. • ik , ' >■ ■ i'K Return this book on or before the Latest Date stamped below. A charge is made on all overdue books. U. of I. Library FEB 2 7 '37 i'tAR -5 '37 m 'i? j 7 fifty -sT JUL 1 5 1339 * i JUL 2 7 1333 * < ; r ' ; _l . 9324-S / / U7 r / ■> .- 7 ^ 1 If // I K 1 \La / * ■ • '•;* / y (A> i A / | \k >. y! />A -\ o THE UNIVERSITY OF ILLINOIS LIBRARY 530.01 Sch 83 m2- Remote storage Y * 'V •; \ . % ■ .S'- .. •• \ ■ Manual of Experiments in General Physics BY William F. Schulz, Ph. D. Assistant Professor of Physics , University of Illinois With Additional Experiments by E. H. Williams, Ph. D. Associate in Physics 9tr . University of Illinois SEP 1 9 1919 Printed for Sophomore students in the laboratory courses in general physics, University of Illinois (Not Published) Rogers Printing Company Dixon and Chicago, Illinois 5 3 0 » o 1 5dk, DEMOTE 1 A v A >v >v Vw F 1 01 Preface- 3 / 5ig This manual is written for the use of sophomore students in the general physics courses at the University of Illinois. The directions are such as apply to the particular forms of apparatus and facilities to be found in this laboratory. The methods of performing experiments, and the designs of special forms of apparatus have been developed under the supervision of Professor A. P. Carman, Director of the Physical Laboratory. In the second edition of this manual, Dr. E. H. Williams, Associate in Physics, has added or has re-written experiments Nos. 4, 6, 10, 18, 19, 20, 22, 25, 30, 48 and 55. The laboratory course runs parallel with the lecture course in general physics. The experiments are per- formed in groups of three, which are more or less related, but independent so far as the order in which they are per- formed is concerned. With ten sets of each apparatus, a class of sixty students is provided for in the laboratory at one time, by letting students work together in pairs and having the three related experiments in operation simul- taneously. The experiments to be performed during the first semester are described in detail; the directions for those designed for the second semester’s work are made less definite, leaving the student some choice in the method of procedure, or giving him a chance to exercise his ingenuity and making him more dependent upon his own knowledge of the underlying principles. Material for the tables in the appendix has been drawn from a large number of sources, and it would be difficult to give credit to all of these, hence such references have been omitted entirely. Physics Laboratory, University of Illinois June 30, 1916. CONTENTS Introduction page Laboratory Directions ..... 9 Units ........ 12 Significant Figures ... . . . .14 Errors . . . . . . , .15 The Plotting of Curves ..... 20 Experiments in Mechanics no. 1 — Density of Solids ..... 27 2 — Testing a Spirit Level ..... 33 3 — Equilibrium of Concurrent Forces. Composition and Resolution of Forces .... 37 4 — Equilibrium of Concurrent Forces by the Triangle of Forces ...... 40 5 — Equilibrium of Non-Concurrent Forces . . 44 6 — Parallel Forces ...... 49 7 — The Fine Balance ..... 52 8 — The Pendulum . .... 58 9 — The Law of Falling Bodies . . . 66 10 — Acceleration of Gravity with a Falling Tuning Fork 69 11— Friction 72 12 — Stretching Wires and Young’s Modulus . . 77 13 — Laws of Bending of Rods. Modulus of Elasticity 82 14 — Laws of. Twisting of Rods. Modulus of Rigidity 87 15 — Moment of Inertia . .... 90 16 — The Torsion Pendulum . . . . . 95 17 — The Spiral Spring Balance. Specific Gravity . 100 18 — Specific Gravity with Hydrometers . . . 105 19 — Specific Gravity by Hare’s Method . . 108 20 — Surface Tension : Part 1. — By Capillary Tubes . . . Ill Part 2. — By Means of the Jolly Balance . 114 21 — Boyle’s Law .... . . . 116 Experiments in Heat 22 — Calibration of Thermometers . . . . 121 23 — Charles’ Law . . . . . .126 24 — Linear Expansion . . . . . .131 25 — Coefficient of Cubical Expansion of a Liquid . 134 Calorimetry . . . . L . .138 26 — Specific Heat . . . . . 139 27 — Mechanical Equivalent of Heat . . .142 28 — Latent Heat of Fusion ..... 147 29 — Latent Heat of Vaporization . . . .149 30 — Hygrometry, Humidity and Dew Point . . 152 Experiments in Magnetism and Electricity 31 — Magnetic Fields of Force . . . .157 32 — Action Between Magnetic Poles . . . 162 33 — Determination of H. and M. . . . . 168 34 — Galvanometers . . . . . . 173 35 — Ohm’s Law and Potential Drop . . . 184 36 — The Wheatstone Slide Wire Bridge . . . 190 37 — The Post Office Box Bridge . . . .199 38 — The Potentiometer . . . . 204 39 — The Figure of Merit of a Galvanometer . . 213 40 — Internal Resistance of a Galvanic Cell by Ohm’s Method ....... 217 41 — Potential Difference at the Terminals of a Cell as a Function of the External Resistance . . 218 42 — Electrolysis and the Copper Voltameter . 224 43 — Electromagnetic Induction .... 229 44 — Joule’s Law and the Electro-Calorimeter . . 233 Experiments in Light v 45 — Photometer ....... 237 46 — Reflection and Mirrors ..... 242 47 — Refraction and Lenses . . - . . 252 48 — Index of Refraction of a Lens . . : 258 49 — Critical Angle and Index of Refraction of Glass in Air ....... 265 50 — Microscopes and Telescopes .... 269 51 — The Spectrometer ...... 273 52 — The Diffraction Grating ..... 277 Experiments in Sound 53 — Graphical Determination of Pitch of Tuning Forks 284 54 — Vibrations of Stretched Strings . . . 286 55 — Resonant Air Columns ..... 288 Appendix I — Common Conversion Factors . . . 292 II — Specific Gravity ..... 293 III — Elastic Constants . . . . 293 IV — Moments of Inertia ..... 294 V — rBarometer Corrections . . . . 294 VI — Heat Constants - . . . . 295 VII — Steam Temperatures . . . f 296 VIII — Hygrometry Table ..... 296 IX — Specific Resistance ..... 297 X — Electrochemical Equivalents . . . 297 XI — Indices of Refraction ..... 297 _ _ + XII — Natural Trigonometric Functions . . 298 XIII — Logarithms ..... 299-300 Introduction. Laboratory Directions. — In order to facilitate the dis- tribution of apparatus, handling of large classes, etc., the following method of procedure is in use. The experiments are performed in groups of three, the classes being divided into three groups, and the particular experiment to be performed by each group is posted on the bulletin board. Experiments are performed in cyclic order; that is, group one will perform the experiments in the order 1-2-3, 4-5-6, etc., group two in the order 2-3-1, 5-6-4, etc., group three in the order 3-1-2, 6-4-5, etc., the order remaining the same throughout the semester. Students generally work in pairs, the assignment of partners being made at the beginning of each period. Stu- dents are required to read carefully the description of the assigned experiment so that they are prepared to perform it without referring to the manual. The first ten or fifteen minutes of each period are devoted to a discussion of the experiment to be performed. The instructor calls attention to those parts of the experiment which require particular care, and students are given a chance to ask questions concerning those points which they do not understand. At the beginning of a laboratory period, each student and the partner assigned him for that particular period will be given a key to an apparatus locker, and three blank data sheets on which will be written the number of the experiment assigned, the number of the apparatus locker and the place of working. The laboratory manual gives a list of appa- 10 Manual of Experiments ratus needed, and the directions for performing the experi- ment. This list will be found on the data sheets also. The student will inspect the apparatus in the locker and at the place of working, notifying the instructor at once of missing or damaged apparatus. Students are held responsible for apparatus assigned them. The experiment will be per- formed in accordance with the laboratory manual, and the results obtained recorded on the data sheets. The measure- ments taken may be recorded first in pencil on one of the sheets, but must be copied in ink on the others. The nu- merical calculations should be completed in the laboratory so that the instructor may point out any errors. Apparatus must be returned to the locker, and the place of working be left clean. The students will then return the locker key and data sheets to the instructor, who will stamp- them and return the ink copies to be bound in the written report of the experiment. Alternate periods will be devoted to performing an ex- periment and writing a report of it, until the student has learned how to write the reports properly. After the first few weeks, three experiments will be performed in successive laboratory periods, and followed by a written report of only one of them. The results of calculations with the data taken, and answers to questions and problems, are however always due at the beginning of the period following that on which the experiment was performed. In any case, each set of experiments will be followed by a written quiz on all three of them. Writing of Reports: — The report of each experiment is to be written in ink and bound in one of the flexible covers used for Engineering College reports. The title page should be properly filled out in free hand lettering. In General Physics 11 The data sheet should precede the report and when a curve is plotted it should be placed face to face with the data sheet. The curve must be drawn according to the directions given at the end of this introduction. In writing up reports the following topics should be treated in logical order — (a) Object of experiment; (b) Theory; (c) Method; (d) Conclusions and discussion; (e) Questions and problems. Under “Object of experiment ” give a brief and accurate statement of what you think is the object of the experiment. Under “ Theory ” give briefly the fundamental theory underlying each experiment and definitions of the physical quantities which have been determined, as well as the equa- tions used in calculating results. Under “Method” draw (in ink and with instruments) a diagram of the essential parts of the apparatus used whenever this will aid materially in describing the experi- ment, and give an abstract of the process of performing the experiment. Avoid the use of personal pronouns. Under “Conclusions and Discussion” tell briefly but accurately all that you have learned about the phenomena illustrated by the experiment, and discuss the accuracy of your results, stating the sources and magnitude of errors which may occur and methods of avoiding them, as well as the magnitude of the errors made by neglecting any fac- tors in the computation. Answer the questions and solve the problems assigned for each experiment, indicating the method of solving and giving the final result. References: — References will be given to sections in the following books: D. — Duff’s Textbook of Physics, Third edition. G. — Ganot’s Physics (Atkinson) Seventeenth edition. 12 Manual of Experiments K. — Kimball's College Physics. W. — Watson's Text book of Physics, Fourth edition. Units: — The magnitude of every physical quantity ob- tained by measurement or by calculation must include the unit in which this quantity is expressed, and a numeric giving the number of times the unit is contained in the quantity. It is evident that we can have no idea of the magnitude of a physical quantity in general unless it is expressed in terms of a definite unit. Thus to say that a length is two conveys no idea of its size unless we state whether it is two inches or two miles, etc. On the other hand some physical constants are ratios between two physical quantities of the same kind. For example, the specific gravity of a substance is the ratio of the density of the substance to the density of water, and is expressed as an abstract number without a specific unit being mentioned. Specific heat of a substance is the ratio of the quantity of heat required to raise one gram of the substance one degree Centigrade at a definite temperature to the quantity required to raise one gram of water one degree at the same temperature, and a number is all that is necessary to express this ratio. Systems of Units: — D. 149, 150, 154. — All the units of Mechanics may be expressed in terms of three independent units called “fundamental units." All other units are de- fined by reference to these fundamental units and are called “derived units." A system of units in which the derived units bear the simplest possible relation to the fundamental units is called “an absolute system." The three fundamental units in general use are those of length, mass and time. In the C. G. S. absolute system, the centimeter, gram and second are the fundamental units. In General Physics 13 In the F. P. S. absolute system, the foot, pound and second are the corresponding fundamental units. A third system of units, sometimes called the gravi- tational system, is frequently used. In the absolute sys- tems, the unit of force is defined as the force which gives unit acceleration to the unit mass. The C. G. S. unit of force is called the dyne, the F. P. S. unit the poundal. In the gravitational systems the unit of force is the weight of unit mass, or the force with which the earth attracts the unit mass, giving it the acceleration of gravity, 11 g. ’ ’ Since the gravitational unit of force gives to unit mass in unit time “g M times as much acceleration as does the absolute unit, the gravitational unit of force must equal the absolute unit multiplied by the acceleration of gravity “g M . The gram weight is 981 dynes where g = 981 cm. /sec 2 . The pound weight is 32 poundals where g = 32 ft. /sec 2 . In expressing the result of a measurement or calculation the following directions should be followed: Be sure to state the unit in which the result is expressed. Unless otherwise directed use the centimeter, the gram and the second as the unit of length, of mass and of time respectively. Express all fractions in the decimal form. To avoid repetition of cyphers, express all large numbers in positive powers of ten, all small numbers in negative powers of ten. For example, 30,000,000,000 = 3 x 10 10 . 0.0000117 = 11.7 x 10“ 6 Be careful to distinguish between unit mass and unit weight (used as a force).. Use the symbols gm. and lb. for gram and pound mass respectively, the symbols gm. wt. and lb. wt. for gram and pound weight respectively. Where a unit has not been given a name, express it in 14 Manual of Experiments terms of fundamental units. For example, in the C. G. S. system, the unit of velocity is one cm. per sec. (cm. /sec.); unit acceleration, one cm. per sec. per sec. (cm. /sec 2 .) etc. Significant Figures: — The accuracy of physical mea- surements depends amongst other things upon the instru- ments or apparatus used. For instance in measuring a length, the vernier is correct to 0.01 cm. and the micrometer to 0.001 cm. But it is possible to estimate fractional parts of a scale division on these instruments, so that we can ex- press the length to one more decimal place than that indi- cated above, although this last figure is somewhat in doubt. It is customary to retain this last figure in the measurement. If however all the doubtful figures are retained in calcula- tions with these measurements, we shall waste considerable time in unprofitable calculations and also give a false idea of the accuracy of the final result. For instance if we are determining the area of a rec- tangle and the measurement of the sides gives us 4.35 and 11.51 respectively, the last figure in each being doubtful, (which is indicated here by underscoring them), the product is 50.0685, but the underscored 0 and all the figures after it result from multiplication with a doubtful figure, and are therefore doubtful themselves. These figures should there- fore be discarded in any further calculations, to save time, the area being written 50.1 sq. cm. If retained they would give the false impression that the area as calculated is correct to the ten thousandth decimal place. Again if we are determining the density of a body (or mass per unit volume) and find its mass and volume to be 49.2gm. and 4.352cc. respectively, the quotient of these two will give 11.305, but the underscored 3 and the following figures are the result of operations with doubtful figures and In General Physics 15 are themselves doubtful, so that the density should be written 11.3 gm./cc. All figures up to and including the first doubtful figure are called significant figures. It is readily seen from these examples that the number of significant figures in the final result is the same as the least number of significant figures in any one of the factors. If the second doubtful figure is less than five it is to be discarded. If it is five or more the first doubtful figure is to be increased by one, as in the first example given. In computations with logarithms, as many places should be retained in the mantissae as there are significant figures in the data. Thus log 3.1416 = 0.49714. The characteristic is not to be considered in counting the figures. When the purpose of a zero is simply to fix the decimal place in front of significant figures it is not counted as a significant figure. Thus .0345, 1.03, and 32.0 each have three significant figures. Where the omission of the second doubtful figure would make a difference as great as, or greater than the experi- mental error, it should be retained. Errors: — Since the accuracy to which physical measure- ments are made depends amongst other things upon the construction of the instruments used, there is always a calculable error in every measurement. Thus a micrometer may indicate a length correctly to 0.001 cm., but by estimat- ing the fractional part of a scale division, the length may be expressed to 0.0001 crm If the scale divisions are sufficiently large and the eye of the observer sufficiently trained, the measurement may be correct to 0.0001 cm. Usually the readings with the particular micrometer calipers to be used are correct to only 0.0003 cm. or even 0.0005 cm. The possible error in the length from this source alone would 16 Manual of Experiments then be 0.0005 cm. and a length X measured with such a degree of accuracy would be recorded 0.0005 cm. It is sometimes desirable to know the effect that such errors have on the final result of computations with these measure- ments. We shall develop approximate rules for the cal- culation of such errors. , (i a ) — Rule for sums and differences . — If the only operations performed with the measurements are addition and subtraction , the maximum possible error in the final result will be the sum of the errors in the measurements . For instance if two measurements X and Fhave errors =*= a and =*= b respectively, their sum will be (X-\-Y) =±= (a=*=b) and their difference (X— Y) ± (a=Fb). In each case the maximum error is the sum of a and b, the minimum error their difference. If now the operations involve multiplication or division, it is convenient to use the percent error rather than the numerical error. A percent error means the error per hun- dred parts. Thus an error of 2 parts in fifty is an error of 4 parts in 100 or 4%. An error of 0.004 parts in 0.020 is (0.004/0.020) X 100 or 20%. The percent difference be- tween readings of the same quantity is calculated from the mean or average. Thus if two measurements give 49 and 51 units respectively, their mean is (49 + 51)/2 = 50, their difference 51—49 = 2, the part difference is 2/50 and the percent difference (2/50X100) =4%. The error from the mean is sometimes asked for. This is the difference between any one reading and the mean. In the case of two numbers the percent error from the mean of the two readings is the same, namely 50-49 50 51-50 x 100, or — ' 50 x 100 = 2% Or in general, if X and Y be the numbers, the % error In General Physics 17 from the mean will be X-Y X+Y X-Y — — _ _i_ _ x 100, or — x 100. 2 2 X+Y (b) — Rule for products and quotients.: — The maximum possible percent error in a product or in a quotient is the sum of the percent errors in the separate factors provided all of the errors tend to increase or all tend to decrease the product or quotient. That is, if u, v, and w are respectively the percent errors in X, Y, and Z, then the error in any multiplication or division between these factors (as for instance XY/Z) will be ( u+v+w ) %, under the condition mentioned. (It should be stated that this rule is only approximate, but that it gives a good idea of the maximum possible per cent error and saves considerable calculation that would be necessary if the numerical errors were carried through the whole calculation.) To derive this rule let the measurements X and Y have the numerical errors =±= a and =±= & respectively multiplying (X+a) by (Y+b) we get {X+a)(Y +b) = XY +aY +bX+ab. a and b are small quantities and their product ab is usually a very small fraction which may be neglected in comparison with aY and bX. The error in this product is therefore aX+bY., for if X and Y were accurate, their product would be XY. The fractional error is therefore the difference between the calculated and true values of the product divided by the true value, or (aY+bX)/XY = a/X+b/Y. The percent error is this fraction of 100. The same result is obtained when the negative sign is used with a and b. But 100 a/X is the percent error in X and 100 b/Y is the percent error in Y. Hence the percent error in the product is the sum of the percent errors in the factors. If now we divide (X+a) by ( Y—b ), (using opposite signs so that both errors affect 18 Manual of Experiments the quotient in the same way) we may write (X-\-a)/(Y—b) = (X+a) ( Y-b)-\ Expanding the last term and neglecting products and powers of the errors which become very small, we have (X+a) (l/Y+b/Y 2 +etc:)=X/Y+a/Y+bX/Y*+etc. The fractional error is therefore (a/Y+bX /Y 2 ) /{X /Y) = a/X+b/Y and we see as before that the percent error in the quotient is the sum of the percent errors in the numerator and denominator. The same result is obtained when the signs of a and b are interchanged. (c) — Rule for powers and roots — If the measurement be raised to any power, the percent error in the power is the percent error in the measurement multiplied by the index of the power, i. e. if squared, the percent error is multiplied by two, for the square of a number is that number multiplied by itself and by rule b) the percent error in a product is the sum of the percent errors in the factors. If the root of the measurement is extracted, the same rule applies, for the square root is the measurement raised to the Y /i power, etc. Hence the rule for a number raised to a power may be used, and if u be the percent error in X. the percent error in V X will be uf 2. etc. Multiplying or dividing the measurement by a constant quantity (in which there is no error of measurement) does not alter the percent error. If K be a constant and X a measurement with a numerical error ±< 2 , then K (X=±a) = KX^=Ka. But Ka is the same percent of KX that a is of X. The same reasoning holds if we multiply by 1 /K, or divide by K. If on the other hand we divide the constant K by the measurement X with an error of b%, since the error in K is 0 , the error in the quotient is by our rule (o-\-b)% or the same as the error in X. In General Physics 19 Percent errors are also useful in judging the relative importance, so far as the final result is concerned, of the errors in the measurements which enter as factors in the calculation of this result. It is evident that a given numeri- cal error is a larger percent of a small measurement than it is of a large measurement. For instance if a micrometer caliper is accurate to 0.0005 cm. then a measurement of a length of 5 mm. made with it will have a possible error of (0.0005/0.5) X 100 or 0.1% whereas a measurement of 10 cm. would have a possible error of (0.0005/10) X 100 = 0.005%. It is therefore advisable to make the small meas- urements more carefully than larger ones and with instru- ments capable of greater precision. But this is not the only thing to be considered, for the factors in a computation do not all have the same importance. Some of the factors enter to a higher degree than others and it should be remembered that a measurement which is cubed has its percent error multiplied by three, etc. Take for example, the volume of a cylinder of length / and radius r. The volume is wr 2 l , and it is evident that the relative importance of the errors in r and / depends upon their magnitude and the degree to which they are involved. If r and / are equal and are measured with the same instrument, the percent error will be the same in both, but since r is squared while l is not, the error in r has twice as much effect upon the accuracy of the result, as the error in /. If l = 2r the percent error in r will be twice the percent error in / and the error introduced into the calculated volume by r 2 will be four times that due to /. If r = 2l the percent error in r will be 3^2 the percent error in / and the percent error in r 2 will equal that in /, so that both are of equal importance in the calculation, etc. In this way the relative importance of the different factors in a computation may be studied. It is always ad- 20 Manual of Experiments visable on beginning an experiment to obtain some idea of the relative magnitudes of the measurements made, (by a preliminary trial if necessary), and the degree to which they are involved in the computation, to determine which should be measured with the greatest care and the most ac- curate instruments. It must be remembered that the num- ber of significant figures m the result of the computation depends upon the least number of significant figures in any one of the factors. To avoid useless computation it is advisable to inspect the formulas used in calculating results and to disregard all terms wdiose omission would introduce negligible errors only. For instance when a shunted galvanometer is used in series with a large resistance, it is well to calculate what percent- age of the total resistance is included in the galvanometer and shunt. If their omission introduces a small fraction only of one per cent error in the final result, the term for the shunted galvanometer should be discarded in the calculation. The Plotting of Curves: — The relation between two physical quantities which vary, the one with the other, is often more clearly seen from the curve or graph obtained by plotting on coordinate paper a series of simultaneous ob- servations of the two quantities. Such a curve has other uses besides presenting clearly to the eye the relative varia- tions of the two quantities. For instance we may, by inter- polation, determine corresponding values of the two quan- tities, other than those for which the data have been taken. Again when one or two plotted points deviate very markedly from a smooth curve drawn through the remaining points, we may discard the data for the former as being in error. Usually such curves are plotted on rectangular coordinate paper having centimeter of half inch spaces divided into tenths. Sometimes however it is advantageous to plot polar In General Physics 21 graphs on circular and radial coordinates, or logarithmic curves on specially ruled logarithmic paper. In the follow- ing discussion only rectangular coordinates will be used. Two intersecting lines, usually the left hand and lower edge of the ruled space, are chosen as coordinate axes, the hori- zontal one being called the axis of abscissae, or X, the vertical one the axis of ordinates, or Y. To plot the curve, a measured value of the independent variable is laid off along the axis of abscissae from the intersection of the axes. From this point on the axis the corresponding value of the dependent variable is laid off parallel to the axis of ordinates. The point reached in this way is one point on the curve. This is repeated for each pair of coordinate values, and a smooth curve is drawn through the plotted points. As a result of many experiments we find that one of the measured quantities varies directly as some other, or the ratio of their corresponding values is a constant. If we plot corresponding values of such quantities, they will lie along a straight line, generally inclined at an angle to the axes. The general equation of such a curve is y = a-\-bx where y is a value of the ordinates, x the corresponding value of the abscissae, a is a constant representing the portion of the axis of ordinates intercepted by the curve, and b a con- stant denoting the slope of the line. As an example we may take corresponding values of the mass and volume of a sub- stance. The ratio between corresponding values of mass and volume is a constant, the density of the substance, hence in symbols, if m represents mass, v volume and d density, m = dv. If we plot corresponding values of m and v they will lie along a straight line, inclined at an angle with the axis of x whose 22 Manual of Experiments tangent is d; for here y = m , x = v , b = d; also, since where i is the current strength, the deflection and k the reduction factor of the instrument, a constant. If we plot i and we get a curve. We may judge by inspection that it is a tangent curve, and plot corresponding values of i and tan 0, which will lie on a straight line. The above equation is then of the form y = kx. As another example of this process we may take the results of the experiment on Joule’s law. Here the heating effect of an electric current is proportional to the square of the current strength, or in symbols H = (i 2 R)/J where H is the heat energy, i the current strength, J the mechanical equivalent of heat, a constant, and R the re- sistance of the circuit, another constant. On plotting H and i, we get a parabola, but corresponding values of H and i 2 when plotted yield a straight line, the equation being of the form y^={R/J)x. Where higher powers of y and x are involved in the equation, it is generally advisable to proceed in a different manner to establish a linear relation in its place, namely, 'by taking the logarithms of both sides. Taking the general relation * . m n y = a x . where m and n are any powers whatever, and passing to logarithms, we have m log y = log a + n log x which is a linear relation between log y and log x. A simple In General Physics 25 example of this is the one already given, namely Boyle’s law. This may evidently be written p — c v~ l or in logarithms log p = log c — log V. If now we plot corresponding values of log p and log c — log v, as ordinates and abscissae respectively, we shall find that the plotted points lie along a straight line. The following rules should be observed in plotting curves : — Use coordinate paper with millimeter or 1/20 inch spacings on a sheet of the same size as that used in the body of the report. In order to have the curve facing the data sheet, so that we may refer readily from one to the other, the coordinate paper should be laid with the binding edge on the right hand side while plotting the curve. Give the curve a title, which where possible, indicates the relation between the two plotted quantities. Draw two heavy black lines to represent the coordinate axes, and on each of these mark the scale and name of the plotted quantities . Plot values of the independent variable as abscissae and those of the dependent variable as ordinates. Make the value of a scale division correspond, as nearly as possible, to the least quantity which you can measure. If this be impossible choose such values for scale divisions, as will make the curve most nearly fill the page with the observations to be plotted. All lettering and numbers should be placed in such a position that they can be read without turning the paper. Around each plotted point as a center draw a small circle about one mm. in radius. 