meter a variable resistance, and a generator. (See Fig. 45). If a storage battery or a dynamo is used as generator, in- candescent lamps may be used as the variable resistance. (Six different resistances may be obtained with three lamps. (See Experiment 100.) Observe galvanometer deflections for several different currents and at the same time read the anuneter. Tabulate data and results as follows : Galvanometer No Coil No. of turns Mean radius Ammeter No... Current by ammeter GSalvanometer readings Deflections Average deflection Tangent of deflection //tan e Direct Reverse i. Mean H= TM * ,-f Plot a curve using currents (in amperes) as abscissas and tangents of deflections as ordinates. Draw conclusions from the curve. Express them in words, also in form of an equation involving a proportion- ality constant. What is this factor and upon what does its value depend ? Show whether the experiment verifies the tangent law or not. II. Having the mean radius of the coil given, compute from the slope of the straight part of the curve the value of H at that place. From the given dimensions compute the constants of the different coils and the values of the reduction factors. Tabulate the results as follows : Constants of galvanometer .used at station .where the horizontal component of the earth's field is 168 ELECTRICITY AND MAGNETISM Coil No. No. of turns Mean radius G. h. III. To illustrate the effect of varying the number of turns n, observe the deflections where the same current is made to pass successively through the various coils of the galvanometer. Use four combinations. Tabulate data and results as below : No. of turns, n / Deflections Direct / ^9.y|3 4 a' -Hr9ryHt#^^ Reverse Tangent of deflection Tan d/n. f\3L. Plot a curve with values of n as abscissas and tangents as ordinates. Draw conclusion. Why should the points theoretically not all lie on a straight line in this particular case ? Experiment 102 Calibkation of a d'Ahsonval Galvanometek* The d'Arsonval galvanometer is an instrument which is coming into general use in commercial as well as scientific work. The underlying principle is practically that of the electrodynamometer. In the galvanometer one coil is replaced by a permanent magnet. This feature makes it *See Nichols and Franklin, Elements of Physics, Vol. II, pp. 40-41, 45; and Nichols Laboratory Manual, Vol. I, (Edition of 1912) p. 242. ELECTHICITY AND MAGNETISM 169 independent of the earth's field and therefore much more convenient than the tangent galvanometer for many pur- poses. Many types of commercial ammeters and volt- meters are modified forms of the d'Arsonval galvanometer. The theory is briefly outhned as follows : Whenever a wire carrying a current is in a magnetic field and has a component at right angles to that field a force acts on the wire tending to push it in a direction at right angles to the field and to the direction of the current. The direction of the force depends on the directions of the field and the current. If a coil of wire be suspended in a horizontal magnetic field so that it may turn about a vertical axis, and a current be sent through the coil, it will turn about the axis until the return torque due to the wire suspension is equal to the magnetic torque. The magnetic torque depends on the strength of the magnetic field, on the number of turns in the coil, the di- mensions of the coil, and the strength of the current flowing in it. The equation connecting these quantities varies with the type of instrument used. In some types of instruments the current is proportional to the angular deflection and in other to cos 6; that isJ = fc^or7 = fc cos 6. In most cases, for small deflection the current is propor- tional, to the number of scale divisions of deflection such that I = ks where s is the number of scale divisions of single deflection. Connect a sensitive galvanometer whose constant is known in series with a variable high resistance, the proper reversing keys, a battery and the d'Arsonval galvanometer to be tested. One set of readings will consist in reading deflections 170 ELECTRICITY AND MAGNETISM of each galvanometer for direct and reverse current for a given resistance. The readings are to be repeated as a check and the means used in computing. Make at least three sets of readings using such resistances as to give a good range of deflections. If the galvanometer whose constant is known is non- sensitive then it will probably be necessary to connect the d'Arsonval galvanometer in shunt with a low resistance, I ^1 Fig. 47 and if the galvanometer resistance be not high enough to keep the deflection on the scale, a resistance will have to be put in series with it. The connections are outlined in Figure 47. Adjust the resistances until good deflections are ob- tained on each galvanometer and proceed as in case one making three sets of readings. The current in the main circuit may be found from the constant