26 Manual of Experiments Through and between the plotted points draw a smooth regular curve, such that if the points do not all lie on the curve, there will be approximately as many on one side as on the other, alternately. Any point which deviates markedly from this curve may be assumed to be due to an accidental error and should be discarded. In General Physics 27 Experiments in Mechanics 1. DENSITY OF SOLIDS The object of this experiment is to determine the density of various solids and to become familiar with the methods of using the coarse balance, the vernier calipers and the mi- crometer calipers. (References: D. 162. G. 10, 11, 27, 73. K. 69, 159. W. 16-18, 95, 129.) Apparatus. — Coarse balance and set of weights, ver- nier calipers, micrometer calipers, cylinders of wood, brass and steel. Description and Theory. — The density of a substance is defined as its mass per unit volume. The mass of the substance is to be found by comparison with known masses on a coarse balance, and its volume to be calculated from its dimensions measured with the vernier and the micrometer calipers. 28 Manual of Experiments an accuracy greater than 0.1 gram is not necessary. It is essentially a lever of the first class with equal arms, and when both arms are not equally loaded it is in equilibrium under the action of two equal and opposite moments about the ful- crum. It consists of a beam supported on knife edges at its center, with a pan supported on knife edges at each of its ends. (See Fig. 2.) Two nuts (N) working on a horizontal screw at the center of the beam, serve to bring it into its equilibrium position when there is no load in either pan. A vertical pointer (P) at the center, indicates when this balance position is reached. The unknown mass in one pan is balanced as nearly as possible by known ‘ ‘ weights ’ ’ placed in the other pan. By means of the slider (S) moving along a scale on the front of the beam, the known weights can be adjusted to 0.1 gram. The vernier calipers consist of a straight scale having a fixed jaw at right angles to it at one end, and a movable jaw bearing a vernier scale. The vernier is a device for measuring fractions of a scale division. The ordinary ver- nier scales have a space covering N— 1 divisions of the main scale, divided into N parts, so that each vernier division is shorter than a main scale division by one Nth. of the latter. 1/A is called the vernier fraction. When the two jaws of In General Physics 29 the vernier are in contact the zeros of the two scales coincide. If the jaws are separated until the first mark after the zero mark on one scale, coincides with the first mark on the other, the distance between the two zeros (and therefore between the two jaws also) is 1/N of a scale division. If separated until the second marks of the two scales coincide, the distance between the jaws is 2 / N of a scale division; if the third marks coincide 3/N of a division, etc., so that when the Nth. marks coincide the jaws are separated by a whole division. To de- termine any distance between the jaws, for instance the length of an object placed between them, we note the whole number of divisions on the main scale up to the zero of the vernier. If this zero does not coincide with a main scale division, the distance is greater than the whole number by a fraction of a division which can be determined by ob- serving the number of the vernier division which coincides (or most nearly coincides) with a main scale division (no matter which one.) In figure 3, the vernier reads 18.8 mm. The micrometer calipers consist of a curved or rectangular frame, one end of which serves as a fixed jaw, and an accurate screw which serves as a movable jaw. The other end of the frame is^threaded so that the screw can be turned in it, 30 Manual of Experiments toward or from the fixed jaw. It bears a straight scale whose divisions correspond to the pitch of the screw. The screw is turned by means of a milled head attach.ed to a sleeve which moves over the straight scale and which has on its edge a circular scale whose equal divisions represent fractions of a turn. Usually the pitch of the screw is a half millimeter so that one turn of the screw advances it a half millimeter, the edge of the sleeve moving over one division of the main scale. In this case there are 50 divisions on the circular scale, each division representing one fiftieth of a half millimeter of 0.01 mm. In some cases the main scale is divided into millimeters, instead of half millimeters, each division corresponding to two complete turns of the screw. The object to be measured is placed between the fixed jaw and the tip of the screw. When these two are in contact the zeros of the two scales should be together. As the screw is turned the distance between them is in- dicated on the two scales, the number of millimeters on the straight scale, the fractional part of a millimeter on the circular scale. In figure 4 the reading of the micrometer scales is 7.55 mm. Directions. — Place the slider of the balance at 0 on the scale and adjust the nuts on the horizontal screw so that the balance is in equilibrium, i. e. the pointer swings equidistant on each side of the center. Place the object to be weighed in one pan and add known weights to the other until the beam is very nearly balanced. Then the slider can be moved over the scale until equilibrium is again established. While ad- justing the weights in either pan, the- plunger under one of them should be held between thumb and fore finger in such a way as to form a cushion between pan and support, to pre- vent any jarring between them, otherwise the knife edges In General Physics 31 may be thrown out of position and a readjustment for equi- librium will be necessary. The slider reading is to be added to the known ‘ 1 weights ’ ’ when they are in the right hand pan, but subtracted when they are in the left pan. Weigh each of the three cylinders to 0.1 gram and record their masses in the proper place on the data sheet. Measure the length of each cylinder with the vernier cal- ipers, and its diameter with the micrometer calipers (if possible). The cylinders may not be perfectly circular and their ends may not be perpendicular to their axes so that measurements of the same dimensions taken at different parts of the same cylinder may differ slightly from each other. To get a good average measurement, the values recorded on the data sheet should be the mean of about 10 measurements taken at different parts of the cylinder. To measure the length of the cylinder, first note the main scale divisions and determine the vernier fraction. If the zeros of both scales do not coincide when the jaws are to- gether, note the zero error (the distance between them) and determine whether it is to be added to or subtracted from the readings to be taken. The cylinders should be placed with one- end flat against the inner face of the fixed jaw, and its side in contact with the main scale, and the movable jaw should then be moved up flat against the other end, just touching it, and clamped in position while the reading on the scales is taken. To take a measurement with the micrometer calipers, the scale divisions are first noted. Then the jaws are brought together, the milled head being held between the forefinger with slight pressure and turned until it just slips between the fingers instead of rotating. After noting the zero error as was done with the vernier calipers, the cylinder 32 Manual of Experiments is placed between the jaws and in contact with them, the screw being turned as before with slight pressure until it just slips through the fingers. Note : — Never hold the milled head tight while turning the screw into contact as this will jam the threads and injure them, besides causing errors in the determination. Care should be taken to measure the diameter and not a shorter chord of the cross section of the cylinder. Where the main scale is divided into millimeters, care must be ob- served to read fractional part of a division correctly, for evi- dently a reading on the circular scale may be so many hun- dredths of a millimeter or, 1/2 millimeter plus so many hun- dredths. In this case observe whether the edge of the cir- cular scale is more or less than half a division from the last preceding millimeter mark. If this can not be determined by eye, remove the cylinder from between the jaws and turn the screw back until the circular scale reads zero. Then observe whether the edge stands at the millimeter mark or half a millimeter from it. From these measurements the volume and the density of each cylinder are to be calculated and recorded on the data sheet. The error in measurements due to the graduation of scales on instruments should be estimated in each case and recorded on the data sheet, and the maximum possible per- cent error in volume and in density, due to these errors in measurements, should be calculated. Questions: — 1 — Determine from an examination of your calculations, which measurement causes the largest error in the density of each cylinder, and explain why. 2 — A' circular scale has divisions equal to 1/6 of a degree. In General Physics 33 A vernier attached to it has a total length of 59 main scale divisions, and is itself divided into 30 parts. Show what the vernier fraction is in this case. 2. TESTING A SPIRIT LEVEL. In this experiment we are to determine the radius of curvature of a level tube and to find the angle of tilt necessary to move the bubble through one division of the scale on the tube. (D. 183. G. 109. K. 175. W. 144.) Apparatus. — Level tube, level tester and meter stick. Description and Theory. — A spirit level is an instru- ment for detecting the deviation of a plane from the hori- zontal, and for measuring small angles. It is a glass tube having its inner surface accurately ground in an arc of a circle with a long radius of curvature. It is filled with a mixture of ether and alcohol except for a small bubble of air which always tends to move toward the highest point of the tube. A scale on the upper surface of the tube has divisions equal to 1/10 inch. When the tube is in a hori- zontal position the bubble is in the center of the tube, but when inclined the bubble moves through a distance which depends upon the angle of inclination and the radius of curvature. * / D - L I J !- -i::i TW S G Mil lannnui t in ijnr ii Lj Fig. 5 The level tester consists of an iron arm DG hinged at one end into a heavy base and carrying a micrometer screw M at the other end. The glass level tube T is mounted in V grooves on the iron arm and may be tilted by turning the 34 Manual of Experiments micrometer screw. (See Fig. 5.) Attached to the level tester and parallel to the axis of the screw is a millimeter scale. A circular disc having its upper edge divided into 50 equal divisions is attached to the screw and moves along the straight scale as the screw is turned. It serves to indicate fractions of a turn. The pitch of the screw is a half millimeter, so that each of these divisions represents 0.01 mm. 6 In figure 6, let BH represent the inner curved surface of the tube and AC the vertical radius of curvature through the highest point (center of the bubble) when it is in the horizontal position, A being the center of curvature. If the In General Physics 35 tube is inclined at an angle with the horizontal FD, the bubble moves from C to B , and the vertical radius through its center will be AB. The radius AC makes an angle with its former position. If the angle be measured in radians we have cf ) = BC — 5“ AB (1) We have also the relation 0 = FG -r- GD, or if is so small that the arc FG and the distance EG, through which the mi- crometer screw raises the end of the level tester, may be considered practically equal, 0 = EG -5- GD (2) It follows from (1) and (2) that AB = BC/4> = BCXGD/EG (3) » or the radius of curvature equals the distance the bubble moves multiplied by the ratio of the length of the iron arm to the vertical height through which its end is raised. Since the bubble moves through BC divisions when the tube is tilted through an angle <£ radians, the angle necessary to move it one division will be 6 = -r- BC, but by equation (1) 0 = d>/BC = 1 /AB (4) when AB is in scale divisions. Therefore the reciprocal of the radius of curvature gives the angle of tilt necessary to move the bubble through one scale division. It should be noted that equation (4) can be true only when AB is expressed in scale divisions, hence the radius of curvature should be expressed in tenths of an inch for this calculation. Directions. — Measure carefully the length of the iron arm from the center of the micrometer screw to the center of the hinge with a meter stick. As* the ends of a straight scale become worn, the end divisions are usually inaccurate and measurements should be taken from some inside division near the end. The length should be estimated to 1/10 mm. 36 Manual of Experiments Next place the level tube in the V grooves and adjust the micrometer screw until one end of the bubble is just under a division mark at the end of the scale. Read the position of both ends of the bubble and of the micrometer and record them on the data sheet. Then turn the micrometer until the end of the bubble is at some division near the opposite end of the scale, and take readings as before. Repeat this operation five or six times, moving the bubble from one end of the scale to the other, and taking readings in both po- sitions, but be careful to move the end of the bubble over the same number of divisions each time (or between the same two divisions). Calculate the mean displacement BC of the center of the bubble, and the corresponding mean height EG. By means of equation (3) calculate the radius of curva- ture and express it in meters and in feet. By equation (4) calculate the angle of tilt for a movement of one scale divi- sion and express it in radians and in degrees. (One radian = 57° 17' 45" = 206265") Note: — Be careful not to lean on the table, bench, or apparatus, nor to breathe on the level tube, nor to handle it more than is absolutely necessary and thus avoid errors due to a change in the size or position of the bubble.* In order to diminish the error due to lost motion, the micrometer should be turned so as to raise the iron strip in each final ad- justment before taking a reading. Questions. 1 — Which is the more sensitive, a level with a short radius of curvature or one with a long radius, and why? 2. — A carpenter’s level with radius of curvature of 30 feet is placed on a board 10 feet long and the bubble position In General Physics 37 noted. The spirit level is then reversed and the bubble moves 1.5 cm from its former position. How much in cm. must the end of the board be raised to level it? 3. EQUILIBRIUM OF COPLANAR CONCURRENT FORCES. COMPOSITION AND RESOLUTION OF FORCES. (, a ) To show that when three concurrent forces are in equilibrium, each is equal in magnitude and opposite in di- rection to the resultant of the other two. {b) To show that any one of these forces may be replaced by a pair of rectangular components. (D. 95, 96, 102-105. G. 33-38. K. 46-49. W. 65-67, 71, 72.) Apparatus. — Course balance and set of weights, 3 uni- versal pulleys, 2 small buckets, can of shot, 4 iron weights, 4 strings with hooks attached, small ring, sheet of paper, 4 thumb tacks, dividers, and block of wood, vertical board and meter stick. Theory and Method. — ( a ) Three weights are allowed to act simultaneously at a point. The resultant of two of these forces is found by parallelogram construction. The parallelogram law states that when two concurrent forces are represented in direction and magnitude by the direction and length of two straight lines drawn from one and the same point , their resultant is similarly represented by that diagonal of the parallelogram constructed on these two lines as sides , which starts from the same point. The resultant so found is compared in direction and amount with the third force. (b) One of the forces is then replaced by a vertical and a horizontal force, which act at the same point as the original forces. These two new forces are adjusted in magnitude un- 38 Manual of Experiments til equilibrium is again established without altering the two remaining original forces. They are then compared with the vertical and horizontal components of the third force, as found by the parallelogram construction. We now have four concurrent forces in equilibrium, but it should be noticed that the resultant of the two rectangular components (the original force) is equal and opposite to the resultant of the other two forces. Directions. — Find the mass of the weights on the coarse balance. Clamp a pulley on each side of the vertical board and fasten a sheet of paper to the latter with thumb Fig. 7 tacks. Suspend the smallest and one of the larger weights {A & B) over the pulleys by means of the strings, and hook the other ends of the latter to the small ring. Suspend a third weight from this ring so that it hangs freely. The pulleys should be adjusted so that they turn in a plane parallel to the board and so that the weights do not touch the board or bench. Adjust the height of the pulleys so In General Physics 39 that the center of the forces is in a convenient position for constructing the parallelogram of forces on the paper. Displace the weights slightly and allow them to move freely into a position of equilibrium. Then draw lines representing the forces under each of the strings, taking care not to alter the position of the latter by pressing against them. This may be done by carefully sliding a small rectangular block of wood along the paper until one face is in contact with the string near one end, without pressing against it, and drawing a short mark along this face with a sharp pencil. This is repeated at the other end of the string and a line is drawn through these two marks. The three lines drawn in this way should intersect at a point. Begin- ning at their intersection c, lay off on these lines to some convenient scale (say 1 mm. for each 5 grams) the cor- responding forces. On the upper two lengths, ca and cd, as sides, construct a parallelogram abed. The resultant of these two forces is the diagonal cb. Measure its length, calculate the value of the resultant in grams weight and compare it with the third force. Note also the angle be- tween be and the vertical. (cC prolonged upwards). In the same way find the resultant of A and C and com- pare it with B. It should be noticed in this construction that the sides of the triangle bed are parallel to and proportional to the three forces. Whenever three concurrent forces are in equilibrium , they are proportional to the sides of a triangle , the sides taken in order having directions parallel to the forces. (b) Lower the pulley q until its upper rim is in the hor- izontal line ce, and replace the weight B by a bucket partly filled with shot. Clamp a third pulley to the top of the board, adjust so that its rim is vertically above c and suspend a bucket and shot over it by means of a cord hooked to the 40 Manual of Experiments ring. Adjust the shot in these two buckets until the ring returns to its former position c , the strings be and ce being respectively vertical and horizontal. Next weigh each bucket with its shot. Draw the horizontal line ce and drop the vertical de from the end of the line representing the force B. ce and ed now represent respectively the horizontal and vertical components of B. Measure them, calculate the value of the components in grams weight and compare these with the corresponding weights of bucket and shot. Note: — For accurate results the parallelograms must be carefully drawn with a sharp pencil, and as large a scale as possible should be used in laying off the forces. Question. — If you had more than four concurrent forces in a system, how would you show that they were in equili- brium? 4. EQUILIBRIUM OF COPLANAR CONCURRENT FORCES BY THE TRIANGLE OF FORCES. To verify the laws for three forces acting upon a point. (D. 52; G. 38; K. 48, 49; W. 72.) Apparatus. — Force table and accessories, known masses. * Description and Theory. — The force table consists of a cast iron disk whose edge is divided into degrees and which is mounted upon a heavy tripod (See Fig. 8). Three pul- leys, working very free of friction in cone bearings, can be clamped to any part of the rim of the table. Known masses are fastened to light, flexible cords which pass over the pul- leys and are attached to a small ring in the center of the table. While the masses are being adjusted the ring is held in place by a pin. If suitable masses M, N and 0 are placed on the holders at the ends of the strings, the pulleys In General Physics 41 can be arranged so that the center of the ring remains over the center of the table when the peg is removed. If a given point, say the center ofthe above ring, is held in equilibrium by three forces, the condition that must be satisfied is, that the forces are proportional to the sides of a triangle, the sides of the triangle being taken in the order of the forces and having directions parallel to the forces; or, in other words, the sum of the components of the forces resolved in any given direction must be zero. Suppose for equilibrium, the forces take up positions as shown in Fig. 9 a. Let the forces be X = Mg, Y = Ng and Z = Og, and the angles opposite the forces be cc , 0, respectively. The condition for equilibrium states that the forces must form a triangle when lines proportional 42 . Manual of Experiments to the forces are drawn parallel to the directions in which the forces are acting. If the position of the forces be rep- resented by Fig. 9 a , the state of equilibrium is represented by Fig. 9 b. From Fig. 9 b ) we obtain, by trigonometry, the relation, X Y Z sin a sin b sin c or Mg Ng Og sin a sin b sin c Since the acceleration of gravity “ g” enters each ratio, we can neglect it and use the values of the masses instead Figs. 9a and 9b of the forces, they being proportional to each other. The angles oc f ft, are supplements of the angles a, b, c, respec- tively, and hence the final relation can be written M N 0 sin ft sin cc sin 7 (3) In General Physics 43 Directions. — See that the force table is level. Place the small ring in the center of the table and put the pin through it to hold it in place. Fasten a flexible cord to each of the three weight holders, and, having run the strings over the pulleys, fasten them by means of the small hooks to the ring in the center of the table. Add masses of from 300 to 600 grams to each holder, but do not have the same mass on any two holders. Place one of the pulleys at zero and adjust the other two until approximate equilibrium is obtained, after which remove the pin from the center of the ring. Continue to adjust the pulleys until the ring remains over the center of the table when the latter is struck lightly with the hand. When equilibrium is obtained and the center of the ring is over the center of the table, read the angles between the lines of action of the forces, or the angles formed by the * strings with each other, and designate them as °c y and y (Fig. 9 a). Determine the value of the masses at the end of each string (known masses plus the holder) and test the re- lationship expressed by equation (3). Repeat the determination four more times using different combinations of the masses suspended each time. Find the mean value of the constant (mass divided by the sine of the angle opposite) for each of the five determinations. For example, — X sin cc Y First Determination — = sin Z sin 7 Mean = 44 Manual of Experiments Question. — (1) What is the maximum percent error from the mean, (a) in your best determination, (b) in your poorest? 5. EQUILIBRIUM OF COPLANAR NONCONCUR- RENT FORCES. PARALLEL & RANDOM FORCES. V The object of this experiment is to become familiar with the laws of equilibrium (. a ) for parallel forces acting in a plane. (b) for random forces acting in a plane. (D. 81, 95-99, 102-105. G. 39, 40. K. 53-63. W. 68-73). 1 Apparatus. — 4 spring balances, 3 steel balls, 4 iron pegs, 2 boards, pair of dividers, 4 iron blocks, meter stick, 2 sheets of paper. { Theory and Method. — In order that nonconcurrent forces may he in equilibrium they must tend to produce neither translation nor rotation. Hence there are two rules for the equilibrium of such forces. We are going to consider only forces which act in the same plane. In this case — I If the forces are all parallel , the algebraic sum of the forces must equal zero , or the sum of the forces in one direction must equal the sum of the forces in the opposite direction. If they are not parallel we may consider their components taken parallel to two rectangular axes, in which case the sum of the components in one direction must equal the sum of the components in the opposite direction. II In either case the algebraic sum of the moments of all the forces taken about any point in the plane must equal zero in order that there may be no rotation. The moment of a force about a point is a measure of the turning tendency of In General Physics 45 the force about an axis through that point. It is numerically equal to the product of the force into the perpendicular distance from the point to the line of direction of the force. A clockwise moment is one which tends to turn about the point in the same sense as the hands of a clock. An anti-clockwise moment tends to turn in the opposite direction.. For equilibrium the sum of the clockwise moments must equal the sum of the anti-clockwise moments. Both of these laws must be fulfilled by every system of coplanar forces when in equilibrium. We may find that a certain system does not produce rotation about a certain point. This does not necessarily mean that there is no rotation about some other point, or that there is no trans- lation, hence we cannot say that the system is in equilibrium from this single test, but in general must apply both tests. Translation may be considered as rotation about an in- finitely distant axis, so that the first law may be considered a special case of the second. The apparatus consists of a base board with leveling screws, and a glass plate on its upper side, and an upper board having a series of equidistant holes in parallel rows. The upper board rests on three steel balls placed on the glass plate. The object of this arrangement is to avoid friction as far as possible. Small spring balances with their scales graduated in ounces are held by pins on the iron blocks and may be moved to any position relative to the board and to each other. By means of strings they may be fastened to pegs placed in any of the holes in the upper board and the tension on them may be adjusted by moving the iron blocks. The spring balance consists of two cylinders one of which slides within the other, and which are held together by a spring. The extension of this spring is by Hooko’s law pro- 46 Manual of Experiments portional to the weight suspended from it, and a pointer at- tached to the inner cylinder indicates this weight directly on the graduated scale of the outer cylinder. In a vertical po- sition the inner cylinder is supported by the spring, whereas in the horizontal position it is not. Therefore the no load reading will be different in the two cases, and this difference must be added to the zero error if there is any, when the balance is used in the horizontal position. Directions. — ( a ) — Assume the balance marked A to be correct (without zero error) and find the zero error of the other balances by placing each on an iron block, joining it to A with a string, and pulling the two against each other in a straight line until a force of one or two pounds is indicated. Be careful in using the balances to have the inner cylinder in line with the outer cylinder to avoid errors due to friction between them. Note the zero error of the balance (the difference between the reading on it and that on the standard A), and mark it + or — according to whether it is to be added to or subtracted from future readings of the scale. Place the three balls between the glass plate and the upper board (as shown in figure 10) and adjust the leveling screws until the board stands at rest without touching the side stops. Arrange three balances as shown, fastening the strings to pegs which may be placed in any of the holes of In General Physics 47 the upper board. Adjust the positions of pegs and balances until the pull on the single balance A is as large as possible and the strings and balances are in line and in the same horizontal plane, the strings being parallel to each other as shown by sighting along the parallel rows of holes. The upper board should rest on the three balls and be free from the side stops. Note the pulls on the balances, correct them for the zero errors, and record them. Next choose an arbitrary point on the board as a center and measure the perpendicular distance in inches from this point to the line of direction of each force, noting at the same time whether the force tends to rotate the board in a clockwise or an anticlockwise direction about the arbitrary center. Then calculate the moment of each force and record it. On the data sheet compare the sum of the forces pulling in one direction, say east, with the sum of the forces pulling in the opposite direction, and the sum of the clockwise mo- ments with that of the anticlockwise moments, giving the percent difference between them (calculated from their mean) in each case. (b) Place a sheet of paper on the top board and arrange four balances so that the forces acting on the board are not parallel, but are placed at random, the iron pegs being punched through the paper. When equilibrium has been established (the balances being in line with their strings and the top board free from sidestops), rule a line under each string as in Exp. 3, and record the correct values of the forces pulling on the balances beside each line. Remove the pegs and paper and draw a pair of rectangular axes, say east and west, north and south. Lay off on each of the four lines to some convenient scale, the corresponding forces, and from the ends of the lines so laid off draw the components of the forces parallel to the two axes. Record 48 Manual of Experiments the values of the components and their direction. (E. N., etc., for east, north, etc.) Choose an arbitrary point p on the paper as a center and measure the perpendicular distance in inches from this point to each of the force lines (extended if necessary as in Fig. 11). The arbitrary center should be so chosen that the perpendicular distances are as large as possible, to reduce errors in measurements, and the distances should be es- timated to 1 64 of an inch. Note which forces give clock- wise and which anticlockwise moments and record them properly. On the data sheet compare the sum of the east compon- ents with the sum of the west components; the sum of the north with the sum of the south components; and the sum of the clockwise with the sum of the anticlockwise moments, and calculate the precent difference between them in each case. Questions — 1 — A beam 10 feet long weighing 200 lbs., along which a pulley and chain travels, is supported at one In General Physics 49 end by a wall, at the other by a column which cannot support more than 600 lbs. If the pulley with its load weighs 800 lbs. how close can it be drawn to the column without over- loading the latter. Apply both laws of equilibrium. 