and the deflections of the galvanometer therein. The current through the d'Arsonval galvanometer may be computed from the theory of shunts circuit. ELECTEICITY AND MAGNETISM 171 Experiment 103 Ohm's Law foe a Simple Ciecuit Ohm's Law states that the current I flowing in any closed circuit is proportional to the electromotive force E.MF. in the circuit and inversely proportional to the resistance B of the circuit. From this it follows that in a circuit in which the E.M.F. is a constant the current will be inversely proportional to the total resistance. That is, in a simple circuit E 1 ^ = ¥ =^E where B is the resistance of the circuit. The term B in- cludes all the known and unknown resistances of the cir- cuit such as the resistance of galvanometers, batteries, resistance boxes, connecting wires, etc. If we let B repre- sent the known resistances and Bo the unknown but con- stant resistances, equation (1) may be written I = ^T-r^r- =E- xi-|-ito B-\-Bo Put a galvanometer of known constant in series with a known variable resistance, and a constant source of E.M.F. A reversing switch to change the direction of the current through the galvanometer should also be in the circuit. Make a working diagram of the connections in your note book at the time of performing the experiment. Make a series of readings for direct and reverse currents due to eight or ten different values of the known variable resistance. From the values of tangents of the angles of deflection and the 7o of the instrument if it be a tangent galvanometer or from the deflections and the constant per scale division 172 ELECTBICITY AND MAGNETISM <§> Fig. 48 k of single deflection if the instrument be a d'Arsonval gal- vanometer compute the current flowing fpr each resistance. Draw a curve using values of the known resistances for abscissas and the corresponding values of the reciprocals of the current for ordinates. Be careful to use a good scale for plotting. In drawing the curve, leave plenty of space to the left of the origin, as there is usually a considerable negative x-intercept. The slope of the curve in terms of the scale used is numeri- cally equal to the reciprocal of the E.M.F. of the battery. The negative intercept on the x axis gives the value of J?o. Get the E.M.F. from the curve. Draw conclusions. Data may be tabulated as follows : Station No Cell No Kind of cell Galv. No Coil or shunt No Jo or k = H = Resistance box No. - Resistance Galv. Readings Direct Reverse Tan or mean rf if a d'Arsonal valvanom- otcr Value of E from curve = . Value of Ro from curve = ELECTRICITY AND MAGNETISM 173 Experiment 104 Determination op the E. M. F. of a Battery E From Ohm's Law R-\-Ro for a complete circuit, where / is the current flowing when E is the total effective E.M.F. and R+Ro is the total re- sistance, R being the known resistance in the box and iBo the constant unknown resistance. Let Ii be the current for a box resistance of Ri and /a be the current corresponding to a known resistance of R2; then the expressions for the two currents will be E E I\ = _ , _ and h = Ri-\-R(s R2-\-R(i If the /o of the galvanometer be known the current may be computed from the expression already obtained. /=7o tan d or ks. Then as Ro is the constant resistance outside the box in any case it may be eliminated between two equations and the value of E may be determined. Put a galvanometer of 50 turns in series with a reversing key, a resistance box, and two gravity cells. (See Fig. 48.) Take four or more galvanometer readings, for direct and reverse currents, first making R = then of such values as to reduce the galvanometer deflections to about ^/t, V2 and V4 the original deflection. Compute values of I from the formula I = lo tan d and then by eUmination, determine E. 174 ELECTKICITY AND MAGNETISM Galvanometer No. Jo = Station No. R Galvanometer deflections tan 6 I Direct Reverse Average Plot a curve using values of -=- as y's and corresponding values of R as x's. If Experiment 103 has been done, use the data and curve of that experiment; compute E and Ro by eliminating be- tween three pairs of observation equations. Experiment 105 Measurement of Resistance by Comparison From Ohm's Law 7 = 2E 2R (1) If a battery of constant E.M.F. E of which the internal resistance is Ri, be put in series in a circuit with a known variable resistance R, an unknown resistance X and a gal- vanometer whose resistance is Rg the current flowing will be /= (2) R+X+R^ where Ra is the resistance of the battery plus the galvanom- eter and connecting wires. For a tangent galvanometer 7=7o tan d (3) E h tan 6 = - R+X+Ro (4) ELECTRICITY AND MAGNETISM 175 Now if the unknown resistance X be removed from the circuit and B be increased to such a value, Ri that the gal- vanometer deflection is the same as before then I will have the same value and therefore the denominator of equation (4) will be unchanged. .• . R+Ro+X = Bo+Bi, from which X = Ri-R. It is not often possible to get the same deflection but nearly the same deflection may be obtained. Then there may be obtained two equations of the form of equation (4) . E and 7o tan di = — (6) Ki)-\-Ki tan Ro+Ri Dividing (5) by (6)— — = — ^J-i- tano'i R + Ro+X Ra is constant but is perhaps not known. It may be computed as follows : Leaving out the unknown X connect the battery, gal- vanometer and resistance box in series. Adjust the box resistance to get a deflection of about 45° if a tangent or approximately a full scale deflection if a d'Arsonval gal- vanometer is used. Call this resistance R'. Read the de- flection then change the box resistance to R" getting a de- flection about V2 as great as before. The two equations obtained will be from these two equations tan 6' Ro+R" tan^" Ro+R' ■Ro being the only unknown may be readily computed. 176 ELECTRICITY AND MAGNETISM Experiment. Find Ro as outlined above then make two independent determinations of the unknown resistance X using different values of R. Experiment 106 The Slide-Wire Bridge For the theory of the bridge see any text or advanced laboratory manual. Briefly outlined the theory is as follows : Fig. 49 In the circuit shown above, which is the most sensitive arrangement of the bridge, the current from the battery divides at a flowing to c by two paths a d c and a b c but none flowing through the galvanometer when the bridge is "balanced." From Ohm's Law pd = i r. Let z'l be the current in the branch a d c and ii be the current in the branch ab c. Let pdi be the fall in potential between a and d and also be- tween a and b. Let pdi be the fall in potential from cZ to c and also from b to c. Then pdi = iiR = C2ri (1) also pd2 = i\X = Ciri (2) Dividing (2) by (1) ^=—ovx = —R (3) R n n Since the slide wire is supposedly of uniform cross- section the resistance is proportional to the length, there- ELECTRICITY AND MAGNETISM 177 fore for n and r^ we may substitute h and k the correspond- ing lengths and equation (3) becomes k The best ratio of the ratio arms ra and n is one to one. Measure two resistances separately, in series and in multiple. Do not slide the knife-edge along the wire, as it is apt to knick it. Make your connecting wires short and all junc- tions tight; this latter is very important as the method, is a delicate one. First, change the known resistance R until with the slider h near the center of the slide wire, a change of the smallest unit in the box R will change the direction of the deflection of the galvanometer. Then with this resistance fixed change the position of the slider until a position is found at which no deflection of the galvanometer is obtained on closing the battery key. Read the lengths of slide wire on the two sides of the slider 6. The resistance of the unknown may be computed from these lengths and the value of R. For more accurate results, after obtaining one balance with the slide near the center, interchange the resistance- box and the unknown resistance and balance again. Add (together the two segment lengths corresponding to the box, also those corresponding to the unknown, and use the ratio of these two sums instead of the ratio of the segments. For explanation, see Carhart and Patterson, Electrical Meas- urements, p. 37. Tabulate data and results as indicated in the following table : 178 ELECTRICITY AND MAGNETISM Description of resistance Segment ab Segment be Resistance in box Resistance determined Compare the series and multiple resistance obtained from observations with those computed from the averages of the determined values of the separate resistances, ac- cording to the laws for resistances in series and in multiple. How small a movement of the slider will produce a no- ticeable deflection ? Name two or more probable sources of error. Experiment 107 Wheatstone Bridge Box Pattern The box form of the Wheatstone bridge differs from the slide-wire bridge described in Experiment 106 (q. v.) only in form, the same principle applying. The slide- wire is replaced by two sets of resistance coils which form the ratio arms of the bridge. Each set may have such re- sistances as 1, 10, 100 and 1000 ohms so that ratios of 1000 to 1, 100 to 10, 10 to 1000, for example may be used. Generally within the same box will be found the known resistance which has the same function as the known re- sistance in the slide wire bridge. The resistance to be measured forms the fourth arm of the bridge. There are generally two keys supplied, one in the battery branch and one in the galvanometer branch which should be closed in the order named in order to get rid of the effect of self- induction in any arm of the bridge. To measure a resistance take some unit ratio such as ELECTEICITY AND MAGNETISM 179 1000 to 1000 or 10 to 10 and change the value of the re- sistance in the rheostat until a balance shall have been obtained or until a change of the smallest unit in