2. — In performing this experiment with 4 random forces, it was found that two of them intersected in a point 0; namely a force A of 10 oz. directed 30° W. of N., and B a force of 20 oz, directed 60° E. of N. A third force C of 30 oz. directed 45° E. of S. had a moment arm 3 inches long in a direction S. W. from O. Find the direction and magni- tude of the fourth force and its distance and direction from O. 6. PARALLEL FORCES. {a) To verify the law for the equilibrium of moments. (, b ) To determine the value of an unknown mass, (c) To find the mass of a meter stick. (D. 97-100; G. 39-41. K. 59-64; W. 69, 70.) Apparatus. — Clamp stand, meter stick, set of known masses, three lever holders, coarse balance, unknown mass. Description and Theory. — Moment of a force is the term used to express the value of a force times its arm, where the arm is the distance measured at a right angle from the line of action of the force to the axis about which rotation is capable of taking place. A body, free to rotate about an axis, is in equilibrium only when the two following conditions are satisfied: first, when the sum of the components of the forces in any direction is equal to the sum of the components in the opposite direction; and second, when the sum of the moments of the forces tending to produce rotation in one di- rection is equal to the sum of the moments of the forces tend- ing to produce rotation in the opposite direction. The latter 50 Manual of Experiments condition, or law, may be verified by a simple arrangement consisting of a meter stick supported on a knife-edge at its center of gravity and having known masses supported on knife edges at such distances from the center of gravity that the product of one known mass plus the mass of the support- ing knife edge times their distance from the center of gravity is equal to the corresponding product for the other mass. The law can be applied to determine an unknown mass by replacing one of the known masses by the unknown. The product of the unknown mass times its distance from the cen- ter of gravity of the supporting meter stick is equal to the product of the known mass times its distance from the cen- ter of gravity. In all cases, the masses of the supporting knife edges which is stamped on the holder must be taken into account. The mass of a meter stick can be determined by first finding the center of gravity of the meter stick by balancing it alone and then, having placed the center of gravity a short distance from the supporting knife edge, putting a known mass at such a distance from the supporting knife edge as to obtain equilibrium. The product of the mass of the meter stick times the distance of the center of gravity of the stick from the supporting knife edge must equal the product of the known mass times its distance from the point of support. Directions. — To verify the law for the equilibrium of moments, support the meter stick by one of the lever holders and adjust the same till equilibrium is obtained. Put a lever holder on each end of the meter stick and hang masses A and B (Fig. 12), of 200 and 300 grams on the holders. Adjust the position of the holders (at least 20 cms. from the supporting knife edge) with the suspended masses till equilibrium is obtained. Note the distance from the supporting knife edge to the knife edge of each of the holders. In General Physics 51 Test the relation for the equilibrium of moments by mul- tiplying the mass of the holder and the suspended weight by its distance from the supporting knife edge. The product of one suspended mass plus the mass of the holder Fig. 12 supporting it times their distance from the supporting knife edge should be equal to the corresponding product for the other mass. Make a second adjustment of the masses A and B for different distances from the supporting knife edge and test the relation for the equilibrium of moments. Use another combination of masses 300 and 400 grams, and obtain the products of mass times the distance from the point of support for two different positions of the masses when in equilibrium. In order to determine the value of an unknown mass it is only necessary to substitute the unknown mass X for either of the masses A or B above, and find the position for equilibrium. The unknown mass X is determined from the fact that the product of X times its distance from the point of support must be equal to the product of the known mass times its distance from the supporting knife edge. Make three different determinations of the value of X and find their mean. Remember that the masses of the knife edges supporting the known and . unknown masses must always be considered. 52 Manual of Experiments The mass of the meter stick is found as follows: Find the position of the center of gravity of the meter stick by suspending it from one of the knife edges. Move the stick so that its center of gravity is from 15 to 20 centimeters from the supporting knife edge and suspend a 100 gram mass by means of a lever holder at such a position as to produce equilibrium. Note the distances of the 100 gr. mass and the center of gravity from the supporting knife edge. Then, if M is the mass of the meter stick, M times the distance of the center of gravity of the stick from the point of suspension is equal to 100 grams plus the mass of the holder supporting it times its distance from the knife edge of the holder sup- porting the system. Make two more adjustments for equilibrium using dif- ferent distances and determine the mass of the meter stick in each case. Calculate the mean of the three determina- tions of the mass of the meter stick. Question. — What two conditions must be satisfied for equilibrium of non-concurrent parallel forces? Show how, if the sum of all the forces downward did not equal the sum of the forces upward, the other condition could be fulfilled and still not haye the bar in equilibrium. What kind of motion would we then obtain? 7. THE FINE BALANCE. In this experiment we are to become familiar with the vibration method of using the fine balance. (D. 135; G. 73- 76: K. 69, 70. W. 95.) Apparatus. — Set of weights (2mg. to 100 g.,) lead plate to be weighed, spirit level, and fine balance. Description and Method. — The fine balance is to be In General Physics 53 used whenever an object is to be weighed to less than 0.1 gram. It consists of a rigid beam with a central knife edge resting on a polished agate plate at the top of a pillar, and two equidistant knife edges KK from which the balance pans are suspended by means of stirrups with polished agate plates. (See Fig. 13.) Agate is used in order to make the friction in swinging very slight. To protect the balance from disturbing air currents and dust, it is enclosed in a glass case having a heavy glass base and supported on leveling screws. To avoid jarring and chipping of the knife edges when the balance is not in use or while ‘ ‘ weights ’ ’ or objects are being added to or taken from the^pan, an arrestment ■ Fig. 3 54 Manual of Experiments device is used. This is operated a thumb nut (N) at base of balance which, on turning, lifts the beam off the agate plate and the stirrups off the knife edges. Two stops rest against the pan to prevent them from swinging. To release them a button ( B ) is pushed in until it is caught by the lever ( A ). A long pointer ( P ) attached to the beam in- dicates the position of rest on a scale at the base of the pillar. As the milligram weights are quite small and incon- venient to handle, a rider ( R ) is often used in place of weights less than 10 mg. It is a piece of wire which weighs exactly 6 mg. and is so shaped that it may be placed astride of a scale on the beam. By means of a system of levers operated . from the outside by a thumb nut (L) at the right of the case, this rider may be moved to any point on the scale. When just above one of the end knife edges its effect is the same as if it were placed in the pan. When placed at one sixth this distance from the central knife edge its moment about the fulcrum will equal that of one sixth of its mass, or (1 mg.), placed in the pan. The marked divisions on the scale represent milligrams, the smallest divisions tenths of a milligram. Adjustable nuts working on horizontal screws at the ends of the beam, make it possible to balance the latter alone. A good balance should fulfill the following conditions: (1) The balance must be true, that is, the beam must re- main horizontal whenever equal masses are placed in the scale pans. (2) It must be sensitive, i. e. small differences in the masses in the two pans should cause an appreciable deviation of the beam from its horizontal position. (3) It must be stable; that is, after the beam has been disturbed from its equilibrium position, it must return to it. (4) It In General Physics 55 should have a short period; i. e. when the beam has been dis- turbed and oscillates before coming to rest, the time re- quired for it to make a complete swing should be short. In order that the balance may be true, the beam should be inflexible and its arms of equal length. To be sensitive the knife edges should be parallel and in the same plane, the arms should be long, and the center of gravity of the beams should be just below the central knife edge. In order that the beam may quickly return to its equilibrium position when displaced, the arms should be short and the distance between center of gravity of beam and knife edge large. These conditions conflict with those for sensitiveness and in • practical balances a compromise between them must be chosen. The approximate weight of an object may be obtained by the equal swing method. Small bits of paper are added to the pans until the pointer swings equidistant on either side of the center division. Then the object is placed in the center of one pan and weights are added to the other until the pointer swings as nearly as possible as before, equidistant on either side of the center division. Then the object and known weights will have approximately the same mass. To find the mass more accurately, it would be necessary to find the true resting point (i. e. the point on the scale at which the pointer would finally come to rest if allowed to swing freely). As the period of the balance (the time required for a complete vibration of the beam) is usually long, it is preferable to use the following “vibration method of weigh- ing”. The resting point with no load in the pans is found by observing an odd number, say five, consecutive turning points of the pointer, estimating to tenths of a scale division. The reason for taking an odd number of turning points is the following: — The amplitude of swings to either side of 56 Manual of Experiments the resting point is continually decreasing by approximately equal amounts, say a divisions for a few equal swings (see Fig. 14). The mean of the first and second turning points would be approximately a / 2 divisions to one side of the rest- ing point. The mean of the second and third turning point would be about a/2 divisions to the other side of the resting point. The mean of two means would be very near the resting point. If five turning points were observed instead of three, the mean of the means of the five readings, taken two and two consecutively, would be still nearer the point, whereas if any even number of points or an odd number of swings were observed the mean would always be to one side or the other of the true resting point. Therefore, to find the true resting point, average the mean of n (odd) turning points on one side {a, c, e, Fig. 14.) with the mean of (n — 1) intervening turning points on the other (b, d. Fig. 14). As an example, if the pointer swings from 6.0 to 12.8 and back to 6.8, the two means would be 9.4 and 9.8 respectively and the mean of the two means of resting point would be 9.6. This no load resting point we will call S 0 . If now the object to be weighed is placed in one pan and known weights in the other until they balance, the load resting point Si may be found in the same way as S Q and in general will In General Physics 57 be different from S 0 . But it is evident that the masses in the pans can only be equal when the pointer comes to rest at S 0? since this is its equilibrium position. We must there- fore determine what weight must be added to or taken from the known weights in the pan in order to make the pointer come to rest at S Q (instead of Si) when the balance is loaded. To do this we add a small weight, say 5 mg. to the known weights and observe the new load resting point S 2 . Since 0.005 grams moves the resting point from Si to S 2 the number of grams n necessary to move one division will be 0.005 -f- (S 2 — Si). The reciprocal of n is called the sensibility of the balance; i. e. the number of divisions through which the pointer is moved by 1 mg. in the pan. In order that the load resting point may be changed from Si to S Q we must change the known weights by n (Si — S 0 ) g. If the known weights are in the right pan we must add this amount to them when Si is greater than S G and subtract it when Si is less than S c , in order to obtain the true mass of the object. Directions. — Caution. Always keep the case closed ex- cept when it is necessary to adjust the weights. Lower the arrestment slowly and carefully, avoiding so far as possible, all jarring of the balance. Always raise the arrestment be- fore changing the weights in the pans. Handle the weights with tweezers to avoid corroding them or changing their weight. Do not attempt to make any of the finer adjust- ments with the screws on the balance. If such adjustments are necessary call on one of the instructors to make them. Adjust the leveling screws until the base of the balance is horizontal as shown by the spirit level. Lower the arrest- ment until the pointer swings freely and note whether the no load resting point is within one or two divisions of the center. If it is not, drop small bits of paper into the proper pan until it is within this distance from the center. Then 58 Manual of Experiments find the no load point S c accurately, by taking five con- secutive turning points, (estimating to 1/10 of a scale division) and averaging the mean of those on one side with the mean of those on the other side. Next raise the arrest- ment and place the object to be weighed in one pan and known weights in the other to balance. Lower the arrest- ment a trifle and note which way the beam tilts {i. e. which of the pans has the heavier mass in it) ; then raise the arrest- ment and adjust the weights to secure a better equilibrium. Repeat until there is very little tendency to tilt the beam. Time may be saved by trying the weights in the order in which they come in the box, and replacing them in similar order. The finer adjustments may be made with the rider. Record the mass m of known weights. Determine the load resting point Si from five con- secutive turning points. Add 5mg or less to the known weights, determine the load resting point S 2 , and calculate n, the reciprocal of the sensibility of the balance. Remove the weights and object from the pans (while the arrestment is raised, of course) and determine the no load resting point S 3 . If the latter differs by less than one division from first value S 0 the weighing is sufficiently accurate. If the difference is greater than one division the experiment should be repeated. From the data taken, calculate the true mass M of the object. M = m =±= n (Si — S Q ) Question. — If the center of gravity of the beam could be raised or lowered by means of an adjustable nut, how would the balance be affected in each case. ' - * 8. THE PENDULUM. (a) The laws of the simple pendulum. (b) The determination of the acceleration of gravity In General Physics 59 with the Kater’s pendulum. (D. 117, 120, G. 56, 81-84, K. 127, 128, 141, W. 112-118.) Apparatus. — Two lead balls, fine thread, vernier calipers, reversible pendulum and support; beam compass, meter stick, and seconds clock. Description, Method and Theory. — (a) If it were possible to have a material particle concentrated in a point and suspended by a weightless, inextensible thread, this would be a theoretical simple pendulum. These conditions can not be fulfilled in practice, and the practical simple pendulum consists of a small heavy bob (usually spherical) suspended by a light and practically inextensible thread or wire. Fig. 15. The period of such a pendulum is the time required for a complete oscillation, or double swing; that is, the time which elapses between two successive pas- sages of the pendulum in the same direction , through some - point of its path, such as the center of its swing. The length of “the equivalent simple pendulum ” (that is, the length of the theoretical simple pendulum which has the same period) is, for a spherical bob (neglecting the mass of suspension), / = d + ( 2r 2 /5d ) where d is the distance from the point of support to the center of the bob, and r is the radius of the bob. For practical purposes the length of the pendulum is taken as equal to d. The period of such pendulum is T = 2w V // g, where g is the acceleration of gravity. The period is inde- pendent of the mass of the bob and of the arc of its swing, provided that arc does not exceed about 4 degrees. We wish to test the variation of the period of such a pen- dulum with its length and with its mass. In order to find 60 Manual of Experiments the variation of any physical quantity with one of its factors, it is evident that all other factors, must be kept constant. We therefore take two pendulums of equal mass but of different lengths, in order to compare the variation of their periods with their lengths, and two pendulums of equal lengths but of different masses, to note a possible variation of period with mass. If the student is working alone he may determine the period of the pendulum with a stopwatch, by starting and stopping the watch at the beginning and end, respectively, of a definite number of oscillations. If two students work together, an ordinary watch with a seconds hand may be used, one student observing the number of oscillations of the pendulum, the other the number of seconds which elapse. The first student should call ‘ ‘ ready ’ ’ a few seconds before he begins to count the oscillations. Then at the exact instant when the pendulum passes through its center of path (position of rest) he should call 11 start ”, beginning at the same time to count the passages of the pen- dulum through this point in the same direction , starting with In General Physics 61 zero for its first passage. The other student should note the exact time at this instants On the 95th or 96th passage the first student again calls “ready” to warn the second student, and on the instant that the pendulum completes its 100th passage through the position of rest he calls “stop ’ upon which the second student again notes the exact time. If the students have no watch at hand, they may observe the time on the seconds clock in the laboratory. This clock has a pendulum which completes one swing (half oscillation) every second. It has a device for closing an electric circuit every time it passes through its position of rest. In this circuit are included a galvanic cell and a tele- graphic sounder, and every time the circuit is closed, the sounder gives a loud click, thus indicating seconds. When all is ready for the trial one of the students calls “ready”. The second student then observes both the passage of the pendulum through its position of rest and the click of the sounder. At the instant when both events occur exactly in unison he calls “start ’ ’, beginning at the same time to count the clicks of the sounder, zero, one, two, three, etc., while the first student counts the passages of the pendulum in the same direction, beginning zero, one, two, etc. When nearly one hunderd swings have been completed the first student begins to observe the clicks of the sounder as well as the passages of the pendulum and when he sees that the two nearly coincide he calls “ready”, and at the exact instant when both occur simultaneously he calls “stop”, noting the exact number of passages, while the second student notes the exact number of clicks up to this instant. To avoid the tedious counting of oscillations we may proceed as follows : — Note as accurately as possible with a watch or with the seconds clock the time of 10 complete oscillations, and calcu- 62 Manual of Experiments late the approximate period. Then note the exact instant of the transit of the pendulum through its position of rest. Calculate the approximate time required for 100 oscillations, and at the expiration of this time again note the exact instant of transit in the same direction as before. Divide the inter- val of time between these tw r o transits by the approximate period. The whole number coming nearest to this quotient will be the exact number of oscillations in this interval and on dividing the same interval by the exact number of oscil- lations we get the exact period. The length of the pendulum is measured with a beam com- pass which consists of a straight wooden beam along which two trammel points may be moved and clamped at any de- sired position. The distance between the trammel points may be measured with a meter stick. The average time of one complete oscillation should be calculated and a comparison made between the periods for different lengths and for different masses. (b) The compound pendulum differs from the simple pendulum in that its mass is not concentrated at one point. Evidently every physical pendulum is really a compound pendulum. That point in the pendulum where the particles are oscillating with the same period that they would have if they were bobs of simple pendulums suspended from the same point of support as the actual pendulum, is called the center of oscillation, the point of support being called the center of suspension. These two points are interchangeable. That is, if the pendulum is suspended from the center of oscillation, the previous center of suspension becomes the center of os- cillation. If the whole mass of the pendulum were concen- trated at the center of oscillation, the period of the resulting simple pendulum would be the same as that of the actual pendulum. Therefore the length of “the equivalent sim- In General Physics 63 pie pendulum ’ ’ is the distance between the center of oscilla- tion and the center of suspension. The fact that these two centers are interchangeable makes it possible to determine their relative positions accurately and so makes possible the accurate determination of the length of the equivalent simple pendulum. The period may also be determined accurately Fig. 16 i L and consequently an exact determination of the acceleration of gravity is possible from the relation £ = 4t t 2 //P 64 Manual of Experiments (see expression for period of simple pendulum). For this purpose a Kater’s reversible pendulum is used. It consists of a metal rod, carrying two adjustable steel knife edges KK with their edges turned toward each other. See Fig. 16. At one end a heavy mass A is fixed, while a light mass B with adjusting nuts may be clamped at any position along the length of the pendulum. The knife edges are fixed at some definite distance apart and the pendulum is swung first from one knife edge, then from the other, its period being determined each time and the weights adjusted until the periods are exactly equal. The period and the length between the knife edges may now be accurately measured and the acceleration of gravity calculated. Directions. — ( a ) Relation between period T and length h — -Fasten the smaller ball to the thread and suspend it from the support. Adjust the length of the thread until it is about 60 or 64 cm. Adjust the pointer on the support until it is just under the center of the bob when the latter hangs at rest. Place the beam compass in line with the pendulum, one end resting on the floor, and adjust the trammel points until they stand exactly at the ends of the thread. Lay the beam compass on the table and measure the distance between the trammel points by means of a meter stick and record it on the data sheet. Measure the diameter of the bob with the vernier calipers in four or five different positions, and re- cord the mean diameter. Then calculate the length of the pendulum. Set the pendulum to swinging through a small arc. Be careful to have it swinging in a plane; that is, so that the path of the bob is a straight line, not an ellipse. Find the time for one complete vibration by the method explained above, noting the passage of pendulum past the pointer. Repeat the trials until three values of the period are obtained In General Physics 65 which do not differ by more than 0.01 second and then record the mean of these trials. Repeat these measurements with a pendulum about one meter long. Show by your data that the period is directly proportional to the square root of the length, by calculating the ratio be- tween them in each case. Record these ratios and the per- cent difference between them calculated from their mean. Independence of Period and Mass . — Fasten the large lead ball to the thread and suspend it as before. Keeping the po- sition of the trammel points on the beam compass exactly as they were for the last measurement of length, adjust the length of the pendulum until the center of the bob is at the lower trammel point. The pendulum then has very nearly the same length as the previous pendulum with the smaller bob. Proceed to find the period of the pendulum and com- pare it with that of the previous pendulum of the same length. Record these periods and the percent difference between them calculated from their mean. (, b ). The determination of g. — Using the Kater’s pen- dulum let one knife edge and the two weights remain fixed, and the second knife edge K movable. Determine the period of the pendulum about both knife edges for one position of K, and measure the distance il l ,J between knife edges. Then shift K about 5 cm. and repeat the measure- ments. Repeat until you have four periods about each knife edge and the corresponding distances between them. Plot two curves having values of / as abscissae and correspond- ing periods about both knife edges as ordinates. These two curves will intersect and the ordinates of the point of intersection will be the period and length of the equivalent simple pendulum respectively. For accurate results the distance between knife edges 66 Manual of Experiments should be measured with a comparator, but for our purposes we may use the length measured with the beam compass and meter stick. From the period and length of the equivalent simple pendulum calculate the value of g and record it on the data sheet. Question. — In the case of the third simple pendulum used (with the larger bob) what is the percent error in the length due to the assumption that the length equals d? 9. LAW OF FALLING BODIES. The object of this experiment is to observe the time and distance of the free fall of a ball and, applying the laws of uniformly accelerated motion in a straight line, to determine how accurate a value of the acceleration of gravity can be found from such observations. (D. 26-29, G. 51, K. 96-99, W. 35.) Apparatus. — Falling body apparatus, beam compass, meter stick, tray, steel ball, terminals of storage battery and regulating resistance. Theory, Method and Description. — If a body falls freely a distance d in a time t, we have the relation d = 1/2 gf where g is the acceleration due to gravity. With the appar- atus we measure corresponding values of d and t and since g is constant we can test the above law by seeing whether the ratio of corresponding values of d and t 2 is constant. We can also substitute corresponding values in the formula and get a value for g. In the experiment, a small ball B is held by an electro- magnet A (See Fig. 17.) A pendulum KWP is hung from In General Physics 67 the knife edges K just under the ball, so that, when the latter is released, it falls through a slit X in the platform P at the lower end of the pendulum when the pendulum is at rest. The pendulum can be held at one end of its swing by a second electromagnet C. Both A and C are in the same electric circuit, so that when the current is broken the ball and pendulum are released at the same time. The distance of fall can be varied by raising or lowering the rod which carries the magnet A, until it is at such a position 68 Manual of Experiments that the ball drops through the slit X as the pendulum swings through its position of rest. The time of fall then equals the time required for the pendulum to swing from one end of its path to its position of rest, which is one quarter of its period. The corresponding values of d and t are then to be measured. The period may be altered for a second trial by shifting the iron frame W to a different position. Directions.— The battery circuit is connected through a resistance and a snap switch to the binding posts on the fall apparatus. Adjust the current by increasing or de- creasing the resistance until the electromagnets are just strong enough to hold the ball and the pendulum. Slip the frame W to the bottom of the pendulum and adjust the position of the slit X so that the ball drops through it when the pendulum is at rest and the width of the slit is not over one fourth of a millimeter greater than the diameter of the ball. Then by adjusting the height of the upper magnet, find the position for which the ball will slip through the slit of the swinging pendulum when ball and pendulum are re- leased simultaneously by snapping the switch quickly . One jaw of the slit is of lead, the other of brass, so that one can tell by the click on which plate the ball strikes, hence whether it strikes in front of or behind the slit. In the first case the magnet should be raised; in the second it should be lowered. The tray should be placed under the apparatus to catch the ball as it drops. When the height of the ball has been pro- perly adjusted, measure the distance d from the center of the ball to the slit with the beam compass and meter stick. Then set the pendulum in vibration and determine the time of 50 complete swings. Repeat this a second time and cal- culate the mean quarter period of the pendulum. Change the position of the frame W so that the period and distance d are altered, and repeat the experiment. In General Physics 69 With these values of d and t calculate the acceleration of gravity by the formula given. Finally determine by about ten trials the upper and lower limits of the height through which the magnet A can be shifted and still allow the ball to pass through the slit in the swinging pendulum. Question. — What is the maximum possible percent er- ror in the calculated value of g due to this possible variation in the height d determined by the last measurement? 