the rhe- ostat will produce galvanometer deflections in opposite direction. The resistance to produce a balance may be computed by interpolation (for method see Experiment 49) if nieither galvanometer deflection is off the scale. Find other values of the unknown by using two other different ratios. In like manner find the resistance of a second unknown. Then find the resistance of the two unknowns when con- nected in series and also in multiple, using three ratios in each case. Draw a. diagram of connections showing them as they were made in the experiment. Compare the series and multiple resistances obtained from observations with those computed from the average values of the separate resistances. For one of the resistances measured determine what unit ratio gives the greatest sensibility of the bridge. Experiment 108 Determination op the Resistance or a High Resistance Galvanometer A direct method for determining the resistance of a high resistance galvanometer is as follows : Connections are made as in figure 50. R^ is made so small that its resistance may be neglected in comparison with that of the galvanometer. The resistance r and Ri are adjusted until a large deflection of the galvanometer is obtained, then Ri is changed so that the deflection is halved. Call this new value of the resistance R^. Since the resist- 180 ELECTRICITY AND MAGNETISM Fig. 50 ance in the new circuit has been doubled while the potential difference ab remains practically the same, (why ?) we have R2 + Rg = 2 {Ri + Rg) orRg = R2-2 Ri If Ri be zero, the method is further simplified. The resisatnce r is inserted in the battery circuit as an aid in controlling the value of the initial deflection of the galvanometer. If a gravity cell is used, or any other cell of fairly high resistance that does not polarize, the cell may be simply short-circuited by a wire having a resistance of a few tenths of an ohm. Experiment 109 Resistance of a Cell by the Half-Deflection Method The connections are shown in figure 51. A sensitive galvanometer G is shunted with a wire J?, whose resistance is small enough to be neglected. The resistance Ri is varied until a good deflection is obtained. Ri is then changed to Ri, such a resistance that the deflection is halved. ELECTEICITY AND MAGNETISM 181 Since the resistance of the battery circuit in the second case is twice that in the first, (why ?) we have Fig. 51 Rj, = R2~2 Ri — r. where i?, is neglected, and r stands for the total resistance of the connecting wires in the battery circuit. Make three sets of readings using different initial values of i?i. Experiment 110 Variation of Potential between Genekatob Terminals It has already been shown in Experiment 103 on Ohm's Law for a simple circuit that the current / can be expressed in terms of the E.M.F. of the generator and the resistance of the circuit in the form W R -^- Rb in which R and Ri, are the external resistance and the internal resistance of the generator respectively. When current is flowing the fall of potential through the external resistance, or what is the same thing, the dif- ference of potential of the terminals of the generator, may 182 ELECTRICITY AND MAGNETISM be written as the product of the resistance between the two points in question and the current flowing. ^VdB=IR (2) The object of this experiment is to study how this -pd changes as the external resistance changes. It is obvious, from equation (1), that as R increases, I decreases. But because of the internal resistance, these quantities do not change in the same ratio, and the product IR in equa- tion (2) does not remain constant. The difference of potential of the terminals of any generator of constant E.M.F. and internal resistance in- creases as the external resistance increases. Its value is when R is 0, and is greatest when R is infinite, being then equal to the E.M.F. of the generator. Equation (1) may be written in the form E=IR-\-IRi from which it may be said that the E.M.F. may be di- divided into two parts one of which, IR, is used to drive Galv. H.R. rXn I. E I — m/mimmm — ' Fio. 52 ELECTRICITY AND MAGNETISM 183 the current through the external resistance B and the other I Rb to force the current through the generator itself. When R = the whole of the E.M.F. is used within the generator which is said to be short-circuited. Connect agravity cell in series with a resistance box which forms the variable external resistance R. The pd of the terminals of the cell is to be measured by means of a volt- meter or galvanometer of very high resistance connected to the terminals of the cells as in Figure 52. If a galvanometer be used in which the deflections are proportional to the current flowing through it Ohm's Law gives Ig = pd/Rg in which Rg is the resistance of the galvanometer branch and pd is the potential difference at the terminals of that branch, in this case at the battery terminals. From the above relation