10. ACCELERATION OF GRAVITY WITH A FALL- ING TUNING FORK. To determine the. value of the acceleration of gravity. (D. 26-29; G. 81; K. 96-99; W. 35.) Apparatus. — Acceleration apparatus, meter stick. Description and Theory. — The acceleration apparatus (Fig. 18a) consists essentially of a tuning fork attached to a frame arranged to fall about one meter between two vertical guides which offer as little resistance as possible. The fork is held at the top of the guides by a trigger by means of which it can be released and at the same time set in vibration. On one prong of the tuning fork is a stylus for recording the vi- brations upon a long plate of smoked glass in the nature of a curve such as is shown in Fig. 18b. If the point a (Fig. 18b) near the beginning of the trac- ing, where the waves are perfectly formed, is selected and spaces of exactly six wave lengths marked off, it is seen that any set of six wave lengths is longer than the set of six just preceding it. The time required for the tuning fork to pass over the distance is the same, however, because the fork has made six complete vibrations in each case. 70 Manual of Experiments When the fork had reached the point a it had acquired a velocity which may be represented by v 0 . If t represents the time for the fork to make six complete vibrations, it will Fig. 18a and b pass over a distance v Q t during the interval t because of its initial velocity alone. But gravity also acts and will in- crease the space passed over in the interval t by the amount The total space passed over in the first interval a\ob is then, In General Physics 71 ab = v Q t + 3 4gt 2 (1) In two intervals of time t, (i. e. 2 1), the tuning fork will pass over a distance given by ac = 2v 0 t + YigQtY (2) and for three intervals ad = 3v 0 t + 3^g(3*) 2 ' (3) The differences ab , ac — ab and ad—ac are the distances passed over in successive equal intervals of time. The distances are given by the following equations: ab = v Q t + Yzg? (1) ac — ab = bc = v 0 t + | gt 2 (4) ad — ac = cd = v Q t + | gt 2 (5) From these equations it is seen that the increase in the space passed over in successive intervals is bc — ab = cd — bc = gt 2 (6) This states that the difference between the distances repre- sented by any two successive groups of six wave lengths is given by the value of gravity times the square of the time consumed by the tuning fork in making six complete vi- brations. The value of g, the acceleration of gravity, can be determined if the time t required for the fork to make six complete vibrations and the difference between the distances , passed over in two successive six vibration^ intervals are known. The time t is obtained by dividing the number of vibrations made in the time t by the number made in one second which is recorded on the fork, or t =1 (7) where n is the pitch of the fork. The difference between successive distances can be obtained by direct measurement. Directions. — In order that the friction may be as little as possible the uprights should be standing vertically. Put 72 Manual of Experiments the smoked glass plate in position moving it towards one side so that the first tracing made will be near one edge. Fasten the tuning fork on the eccentric at the top of the frame and, having the stylus pressing closely against the glass plate, turn the lever connected to the eccentric so as to release the fork. The fork will fall freely and, in so doing, make a tracing upon the smoked glass. In a similar manner make at least six tracings on the plate, shifting the position of the plate each time so as to have each tracing separate from the others. Remove the plate from the frame and, beginning with the first good wave, mark off spaces of six wave lengths on each tracing. Measure with a millimeter scale all of the six vibration spaces on each tracing that it is possible to obtain, and find the difference between suc- cessive distances represented by six wave lengths. Find the mean value of the differences for each of the tracings. Note the pitch of the fork as recorded upon it and, sub- stituting the pitch n in equation (7), find the value of t. Substitute the mean value of the differences for one of the waves and the value of t in equation (6) and solve for g. In like manner determine the value of g as given by the data of each of the other curves. Calculate the mean of the values of g thus found. Questions. — What effect will friction of the falling frame have on the value of g? Explain your answer. What percent error in g will be caused by an error of one percent in n, the frequency of the fork? ■L- 11. FRICTION. To study the laws of sliding friction and to determine the coefficients of static and of kinetic friction. (D. 126- 130, G. 49, K. 80-84, W. 96-99). -j In General Physics 73 Apparatus. — Friction table, block of wood, strip of al- uminum or glass, coarse balance and weights, small can and thread, metal shot and spirit level. Method, Theory and Description. — By friction we mean the resistance to motion between two bodies in con- tact, caused by their surfaces. This resistance is independ- ent of the area of contact between the two bodies but is di- rectly proportional to the total pressure between them. In general, friction between two surfaces of the same substance is greater than friction between two surfaces of different sub- stances. When the two bodies are at rest, the force neces- sary to make one slide horizontally upon the other must exceed a certain definite value, which is called the 1 1 maximum static friction The ratio of the maximum static friction F between two surfaces , to the normal pressure N between them is called the “ coefficient of static friction After the one body has started to slide over the other a certain force is necessary to keep it in uniform horizontal motion. The ratio of this friction force to the total pressure is called the “coefficient of kinetic friction ” p. When one of the surfaces is a plane inclined to the hor- izontal and the angle of tilt is gradually increased, there is a critical angle tl i” at which the second surface will slide uniformly down the plane when given a start. It may be shown that the tangent of this critical angle is numerically equal to the coefficient of kinetic friction , or tan i = p The friction table Fig. 19 consists of a heavy base board mounted on leveling screws, upon which a lighter board with a very smooth top is hinged at one end. By means of the hinges and a side clamp, the upper board may be fixed at any desired angle to the base board. A small block of wood 74 Manual of Experiments with four smooth sides is made to slide on the upper board. (Two of these sides have twice the area of the other two). It is pulled along by a weight, hung from a thread which passes over a universal pulley clamped at the end of the table. The friction between the two surfaces under the different conditions mentioned above and the corresponding pressures between them are observed, and the coefficient of friction calculated in each case. faces with the hands. The oil of the skin will lubricate the spots touched and spoil the uniformity of the surfaces. Place the top board flat upon the base and get its upper surface level. To do this place the level in the center with its length across the board, and adjust the two leveling screws at one end of the board until the bubble of air in the level is at the center mark. Then place the spirit level lengthwise of the board and adjust the third leveling screw until the bubble stays at the center mark. (In every case below, adjust the pulley until the thread is parallel to the upper board.) Static and kinetic friction . — Fasten one end of the thread to the hook on the block, pass it over the pulley, and fasten In General Physics 75 the other end to the bucket. Draw the block to the far end of the friction table with its flat side on the board, and place a 100 gram mass on top of it. Then carefully place shot in the bucket until the block just begins to slide. The observer should hold one hand under the bucket while dropping in the shot, to prevent spilling the latter if the bucket begins to drop suddenly. Owing to the fact that the friction table is not uniformly smooth all over its surface, or that the re- lative directions of the grain of block and board are not ex- actly the same in successive trials, or to other causes, the amount of shot needed may vary in successive trials. Do not spend too much time in trying to get such trials to agree. Repeat the trial three or four times and take the mean weight of bucket and added shot for the force necessary to over- come the friction. Under the same conditions vary the amount of shot in the bucket and find what weight of bucket and shot (the mean of several trials) is necessary to keep the block moving uniformly after a slight push has been given to start it. Find the weight of block and added 100 grams mass, and from the data, calculate the coefficients of static and of kinetic friction and compare them. Friction corresponding to different areas of contact. — Turn the block on edge and adjust the pulley until the thread is parallel to the upper board. Place the 100 gram mass on the block and find as before what weight is necessary to keep it in uniform motion after a start has been given it. Measure the areas of the two surfaces of the block that were used as friction surfaces. Compare the kinetic friction in this trial with that in the previous trial and note what effect the area of the surface of contact has upon the friction. V ariation of friction force F with total normal pressure N. — Find the friction force when the block is laid flat with 76 Manual of Experiments 100 grams mass on it. Repeat with 300 grams mass on it. Show from your data that F is directly proportional to N. (Remember that N includes the weight of the block in each case). Compare the coefficients of kinetic friction in the two cases. Variation of friction force with nature of surface of con- tact. — Place the strip of aluminum (or glass) on top of the friction board, holding the former by the edges while hand- ling it. Determine the mass necessary to slide the block with 100 grams on it uniformly along the surface of the metal. Calculate the coefficient of friction and compare it with that between wood and wood. To determine the critical angle. — Remove the aluminum plate, and take the thread from the block. Tilt the top board to such an angle that when started by a slight push, the block will slide uniformly down the board. Clamp the board in this position and measure the perpendicular “a” dropped from the lower face of the upper end of the board upon the base, and the distance iC b” from the hinged end of the board to this perpendicular. See Fig. 20. The ratio a/b is equal to the tangent of the critical angle. Compare this tangent with the coefficient of friction 4> found for the flat block and 100 grams mass sliding on the horizontal wooden surface with uniform motion. * In General Physics 77 Question. — Prove with the aid of a diagram the fact that tangent i= , and show that ‘ Y ” is independent of the mass of the block. ■ 12 STRETCHING WIRES AND YOUNG ’S MODULUS To study laws of stretching wires and to determine Young ’s modulus of elasticity. (D. 164-172, G. 20, 87. K. 235-239, 244, W. 122, 17-2.) Apparatus. — Stretching wire apparatus with optical lever, telescope and scale, four 2Kg. and two lKg. weights, micrometer calipers, beam compass, meter stick, weight holder, and three wires, (two of the same diameter but dif- ferent lengths, and a third of the same length as one of these but having a different diameter.) Theory, Description and Method: — The elasticity of a body is its tendency to recover from distortion when the forces which produce the strain are removed. By strain we mean the change of shape or volume produced by the forces. By stress we mean the internal forces between contiguous parts of the body when it is in a state of strain. Hook’s law states that within the elastic limits , stress is pro- portional to strain. The measure or modulus of elasticity is the ratio of the magnitude of the stress to that of the accompany- ing strain. In the case of simple longitudinal strain, as for instance, a stretching wire, this modulus is known as Young’s modul- us. In this case the stress is equal to the external force per unit area of cross section of wire, or =F /a where F is the force and a the area. The strain or stretch is the extension per unit length of wire, or =l/L where / is the total extension and L the length. Then for a circular wire of radius r, the area is 7r r 2 and Young modulus is 78 Manual of Experiments stress F / FL Y = — — — = — = — strain a L t r 2 l The apparatus consists of a supporting frame of two heavy steel rods about 150 cm. long, mounted vertically on a tripod with leveling screws, and fitted with a clamp at the top for bolding the wire to be tested. See Fig. 21. A small bushing is clamped on the wire near its lower end, and an adjustable platform, sliding on the supporting rods, may be clamped at the same height as this bushing. The wire is In General Physics 79 stretched by means of weights which are placed on the sup- port hung from its l6wer end, and the amount of the stretch is measured by means of an optical lever and telescope with vertical scale attached. The optical lever consists of a T shaped base with a leg at each extremity and a small plane mirror mounted on a horizontal axis above the short arm of the T and facing away from the third leg. This leg rests on the bushing clamped on the wire. The other legs rest on the platform. In its normal position the plane of the mirror is vertical and the telescope is mounted directly in front of it at a distance of about 80 cm., so that on looking through the telescope when it is properly adjusted, we see the center of the scale reflected in the mirror. If now the wire is stretched, the leg V is lowered, (see Fig. 22) tilting the lever through an angle c/>. The mirror and its normal N M will also be tilted through this angle . (In the figure M is misplaced from the intersection of lines from S, N and T.) But by the law of reflection of light rays, the angle made by the incident ray (from scale S to mirror M) with the normal to the mirror (MN) is equal to the angle be- tween the normal and the reflected ray (from the mirror M to the telescope T), and by the geometry of the figure we see that these angles equal also. The point S on the scale will therefore be seen in the telescope. For the small angles of tilt to be used, the point of intersection N between 80 Manual of Experiments normal and scale may be considered as halfway between S and T, and NM as equal in length to TM. Then in the similar triangles VCO and NMT we have VO TN VC x TN = — — , or VO = — VC TM TM where VO is the stretch of the wire, VC is the perpendicular distance from the leg V to the line joining the other two legs, TN is half the distance between the points on the scale observed in the telescope before and after tilting the mirror, and TM is the distance from scale to mirror. Directions. — Caution . — In order to prevent bending and kinking of the wires, keep them stretched by a small mass. The wire when not in use should be hung on the supports provided. Preliminary . — Place the optical lever on a sheet of paper and press lightly on it, leaving an imprint of the three legs. Draw a straight line between the two nearer marks and drop a perpendicular from the third mark upon this line, and measure its length. Measure also the diameters of the three wires with the micrometer calipers, and the effec- tive stretching length between bushing and lug with the beam compass and meter stick. After the stretching weight is removed, the wire to be used is slipped through the hole in the platform and fastened in the upper clamp by means of the lug. To free the wire of all bends an initial stretching weight of 2Kg. is hung on it , and all no load readings are to be taken with this weight attached. The platform is placed in line with the bushing on the wire, and the tripod is leveled so that the bushing slips freely through the opening when the wire is stretched. The optical lever is then placed on bushing and platform. The telescope and platform must be adjusted until the middle of the scale is In General Physics 81 seen in the former. The telescope has three tubes which slide within each other, one containing the object lens, the second a pair of cross hairs, and the third the eyepiece. The latter must be drawn out until the cross hairs are seen sharp- ly focussed; then the second tube is drawn out of the first until there is no parallax between the cross hairs and the scale divisions seen in the telescope; i. e. until there is no relative motion between the two when the eye is moved from side to side in front of the eyepiece. This adjustment should not be disturbed during the remainder of the experi- ment. The distance from mirror to center of scale should then be measured. Relation between stretch and length of. wire. — Suspend the shorter wire with a weight of 5Kg. and note the reading at the cross hairs in the telescope. Then carefully remove three kilograms, the stretching weight, and note the no load read- ing in the telescope. Replace the wire by a longer one of the same diameter, and note the no load and load readings for the same stretching force. Then calculate the stretch by the formula given, and the ratio of stretch to original length for both trials, and compare them. Relation between stretch and cross sectional area of wire . — • Replace the last wire used by the third one, having the same length but a different diameter, and take the load and no load readings with the same stretching force as before. Cal- culate the stretch and the cross sectional areas of the last two wires used. Calculate also the product of stretch and area for each wire, and compare these products. Relation between stretch and stretching force , Hook’s law — Using the same wire as in the last case, change the stretching force to 5Kg. and note the load and no load readings. Calculate the stretch as before. Calculate also the ratio of ¥i < 82 Manual of Experiments the stretch to the stretching mass for each of the two trials with this last wire, and compare these ratios. Young 7 s modulus . — From the data taken in the last case calculate the value of Young’s modulus for the wire used (brass or steel) Assume g = 980 cm/sec 2 . Question. — The data taken show that the stretch / = CmL/ A, where m is the mass producing the stretch, and C is a constant. What is the value of this constant? 13. LAWS OF BENDING OF RODS. .Modulus of Elasticity. To study the relation between the bend of a rod and its length, breadth, depth, and material, and the bending force; also to calculate Young’s modulus for brass and steel. (D. 164-173; G. 20, 88; K. 235-239, 244; W. 122, 173.) Apparatus. — Channel iron bench about a meter long, micrometer screw, micrometer calipers, two blocks with knife edges, two weights, weight holder and loop of thread, meter stick, and three rods, two of them having the same square cross sections but one of brass, the other of steel, and the third of steel with a rectangular cross section. Theory, Description and Method. — (Read that part of the theory in experiment 12 which deals with elasticity.) It has been found that when a rod is bent out of its original position, the amount of the bend B is directly proportional to the cube of its length L and to the force F producing the bend, and inversely proportional to the cube of its depth d and to its breadth b. That is FU B = C bd 3 In General Physics 83 where C is a constant depending upon the mode of support and the material of the rod. If the rod be supported at both ends and the force be applied midway between them, this constant C = i/(4Y) where Y is the modulus of elasticity of the material. In this case it is readily seen that the lower layers of the rod are stretched and the upper layers compressed. The modulus of elasticity involved is therefore Young’s modulus. The relation between bend and modulus is FU FU B = , or Y = 4 Ybd 3 4:Bbd 3 To study the laws of bending a comparison is made be- tween the values of the bend obtained for different values of one of the factors upon which it depends, all the other factors remaining constant. The relation between the bend and this factor can then be determined. If for instance we determine the amount of bend corresponding to two or three different values of the length when all other factors are kept constant, and find on examination that the ratio between corresponding values of bend and length cubed is constant, it shows that the bend is directly proportional to the cube of the length, etc. For these observations we use an iron bench at the middle of which is a vertical support for a mi- crometer screw. See Fig. 23. Two knife edges placed across Fig. 23 $ ' $ 84 Manual of Expeeiments the bench, equidistant from the screw, support the rod to be used. The micrometer is adjusted until its tip just touches the rod, first when the latter is unloaded except for the wire frame, then when it is loaded with a known mass placed on the frame. A reading of the micrometer is taken in each of these positions, and the difference between the two readings gives the bend of the rod. The micrometer screw has a pitch of a half millimeter. Its straight scale is divided into millimeters, its circular scale into 100 equal divisions so that each division represents 1/100 of 1/2 millimeter of' 1/2000 cm. To avoid errors in reading the micrometer the following method may be used: Adjust the micrometer until the screw tip is about 1/8 inch above the rod, the zero of the disc coinciding with one edge of the vertical scale. Carefully turn the screw until the tip just touches the rod, counting the number of complete re- volutions and observing the fractional part of a revolution (the number of divisions beyond zero through which the disc is turned), estimating to 1/10 division. Thus for two com- plete turns and 10 and 1/10 divisions beyond zero the read- ing would be 210.1 divisions. Call this reading R 0 . Return the screw to its original position, suspend a mass from the rod, and proceed as before taking the new reading Ri when the screw just touches the rod. The bend in cm. will be (Ri-Ro) / 2000 . It is somewhat difficult to see just when the point of the screw comes in contact with the rod, therefore an electric buzzer B is used as a detector. See Fig. 23. A dry cell is connected by means of wires to the screw and to the buzzer and from the latter to one end of the rod. Hence when the screw comes in contact with the rod, the circuit will^be closed and the current of electricity from the cell flowing through the buzzer will cause it to emit a humming noise. In General Physics 85 The tip of the screw and the part of the rod which it touches must be kept well polished to offer as little resistance to the flow of current as possible, otherwise the detector will not buzz until there is considerable pressure between screw and rod. Directions. — Relation between bend and length of rod , and between bend and force producing the bend. — Polish the middle of the rods where the screw tip will touch them. Place the knife edges 80 cm. apart with the micrometer half way between them and lay the small steel rod flat on them. Lay the loop of thread over the middle of the rod, bring both ends down through the hole in the bench, and hook them together with the wire frame which is to support the li weights. M Adjust the micrometer and have the threads about 5mm. apart so that the screw can touch the rod be- tween them. Turn the screw until its tip just meets its image reflected from the rod. The humming of the buzzer will indicate when this occurs, if it cannot be seen with the eye. Read the micrometer to 1/10 division and repeat twice more, averaging the three readings to get R 0 . Then hang the smaller mass on the wire frame, take an average of three readings for Ri, and calculate the bend for the weight of this mass, wj. In the same way find the bend corresponding to the weight of a larger mass, m 2 . Then move the knife edges closer until they are 60 cm. apart, the micrometer still being midway between them, and find the bend under the weight of the larger mass. From these data show that the bend varies directly as the force applied and as the cube of the length between supports by calculat- ing the ratios of corresponding values of bend and mass pro- ducing the bend for each case tried, and comparing them; and of bend and length cubed for each case and comparing them. 86 Manual of Experiments Relation between bend and breadth of rod , and between bend and depth of rod . — Measure with the micrometer calipers the width and thickness of all three rods. Take several measurements at different parts of the rod and average them. Using a length of 80 cm. and the larger mass de- termine the bend of the larger steel rod, first with its flat side (the breadth) on the knife edges, then with its narrow side, (which was the depth in the previous trial) on knife edges. If time permits, redetermine the bend of the small steel rod for the same length and mass; if not use the data already obtained for this case. Show with your data that the bend varies inversely as the breadth and inversely as the cube of the depth of the rod, by calculating the products of corresponding values of bend and breadth for each case tried, and comparing them; and of bend and depth cubed for each case and comparing them. Variation of bend with material of rod. Y oung ’s modulus . — - Determine the bend of the brass rod for a length of 80 cm. and the weight of the small mass. Compare this with the bend of the small steel rod under the same conditions and assuming that they have the same breadth and depth, find the ratio of the bends, that of steel being taken as 1. From the data for the first, and last trials calculate Young’s modulus for steel and for brass. Assume g = 980 cm. /sec 2 . Question. — Using the formula given for the bend of a rod, what is the simplest expression for the ratio of the bend of one rod to that of another? (Distinguish between the various factors for the two rods by means of subscripts 1 and 2.) In General Physics 87 14. LAWS OF TWISTING OF RODS Modulus of Rigidity. To study the relation between the amount of twist of a rod and its length, its diameter, the force producing the twist, and the nature of the rod; also to find the modulus of rigidity of steel and of brass. (D. 164-170; G. 20, 89; K. 235-239, 243; W. 122 , 171, 174.) Apparatus. — Iron bench, micrometer calipers, two weights, meter stick, and four rods, three steel ones of dif- ferent diameters, and one brass one. Theory, Method and Description. — (See that part of theory in experiment 12 which deals with elasticity.) It has been found that the amount of twist in a metal rod is directly proportional to its length and to the force pro- ducing the twist and inversely proportional to the fourth power of its radius. The rigidity of a rod under the action of a certain force is inversely proportional to the twist pro- duced by the force. That is if R and T are respectively the rigidity and the angle through which the rod is twisted, the relative rigidity of two rods will be Ri/R 2 = T 2 / 1\. When a rod is twisted the distortion is a change of shape only; that is, it has undergone a shear. The ratio of the shearing stress per unit area to the shearing strain per unit length is the “modulus of rigidity ” (sometimes called co- efficient of torsional rigidity or simple rigidity.) It may be shown that this modulus is . 2 LM n = 7 r 0 r 4 where L is the length of the rod, r its radius, M the moment of the torsional couple, and 6 the angle (in radians) through 88 Manual of Experiments which the rod is twisted. If the rod is twisted by a mass m hanging with a moment arm R , we have M = mgR and 2 LmgR n = 7r dr 4 To study the relation between the twist of a rod and any of the factors upon which the twist depends, the twist for different values of this factor are obtained, all other factors remaining constant. From a comparison between a value of the factor and corresponding value of the twist, the law of variation between them can be determined. Keeping all factors constant but using two rods of different materials and observing the twist for a given force, their relative rigi- dity can be determined by calculating the inverse ratio of their twists. 1 ■L\ L. . „ - Si p ® j — oot , — 1 — — — Fig. 24 The torsion bench is about a meter long. See Fig. 24. An iron clamp may be screwed fast at various positions on the bench and serves to hold in a fixed position the rod to be twisted. The end of the rod is held by a clutch mounted on ball bearings at one end of the bench. Fastened to the clutch is a grooved wooden wheel from the circumference of which is to be suspended the mass which produces the twist. In General Physics 89 A circular scale is attached to the wheel and a fixed pointer indicates on the scale the angle through which the rod has been twisted. Directions. — Relation of angle of twist to length and to force producing twist. — Clamp the smallest steel rod in posi- tion for twisting, having distance between the clamps about one meter. (Some of the small steel rods are copper plated and so do not look like steel.) Take the initial or no load reading T 0 to 0.1 division on the scale when only supporting cord and hook hang from the wheel. Then suspend the smaller mass from the wheel and note the load reading T±. As a check against possible slipping the no load reading should always be repeated after a load reading has been taken. The difference between these readings gives the an- gle of twist, T°. Replace the small weight by the larger one and find the twist produced. Then change the position of the movable clamp so that the rod is about half its original length and find the twist produced by the larger mass. Note the weight of the masses and measure the length of the rod between the two lugs in each case and from the greater length subtract the length of the middle lug (since the part of the rod within this lug is not appreciably twisted.) From the data taken, show how the twist varies with the length and with the weight which produces the twist, by comparing the ratios of corresponding values of twist and length in each case, and the ratios of corresponding values of twist and twisting mass in each case. Relation between twist and radius of rod. — Determine the twist for the next larger rod, using a length of about one meter and the larger mass. Do the same for the largest steel rod under the same conditions. Measure the mean diameters of the two rods with the micrometer calipers, and from your data show the relation between the twist and the 90 Manual of Experiments radius of the rod, by comparing the products of correspond- ing values of twist and the fourth power of the radius in each case. Relative rigidity of steel and brass. — Determine the twist of about one meter of brass rod for the larger mass. If time permits redetermine the twist of the largest steel rod under the same conditions; if not, use the previous deter- mination, and calling the rigidity of the steel rod unity, cal- culate the relative rigidity of the brass rod. Modulus of rigidity. — Measure the circumference of the grooved wheel with a thread and meter stick, and calculate its radius R. Reduce the angle of twist in the last two ob- servations from degrees to radians. Then from data taken calculate the modulus of rigidity for steel and brass. As- sume g = 980 cm. /sec 2 . Question. — If one rod has a times the length and 1/b times the diameter of a second rod, and c times its modulus of rigidity, what is the ratio of the twist of the first to that of the second when the masses acting with the same moment arm for both, are in the ratio d to f respectively? 15. MOMENT OF INERTIA. To find the moment of inertia of a rectangular bar of steel. (D. 81-92; G. 89 A; K. 135-140; W. 85.) I Apparatus. — Upright support with two pulleys, ro- tating disk, rectangular bar, two masses (45 and 100 g.) coarse balance and weights, beam compass, two meter sticks, and string. Theory, Method, and Description. — The moment of inertia of a body about an axis is the sum of the products of the mass of each particle of the body into the square of its In General Physics 91 distance from the axis of rotation. If / be the moment of inertia, m h ra 2 , etc., the masses of different particles and ri, r 2 , etc., their respective distances from the axis of rotation, then / = rmri + m^rf + etc. In the equations of nation for a body in rotation its moment of inertia is the equivalent of its mass in the equations for translation. For instance, the kinetic energy of a mass m with linear velocity v is mv 2 /2. Its kinetic energy of rotation with an angular velocity co is loo 2 / 2. If the whole mass of the rotating body be considered concentrated at a distance k from the axis of rotation, such that its kinetic energy is unaltered, since the linear velocity at this distance is v = k, we may write mv 2 / 2 = mk 2 or/2 = 7co 2 /2 Hence I = mk 2 and k is called the radius of gyration of the body. It may be shown that when a body of mass m has a mo- ment of inertia I Q about an axis through its center of mass, its moment of inertia about a parallel axis at a distance d from the first is 7 = 7 0 + md 2 The method of finding the moment of inertia of the bar is to set it in rotation by a known falling mass. Its kinetic energy of rotation is then the difference between the poten- tial energy of the suspended mass and its kinetic energy at the end of its drop. The angular velocity having been de- termined, the moment of inertia is calculated from the kine- tic energy of rotation. 92 Manual of Experiments The apparatus consists of an aluminum disk mounted on a vertical spindle, and a vertical support upon which are fixed two pulleys. See Fig. 25. A known mass is suspend- ed by means of a cord which passes over the upper pulley and under the lower one, and is wound several times around a wooden spool on the vertical spindle, so that when the mass drops, it sets the disk in rotation. The metal bar whose moment of inertia is to be found is placed on the disk and held in position by projecting pins which fit into holes in the disk. The disk is released by tilting a small lever which catches on a projecting pin on the wooden spool. A small adjustable platform on the upright support serves to locate the height to which the mass has dropped after falling a given number of seconds. As the mass falls its potential energy is converted into kinetic energy, and at any instant its loss in potential energy, is equal to its gain in kinetic energy plus that acquired by the rotating disc. If m be the falling mass, h the distance it falls in time t, g the acceleration of gravity, v the final In General Physics 93 velocity of m at the end of its fall, co the angular velocity of the disk corresponding to v, and I the moment of inertia of the rotating disk, spindle, and spool, m g h = }/% m v 2 2 / co 2 If we assume the friction of the apparatus to be constant the mass will fall with constant acceleration a and the dis- tance fallen will be h = at 2 / 2 The velocity at the end of the fall will be v = at. Hence from the preceding equation v = 2 h/t. The linear velocity of the circumference of the spool will at any instant be the same as that of the falling mass since they are connected by a cord, and as the angular velocity of the spool equals the linear velocity of its circumference divided by its radius r } the angular velocity at the end of the fall will be a> = v/r. ' The linear and angular velocities can therefore be determined from measurements of m, h , t and r and the value of / may be calculated from the equation of energy. When the bar is placed on the disk and the latter is set in rotation by th£ larger mass m h the distance fallen h h in the same time t, will be different, and therefore the velocities vi and coi will be different. If A be the moment of inertia of the bar, the equation of energy becomes pxus-^S- Q- sl —> t ^ ^ 17 <2 Y^ 'g fa (yoJTTbi / , /. o. n AS v/ I L ' In General Physics 119 until the surface just touches the ivory tip which represents the zero of the scale. The barometer height should be care- fully read on the attached scale and vernier. There are cer- tain corrections to be made in this reading. Since mercury in a capillary tube is depressed below its free level, the height of mercury in the barometer tube is less than it should be by an amount depending on the diameter of the tube. This correction should be added to the barometer reading. Since on the other hand the brass scale of the barometer, and the mercury expand with an increase of temperature, the density of the mercury will be less, and the reading will be greater at ordinary temperatures than it would be under standard conditions; that is at 0° Centigrade. We must therefore subtract a certain correction from the barometric height as read on the scale. A table of these corrections will be found in the appendix. For very careful work a correction should also be made for variations in the acceleration of gravity at different po- sitions on the. earth’s surface, but the correction may be neglected in the present case. In this particular experiment we are dealing only with relative pressures, and as the barometer is at practically the same temperature as the other mercury columns and the expansion of the brass scale is small compared with that of the mercury, the temperature correction may be omitted. The two glass tubes are of the same bore, hence the cap- illary action is balanced in them and no capillary correction is needed in the heights of the two mercury columns. Directions. — Caution . — Be careful not to lower the open tube so far that the mercury may run out. Also avoid handling the closed tube or in any way heating it, as small changes in temperature produce an appreciable change in 120 Manual of Experiments volume. When the volume of the air column has been changed by altering the level of the mercury, always give the air time to recover from the resulting change in temper- ature before reading the height of the mercury columns. Attach the thermometer to the closed tube in order to see that the temperature of the enclosed air remains constant. Record the temperature to 0.1° and see that it is the same after each change in the height of the mercury before taking a reading. Adjust the height of the tubes until the mercury stands near the top of the open tube. Read to 0.01 cm. the height “a” of the column in the open tube and the height “b” of that in the closed tube. Read also the height “c” of the top of the enclosed air column. Lower the mercury in the open tube about 10 cm. and read the new heights a and b. Continue this process until the mercury in the open tube is near the bottom. Then raise the mercury again by stages of about 10 cm. each, reading the corresponding heights a and b at each stage. Read the barometer height and make the necessary corrections for capillarity. Cal- culate the pressure p = h + (a — b) at each position (“ h ” being the corrected barometric height), and the correspond- ing length c — b. With this data show that Boyle’s law holds true. Question. — -a) Show the form of the graph for Boyle’s law by plotting corresponding values of pressure and volume of your data on coordinate paper. b) Plot a curve having products of p and v as ordinates and corresponding values of p as abscissae. In General Physics 121 Experiments in Heat. 22. CALIBRATION OF THERMOMETERS. (a) To determine the zero and steam points of a ther- mometer. (b) To calibrate the scale of a thermometer. (D. 267; G. 300-304; K. 365-368; W. 180.) Apparatus. — -A 100°C thermometer, a thermometer with an uncalibrated scale, hypsometer, copper calorimeter, ring stand, gas tubing, Bunsen burner, glass funnel, beaker. Description and Theory. — A mercury thermometer consists of a thin glass bulb connected with a capillary tube. The bulb and part of the capillary tube are filled with mer- cury and then the tube is sealed off after all the air has been expelled. There are two definite temperatures that may be used to locate two fixed points on the thermometer. One is that of melting ice which, on the Centigrade thermometer, is marked as 0° and the other is the temperature of vapor formed over water boiling freely under a pressure of 76 cm. of mercury. This point is marked 100° on the Centigrade thermometer. When an ordinary thermometer is used for any sort of accurate work it is generally found to be quite badly in er- ror. Glass is very unstable in its molecular structure. As it gets older the molecules gradually readjust themselves and a sort of annealing process goes on similar to that oc- curing when glass is gradually cooled after being worked in a flame. This change in the walls of the thermometer alters the volume contents of the thermometer bulb and tube, so that the fixed points of the two definite temperatures will be found to have changed. In making standard thermo- 122 Manual of Experiments meters the glass must be cooled very slowly and allowed to become quite old before they are calibrated, in order that the annealing process may be fully completed. Because of the change in the boiling and freezing points of the thermo- meter, there is, of course, a corresponding change in the position of the other divisions of the thermometer. When the space between the two fixed points is divided into equal divisions the bore of the capillary tube is sup- posed to be uniform throughout. This is not really true so that, for very accurate work it is necessary to calibrate and make a calibration curve for the thermometer. The fixed points, the 0° and the 100° points, of a thermo- meter are tested by noting the readings indicated by the thermometer when it is placed in melting ice and dn dry steam. If t 0 is the reading of the calibrated thermometer when the bulb is in melting ice then the correction at zero degrees is C 0 = [0° - to] . . . . (1) Again if the temperature of steam at the given pressure is T a and the reading of the thermometer when placed in the steam is t a , the correction at the boiling point is “ [Tg : ~ t a ] . . . . (2) Assuming the bore of the calibrated thermometer to be uniform, which is allowable for ordinary work, the correction for any intermediate temperature t t is given approximately by x C t = [C 0 + c s -c o 100 Therefore the correct temperature t c , when the thermometer reads t t , is given by 4 = 4 + + C s -C 0 4 ] 100 ( 4 ) In General Physics 123 The zero point and the steam point of the uncalibrated thermometer can be located in the same manner as the corres- ponding points of the calibrated thermometer. The inter- mediate points can be calibrated by placing the two ther- mometers in the same vessel of water at various tempera- tures and, having observed the readings of both, by de- signating the point on the uncalibrated thermometer by the corrected reading of the calibrated thermometer for each temperature of the water in which they are immersed. The hypsometer, Fig. 31b, used in locating the steam point, is a copper vessel having a reservoir in the lower part for holding water and in the upper part having an inner and 124 Manual of Experiments an outer jacket. The two jackets are so constructed that the steam generated in the reservoir has to pass up through the inner jacket and down through the space between the two jackets before escaping through the outlet tube. If the thermometer is inserted in a cork in the top of the hypso- meter it is thus surrounded by dry steam. Directions. — The freezing point should be tested first. To do this, arrange a glass funnel in a ring support as shown in Fig. 31a, with a beaker beneath to catch the melted ice. Fill the funnel with snow or cracked ice and insert the ther- mometer so that the entire thread of mercury is beneath the surface of the melting mixture. After the thermometer has been in the mixture long enough for its reading to reach the melting temperature, take readings t Q , every minute until three readings have been obtained. Read carefully to tenths of a degree. At the time of each reading raise the ther- mometer so that the top of the thread of mercury may be seen just above the surface of the cracked ice and then quick- ly take an accurate reading. In order to test the boiling point, pour water into the reservoir of the hypsometer till it is about two-thirds full and place a Bunsen flame under it. This may be done while the freezing point is being tested. Do not let the hypsometer boil dry. Insert the thermometer in the hypsometer as shown in Fig. 31b, so that the 100° point may be seen just above the top of the cork. After the steam has been escap- ing freely for some time from the tube leading from the outer jacket, and the mercury in the thermometer has reached a steady point, take readings t s every minute until three readings have been obtained. Read carefully to tenths of a degree. Read the barometer and, having corrected the reading In General Physics 125 according to the table of corrections in the Appendix, find the boiling temperature T a , corresponding to that corrected pressure, by referring to the steam table in the Appendix. It is found that after the thermometer has been ex- posed to the steam the glass bulb does not contract to its former volume for some time afterwards. So that, if the thermometer is placed again in the melting ice (after being allowed to cool 30° or 40° below the boiling point) the po- sition of its freezing point will have changed slightly. After the thermometer has acquired the temperature of the freezing mixture take readings every minute until three readings have been obtained. In the calibration of the uncalibrated thermometer the average of these readings will be taken as the freezing point t Q of the calibrated ther- mometer instead of that obtained before the thermometer was heated. Substitute in equations (1) and (2) and find the corrections C Q and C s . In a similar manner note the readings indicated by the uncalibrated thermometer when placed first in dry steam and then in melting ice. Draw a scale in your note book similar to the one upon which the thermometer is mounted and locate upon the scale the position of the temperature of steam and the zero point just observed. In order to fix intermediate points on the uncalibrated thermometer, place the two thermometers in a calorimeter containing water at about 20°C. Observe the position of the mercury column when they have become stationary, and having corrected the reading t t of the calibrated ther- mometer according to equation (4), record this corrected reading on the scale in your note book at the position indi- cated by the mercury column on the uncalibrated thermo- meter scale. In a similar manner locate and determine the value of the position on the uncalibrated scale when the filler- 126 Manual of Experiments mometers are immersed in water at temperatures of about 40°, 60° and 80°C. Questions. — What is the difference between the aver- ages of the two sets of readings obtained for the freezing point before and after the boiling point readings? Explain the cause of this difference. 23. CHARLES J LAW. Pressure Coefficient of Air. To show that the volume of a given mass of gas remain- ing constant, the pressure varies directly as its absolute tem- perature. (D. 264, 265, 279, 293, 294; G. 332, 333; K. 376, 393-397; W. 195, 197.) Apparatus. — Air thermometer, hypsometer, tripod, Bun- sen burner, copper can, stirrer, mercury thermometer, and cloth. Theory, Method and Description. — (See theory under experiment al.) Charles’ law (also called Gay Lussac’s law) is a special case of the gas law, and holds only when the pressure of a gas remains constant while its volume and temperature vary. Under these conditions the ratio of vol- ume to absolute temperature is v/T = R/p (a constant). % The law may therefore be stated as follows: — The pressure of a given mass of gas remaining constant , its volume is di- rectly proportional to its absolute temperature. It is quite evident that, when the volume of the gas is kept constant, In General Physics 127 its pressure is directly propor+ional to its absolute temper- ature, or p /T = R /v { a constant). This may be looked upon as another way of expressing Charles ’ law, and it is this law which we wish to verify. The gas constant R may be shown to be equal to a p 0 v 0 where p c and v Q are the pressure and volume respectively of the given mass ,of gas at zero degrees centigrade and a is the pressure coefficient , or that fractional part of the pressure at zero degrees Centigrade by which it increases for each degree Centigrade rise in temperature . That is to say, the pressure of the given mass of gas at t degrees Centigrade will be Pt ~Po (1 + at) and a = (pt — po) / (po t) or the pressure coefficient will be the change in pressure per unit pressure at zero degrees Centigrade per degree Centi- grade change in temperature. From the relation p t = p Q (1 + at) we see that at the temperature t = — 1/a, p t will equal zero. This point in the scale of temperatures ’ at which there is no energy in the gas , is called the absolute zero ) and temperatures measured from this point are called absolute temperatures. On the Centigrade scale the ab- solute zero is — 273°, and T° (abs,) = (*+273)°C. * The apparatus used to verify this law is similar to the Boyle ’s law apparatus, but the closed tube leads into a hori- zontal capillary tube a foot or more in length which ends in 128 Manual of Experiments a vertical elongated bulb, B. This arrangement makes it possible to place the bulb in baths at different temperatures, and also keeps these baths at some distance from the mer- cury columns. See Fig. 32. The volume of air can be kept constant by adjusting the mercury each time after the air has come to the temperature of the bath, so that it stands at a mark on the apparatus just at the beginning of the cap- illary tube. The temperature of the air in this tube will be \ i Fig. 32 slightly different from that of the bath, but its volume is so small in comparison with the volume of air in the bulb, In General Physics 129 that we may consider all of the air to be at the same temper- ature. The volume of the bulb does not remain absolutely constant but increases as the glass expands with a rise in temperature, and as the pressure of the confined air increases. By observing the temperatures of the baths, and the cor- responding pressures due to the atmospheric pressure on the open tube and the difference in level of the mercury columns, we obtain the data for verifying the law and calculating the pressure coefficient. The law holds only approximately for gases which do not obey Boyle ’s law, but for air the change in the ratio p/T is very slight and the pressures may be used to find the cor- responding temperatures over a wide range by means of a calibration curve. The apparatus is therefore often called an air thermometer. Directions. — Adjust the mercury columns until the height of that in the open tube is at least 15 cm. less than that in the closed tube. Place the copper can (without stirrer) on a stand so that the bulb projects into it, being care- ful not to break the latter nor the capillary tube. Fill the can with snow or finely crushed ice until the bulb is covered. Fill the interstices of the ice with cold water. Allow several minutes for the air in the bulb to come to the tem- perature of the ice bath, i. e. 0° C. Adjust the tubes until the mercury is at some well defined mark near the begin- ning of the capillary tube, being careful that it does not run into the capillary tube and the bulb. Note the height of the mercury column 11 a” in the open tube and the height il b” in the closed tube and record the difference in level (a — b). Remove the ice from the can and replace it in the ice box at the sink. Fill the can with water and heat it to about 25 degrees C. Then remove the burner and allow the water and the air in the bulb to come to a common tern- 130 Manual of Experiments perature, using the stirrer to equalize the temperature of the water. Note this temperature. Adjust the height of the mercury so that it stands at the same mark on the closed tube as before, in order to make the volume of air the same, and observe the new difference in level of the mercury col- umns. Then heat the water to about 60 degrees and pro- ceed to find the nev difference in level at this temperature. Then carefully lower the level of the mercury column until its height in the open tube is again the first value of “a”, before removing the hot bath, so that when the air in the bulb cools and contracts, the mercury will not run over into the bulb. Next fill the can of the hypsometer about one- half full of water, place it on the tripod so that the bulb of thermometer projects into the hypsometer, and adjust a cloth around the neck of the bulb to prevent steam from escaping freely. Then heat the hypsometer until steam is generated freely and. again adjust the mercury columns until the vol- ume of air is the same as it was before. Observe the differ- ence in level of the mercury columns and also the corrected barometric pressure h. (See experiment 21 for corrections to barometric readings.) Before lowering the temperature of the enclosed air by removing the steam bath, be sure to lower the level of the mercury to the original value “a”, so that no mercury flows into the bulb when the air contracts. Obtain the temperature of the steam at the barometric pressure from the steam table in the appendix, and calculate the total pressure corresponding to each temperature used. With the data taken show that Charles 1 law holds true, and calculate the pressure coefficient of air for constant vol- ume, and the absolute zero. Comparison of Temperatures by the Air and the Mercury Thermometers . — With the data taken for melting ice and for steam, plot a straight line curve having total pressure of In General Physics 131 enclosed air as ordinates and corresponding temperatures as abscissae, assuming the pressure coefficient to be a con- stant throughout this range of temperatures. On this curve find the temperatures corresponding to the pressures ob- served when the bulb was immersed in the baths at about 25 and 60 degrees C. respectively. Compare each of these temperatures with the corresponding reading on the mercury thermometer. Find also the absolute zero on the Centigrade scale by prolonging the curve until it meets the axis of tem- peratures. Question. — What are the reasons for and against the use of the air thermometer as compared with the mercury in glass thermometer? 24. LINEAR EXPANSION. To determine the coefficient of linear expansion of a met- al rod. (D. 273-275; G. 314-321; K. 379-388; W. 184-186.) Apparatus. — Linear expansion apparatus and metal rod, steam generator, Bunsen burner, tripod, thermometer, mi- crometer screw, funnel and waste water jar. Theory, Method and Description. — The coefficient of linear expansion of a substance is the change in length of unit length of the substance for a change in temperature of one degree Centigrade . For instance if a metal rod changes from a length L x cm. to L 2 cm. when the temperature changes from h to t 2 , the total change in length per cm. of its original length is (L 2 — L{) /Li and the change per degree change in temperature will be _ L 2 —Li a ” Uih-h). For isotropic bodies this coefficient is the same in all di- rections. Strictly speaking it is not a constant but depends 132 Manual of Experiments upon the temperature, and in order to define it precisely we should refer to the change in length of unit length of the substance at zero degrees Centigrade. For all practical purposes however, (since the change with temperature is very small) a is calculated by the formula given above. The method used is to place a metal rod about one meter long in a bath of cold water, noting the temperature of the water and measuring the length of the rod when it has come to the same temperature; then to replace the water bath by one of steam, again noting the temperature and meas- uring the length of the rod after allowing sufficient time for it to reach the temperature of the steam. With this data the coefficient of linear expansion may be calculated by the formula given. The metal rod is surrounded by a sheet copper jacket, its ends projecting through rubber corks which fit tightly about the rod and within the ends of the jacket. See Fig. 33. The jacket is covered with sheet asbestos to prevent loss of heat by radiation,* etc., and rests upon a wooden support. At one end of this wooden support is an adjustable screw S which may be turned into contact with the rod. At the In General Physics 133 other end is a micrometer screw M which can also be turned into contact with the rod and serves to measure its elong- ation. To measure the temperature of the bath and rod a thermometer T is passed through a cork in an opening at the middle of the upper side of the jacket. A second opening at one end serves to introduce the water and steam and a third opening at the other end of the under side is fitted with a rubber tube for carrying off the water or steam and may be closed by means of a pinch clamp. Directions. — To save time the water may be heated to the boiling point while the rest of the experiment is being performed. Fill the steam generator about one-half full of water, and heat it with a Bunsen burner, placing it far enough from the rod to prevent heating the latter. Mean- time close the lower outlet with the pinch clamp and fill the jacket with cold water, being careful not to cause an over- flow. Adjust the thermometer in the upper opening so that it rests on the rod and allow sufficient time for the rod and the water to come to a common temperature before noting the latter. Adjust the brass set screw so that it touches the end of the rod, then adjust the micrometer screw until it is about five millimeters from the end of the rod and the zero of the graduated circular disc is at the edge of the straight mm. scale. Then turn the screw into contact with the rod, counting the number of turns and noting the number of divisions on the last fraction of a turn. (Since there are 100 divisions on the circular scale 9 complete turns and 45 divisions over would give 945 divisions, etc.) In order to tell when the screw and rod come in contact an electric buzzer is used. (See experiment 13.) Turn the micro- meter back to the starting point but leave the set screw as it is. Measure the length of the bar in cm. with the beam compass.’ Allow the water to run out into the sink and 134 Manual of Experiments leave the lower outlet open. When steam is being freely generated, connect the generator to the upper end-opening by means of rubber tubing and allow it to flow through the jacket, noting the change of temperature on the thermometer. When steam issues freely from the lower outlet and the temperature remains constant at about 98°, see that the set screw is in contact with the rod. If it is not in contact, do not move the screw but press the rod gently into contact. Then turn the micrometer screw until it just touches the end of the rod, noting the number of divisions through which you turn it. Then record the temperature of the bar in the steam. The pitch of the micrometer screw is 1/2 mm. hence there are 2000 of the divisions on the circular scale in a cm. Find the expansion of the rod, the difference of the micrometer readings in cold water and in steam. From the data obtained calculate the coefficient of expansion of the rod. Repeat the experiment, and calculate the mean coefficient from the two determinations. Question. — a) How would your result be affected if the rod were not in contact with the screw S after expansion. b) What force in pounds weight would be needed to stretch the rod the same amount as in this experiment? 25. COEFFICIENT OF CUBICAL EXPANSION OF A LIQUID. To determine the coefficient of cubical expansion of a li- quid. (D. 277; G. 327, 328; K. 389; W. 189-191.) Apparatus. — Specific gravity bottle, liquid, thermometer, fine balance, set of known masses, calorimeter, tripod, Bunsen burner. In General Physics 135 Description and Theory. — The specific gravity bottle (Fig. 34.) is simply a small bottle which is provided with an accurately fitting ground glass stopper. A very small hole through the center of this stopper leads to the interior of the bottle, its object being to allow the bottle to be completely filled with any liquid. The method used in this experiment to find the coefficient of cubical expansion of a liquid, depends upon the relative expansion of the liquid and glass. Let m 0 be the mass of the bottle and liquid at 0°C, d 0 the density of the liquid at the same temperature and m the mass of the bottle. Then the volume of the liquid at 0°C will be Mo — m Vo = a) If j3 is the coefficient of cubical expansion of the liquid, its volume at t°C will be Mo — M V = Vo(l +(lt) = (1+180 ( 2 ) 136 Manual of Experiments Similarly, the capacity of the bottle at 0°C is Mo — m Vo . ( 1 ) and, if the coefficient of cubical expansion of glass is a , its capacity at f C will be Mo — M v f — Vo (1 + at) — (1 at) do (3) The volume, at f C, of the liquid expelled is therefore Mo ~ M Mo — M Mo—M ■(1+00 U— (1 + at) = ; 1 (fi-a). (4) dc. do 'YYLq — This volume is also given by — — — (1 where Mi is the d Q mass of the bottle and liquid at the temperature t. There- fore, Mo — Ml Mo — M — — — (1 +00 = ■ ■ t (13 — a ). do do Solving (5) for j3, we find (5) Mo — Mi Mo — M 0=7 H oi (6) (Mi — M)t Mi~M Since m 0 — m and mi — m differ very little, equation (6) may be written. Mo ~ Mi (3 = — — + a . . (7) (Mi~M)t This equation expresses the coefficient of cubical ex- pansion of the liquid in terms of the coefficient of cubical expansion of glass, the original mass of liquid in the bottle at zero degrees, the final temperature \ f, and the mass of the. liquid that is expelled from the bottle when its temperature is changed from 0° to t° C. In General Physics 137 Directions. — Clean and dry the specific gravity bottle very carefully, then find its mass m on the fine balance. Fill the bottle with the liquid whose coefficient of expansion is to be determined, place it in the calorimeter and pack snow or shaved ice around it. See that none of the ice gets into the bottle. Place the stopper also in the ice, and after leaving the bottle and stopper in the ice for 10 minutes, dry the stopper quickly, but thoroughly with a cloth and put it in the bottle. See that no air bubbles cling to the stopper. Leave a small drop of the liquid on the top of the stopper so as to have the capillary tube filled in case there is a further contraction of the volume of the liquid in the bottle. Leave the bottle in the ice at least 5 minutes longer, then wipe off the top of the stopper with a cloth. Remove the bottle, dry it carefully, and then find the mass m Q of the bottle and contents on the fine balance. The weighing should be done as quickly as possible, as a precaution against the loss of weight through evaporation of the liquid that may escape as its temperature begins to rise. Heat some water (not closer than to within 10 or 12 degrees of the boiling point of the liquid whose coefficient is being determined,) and place the bottle in it. Do not let the water extend above the neck of the bottle. After leaving the bottle in the water 10 or 15 minutes, note the temperature t of the water, wipe off the liquid from the stopper, remove the bottle, and having dried it thoroughly, find its mass mi on the balance. Taking the value of a the coefficient of cubical expansion of glass, as 0,000025, substitute the values found for m , m Q , mi and t , in equation (7) and calculate the value of p. In a similar manner make a second determination of the value of p. * Questions. — If a bubble of air is trapped in the bottle 138 Manual of Experiments when making the weighing m Q , what effect will it have on the value of (3 found? Explain your answer. CALORIMETRY. * (D. 280-284; G. 452-458; K. 398-401; W. 199-201.) In each of the following four experiments a quantity of heat is to be measured. The calorimeters in which this measurement is to be made, are metal vessels, usually cov- ered with insulating material to prevent the loss of heat by radiation, and supplied with a stirrer for equalizing the tem- perature of the water which they contain. The method usually employed is known as “the method of mixtures, ” which is based upon the principle of equal heat exchanges. That is, when two or more bodies, orig- inally at different temperatures, are placed in thermal con- tact, and the heat exchanges take place exclusively between these bodies, the quantity of heat lost by one part of the sys- tem of bodies is equal to the heat gained by the other part. In this method a given mass M grams of the substance whose heat is to be determined is heated to a temperature t 2 ° C. and is quickly plunged into, or mixed with W grams of water at a lower temperature t° C. The mixture will then come to a uniform temperature f C. It is evident that besides the gain in heat by the water in rising to the temperature of the mixture, the calorimeter containing the water and any other bodies such as the ther- mometer and stirrer, which have a common temperature with the water will also gain in heat. The simplest way of taking these bodies into account is to treat them as so much extra water beyond the amount W) that is, we must find their water equivalent. The water equivalent of a body is the amount of water which requires as much heat to raise In General Physics 139 its temperature i°C. as does the body in question. It is equal to the mass of the body multiplied by its specific heat. In order to avoid errors due to gain or loss of heat by radiation from or to . neighboring bodies, the following method of compensation is generally used. On starting, the water and calorimeter are brought to a certain tempera- ture below that of the room and the process of heating is continued until they are at a temperature as much above that of the room, as they were originally below it. In this way approximately as much heat is gained by the calorimeter from the surrounding air until it comes to the same tem- perature as the latter, as is lost by it to the air, while it exceeds this temperature. 