knowing the current constant of the instru- ment a potential constant pdo may be computed such that pd = pdo s in which s is the number of scale divisions of single deflec- tion of the galvanometer. Thus knowing the potential constant and observing the galvanometer deflections the potentials at the battery terminals for various values of the external resistance R may be computed. Since the resistance of the galvanometer circuit is very large the current flowing through it is very small in any case and so may be said not to appreciably affect the potential dif- ference between the terminals of the cell. Make readings of the galvanometer for both directions of the current through it and compute the pd at the cell terminals for the following box resistances, infinite, 100, 60, 30, 20, 15, 12, 10, 8, 6, 4, 2, 1 , and 0.5 ohms. 184 ELECTRICITY AND MAGNETISM Resistance Galvanometer Readings Deflections proportion- ate to pds. Potential differences in Volts Direct Reverse Pdo per scale Station No Cell No division. Plot a curve using external resistances R as abscissas and terminal pd's in volts as ordinates. Choose the scale so that the a;-axis runs to 30 ohms. Draw a straight line parallel to the a;-axis at a height equal to the E.M.F. of the cell. Find the values of the external resistance for which the -pd of the terminals is V2, V* and V'o of the EMI'. Find from the curve the internal resistance of the cell. Note that equations may be combined into I = E/{R+Ri)='pd/R hence R}, could be found from any point on the curve. Perhaps the most convenient point is that for which the pd. = V2 the E.M.F. State the relation between R and Rj, for this point. Experiment 111 Ohm's Law for a Series Circuit The potential difference between two points in a circuit is defined as the work done in transferring a unit quantity of plus electricity from one point to the other. From Ohm's Law it follows that the potential difference between two points between which there is no E.M.F. along the path ELECTRICITY AND MAGNETISM 185 Fig. 53 traveled may be expressed as follows : pd=Ir in which pd is the potential difference, I the current flowing and r the resistance between the points considered. Whenever a current flows through a resistance there is always a dis- appearance of electrical energy. The energy reappears in some other forms as in heat. The E.M.F. of a battery or a dynamo is equal to the greatest possible potential difference between its terminals; that is, when the generator is allowed to give no current, being on "open circuit." As soon as the battery or dynamo is allowed to give current, the circuit being cloesed, the difference of potential between its terminals no longer equals its E.M.F. but is less owing to a loss of potential due to the work done in sending current through the gen- erator itself, the generator having resistance. The loss of potential inside the dynamo or battery will be equal to the internal resistance multiplied by the current flowing. From the above we get the following expressions for a simple circuit. When no current is flowing pd = E between the generator terminals. 186 ELECTRICITY AND MAGNETISM When a current is flowing then pd = E—Iri where pd is the potential difference between the generator termi- nals, E the E.M.F. of the generator, r^ its resistance and I the current flowing through it. If there is more than one source of E.M.F. in a circuit it may be that the current will flow through one generator in a direction opposite to that in which it would send cur- rent if free to act alone. If this is the case the pd between its terminals will be expressed by the following relation : pd = E+Iri„ because the pd must not only be sufficient to send current through the cell owing to its having resistance, but also to overcome an opposing E.M.F. Instruments used to measure E.M.F. and pd are called potential galvanometers or voltmeters. Such instruments must have comparatively high resist- ances and must be sensitive since very little current is supposed to flow through them. If such an instrument be placed in series with a generator the current flowing is very small, consequently the loss of potential within the generator is also very small and the pd is very nearly equal to the E.M.F. of the generator. If a potential galvanometer or a voltmeter be shunted around a resistance, very low in comparison, the current flowing through the instrument will be negligible in com- parison to that flowing through the resistance and the pd between the terminals of the resistance will be practically undisturbed by the presence of the potential measurer. Suppose a high resistance galvanometer of the tangent or d'Arsonval type be in shunt with a resistance, around which the pd is wanted, then pd=Ir. (1) Also the current through the galvanometer is ELECTRICITY AND MAGNETISM 187 pd Ig = — - — = la tan or ks. (2) Kg according to the type of galvanometer used, Rg being the total resistance in the galvanometer branch of the circuit. From equation (2) pd = Rg 7o tan 6 or Rg ks. (3) Equation (3) shows that the pd is proportionate to tan or the number of scale divisions of single deflection, the products Rglo or Rgk being the potential constant of the instrument. If an ammeter and voltmeter are supplied Part I. may be omitted. I. Arrange a circuit so as to send current from one or two cells through a tangent galvanometer or ammeter and a resistance box. Connect the potential galvanometer to the terminals of the resistance, or a part of known value. Observe the deflections of both galvanometers for five or six different parts of the scale. Tabulate observed data and the corresponding computed currents and potential differences. Plot a curve with volts as ordinates and scale readings of the potential galvanometer as abscissas. From this calibration curve the number of volts corresponding to any given deflection may be easily found. If the line is straight find the fraction of a volt represented by each large scale division. Are scale readings proportional to potential dif- ferences ? Why should the resistance of a voltmeter be comparatively very high and of an ammeter comparatively very low ? Give a working diagram of the connections. 188 ELECTKICITT AND MAGNETISM Potential-Resistance Diagram of Circuit. II. Connect four cells, three resistances, and an am- meter or non-sensitive galvanometer in series as shown in figure 53. Note that one of the cells is so connected as to oppose the other three. Connect the point a to the gas pipe. Its potential will then be zero. Connect the volt- meter to points a and b, and observe deflection; then con- nect to points b and c and observe the deflection as before noticing particularly whether the potential rises or falls from b to c. Remember that the potentials can only fall in a resistance. In like manner observe the voltmeter read- ings when connected successively to c and d, d and e, etc., to h and a. The deflection for current in the main circuit should be observed occasionally to determine whether the current remains constant or not. When the pd and cur- rent have been measured for closed circuit, break the cir- cuit and measure the E.M.F. of each cell. Tabulate the observed and computed data as indicated below. Plot a potential-resistance diagram with resistances as abscissas, taking values from the last two columns of the table. This curve must show the current in the cells as well as in the resistance coils. Draw conclusions from the diagram in regard to (1) the fall of potential in the resistances, (2) the fall or rise of potential through the voltaic cells. (3) What is represented by the slope of the lines of the diagram ? (4) What are the tests of accuracy ? ELECTRICITY AND MAGNETISM 189 Table Volmeter No Scale Ammeter No .Scale.. ..volts. ..amps. Ammeter Readings Parts of circuit Voltmeter Readings Resist- ance in Ohm's BMP of cells* Total from o Volts rise in ' potential Volts fall in potential Volts To Volts Ohms 0.218 CeU No. 16 ft Ri c c Cell No. 2d d R2 e *Cell No. 3/ «3 g 'ceU No. 4A Totals 1.582 .128 1.291 .436 .654 .872 1.018 .022 1.0 2.0 4.0 3.0 .5 4.0 1.0 .1 1.80 1.00 1.40 .80 b C d e f 9 h a 1.582 1.146 1.274 .620 1.911 1.039 .021 -.001 1.0 3.0 7.0 10.0 10.5 14.5 15.5 15.6 •Voltmeter readings on open circuit. In plotting the diagram it is well to arbitrarily assmne that the E.M.F. of the cell consists of two parts; half be- ing between the zinc and liquid and half between the liquid and copper (or carbon), and the liquid itself offering a resistance to the current which causes the pd to be less when the current is flowing than on open circuit. Thus pd = E — In where n is almost entirely the resistance of the liquid in the cell. Following this statement plot one-half the E.MJP. of the first cell on zero resistance, 1.8/2 = 0.9 volt in the dia- gram. Then the fall in potential through the cell resistance of 1 ohm is 1.8-1.582 = 0.218 volt. 0.9-0.218 = .682 volt which is the potential to be plotted at 1 ohm. Then we 190 ELECTRICITY AND MAGNETISM ' If f ( L ( ^ . 5 X s -N. \ ( i E ^ N '\ ^ S ^ 1 S,. ^ l* • «i t /• /x ' 7 LU 4- 77«sia't;«.»e«. Fig. 54 have the other half of the E.M.F. of the cell (0.9) to plot vertically above the second point, bringing the pd up to 1.582 for 1 ohm as shown in the last two columns of the table and to the point b of the diagram. From point 6 to c there is no E.M.F. but there is a fall of potential of 0.436 volt through a resistance of 2 ohms hence 1.582-0.436 = 1.146 volts which is plotted at the total resistance of 3 ohms, the point c of the diagram. Here the potential rises again by one-half the E.M.F. of the sec- ond cell, then falls 0.872 volt through the cell liquid resist- ance of 4 ohms and then rises the other half of the E.M.F. to the point d. The same process is followed for the rest of the circuit excepting cell g-h