26. SPECIFIC HEAT. To determine the specific heat of copper by the method of mixtures. (D. 280-284; G. 452-454, 457-466; K. 398-406; W. 199-207.) Apparatus. — Calorimeter, steam generator, steam heater, two thermometers, copper or aluminum pellets, drying cup, tripod, wooden paddle, paper funnel, cloth, coarse balance and set of weights. Theory, Method and Description. — Every substance requires a definite amount of heat to raise a given mass of the substance through a given temperature. The quantity of heat necessary to raise the temperature of one gram of the substance through one degree C., at any given temperature , is called the specific heat of the substance at that tempera- ture. For scientific purposes the quantity is measured in terms of a unit called the calorie. The calorie may be de- fined as the quantity of heat required to raise the tempera- ture of one gram of water from 15° to 16° C. (This is the 140 Manual of Experiments mean heat capacity of water between its freezing and boiling points). The specific heat of water at 15° C. is therefore unity. In engineering practice quantity of heat is measured in British thermal units. The B. T. U. is the heat required to raise the temperature of one pound of water , one degree Fahrenheit. The specific heat of copper or of aluminum is to be de- termined by the method of mixtures. Let W be the known mass of water in the calorimeter, at a temperature h, M the mass of the substance whose specific heat C is to be deter- mined, h its temperature, t the temperature of the mixture, and w the water equivalent of the calorimeter. Then taking the specific heat of water as unity, the heat gained will be (W-\-w) (t — ti) calories. In falling to the temperature of the mixture, the heated body will lose an amount of heat C M (t 2 — t) calories. Equating the amounts of heat lost and gained, we have C M (fk — t) = (W+w) ( t-h ) from which to calculate the specific heat. The water equivalent of the thermometer is small and may be neglected, and if the calorimeter cup and stirrer are of the same material as the pellets, we may assume that they have the same specific heat. If their joint mass is m, their water equivalent will be Cm and the equation becomes C M (t 2 -t) = ( W+Cm ) (t-h) from which we calculate C To prevent heat becoming lost by radiation the copper calorimeter cup is surrounded by a hair-felt jacket enclosed within a second copper vessel. The calorimeter has a wood- In General Physics 141 en lid which holds a copper stirrer for equalizing the tem- perature of the water. A thermometer passes through a rubber stopper in the middle of this lid and through the stirrer. The metal pellets are heated in a cup which is surrounded by a steam jacket. Directions. — Fill the steam generator about half full of water and heat it with a Bunsen burner. Take enough metal pellets to fill the inner calorimeter cup about one- third full, and pour them into the small cup of the steam heater. The pellets should be perfectly dry. # If they are not, pour them into the brass cup, and heat them over the flame, stirring meanwhile with the wooden paddle until they are dry. When they have cooled somewhat pour them into the small steam heater cup and insert the cork and thermo- meter. Work the bulb of the thermometer into the pellets, being careful not to break it. Connect tU^ojjrer opening of the steam heater to the steam generator^tu allow steam to come out through the opposite upperflTole, so that it passes all around the small can. Close the outer lower opening with a rubber hose and pinch clamp. After the steam be- gins to issue shake the heater occasionally and invert it, so that the pellets are thoroughly mixed and the temperature uniform throughout, being careful to hold the cork and ther- mometer in place. After a time the temperature will be- come stationary at 97° or 98°C. In the meantime weigh the inner calorimeter cup (dry) and record the mass in grams. Fill the cup a little more than half full of ice water and weigh again. Place cork and thermometer in place and adjust the length of the thermometer so that it will reach within about two centimeters of the bottom of the cup. Place the cup in the calorimeter and see that there is no ice in the water. Shake the steam heater once more and read the temperat ure to one-tenth degree. Then remove the cork and with the 142 Manual of Experiments help of the paper funnel pour the pellets into the ice water with one swoop. Do not delay to collect stray pellets but immediately insert the cork and thermometer, being careful not to break the latter by forcing it into the pellets. Shake the calorimeter for a few seconds, being care- ful not to spill any of the water, and watch the thermometer. Just as it ceases rising and begins to drop read the temper- ature to one tenth degree. Weigh the inner calorimeter cup again, with water, and pellets, included. From the data taken calculate the specific heat of the metal pellets. Before repeating the determination note whether the temperature of the mixture was as much above that of the room as the original temperature"" was below it, and if not, determine by an inspection of the data whether more cold water is needed or more hot pellets, to obtain this temper- ature. With the altered amount of water or pellets, repeat the experiment and calculate again the specific heat of metal. Questions. — a) What is the effect on your result, of spilling some of the water from the calorimeter while shaking it? . b) Which metals heat up the quicker, those with high or those with low specific heat? 27. THE MECHANICAL EQUIVALENT OF HEAT. To determine the mechanical equivalent of heat by means of Callendar’s apparatus. (D. 280-282, 290; G. 452-454, 505, 506; K. 409-411; W. 199-201, 249-251, 494.) Apparatus. — Calendar’s apparatus with two 2000 gram weights, coarse balance and set of weights, beaker, funnel, bent thermometer, straight thermometer, six 50 gram weights and hose and nozzle for draining. In General Physics 143 Theory, Method and Description. — The first law of thermodynamics may be stated as follows : — Whenever me- chanical energy is converted into heat , or heat into mechanical energy , the ratio of the mechanical energy to the heat is constant. This constant ratio is called the mechanical equivalent of heat. It is the number of units of work which must be done in order to produce unit quantity of heat energy. To de- termine this constant we shall use the following meth- od: — A cylindrical brass calorimeter containing a known quantity “ W ” of water at a known temperature (C ti” is rotated against the friction of silk bands which bear upon it owing to known weights which are suspended from them. On starting, the water and calorimeter are at a certain temperature below that of the room, and the cal- orimeter is turned until it is at a temperature as much above that of the room as it was below at the start. The total number of revolutions tc n” of the calorimeter and the new temperature li W’ of the water are noted and the circum- ference “ c v of the calorimeter measured. Since the calori- meter also absorbs heat in rising to this temperature we must calculate its water equivalent li zv” or the amount of water which would require just as much heat as the calorimeter does to be raised through unit temperature. If the aver- age force at the circumference of the calorimeter is the weight of a mass m, the work done in overcoming this resistance through a distance equal to n times the circumference c is n c m g. The quantity of heat required to raise the water and calorimeter from the temperature h to the temperature h will be {W -\-w) ( h~h ). The ratio of these two quanti- ties will be the mechanical equivalent of heat “J,” or J (W -\-w) (h — tf) = n c m g. The apparatus (See Fig. 35) consists of the mounted cylindrical brass calorimeter C rotated by hand by means of 144 Manual of Experiments a crank wheel, the number of revolutions being registered by a counter on one side. Unequal weights Mi , M 2 are sus- pended from the ends of a silk belt which bears on the cyliji- Fig. 35 der and makes one and a half complete turns around it. A light spring balance B is arranged so that it acts in direct opposition to the lighter weight. The motion of the belt is limited by means of stops, and the weights are adjusted and the cylinder rotated so that the weights are held suspended, In General Physics 145 the final adjustment being made automatically by the spring balance. The temperature of the water is read on a bent thermometer T inserted through a central opening on the front end of the cylinder. Directions. — Hang the 4000 grams from the end of the belt opposite the spring balance and add small brass weights to the other end until “floating equilibrium M is maintained when the cylinder rotates; that is until the weights are not supported at the stops, but hang freely from the belt. Level the apparatus so that no part of this 146 Manual of Experiments floating system touches the frame. The student chosen to rotate the cylinder should start it a number of times to be- come familiar with its behavior. Hang the straight ther- mometer near the calorimeter, allow it to come to room temperature and note this temperature. Place about 200 grams of distilled water in a beaker and cool it to about 7° C. below the room temperature. Weigh funnel, beaker and water, then pour the water into the calorimeter through the central opening, and reweigh funnel and beaker with whatever water may be adhering to them. Pass the bent thermometer through the central opening (being careful not to put any strain on it by pressing against the walls of the cylinder, )and fasten it so that its bulb is immersed in the water. Remove the weights from the belt and turn the cylinder until the thermometer indicates a steady tem- perature, after which replace the weights. Then rotate the cylinder with a steady motion. A second student should note the number of revolutions on the counter and at every 45th and 95th revolution he should call “ ready ”, and at every 50th and 100th revolution he should call “read”. A third student should note the temperature of the water at the word “read” and a fourth student should record all data. The cylinder should be rotated until the tem- perature of the water is about 7° C. above the tempera- ture of the room. The water should then be drawn from the calorimeter by removing the flat headed brass screw and re- placing it by the nozzle and hose. Calculate the length of the circumference of the cylinder by winding a string sev- eral times around it, measuring its length, and finding the length of a single turn. The average force exerted at the circumference c will be the weight of the 4000 grams M\g plus the spring balance reading Msg minus the small brass weights M 2 £. (See Fig. 36.) In General Physics 147 With the data obtained calculate the mechanical equiv- alent of heat, assuming g = 980 cm/sec. 2 . Question. — If it requires 4.2 joules to raise one gram of water 1°C. in temperature, what is the value of “J” a), in foot pounds per calorie? b). In foot pounds per B.T.U? c) In foot poundals per B. T. U.? 28. LATENT HEAT OF FUSION. To determine the latent heat of fusion of ice. (D. 280- 283, 307, G. 452-458, 467, K. 398-401, 424-426, W. 199-201, 208-211.) Apparatus. — Calorimeter with lid and stirrer, hot water reservoir with lid and stirrer, water heater, thermometer, ring stand, tripod, Bunsen burner, coarse balance, set of weights and drying cloths. Theory, Method and Description. — If a solid be heat- ed slowly, when its melting point is reached it will be found that the temperature remains constant at this point until all the substance is changed into the liquid state, all of the heat supplied being used up in this change of state. Each substance requires a definite amount of heat to transform a given amount of the substance from the solid to the liquid state at the temperature of the melting point. The latent heat of fusion of the substance is the number of calories of heat required to change one gram of the substance at the melting point from the solid to the liquid state. To determine the latent heat of fusion of ice we may use the method of mixtures (see Calorimetry.) A mass M grams of ice at 0° C. is added to a mass W grams of water at h°C. in a copper calorimeter. The temperature t of the water must be noted at the instant when all of the ice is melted. The specific heat of copper may be taken as 148 Manual of Experiments 0.093 calories per gram per degree Centigrade, and if the mass of the calorimeter and stirrer is m grams, their water equivalent w will be w = 0.093 m grams Since the specific heat of water is unity the quantity of heat gained by the ice in melting and rising to the temperature of the mixture after it is melted, will be M — ) calories, where L { is the latent heat of fusion of the ice. The heat lost by the water and calorimeter in dropping to the temperature t will be (W-\-w) ( fa — t ) calories. Equating these two quantities we have M ( L f +t ) = (W+w) (fe-0 from which we calculate L f . To avoid error in the result, the ice should be carefully dried before placing it in the calorimeter. To avoid error due to radiation, the final temperature of the water should be as much below, as its original temperature was above that of the room. The calorimeter consists of a copper cup with a wooden lid and copper stirrer, surrounded by a hair felt jacket en- closed within a second copper vessel. The hot water reservoir is a copper vessel with lid and stirrer, covered with a hair felt jacket and having at the bottom an outlet for the water. The lids and stirrers of the calorimeter and reservoir are pierced with holes to ac- commodate the thermometers. Directions. — Fill the steam heater about three fourths full of hot water and heat to about 60°. Run this hot water into the reservoir, being careful to have the lower opening closed with the rubber tubing and pinch clamp. Weigh out about 60 to 80 grams of ice in lumps about the size of wal- nuts, and place them on a strip of cloth. Note the temper- ature of the room. Dry the inner calorimeter cup and stirrer In General Physics 149 with the hot air blast and weigh them. Then run hot water into the cup and add cold water until the temperature is about fifteen degrees above that of the room. Have the water fill the cup about four-fifths full. While one student weighs the calorimeter with water and stirrer, as quickly as possible, the other should dry the lumps of ice with the strip of cloth. After stirring a few seconds to equalize the tem- perature of the water, read the thermometer to 0.1° and quickly insert the lumps of ice under the stirrer. Stir the water and note the change of temperature as the ice melts. If the ice is all melted before the temperature is as much below, as it originally was above that of the room, or if there are still lumps of ice present when this temperature is reached, treat this as a preliminary trial from which to determine the proper amount of ice to use. With this amount of ice pro- ceed as before and when the ice is just melted read the ther- mometer quickly. Weigh the calorimeter with the stirrer, water and melted ice and from the data taken calculate the latent heat of fusion of ice. Repeat the experiment, using more or less ice, or colder or hotter water to fulfill the given conditions if necessary. Question. — How would your results be affected a) if the ice were not all melted when the temperature of the mixture was taken? b) If an appreciable amount of the ice had melted between weighing it and placing it in the calorimeter? Give reasons for your answers. 29. LATENT HEAT OF VAPORIZATION. To determine the latent heat of vaporization of water. (D. 280-283, 308, 314; G. 452-454, 457, 458, 468; K. 398- 401, 432-442; W. 199-201, 213, 214.) Apparatus. — Calorimeter with stirrer and glass tube, 150 Manual of Experiments water trap, connecting hose, steam boiler, thermometer, bar- ometer, tripod, Bunsen burner, brass cup, coarse balance, set of weights, and cloth. Theory, Method and Description. — -If a liquid be heated slowly, when its boiling point is reached it will be found that the temperature remains constant at this point until all the liquid has been changed to a vapor, all of the heat supplied being used up in this change of state. Each substance requires a definite amount of heat to transform a given amount of the substance from the liquid to the vapor state at the temperature of the boiling point. The latent heat of vaporization of the substance is the number of calories of heat required to change one gram of the substance at the boiling point from the liquid to the vapor state . To determine the latent heat of vaporization of water, we may use the method of mixtures. Steam is passed into W grams of water at a temperature t\ degrees (below that of the room) in a calorimeter until the temperature t of the water is as much above, as it originally was below that of the room. The mass M of the condensed steam is obtained from the increase in mass of the water. The calorimeter consists of a copper cup surrounded by a hair felt jacket and enclosed within a second copper vessel. It has a wooden lid and a copper stirrer for equalizing the temperature of the water which it contains. If the mass of the calorimeter cup and stirrer is m and their specific heat 0.093 calories per gram per degree Centigrade, their water equivalent w will be w = 0.093 m grams. The temperature h of the steam may be obtained from the barometric pressure at the time of the experiment by referring to the steam table in the Appendix. Since the specific heat of water is unity, the heat gained by the calorimeter and water will be {W + w) {t—tf) calories, In General Physics 151 and the heat lost by the steam in condensing and by the re- sulting water in dropping to the temperature of the mixture will be M L y + M (h — t) calories, where L w is the latent heat of vaporization. Equating these two quantities we get M (X v -|-^2 — t) = (JV -{-zv) (t — 1\) from which we calculate L Y . The temperatures t and h are chosen equidistant above and below the room temperature respectively in order to avoid errors due to radiation. Any drops of water which may be carried into the cal- orimeter by the steam will of course produce an error in the result, for the heat of vaporization of these drops has not been available for raising the temperature of the water in the calorimeter. To avoid this error the steam is passed through a water-trap before entering the calorimeter. This trap consists of a glass tube about 2.5 cm, in diameter and 10 cm. in length, closed at the ends by corks. The steam enters and leaves the trap by glass tubes which pass eccentrically through the corks. They are placed out of line so that the steam can not pass directly from one into the other with its full velocity, carrying moisture with it. The outlet tube extends nearly to the top of the trap, the free end passing vertically through the lid of the calorimeter to within about rf one centimeter of the bottom. Any moisture carried over by the steam or condensing in the tube will collect in the lower part of the trap. Directions. — Fill the steam generator about half full of water and heat it. Dry the inner calorimeter cup and stirrer and weigh them on the coarse balance. Note the temperature of the room and the barometric pressure, on the thermometer and barometer in the laboratory. Fill the calorimeter about four-fifths full of water and cool it with 152 Manual of Experiments lumps of ice to about 10° below the room temperature. Then remove any ice which may be left in the calorimeter, insert the stirrer, and carefully weigh the cup with water and stirrer. When steam issues freely from the generator, connect the latter to the steam trap by means of the rubber tube, and allow the steam to flow for a few minutes. Note the tem- perature of the water and calorimeter, remove any moisture that may have collected in the trap, by taking out one of the corks, replace the cork and quickly insert the tube through the lid and stirrer. Stir the water while the steam is condensing and note the rise in temperature. When the water is about 10° above room temperature, quickly remove the tube, stir the water and note the temperature when it reaches its maximum. Then weigh the cup with stirrer, water, and condensed steam. From the data taken cal- culate the latent heat of vaporization of water. Repeat the experiment four or five times, varying the amounts of the different factors as may be found necessary, and using freshly cooled water each time. Question. — How would your results be affected a) if an appreciable amount of solder (specific heat = 0.04) forms part of the weight of the calorimeter cup? b) If the steam were introduced into the calorimeter at a depth of 15 cm. under the surface of the water? Give reasons for your answers. 30. HYGROMETRY.— HUMIDITY AND DEW POINT. To determine the dew point and relative humidity of the air in the laboratory. (D. 309; G. 437-442; K. 447, 448; W. 220, 221; Apparatus. — Nickel plated cup, copper stirrer, vessel In General Physics 153 for holding ice water, two thermometers, cloth, wet and dry bulb hygrometer (mounted in room.) Description and Theory. — The atmosphere consists of air mixed with water vapor. Ordinarily it is not satur- ated though saturation may be brought about either by an increase in the amount (mass per unit volume) of water vapor without change of temperature, or from a lowering of the temperature without change in the amount of vapor present. An hygrometer is an instrument designed for the determination of the amount of water vapor in the air. Two of the most common types are the dew point hygrometer and the wet and dry bulb hygrometer. The dew point is the temperature to which the atmosphere must be cooled in order that it may be saturated by the amount of water vapor actually present. In its simplest form the dew point hygrometer consists of a metal vessel whose outer surface is polished. By adding a cold mixture to the vessel, its tem- perature may be reduced until moisture from the atmosphere is deposited on the polished surface. The temperature, 2 d , at which this occurs is the dew point. Knowing this and the temperature of the air, the absolute and relative humidities may be obtained from hygrometry tables. The absolute humidity is the number of grams of water contained in one cubic meter of air. The relative humidity is the ratio of the mass of water vapor actually present in a cubic meter of air to the mass which a cubic meter of air would contain if satu- rated at the temperature. The wet and dry bulb hygrometer consists of two sensitive thermometers mounted on the same stand, the bulb of one being covered with cloth which is kept continually moist by being connected with a vessel of water. Unless the at- mosphere is saturated with moisture there will be constant 154 Manual of Experiments evaporation from the cloth covering the bulb and this will cause the wet bulb thermometer to indicate a lower tem- perature than the other. The drier the air the more rapidly will evaporation take place and the greater will be the dif- ference between the readings of the two thermometers. With these readings the various hygrometric values such as dew point, relative humidity, etc., may be obtained from tables constructed from the results of experiment. Directions. — -Note the temperature of the laboratory near the place where the experiment is to be performed. Make a mixture of crushed ice and water in the cup for ice water. Clean and polish the nickel cup, pour water from the faucet into it to a depth of about a centimeter and then add ice water slowly until dew forms on the polished surface. In the meantime, stir the mixture with the stirrer. Do not breathe on the polished surface as this will cause a film of moisture to appear. In case the depth of the water in the cup gets to be more than about three centimeters without dew forming, pour out part of the water and continue to add ice water as before. When dew finally forms it will cover the polished surface to the height that the water stands on the inside. Note the temperature of the water at the first appearance of the dew. Now let the water warm up or, if need be, add faucet water in small portions to the mixture, noting the temperature when the dew begins to disappear. The mean of these two readings will give the temperature of the dew point. Repeat the above determination two more times, drying and polishing the can each time. Take the average of the three values for the temperature of the dew point. In case the dew point is below 0°C, the mixture will have to be cooled by the addition of salt and shaved ice. In General Physics 155 Refer to the hygrometry table in the appendix, and determine the saturation humidity at the temperature of the dew point. This is evidently the absolute humidity of the air in the room. Calculate the relative humidity. Now take the thermometer readings on one of the wet and dry bulb hygrometers in the room (the one nearest the place where you have determined the dew point) after first being sure that the cloth is wet. In order to get the proper depression the wet bulb thermometer should be fanned or some other device should be employed for maintaining a rapid motion of the air past the covered bulb. If this is not done the air in the immediate neighborhood of the ther- mometers will soon become more moist than the bulk of the air in the room and evaporation will take place less rapidly than it should. Record the reading of the wet bulb thermometer as t W} that of the dry bulb thermometer as t, and the reading of the barometer as b. From the hygrometry table find p w , the maximum vapor pressure for water vapor at the temperature t w . The actual pressure p exerted by the water vapor present in the atmosphere (at temperature t) depends upon the density of the air into which evaporation takes place, that is, upon the barometric height b } as well as upon the velocity of the air currents. The empirical formula p — p w — 0.00075 b {t — t w ) gives the pressure approximately when the air surrounding the thermometers is in moderate motion. In the above formula p, p w and b are expressed in cm. of mercury and (t — * w ) i n degrees centigrade. Having found p, the dew point td (the temperature at which this pressure produces saturation) may be obtained t 156 Manual of Experiments from the hygrometry table. Compute the relative humidity of the air in the room from the data obtained. Questions. — (1) Explain why drops form on the out- side of a pitcher containing cold water. Does the saying that this 11 indicates an approaching rainstorm ” have any scientific foundation? Explain your answer. (2) What will be the effect on the final result for the relative humidity if the breath of the observer strikes the wet bulb thermometer when the data is being taken? In General Physics 157 Experiments in Magnetism and Electricity. To study the field of force about magnets and about con- ductors carrying a current of electricity. (D. 367, 368, 426, 427; G. 699, 700, 833, 836; K. 486, 487, 674, 676, 679; W. 418-420, 472, 476, 513, 514.) Apparatus. — U shaped magnet, bar magnet, six pieces of cardboard, five paper outlines of size and shape of magnets, six sheets of blue print paper in a closed tube, six clips, pins, iron filings in a sifter, rectangular and circular coils of wire, solenoid, reversing key and small compass, con- necting leads with plug and lamp resistance. Theory and Method. — A magnetic field of force is the space surrounding a magnet or a conductor carrying a current of electricity, throughout which it exerts a magnetic force. We may think of this space as threaded with lines of magnetic force. The latter may be defined as lines drawn in the field, such that at every point of a line, the direction of the resultant magnetic force at that point is tangent to the line. The direction of such a line is the direction in which a free north-seeking magnetic pole would move. Hence the lines pass from the N pole of a magnet to the S pole through the surrounding medium. When iron filings are brought into this field, each small piece of iron becomes a temporary magnet by induction, and places itself tangent to a line of force with its north-seeking pole pointing in the direction of the line. The filings may be sprinkled upon sensitive paper in the field and exposed to sunlight. If the filings are then removed and the paper developed, the image formed 31. MAGNETIC 158 Manual of Experiments by the shadow of the filings will show the shape and density of the lines of force. The lines of force about a straight conductor carrying a current of electricity are concentric circles. If the con- ductor be grasped in the right hand with the thumb point- ing in the direction of the current, the fingers will encircle the conductor in the direction of the lines of force. If the conductor is wound in the form of a flat coil the lines of force about it are no longer concentric circles, but a large number of them will encircle all of the loops. If the conductor is wound in a solenoid (a long cylindrical coil with an axis much greater than its diameter) the field of force is still more distorted and, on the outside, resembles closely that of a bar magnet, the lines passing from one face (or end) to the other face of the coil and back to the first face through the coil. Directions. — Caution . — Do not expose the blue print paper to the light any longer than is necessary but keep it in the closed tube, taking out one sheet at a time. To get the proper time of exposure for this paper, cut a strip about 1/4 inch wide from one of the sheets and expose a piece an inch or two long to bright light (sunlight if possible) until it assumes a bronzed appearance. Then wash for a few minutes in clean water. If the print is a dark blue it has been exposed sufficiently long; if light blue the exposure has been too short; if grey blue it has been too long. In either of the last two cases make a second or third trial and note carefully the time of exposure required to obtain a dark blue print. Fasten a sheet of blue print paper flat upon a cardboard by means of pins at the corners and place it symmetrically over the bar magnet in a place shaded from the sunlight. In General Physics 159 (See Fig. 37). Sprinkle iron filings over the paper from the sifter, holding the latter about a foot above the paper in order to distribute them more uniformly. At the same time tap the cardboard gently with a pencil so that the filings arrange themselves along the lines of force, until the latter are distinctly outlined. If the filings are sprinkled on too thickly the lines will not be distinct. Next pin to the board just over the magnet a piece of thin card cut to the same size as the magnet, with a + and a— cut out at the ends, the + end coinciding with the N pole or marked end of the mag- net and the — end with the S pole. Lift up the cardboard in a vertical direction from the magnet and place it carefully in the bright" light at the window, exposing it for the proper length of time to get a dark blue print. Then pour the filings carefully into the pan provided at the sink, remove the paper from the cardboard, write your name upon the back in lead pencil, and place it face down in the water for about ten minutes or until the unchanged sensitizer has been washed out. In the meantime repeat the performance, using the bar magnet placed about 4 cm. in front of the U shaped magnet with like poles opposite each other. Then reverse the bar magnet and repeat again. Finally make a duplicate 160 Manual of Experiments set of these three prints so that each student has a set to bind in with his written report of the experiment. When the prints are thoroughly washed out, hang them up to dry. Before binding them in your report mark arrow heads on ten or twelve distinct lines of force on each print, showing their direction. Join the loose ends of the lamp cord to the terminals at one end of the reversing switch and place the porcelain plug in the Chapman receptacle in (or by) the bench. Always join conductors to terminals marked with the same sign (+ or — ). The current will flow from the + to the — ter- minal. The reversing switch has its end terminals cross connected in such a way that when the source of current is connected to the middle terminals and the apparatus through (a) (b) Fig. 38a and b which the current is to be sent is connected to the two ter- minals at either end, or vice versa, the current through the apparatus will be in one direction when the switch is closed through one end, and reversed when the switch is closed through the other end terminals. Join the middle terminals of the switch to the rectangular coil as shown in the diagram, Fig. 38a. A lamp joined in this circuit serves to cut down the current which would otherwise flow through the coil and prevents short circuiting when the loose terminals are acci- In General Physics 161 dently brought into contact. Set the rectangular coil with its plane in the north and south meridian, and close the switch. Place the small compass on the shelf at the north' side of the coil and move it around the conductor, noting on the data sheet the directions of the lines of force on each side by means of small arrows. The dots at b, d, etc., on the data sheet represent cross sections of the wire conductors. Do the same for the conductor at the south end of the coil. Then place the compass above and below the horizontal por- tions of the coil at a and at c and note the direction of the lines of force. Next reverse the direction of the current in the coil, by closing the switch in the opposite direction, and re- peat the observation. Replace the rectangular coil by the two circular coils (Fig. 38b.) and connect the coils so that the current of elec- tricity flows through both in the same direction. With the aid of the compass determine the direction of the lines of force at the sides where the coils pierce the supporting table, and note the direction of the current in “the coils at these points (whether up or down). This direction may be determined from the + and — signs on the connecting leads. Change the connections so that the current is reversed in one of the coils and proceed as before to note the relative directions of current and lines of force indicating these on the data sheet by means of small arrows. Replace the coils by the solenoid and repeat the perfor- mance, recording the directions of the lines of force about the horizontal cross section represented on the data sheet. Compare this field with that of the bar magnet, v. Question. — 1. — What are some of the properties of lines of force, illustrated in these blue prints? ^ 2. — Draw the magnetic field of force about a bar magnet and a compass needle, placed as shown in Fig. 42, and ex- 162 Manual of Experiments plain the effect produced upon the needle, by means of the properties of lines of force. 