which being in opposition to the other cells causes the potential to drop instead of to rise. Notice that the lines of the diagram have the same slope. The slope = pd/r = current and since the current remains nearly constant the lines should have nearly the same slope. In general the potential diagrams will not be quite so symmetrical as this one which is made from an ideal case. ELECTEICITY AND MAGNETISM 191 Experiment 112 compabison of e.m.f.'s of batteries by potentiometer Method Apparatus. A sensitive galvanometer, a slide wire bridge and contact key, a resistance, cells to be compared, and cells to supply current. Put a low resistance battery of E.M.F. greater than that of the cells to be tested in series with a wire of a slide wire bridge ah. At the point a connect the two cells to be compared so that their E.M.F.'s will oppose the E.M.F. of the main battery. Connect them to the key k and com- plete the circuit through a variable resistance, a sensitive galvanometer and a flexible conductor to the sliding con- tact key at c. If the fall of potential from a to 6 be greater than the E.M.F. of either of the cells to be compared some point, as c, may be found such that the fall from a to c is equal to the E.M.F. of one of the cells. If the key at k be closed putting one of the cells in the lower circuit in series with ■i(i(.|- U-^^i^^ D Fig. 55 the galvanometer then the slider c may be moved until some point is found, such that no deflection of the galvan- ometer is produced whether the lower circuit is closed or open. Then the plus terminal of the cell will be at the same potential as a and the minus terminal at the same 192 ELECTRICITY AND MAGNETISM potential as c, since no current is flowing; and the pd on the slide wire between a and c will be equal to the E.M.F., E of the cell. But pd=Ir from Ohm's law therefore pd = E=Ir (1) Draw potential-resistance diagram to illustrate this. Treating the second cell in like manner some point of "balance" of the slider may be found such as c', so that no current flows through the galvanometer. Then pd' = I'r' = E' (2) But as no current flows in the lower circuit in either case and the resistance and E.M.F. of the main circuit are con- stant and /=/'. The slide wire is supposed to be of uniform cross section; therefore its resistance is proportional to its length and equation (1) may be written E=Irol (3) and equation (2) becomes Ei=r nV (4) Dividing (3) by (4) E_ Irol I E' ~ rnl'~ v that is, the E.M.F. of the cells compared are to each other as the lengths of wire about which they are shunted to pro- duce zero deflection of the galvanometer. Connect as shown in Figure 55 putting the cells to be compared in the circuit between a and the key k. Close the key k putting one of them in the circuit and change the position of the slider to find zero deflection. Make the proper reading of the length ac. Then put the second cell in circuit cutting out the first and get a new balance, taking the proper reading ac'. Take alternate readings about the two cells until five readings have been made about the first cell and four about the second cell. Find the mean of each set. The ratio of these means will be the ratio of the E.M.F.'s. As a check, read the E.M.F. of each cell by means of a voltmeter. ELECTRICITY AND MAGNETISM 193 Experiment 113 Electholysis ElBCTKO CHEMICAL EQUIVALENT OF CoPPER An electric current may pass through certain liquids call- ed electrolytes. In addition to the usual heating and mag- netic effects produced, chemical changes will take place. The electrolyle is broken up into two parts. One of the products of decomposition will carry positive electricity in one direction and the other will carry negative electricity in the opposite direction. These products of decomposition are called ions, and these ions are thought of as carrying positive and negative charges. The quantity of electricity carried by a given inass of a substance is a perfectly definite amount and on this fact are based laws known as Faraday's laws of Electrolysis. Fig. 56 If the two terminals of an otherwise complete circuit be placed in a copper sulphate CuSOt solution it is found that 194 ELECTRICITY AND MAGNETISM electricity will flow in the circuit and that copper will be deposited on the terminal by means of which the current leaves the solution, called the kathode; and if the terminal by which the current enters the electrolytic cell, called the anode, be also of copper, some of this copper will go into solution. The mechanism by means of which this takes place may be looked on as follows. The CuSOi in solution breaks up into Cu ions which carry + charges to the kathode and (SO4 ions which carry — charges to the anode. The Cu ions give up their charges at the kathode and are deposited on it as metallic copper. The SOi ions give up their charges at the anode and combine with the copper to form CuSOi again, which in turn may be ionized. The amount of copper deposited is proportional to the quantity of elec- tricity that has passed through the cell and to the electro- chemical equivalent of the metal. If m be the mass of metal deposited, q be the quantity of electricity which has passed through the cell, and k be the electrochemical equivalent of the metal; then, m=q k = k 1 1 since the quantity is equal to the product of the cur- rent into the time of flow of current. The object of this experiment is to find the electro- chemical equivalent of copper in CuSOi. Two coils of No. 10 or No. 12 copper wire are to be prepared as follows. Two pieces of wire about a meter long are to be cleaned with sandpaper, the copper dust being removed with clean filter paper. Make coils of these pieces of wire by winding them on forms covered with filter paper, one coil being about 5 cm. in diameter and the other about 10 cm. One end of each coil about 5 cm. long is to be left straight and parallel to the axis of the coil. By means of these ends electrical connections are to be made and the coils are to be handled so that the parts to be in the electrolyte may be clean. Carefully weigh both coils. Arrange them in ELECTRICITY AND MAGNETISM 195 an empty jar so that they will not touch and connect them in series with a galvanometer, a battery of constant E.M.F. and low internal resistance, and a regulating re- sistance. (An ordinary resistance box will not answer, as it will not stand so much current as is desirable.) Determine the direction of the current by means of a compass, and adjust the variable resistance to give a de- flection of about 60° on the galvanometer, using a small number of turns. Place the cell in, circuit so that the cur- rent may enter by the large coil. Pour in the CuSOt and close the circuit noting the time. Make a run of an hour taking direct and reverse galvanometer readings so that one reading is made every two minutes, being careful not to break the circuit through the cell when reversing the galvanometer. This may be done by a short circuiting device, cutting out the galvanometer during reversals. When the run is finished remove the coils and wash them under a water tap, roll the coils on filter paper and then rinse in strong alcohol, after which the coils should be weighed as soon as dry. From the Jo and the mean deflection of the galvanometer I may be found; I = Ib tan 6. The time being observed and the mass of copper de- posited being found A; may be computed. If weighings are made with a Jolly balance, find its spring constant as directed in Experiment 24. Great care must be used in this experiment as careless- ness will most Ukely result in the necessity of making more runs. Experiment 114 Measurement of the Capacity of a Condenser* If a condenser be put in a circuit with a battery and a *Nichols Lab. Manual, Edition 1'912. Vol. I, Experiment UJ. 196 ELECTRICITY AND MAGNETISM ballistic galvanometer, and the circuit be closed, it will be found that the moving parts of the galvanometer will be momentarily deflected. The deflection will be caused by a momentary current flowing in the circuit which will supply enough electricity to charge the condenser. A current will flow for a very short time until the potential difference at the condenser terminals is equal to the E.M.F. of the battery. The ballistic galvanometer is one used to measure quantity of electricity. Any galvanometer in which the damping is not too great may be used ballistically, but practical considerations make it advisable to use a gal- vanometer whose period is not so short as to make it hard to read the deflection, or so long as to use up too much time. In accurate work allowance must be made for air and magnetic damping. The quantity of electricity passing through the galva- nometer in a time so short that the moving parts do not get appreciably away from their positions of rest before the whole electric charge has passed, may be expressed as follows : Q = Qo (1 + \/2) S (1) in which Q is the quantity of electricity that has passed I \ II Fig. hi BLECTEICITY AND MAGNETISM 197 through the galvanometer, Qo is the ballistic constant of the galvanometer, A. is its damping factor, and S is the deflec- tion, for small deflections. The capacity C of a condenser is that quantity of elec- tricity that will produce unit difference of potential between its terminals. From this it follows that Q=:Cpd (2) From equuations (1) and (2) we get Cpd = Qo (1 -I- A./2) tf and C = -% (1 -|- A./2) S (3) pa From the above expression it will be seen that if the bal- listic constant of the galvanometer, its logarithmic decre- ment and the E.M.F. of the battery be known the capacity of a condenser may be determined. It will also be seen that the capacities of two condensers may be compared " without knowing the galvanometer constant if the same E.M.F. be applied in each case, since A. will be the same, for C, = -^ (1 + A./2)