3. — Draw the field of force in a horizontal plane through the middle of a vertical coil of wire carrying a current of electricity and having its plane in the magnetic meridian with a compass needle suspended at its center, (tangent galvanometer, see Fig. 43) and explain the effect produced upon the needle, with the aid of the properties of lines of force. 32. ACTION BETWEEN MAGNETIC POLES Superposition of Magnetic Fields To study the law of action between magnetic poles and to show the effect of superimposing one magnetic field upon another. (D. 372, 380; G. 716-719, 736; K. 479, 485, 489- 491; W. 416, 421,427.) Apparatus. — Two bar magnets, small compass, stand with stirrup for suspending magnets, sheet of paper. Theory and Method. — The law of action between magnetic poles is that “'like poles repel each other; unlike poles attract each other. J ? This law is readily verified by suspend- ing one of the bar magnets so that it is free to rotate in a horizontal plane about an axis perpendicular to its own, and approaching first one and then the other pole of a second magnet to the same pole of the suspended magnet. A mag- net suspended as described forms a torsion pendulum and it performs harmonic vibrations. The characteristic of such motions is that the acceleration at any instant is proportional to the displacement from the position of rest, and the period of the vibrations is T = 2 t y/ — 4> / a In General Physics 163 where “0 M is the angular displacement and 11 a 11 the angular acceleration. When a magnet of length 11 V' and pole strenght “m” is suspended in the magnetic field of strength 77, making an angle 0 with the direction of the field, the force acting on each pole is 77m, and the moment arm is (I/ 2 ) sin 0 so that taking both poles into consideration, the moment of the couple tending to turn the magnet is 2m77 {1/2) sin may be considered equal to 0 itself, so that - 0 / « = 7/ (717 77). Substituting this value in the expression for the period, we have 2 7 r /_/_ V 77717 From this we see that the period of a vibrating magnet is inversely proportional to the square root of 77, the strength of the field in which it vibrates, and of 717, its magnetic moment. 164 Manual of Expekiments When a second magnet is brought into the neighbor- hood of the suspended one, its field is superimposed upon that of the earth, and the strength of field acting on the latter magnet is decreased or increased according as like poles or unlike poles are placed adjacent to each other. In other words, since the predominating action is that between ad- jacent poles, the action of the second magnet upon the first is, in the second case the same as that of the earth, in the first case opposed to that of the earth. If both magnets be suspended together, they may be considered as forming a single magnet, its magnetic moment being greater or less than that of either alone. If the two magnets have their unlike poles in juxtaposi- tion, and have nearly the same pole strength, the directive force tending to keep them in the magnetic meridian is very much reduced. Such a pair of magnets of very nearly the same strength, suspended with opposite poles adjacent is called an astatic system of magnets, and is very sensitive to influences tending to turn it from the magnetic meridian. Since the earth is a large magnet having a magnetic north pole in the neighborhood of its geographic south pole and vice versa, any magnet placed in the earth ’s field sends lines of force to the earth ’s south magnetic pole and receives lines of force from the earth ’s north magnetic pole. Unless a magnet be perfectly symmetrical about its axis and be in the magnetic meridian with its north seeking pole towards the north, the field of force surrounding it will be distorted. Any mass of iron in the earth ’s field and especially vertical iron pipes and columns, being magnetic by induction, distort the earth ’s field in their neighborhood. The distorted fields due to magnets may be studied by plac- ing a small compass in the field and moving it continuously in the direction in which it points, marking the path followed i In General Physics 165 with a pencil. The magnetic intensity at any point of this field is the resultant of the forces exerted by the poles of the magnet and those of the earth upon a free north pole at that point. By Coulomb’s law these forces are inversely proportional to the square of the distance from the point to each pole. Since the compass needle indicates the direc- tion of the resultant force and is, at every point of the paths described, tangent to the path, these paths represent lines of force. Directions. — Suspend each of the bar magnets with its axis in a horizontal plane by means of the stirrup and fine wire, (See Fig. 39), and allow it to vibrate until it indicates clearly which is the north seeking pole. The north seeking pole should be marked with a file or pencil mark. Action Between Poles . — Suspend one of the magnets and bring it to rest. Then approach the north pole of the other to that of the suspended magnet and record the resulting action between them. Bring up an unlike pole and record the resulting action. Effect of Strengthening or Weakening the Earth’s Field . — Turn the magnet from its position of rest through a small 166 Manual of Experiments angle (about 30°) and allow it to vibrate about a vertical axis through its center of suspension in a horizontal plane. Note the time for five complete vibrations and calculate the mean period. Do the same for the second magnet. Next place the first magnet on the table near the second (suspended) magnet, like poles being adjacent. (See Fig. 40). Let the suspended magnet vibrate through a small arc, note the time of five complete vibrations and calculate the new period. (A second trial with the distance between magnets altered may be necessary to show the change in the period. Do not have the magnets so close together that < \ \ \ \ v Fig. 40 the suspended magnet reverses its direction.) Repeat with unlike poles adjacent. Astatic System of .Magnets . — Suspend both magnets to- gether, having like poles contiguous and find the mean period. Reverse one of the magnets and repeat. (If the reversed magnet has the stronger pole the system as a whole will reverse itself.) Distortion of Earth’s Field . — Place a sheet of paper with one edge parallel to that of the table and lay a bar magnet at the middle of the west edge, with its north pole pointing In General Physics 167 north. Draw an outline of the magnet and label its ends properly + and — . Place a small compass at the middle of the + pole and move it about a quarter of an inch in the direction in which the needle points when it has come to rest, marking the direction of the needle by a short line, 1/8 inch long, behind the compass. ' Continue this operation until the edge of the paper or the other pole has been reached. Starting from six equidistant points about one side of each pole, proceed as before. (A semi-circle may be drawn about each pole with the pole as center, and six equidistant points laid off on each. ) From some point c on one of the lines of force which extends from one pole to the other and near both poles , draw lines in the direction of the force exerted by each of the two poles on a free north pole at that point, and on these lines lay off lengths which are inversely proportional to the squares of the respective distances from the point to the pole exerting the force. With these lines as component forces draw the resultant force and note its position with respect to the line of force. (See Fig. 41.) On the reverse side of the paper proceed to trace the lines of force as before but with the magnet in the center of the paper, its + end west and its — end east . 168 Manual of Experiments Make a reduced copy of these fields of force on the data sheet. Questions. — 1. In the last part of the experiment why would it not be satisfactory to find the resultant force at a point distant from the magnet by this method? 'x 2. — Two bar magnets when suspended as an astatic system have a period of a ( = 5.) seconds, but when the weaker one is reversed in the system the period is b ( = 3.) seconds. If both have the same dimensions find the ratio of their magnetic moments. ^ 3. — A certain magnetic needle has a natural period of a ( = 4.) seconds. When suspended above and parallel to a bar magnet lying in the magnetic meridian, its period is b ( = 20.) seconds. How many vibrations per second will it make if the bar magnet is reversed? 4. — At a place where the earth’s magnetic field has a horizontal intensity a ( = 0.2.) units, two parallel bar magnets each b ( = 12.) cm. long, are placed with their axes horizontal and in the magnetic meridian, with their unlike poles oppo- site each other and c ( = 16.)cm. apart. The poles pointing toward the south are d ( = + 50.) and e ( = — 25) units in strength respectively. Find the total force acting on a pole of strength / ( = +5) units at a point midway between the four poles of the magnets. State the direction of the force as well as its magnitude. , 33. DETERMINATION OF H AND M To determine the horizontal intensity of the earth’s field and the magnetic moment of a magnet. (D. 375-384; G. 723, 724, 736; K. 491, 492, 501; W. 423-428.) Apparatus. — Large compass box with attached meter In General Physics 169 rod, small compass, magnetometer box with suspension, magnet and holder. Theory and Method. — If a magnetic needle is free to rotate about a point in the earth ’s field it will place itself in the magnetic meridian, making an angle with the hori- zontal, known as the angle of inclination or dip. If the needle is free to rotate in the horizontal plane only, it will be acted upon by the horizontal component i7, of the earth’s field and will set itself in the magnetic meridian. By the horizontal intensity of the earth’s field at any point we mean the number of dynes of force acting in the horizontal plane upon a unit north magnetic pole at that point. The period of vibration of a horizontally suspended mag- net about its center is ' T = 2ttV I /MH (See theory of experiment 32). From this equation we get MH = 7T 2 / y2 We have also a method of determining M/H and are therefore able to calculate both M and H. Let a magnet whose period has been determined be placed with its axis perpendicular to the magnetic meridian and in a line with the center of a suspended magnetic needle, the distance between their centers being “a’ ^(See Fig. 42). The earth ’s field, H , and that of the magnet, H' will be approximately uniform and perpendicular to each other at the magnetic needle, and will cause it to be deflected through an angle ( / >, its axis lying in the direction of their resultant, so that tan

tan4>= (2M)/(a z H) or M i 3 , ~^zr — — a 6 tan By combining this ratio with the expression for the product of H and M so that first one, then the other of these terms is eliminated, we get expressions for both H and M in terms- of measurable quantities. Directions. — To determine MH. Place the magnetometer box with the glass sides facing north and south respectively. Suspend the magnet in the stirrup so that it lies in the horizontal plane when it comes to rest in the magnetic meridian. Then start it to oscil- lating through an angle of about 10°, preventing any motion of the center of the magnet by means of the wire stop at the side of the box. Determine the period of the motion from the time required for ten complete oscillations. Repeat this determination as a check. The magnets are marked with Roman numerals. Note from the bulletin board the mass, length and radius of the magnet you used, and calculate its moment of inertia and the value of MH. (See Appendix for formulae of moments of inertia). To determine M/H. The large compass box has a small magnetic needle with a long aluminum pointer attached perpendicularly to it, moving over a circular scale on a glass mirror. A meter stick is fastened symmetrically under the center of the box. The compass is turned until the needle hangs at rest, with the pointer in line with the meter stick, and at the same time the leveling screws are adjusted so that the pointer may swing freely about the center of the scale. Be sure the mag- 172 Manual of Expeeiments net is not close enough to affect the needle. Note on the scale the position of the ends of the pointer. To avoid errors due to parallax in taking this reading, place the eye vertically above each end, so that the pointer is seen covering its image in the mirror. Place the magnet symmetrically in the holder and put them on the meter stick with centers 25 cm. east of the needle. Note the direction of both ends of the pointer. Then reverse the magnet and when the needle has come to rest note the deflection of both ends from their original position. Repeat for 25 cm. west of needle. Do the same for 26 and 27 cm. Calculate the mean de- flection in each case and the values of the tangent of this angle times the distance cubed. From the average of these products calculate the value of M/H , and combining with the value found for M77, determine both M and H. Questions. — 1 . Explain how the following arrange- ments of the compass box would introduce errors in the measurements and how the errors were eliminated in each case: a) The center of the scale did not coincide with the center of suspension? b) The center of the meter rod was not under the center of suspension? c) The poles of the bar magnet were not equidistant from the center of the magnet? ^ 2. Find the least distance between magnet and needle in the above experiment for which the approximate formula H' = 2M/a s can be used with a magnet ^( = 10)cm long without making an error greater than b{ — 1.)% in the value of H'. ^3. A rectangular bar magnet having a length a{ = 8.)cm., width b{ — 2.) cm., thickness c( = 1.) cm, and mass ^( = 120). In General Physics 173 grams makes the moment of the deflecting couple will be IGMcosfy and the moment of the balancing- couple due to the earth’s field of strength H will be HM sincf). These moments must be equal when the needle is at rest or, IGMcos.4> = HMsin.cf) and H Hr — tan. 6 = G 2irn tan.4> 176 Manual of Experiments H and G are constants for a given galvanometer in a given position and their ratio K is called the reduction factor of the galvanometer, the current strength in absolute units being K times the tangent of the angle of deflection. Since the practical unit of current strength is one tenth as great as the absolute unit, the number of practical units C will be ten times as great, or the current strength in amperes is 10# 10 Hr C = tan d>— — ■ — tan ' and 0.) When the needle is deflected through an angle and comes to rest, the moments of the deflecting and the restoring couple will be equal. Hence (Ft =*= FsinO) Msin4> = F cos 6 Mcos } and In General Physics 177 tan 0 = F cos 6 H =±= F sm 0 In order that the deflections may be equal for direct and reversed current, however, the plane of the coil must be in the magnetic meridian, or 6 = o. Under this condition sin 0 — o and cos 0=1, hence the previous equation becomes F IG 2t nl tan 0= — = — — = — - — — - H H Hr and 178 Manual of Experiments H Hr I = — tan 6= tan d> G 2tt n as before. In order to read the deflections of the needle (of the tan- gent galvanometer to be used) a long aluminum pointer, P, fastened at right angles to it, moves over a circular scale etched on glass. The lower side of the glass is silvered, forming a mirror so that errors due to parallax may be avoided by reading the position of the pointer when it and its image are in the same lind of vision. It can be shown that the results are least affected by errors in these readings when the angle of deflection is about 45°. The galvano- - meters to be used have three coils the number of turns in each being different. That particular coil should be used in general, which with the available current strength gives a deflection of from 30° to 60°. From the above equation for the current strength it is seen that the sensitiveness of such a galvanometer to weak currents is increased by increasing the number of turns of wire in the coil or by decreasing the radius of the coil or the directive force of the earth upon the suspended magnet. In the astatic galvanometer all three of these methods for increasing the sensibility are used, The needle system is astatic (see exp. 32), the number of turns of wire is large, and the coil is much smaller and has the shape of a flat oval, the conductors lying closer to the needles, so that they are in a stronger deflecting field. In some galvanometers each of the two needles is at the center of a coil, the two coils being wound in opposite directions, so that the deflecting force acting on the needles is in the same direction for both. In others the directive force of the earth is weakened by adjusting a permanent magnet above the galvanometer in such a position that it opposes the earth’s field. In General Physics 179 Both the tangent and the astatic galvanometers are pro- tected from air currents by glass covers, but both are subject to the disturbing influences of surrounding magnetic fields. In the D’ Arson val galvanometer, the small rectangular coil which carries the current is suspended by means of a phosphor-bronze or steel strip between the poles of a fixed permanent U shaped magnet. A fixed cylinder of soft iron projecting within the coil, serves to make the magnetic field in which the coil moves, stronger and more uniform so that within certain limits the deflections are directly proportional to the current flowing in the coil. The magnet is sufficiently strong to make the influence of the earth and of neighboring magnetic fields negligible, and the galvanometer does not have to be fixed in any definite position. In order to observe the small deflections of the coil accurately, a small plane mirror is fastened to it. A millimeter scale is fixed at about half a meter in front of the mirror and its image in the mirror is observed by means of a telescope fastened to the scale. The apparent deflection of the image of the scale as observed through the telescope is twice as great as the actual deflection of the coil and mirror. (See Fig. 22). The telescope may be dispensed with, if the plane mirror is replaced by a concave one. In this case the scale is fastened to a plate of ground glass or other semi-transparent substance, and an illuminated slit is placed at its center. The distance between scale and mirror is adjusted until the image of the slit or of the lamp filament is sharply focussed on the ground glass, so that the deflections of the image may be observed. Either of the above two types of galvanometers may be designed for use in three different kinds of measurements. a) We may wish to measure a current of definite strength which produces a steady deflection of the suspended 180 Manual of Experiments coil or needle. The tangent and the D’Arsonval galvano- meter are frequently used for this purpose. The galvano- meter should be aperiodic or dead-beat; that is, its suspended needle or magnet should come to rest quickly, without vibrating back and forth. In the first type of galvanometer described, this damping is produced by surrounding the needle (usually of an inverted U shape) with a mass of copper. In the D’Arsonval type the damping is produced by winding the coil on a metal frame or placing it in a metal tube. In either type eddy currents are induced in the solid metal and prevent vibrations. (See Exp. 43). b) We may wish to detect the presence of very small currents, in which case a very sensitive galvanometer such as an astatic or a D’Arsonval is needed. This is the case in all measurements where a null method is used; that is, where the apparatus is to be adjusted until no current flows. (See Exp. 36, 37, 38, etc.) c) We may wish to measure, not the strength of a current, but the total quantity of electricity flowing through the galvanometer during a short interval of time, for in- stance, the discharge of a condenser, or an induced current. (See Exp. 43.) A galvanometer adapted to such measure- ments is called a ballistic galvanometer. It should have as little damping as possible, and a long period of vibration so that the needle or coil does not move appreciably from its position of rest, before all of the electricity has passed through the coil. Under these conditions the quantity of electricity is proportional to the sine of half the angle of the first swing (or throw.) The resistance box contains several coils of wire of dif- ferent lengths and cross sections, each coil having a definite electrical resistance of a whole or fractional number of ohms. (See Fig. 45.) The coils are wound non -inductively by re- In General Physics 181 turning the wire upon its own path, and each end of a coil is connected to a separate brass block on the upper surface of the box. Two coils are connected to the same block in such a way that a current of electricity may be passed suc- cessively from one coil into the other, and the resistance of the circuit may be varied by including more or less of these coils. But when a brass plug is inserted between two blocks, the current will pass partly through the coil, and partly through the plug. (See Fig. 46.) The cross section of plugs and blocks is made so large that their combined re- sistance in parallel with the coil is negligible (See Exp. 36) and for all practical purposes that coil is cut out of the circuit. The resistance of each coil, in ohms, is marked beside the gap between the blocks to which the ends of the coil are attached. Unless there is very good contact between the plugs and blocks, the resistance to the flow of current through these surfaces of contact will be appreciable and may cause considerable error in the total resistance. 182 Manual of Experiments It is advisable to use a few large resistance coils rather than a large number of small ones. Directions.— Adjust the level of the tangent galvano- meter so that the pointer swings freely over the whole scale and turn the galvanometer until the plane of the coil is in line with the magnetic needle when at rest. Remove the W W A/j 5lct ion on RB Section of Rtsismuct Box WIT H 4 OHMS IN CIRCUIT fliro w* Show path of cvrrtnt Fig. 46 short circuiting coil from the gravity cell and connect the apparatus as shown in the diagram, Fig. 47, G being the galvanometer, B the galvanic cell, i? the resistance box, and S the switch. Note the position of both ends of the pointer when the switch is open in which case the resistance of the circuit may be considered infinitely great for all practical purposes. Close the switch and when the needle comes to rest again note the position of the pointer. Then reverse the current and note the position of the pointer. If the mean deflection is not the same in both cases it means that the plane of the coil is not in the magnetic meridian and it must be turned/ toward the magnet (not the pointer) in the position of smaller deflections, until the deflections become equal for reversed currents. The box resistance during this trial has been zero, and when the deflections are equal they should In General Physics 183 \ be recorded for both ends of the pointer and for both direct and reversed currents. The corresponding deflections for 1, 2, 5, and 10 ohms of box resistance in the circuit should be observed and the mean deflection calculated. This box resistance is only a part of the total resistance of the circuit,- the latter including the resistance of cell, galvanometer, and connecting wires. To determine this resistance exclusive of the box resistance, plot a curve having for ordinates the cotangents of the angles of deflection and for abscissae, the corresponding box resistances. The intercept of this curve on the axis of abscissae between the origin and curve gives the required resistance to the same scale as the plotted box resistance. Calculate the total resistance in each case. • According to Ohm’s law, the current strength in the cir- cuit equals the electromotive force (e. m. /.) in the circuit divided by its total resistance. (See Exp. 36.) Note the e. m. /. of the cell, and calculate the current strength and the reduction factor K of the galvanometer in each case. Examine one of the D’Arsonval galvanometers on the wall and include a sketch of the essential parts in your written report. Do the same for the astatic galvanometer. * Questions — 1. — a) With the data obtained in the above experiment plot a curve between corresponding values of current strength (as abscissae) and galvanometer deflections. 184 Manual of Experiments b) On the same sheet of coordinate paper plot a second curve between corresponding values of current strength and tangents of the angles of deflection. 2. — Explain why the intercept of the above curve gives the resistance of the circuit exclusive of box resistance. 3. — a) What errors due to the lack of symmetry in the apparatus, are eliminated by the method of taking deflec- tions described above. b) Would the readings be affected if the pointer were bent, or if it were not fixed at right angles to the magnetic needle? Give reasons for your answers. 4. — The coil of a tangent galvanometer with a steady I current flowing through it is placed at right angles to the magnetic meridian. The needle when set in vibration makes a( = 5) oscillations per second, but when the current is reversed it>makes b( = 3) oscillations per second. a) How does the field of the coil at its center compare with that of the earth? b ) What will be the deflection of the needle if the plane of the coil is placed in the magnetic meridian? 35. OHM’S LAW AND POTENTIAL DROP. To test Ohm’s law in a simple circuit, and to show that the potential drop in a conductor is proportional to the resistance of the conductor. (D. pp. 395, 403, 411; G. pp. 866-870, 953; K. pp. 415, 433, 434.) Apparatus.— D’Arsonval galvanometer with shunt box, resistance box (10,000 ohms), high resistance wire with sliding contact, gravity cell, reversing switch, and seven connecting wires. Theory and Description. — We may define the elec- trical potential at a point as the work done in bringing unit In General Physics 185 positive quantity of electricity from an infinite distance up to the point in question. It is shown in books on advanced physics that the work required to bring unit quantity of electricity from an infinite distance to any point is the same whatever path may be followed. In bringing this quantity up to a second point we may therefore follow a path which passes through the first point. It follows that the difference of potential between two points in an electric circuit is the work done upon unit quantity of positive electricity in bringing it from the point of lower potential to the point of higher po- tential; or it is the work done by unit quantity of positive electricity as it flows from the point of higher potential to the point of lower potential. A “current” shall be understood to mean a flow of positive electricity. (Unit quantity of elec- tricity and unit current strength were defined in experiment 34 .) Unless there is a difference of potential between two points of a circuit no current can flow between them, and positive electricity always flows from a point of higher to a point of lower potential. The difference of potential between two points is therefore often spoken of as the potential drop between them. When a current of electricity flows in a simple closed circuit consisting of a source of electricity (such as a gal- vanic cell or dynamo) and a conductor which connects the terminals of the source, the current after flowing from one terminal through the external conductor must flow back to the first terminal through the conductors within the source, (the plates and liquid of the cell, or the wire coils of the dynamo armature), and there must be a potential drop within the source as well as through the external circuit. The electromotive force of the source may be defined as the sum of the external and internal drops of 186 Manual of Experiments potential in the circ uit; or in general the electromotive force in a circuit is the total drop of potential ( or the sum of all the potential differences ) met with in passing from one terminal of the source of electricity through the complete circuit and hack to the same terminal. The absolute unit of potential difference or electromotive force , in the C. G. S. system , exists between two points in a circuit when one erg of work must be done in order to move unit quantity of positive . electricity from the point of lower to the point of higher potential. The practical unit of potential difference or electromotive force is called the volt and is c 10 8 times as large as the absolute unit. The volt may also be defined as the potential difference existing between two points in a circuit when one joule (10 7 ergs) of work is done by one conlomb of electricity ( one ampere flowing for one second) as it flows from the point of higher potential to the point of lower potential. Ohm’s law states that when a constant electromotive force is impressed upon a closed circuit , there is a constant ratio between the electromotive force and the resulting current strength. This constant ratio is called the resistance of the circuit. Ohm’s law is more commonly stated in this form: The strength (/) of the electric current flowing in a closed circuit equals the total electromotive force (E) in the circuit divided by its total resistance (R), or in symbols E The total electromotive force is the summation of the e. m. f.’s of all sources of electricity in the circuit, each with its proper sign. (See Exp. 38.) The total resistance of the circuit is the summation of the resistances of all the conductors in the circuit, the method of summation depend- In General Physics 187 mg upon the manner in which the conductors are connected. (See Exps. 36 and 37.) The absolute unit of resistance in the C. G. S. system is the resistance of a conductor in which a current strength of one absolute unit will flow when the absolute unit of potential difference exists between its ends. The practical unit of resistance is called the ohm and is 10 9 times as large as the absolute unit. The ohm may also be defined as the re- sistance of a conductor in which a current strength of one ampere flows when one volt of potential difference exists between its ends. . . . Ohm’s law is true not only for any closed circuit as a whole, upon which a constant e. m. f. is impressed, but it holds for any part of the circuit as well. The potential drop between any two points of the circuit must equal the resistance between these two points multiplied by the strength of current flowing between them, and the sum of all the potential drops met with in passing completely around the circuit, must equal the total electromotive force in the circuit. In order to test Ohm’s law we may connect a source of constant e. m. f. such as a gravity or storage cell in a circuit containing a box with variable resistance, and a galvanometer arranged to measure the current strength in the circuit. (See Exp. 39.) If the box resistance is made very large in comparison with that of the cell and galvanometer circuit, the latter may be neglected. On changing the box resistance the current will change in such a way that the product of current strength and box resistance will remain equal to the electromotive force. When the galvanometer is arranged so that its deflections are directly proportional to the current strength producing 188 Manual of Experiments them, the product of box resistance and corresponding deflections will be constant also. To show that the potential drop through different parts of a circuit is directly proportional to the resistance of these parts we may connect a long uniform high resistance wire in circuit with a source of constant current (gravity or storage cell), and arrange a galvanometer to show the potential drops through different lengths of the wire. (See Exp. 39.) The ratio of the potential drop through any section of the wire and the resistance of this section will be constant. Since the deflections of the galvanometer are directly proportional to the potential difference measured and since the resistance of any section of a wire of homo- geneous material and uniform cross-section is directly proportional to the length of this section, the ratio of de- flections and corresponding lengths will be constant also. Directions. — a) To show that for a constant e. m. /, in circuit the current is inversely proportional to the resistance . Connect the apparatus as shown in figure 47, where B is a gravity (or storage) cell, R a resistance box with at least 10,000 ohms, and G a D’Arsonval galvanometer with a 1 ohm shunt (See Exp. 36.), connected in the circuit by means of a reversing switch. The deflections of the galvanometer, if they are not too large, will be proportional to the current strength in the circuit. Remove the necessary plugs from the resistance box to insert 10,000 ohms in the circuit. Close the switch and note the deflection of the galvano- meter. Reverse the current by means of the switch and again note the deflection. Calculate the mean of the two deflections and the product of this mean and the box re- sistance. Change the resistance from 10,000 to 3,000 ohms in steps of 1,000 ohms, and note the deflections for direct In General Physics 189 and reversed current in each case. Calculate the products of box resistance and corresponding mean deflection for each case. Compare these products and see that they are practically constant. b) To show that for a constant current flowing in a circuit the potential difference through any section of the circuit is directly proportional to the resistance of this section. Connect the apparatus as shown in figure 56, where B is a gravity (or storage) cell connected through a reversing switch to the ends of a high resistance wire R. This wire is of homo- geneous material and uniform cross section, 10 meters long, and is stretched back and forth in one meter lengths on a board (See Fig. 53.) The D’Arsonval galvanometer G with a 10 ohm shunt S is connected through a 10,000 ohm resistance R x to the end D of the long wire R. The other terminal of the galvanometer is connected to a sliding contact which may be placed at any point on the long wire. The deflections of the galvanometer when so ar- ranged, are proportional to the potential drop through the section of wire across which it is connected. Place the sliding contact at 50 cm from the end D of the wire and note the deflections of the galvanometer when the switch is closed, first with the current flowing in one direction, then with the current reversed. Calculate the mean deflection and the ratio of this mean to the length of wire (50 cm.) Do the same with the sliding contact at 250, 450, 650, 850 and 1000 cm from D respectively and cal- culate in each case the ratio of the mean deflection to the corresponding length of wire. Compare these ratios and see that they are practically constant. Questions, etc. — 1 . — a) What kind of curve would you obtain if you plotted total resistance (in part a) of the 190 Manual of Experiments above experiment) as abscissae and corresponding de- flections as ordinates. b) Plot a curve having lengths of wire (in part b above) as abscissae and corresponding deflections as ordinates. 2. — A storage battery with an internal resistance #( = 0.5) ohms sends a current £( = 1.5) amperes thru a resistance coil of c( = 2.3) ohms in series with an incandescent lamp whose resistance is ^( = 1.8) ohms. Find the potential drop in the lamp and the e. m. /. of the battery. 3. — If in part #) of the above experiment the products obtained were used in calculating the e. m. /. of the cell, the deflections being #( = 1.) mm. per £( = 0.001) ampere, the resistance of the cell c( = 10) ohms and the combined resistance of galvanometer and shunt J( = 0.98) ohm, find the % error in the calculated e. m. f. for a box resistance of ^( = 3000) ohms. 4. — A battery having an e. m. /. #( = 5.6) volts and internal resistance £( = 2.) ohms sends current through a wire of length c( = 20) meters and resistance ^( = 38) ohms. What is the potential drop through ‘ C» -^VW\AAA/WW— a Q- A — / WWWWW\r~ ~ / W\MAAAA/W — — wvwwvww— D D D OJ B b .Parra//c/. Fig. 49 etc., and let the total drop through all of them be V . Then V = vi +^2+^3+ etc., and since the current strength / is the same throughout the circuit, we have on dividing through by / V * 1~~I v 2 1 etc., but by Ohm’s law V/I = R , the total resistance, vi/I = r h etc. Therefore this equation may be written R = fi+f 2 +r 3 +etc. or in words, the total resistance of several conductors joined in series is equal to the sum of their separate resistances. This holds for any number of resistances in series. In figure 194 Manual of Experiments 49a, if the current enters by the terminal A and leaves by the terminal B , it must pass through the four coils in series. If several resistances are connected so that they form a divided circuit, by connecting one end of each together and to one terminal of the source of current, the other ends all being connected together and to the other terminal, the current will pass simultaneously through all of them, and they are said to be in parallel , (or in multiple arc.) In figure 49£, if the current enters by the terminal A and leaves by the terminal B , it will pass through the four coils in parallel; that is, it will divide and part of it will pass through each of the four branches. Those termi- nals of the resistances which are connected together must be at the same potential, or in other words, the drop of po- tential V in any one resistance must equal that in any other resistance and also that in their combined resistance R. Let the current strengths in resistances r h r 2 , r 3 , etc., be i h i 2 , is, etc., respectively. The total current, I, will divide among the various branches so that I — i iT _ f 2 ~f _ 23T-etc. or by Ohm’s law, since V V V I — — .? i\ — — . 72~ — • etc. R ri r 2 V V V V — .= 1 1 (- etc. R T\ r 2 7*3 Dividing through by V gives 1111 — = 1 1 \- etc. R Ti r 2 r 3 In General Physics 195 or the reciprocal of the total resistance equals the sum of the reciprocals of the separate resistances. The reciprocal of the resistance of a conductor is called its conductance, and the law may be stated as follows: — When conductors are joined in parallel their joint conductance is the sum of their separate conductances. Conductance is expressed in terms of a unit called the mho (the reverse of ohm.) From the equation given it is seen that for two conductors in parallel, the last equation may be written in the form H r 2 r i r 2 r s R = , for three, R = etc. r±+r 2 rir 2 +r 2 r z +r 3 ri It should be noted that the total resistance of a number of conductors in parallel is always less than the resistance of any one of them alone. ^ Shunts — Since for parallel circuits the drop of potential is the same through all of the branches, in the case of two branches Therefore V = IR — iiri = i 2 r 2 ii : H = r 2 : n or in words, the currents in branched circuits are inversely proportional to the resistance of the branches, so that the * larger part of the total current flows through the smaller resistance and vice versa. From the equations obtained for R and V we derive the following expressions for ii and i 2 : 11 = IR/ n = Ir x r 2 / (n + r 2 ) n = Ir 2 / (n + r 2 ) 1 2 = IR/ r 2 = Ir x r 2 / ( r x +r 2 )r 2 = In/ (n+r 2 ) . Therefore ii : I = r 2 : (ri+r 2 ) 196 Manual of Experiments and k : I = n : Oi +r 2 ) or in words, the current in one branch of a parallel circuit is to the total current as the resistance of the others branch is to the sum of their resistances. The sudden rush of a strong current through a galvano- meter causes a sudden large deflection of the movable sys- tem, which is likely to break the suspension. A large cur- rent may also overheat the coil and injure the insulation of the wire, etc. To avoid these effects a low resistance is joined in parallel with the galvanometer, so that the larger part of the current is diverted from it. Such a low resistance is called a shunt, and sensitive galvanometers are frequently supplied with three shunts of such resistance that only 0.1, 0.01, or 0.001 of the total current may pass through the galvanometer when they are joined in parallel with it. If S be the resistance of the shunt, G that of the galvano- meter, / the total current and Ig that through the galvano- meter, we see from the above theory of parallel circuits that Ig \ I — S \ S-\-G or S I g = I— S + G • so that for values of S that are small compared with G, the galvanometer current is a small fraction of the total current. The shunt box shown in Figure 50, is connected to the galvanometer by means of the binding posts A and B. Any one of the three shunt coils may be placed in parallel with the galvanometer coil by putting the plug in the gap between that coil and the block of metal to which the binding post B is connected. In General Physics 197 Directions. — Connect the apparatus as shown in the diagram (Fig. 48). The unknown resistance consists of four coils, each having its ends joined to binding posts at opposite sides of the box. (See Fig. 49). Connect the coil marked No. 1 with the bridge and, 'with the smallest value of R in circuit and the key S at the middle of the bridge wire, note the deflection of the galvanometer when both K and S are closed. K should always be closed first, and since an open circuit cell is used it should not be kept closed longer than necessary. S should never be pressed down on the wire while it is being moved. The same pres- sure should be used on the keys at all times. Increase R and note whether the direction of the deflec- tion is opposite to what it was before. If not, try a still greater value of R. In this way find two values of R for which the deflections are in opposite directions. The value 198 Manual of Experiments of X must evidently lie somewhere between these two values of R , and when the deflection is in one direction R is too. small, when in the opposite direction R is too large. With these relations fixed in mind, vary R by smaller and smaller steps until the minimum deflection is reached, then shift the contact key S to one side or the other in steps of a half millimeter or less, until there is no further decrease in the deflection of the galvanometer when both K and S are closed. S should be kept as near the middle of the bridge wire as possible. During all of the preliminary trials a low value shunt should be connected across the terminals of the galvanometer but when the balance position has been approximately determined the shunt may be removed, in order to make the final adjustment. The balance position may also be determined by sliding S from one end of the bridge toward the center until equilibrium is obtained, then from the other end of the bridge toward the center until no deflection is obtained. If the two balance positions do not coincide, their mean should be taken as the balance position. As a check X and R may be interchanged and balance again obtained. Of course x and l — x should inter- change their values also. Note the value of R , x and / and repeat the performance with each of the other three coils. Then do the same with all the coils in series, with 1 and 2 in parallel, and finally with all of them in parallel. Calculate the resistance and conductance in each case, and show that the laws for parallel and series connection of resistances hold. Questions. — 1 . — From the general expression for the total resistance of a number of resistances connected in parallel, show that the total resistance of n coils in parallel, each of resistance r will be r/n. 2. — What resistance would be required in series with In General Physics 199 a galvanometer whose resistance is < 2 ( = 200) ohms, in a circuit with an electromotive force of b( = l) volt, in order that the current through the galvanometer may be c( = 0.0001) ampere? 3. — If in place of the resistance calculated in question 2 we put <7( = 160) ohms in series with the galvanometer? what resistance used as a shunt to the galvanometer, would give the same galvanometer current with the same electro- motive force? ~ 4. — a) Given three resistance coils A, B and C, with resistances a( = 1), b( = 2) and c( = 3) ohms respectively, describe (or sketch) the eight possible combinations of all three coils (arranged in series, or parallel, or combinations of the two)which will give a different total resistance. b) Calculate the total resistance in each case. 37. THE POST OFFICE BOX BRIDGE To determine the specific resistance of several wires and the variation of their resistance with length and cross section (D. 442-447, 456; G. 923, 844, 845, 850, 856; K. 641, 642, 646-649, 651; W. 480-484, 486, 487.) Apparatus. — Post office box bridge, dry cell, D ’Arsonval galvanometer, board with unknown resistances of man- ganin and German silver, six connecting wires. Theory and Description. — The resistance of a con- ductor may be defined as the ratio of the e. m. f. applied to its ends and the resulting current strength flowing through it. It may be shown that the resistance of a conductor is directly proportional to its length /, and inversely proportional to its cross section A, and that it depends upon the material of the conductor, or in symbols R = p {1/ A) 200 Manual of Experiments The factor p is a constant for any one substance at a given temperature, and is called the specific resistance of the sub- stance, or its resistivity. If both / and A are unity, R will equal />, hence the specific resistance of a substance may be de- fined as the resistance of a conductor of that substance having a length of one centimeter and a cross section of one square cen- timeter. The Post Office box is a compact form of the Wheatstone bridge, in which there are two identical sets of resistance coils ( A to G in Fig. 51) generally of 10, 100 and 1000 ohms', called the ratio arms of the bridge, and the usual set of known resistance coils R, (between A and X ) with two contact keys, all contained in the same box. The con- In General Physics 201 nections are somewhat different from those of the slide wire bridge. There is a fixed connection from the junction of the two ratio arms to the left hand key, and another from the block A to the right hand key, by means of wires within the box. The galvanic cell is to be connected to the binding posts B and X, the galvanometer to the bind- ing posts G and Gi, and the unknown resistance X between the binding posts marked X and G. If we let R\ be the ratio arm adjoining G and R2 the one adjoining A, and represent Ri, R 2 , R and X (in the order given) as the sides of the diamond shaped figure which is the usual diagrammatic representation of the Wheatstone bridge, we find on tracing out the connections in Fig. 51, that the galvanic cell is connected between the junction of the two ratio arms and the junction of the known and unknown resistances; and that the galvanometer is connected across the two ratio arms. It is readily shown by applying the same train of reasoning to this diagram as was used in the case of the slide wire bridge, that for values of R , R h and R 2 which balance the bridge (give zero deflection of the galvanometer) the unknown resistance may be cal- culated from the proportion X : R = Ri : R 2 From this proportion it may be seen that if X be greater than R , then Ri must be greater than R 2 , and vice versa. Since X = RxRi/ R2 the smallest unknown resistance which can be measured with this bridge is the smallest fraction R\/R 2 (which can be formed with the fixed values of R\ and R 2 ) of the smallest of the known resistances R. In other words if Ri and R 2 are set at 10 and 1000 ohms re- spectively, and the smallest of the known resistance coil is 1 ohm, the unknown resistance can be determined to 202 Manual of Experiments 1/100 of an ohm, whereas if Ri and i? 2 are set at 10 and 100 ohms respectively, the unknown resistance may be deter- mined to only 1/10 ohm. In any form of Wheatstone bridge there are six re- sistances to be considered, namely the known and un- known resistances, that of the galvanometer, that of the cell, and those of the ratio arms (or of the two sections of wire in the slide wire bridge.) In books on advanced Physics it is shown that the most sensitive arrangement of the bridge is to have the greater of the two resistances, that of the galvanometer and that of the galvanic cell, connected between the junction of the two larger remaining resistances and the junction of the two smaller ones. In general the galvanometer has a greater resistance than the cell, and since J and are either greater or less than R and i? 2 respectively the arrangement of the bridge shown in fig. 51 is the most sensitive one. Directions. — Connect the apparatus as shown in the diagram. The unknown resistance consists of four wires. Three of these are of manganin, two of which have the same length but different diameters, while two others have the same diameter but different lengths. The fourth wire is of German Silver. The resistance of each is to be de- termined. Having connected one of the unknown resistances to the blocks X and G, find the approximate resistance of X by making the ratio arms equal (say 100 ohms each) and varying R until the galvanometer deflection reaches its minimum value. Note that if R be started at a small value and gradually increased, the galvanometer deflections will decrease to zero when the bridge is balanced. If R be still further increased the galvanometer will be deflected in the opposite direction from what it was at first. De- In General Physics 203 flections of the galvanometer in the first direction then indicate that R is too small for balance, whereas deflections in the opposite direction correspond to values of R too large for balance. These relations should be borne in mind throughout the experiment. On account of the fixed re- sistance of the coils in R it may not be possible to obtain perfect balance (zero deflection), but when the deflection of the galvanometer is a minimum., R is approximately equal to X. Now find the approximate ratio between the approximate value of X and the total available resistance of R and adjust the ratio arms, R\ and R 2 , to approximately the same ratio. Vary R until balance is obtained (minimum deflection of galvanometer), when both keys are closed. Note the values of R , R\ and R 2 , and calculate X. In the same way determine each of the unknown resistances. Note the length, diameter and material of each wire and calculate the resistance per centimeter, and the area of cross section . To find the relation between the resistance of a wire of any given material and one of its dimensions, all of the other dimensions must be kept constant. Keeping this in mind show with your data that the resistance varies directly as the length and inversely as the cross section of the conductor by calculating the ratios of X and L for wires of the same material and diameter, and the products of X and A for wires of the same material and length. Also calculate the specific resistance of manganin and of German silver. Questions. — 1 . — In Electrical Engineering the ex- pression “resistance per mil-foot” is often used, meaning the resistance of a conductor one foot long having a cross sectional area equal to that of a circle whose diameter is one mil (one thousandth inch). Find the ratio between specific resistance and resistance per mil-foot of a substance. 204 Manual of Experiments 2. — If a cylindrical conductor were stretched to a( = 2) times its original length, assuming its volume to remain constant, how would its resistance compare with its original resistance? 3. — An all metal telephone line (two wires) has a re- sistance of a( = 2A0) ohms when measured at the station, indicating that it is short circuited somewhere. If the wire is of copper with cross section £( = 0.1) sq. cm. and specific resistance 1.5X10- 5 ohms per c. c. what is the distance of the short circuit from the station? 4. — In measuring an unknown resistance with a P. 0. box bridge, the ratio arms were set at a( = 1000) and £( = 10) ohms respectively, and balance was obtained when the adjustable resistance (connected to the smaller ratio arm) was c{ = 750) ohms. The P.D. of the cell was ^( = 1.4) volts. Compute a) the unknown resistance, b) the current through it, c ) the P. D. across it, and d) the current through the cell. 38. THE POTENTIOMETER. To determine the electromotive force of several galvanic cells, singly and in combination, by means of the poten- tiometer. (D. 441, 454, 455 469-476; G. 812-831, 844-849, 925, 927; K. 622-642; W. 480, 490, 550, 557.) Apparatus. — Pye potentiometer with two storage cells, D’Arsonval galvanometer, dry cell, salammoniac cell, Daniel cell, two gravity cells and eight connecting wires. Description and Theory. — A galvanic cell consists in general of two plates of different metals (or a plate each of a metal and some other substance such as carbon) placed a short distance apart in an acid or a solution of a salt, such that a difference of potential exists between the plates, and a current of electricity will flow from one to the other when In General Physics 205 their external ends are joined by means of a conductor. The electromotive force of such a cell is equal to the difference of potential existing between the external ends of these plates when they are not connected , (i. e. there is an open circuit) or practically when they are joined by a very high resistance. The electromotive force of cells may be compared by the potentiometer method. The simple potentiometer con- sists of a long homogeneous wire AD (Fig. 52) of uniform cross section. A constant fall of potential is produced in this wire by means of a source of constant current (in this case a storage battery S. B.) By Ohm’s law the potential drop through a wire is equal to the current through it times its resistance. Since our wire is uniform and homogeneous, its resistance per unit length is the same throughout its length, and since the electromotive force ( e.m.f .) of the stor- age cells is practically constant, the current strength is constant. It follows that the potential drop per unit length of wire is the same at all points. A galvanic cell B is now connected in series with a sen- sitive galvanometer G to one end A of the wire and by means of a sliding contact key S to another point on the wire. This second cell also tends to produce a constant potential drop through AS. But AS is in parallel (See Exp. 36) with both of the circuits ' containing cells, and if like terminals (either -f- or — ) of both cells are connected to the end A their e.m.fd s will be opposed. The current from cell B will tend to divide at A part going through AS and part through the circuit A(SB)DS. The current from S.B. will tend to divide at the same point, part going through AS and part through the circuit ABGKS. Each branch current will be opposed by the electromotive force of the other cell and if the drop of potential through AS is not the same for both cells, a current will flow through it in one direction or the 206 Manual of Experiments other, depending upon which cell produces the greater drop, and there will be in general a deflection of the galvanometer. If now the sliding contact key S be moved along the wire a point may be found where there is no deflection of the Fig. 52 galvanometer when the key is closed, because there is no current flowing in the galvanometer circuit. This indicates that the potential drop between A and S due to cell B is exactly equal to that due to cells S.B . , or in other words the tendency for cell B to send current through the galvano- meter circuit from S to A is exactly neutralized by the ten- dency of cells S.B. to send current through the same circuit. Since there is no current through the galvanometer circuit there will be no potential drop between the cell B and the points A and S so that the point A must be at the same potential as the positive terminal of B and the point S at the same potential as the negative terminal. Con- sequently the e.m.f. of the cell B must be equal to the po- tential drop in the section of wire AS due to the cells S.B., and is therefore directly proportional to the length of wire AS. A second cell C of known e. m. /., such as a standard or a Daniel cell, may now be inserted in place of B and the contact key moved along the wire to some point S' where there will be no deflection of the galvanometer when the key is closed. The e.m.f. of cell C is then directly propor- tional to the length of wire AS'. In General Physics 207 If E be the e.m.f. of a cell, L the length and R the re- sistance of the stretch of wire across which it is balanced, P. D. the potential drop through it, and I the current flowing in it, each with their proper subscript, we may write Therefore E b P.D. b IRb Lb Ec P.D. c IRc Lc Lb FjB = Ec — — Lc The Pye potentiometer has a uniform high resistance wire, ten meters long, wound back and forth for convenience sake, in ten lengths of one meter each, on a board supplied with a millimeter scale. (See Fig. 53). The scale is calibrated from both ends so that lengths of wire from either end of a loop may be measured directly. The key S, is carried by a sliding triangular bridge, so that contact may be made at any point on any one of the ten lengths of wire. The storage batteries are connected to terminals A and D of the potentiometer wire, the primary cell circuit to A and to E or F, like poles of both being connected to A. Terminals E and F are at the ends of a metal strip on which one leg of the bridge rests, so that the current may pass from E or F to S through the bridge. The other two legs of the bridge rest on the wooden base and are therefore insulated. Care must be taken that the bridge does not come in contact with the binding post A, as this will cause a shortcircuit. When the potentiometer is balanced, the length of wire from binding post A to key S may be read on the attached scale. The Daniel cell has a sheet of copper bent into cylin- drical shape and immersed in a copper sulphate solution. A small pocket on the plate contains crystals of blue vitriol 208 Manual of Experiments for keeping up the concentration of the solution. Within this cylinder is placed a porous cup containing a zinc rod in dilute sulphuric acid. When properly set up this cell has an e.m.f. of 1.08 volts, which with proper care remains con- stant for a long time. The Daniel cell may therefore be used as a standard for comparison. The gravity cell has the same elements as the Daniel cell, without the porous cup, depending upon the difference in the specific gravities of the copper sulphate and the zinc sulphate formed to keep the liquids apart. The salammoniac cell has carbon and zinc plates in a solution of ammonium chloride, and the dry cell is practically the same except that the carbon plate is packed in some material that absorbs the solution and prevents polariza- tion, and is contained in a zinc cylinder. Fig. 53 The storage cells, in their common form, consist of two lead grids having their interstices filled with lead sulphate formed by making a paste of red lead and dilute sulphuric acid. These plates are immersed in a weak solution of sul- phuric acid and are charged by passing a current from one to the other. On connecting the two together externally a current will flow in the opposite direction to that of the charging current. They are charged and discharged in this way a number of times to form them. The Daniel and gravity cells, and the storage cells may be left in a closed circuit for some time without their e.m.f. depreciating markedly, i. e. they are closed circuit cells and In General Physics 209 do not polarize easily. The salammoniac and dry cells are however open circuit cells, and can be used only for short intervals at a time, because their e.m.f. drops when they are in use due to polarization, and only recovers when they have been left on open circuit for some time. In order that we may have the proper current strength or electromotive force in a given circuit, it is sometimes necessary to connect several cells in series, or in parallel, or in combinations of the two. (See Fig. 54.) To connect them in series the positive terminal of one is joined to the negative terminal of the next, etc., leaving one free positive and one free negative terminal to be connected to the given circuit. The potential difference from one end of the series to the other is evidently the sum of the potential drops through each cell, so that the combined e.m.f . of the cells in series is the sum of their separate e.m.f ’s. But it must be remembered that the effective (internal) resistance of such a cell is that of the liquid through which the current passes from one plate to the other (the resistance of the plates Fig. 54 themselves being negligible), and when several cells are in series the current must pass through their respective liquids successively, or the total resistance of the cells in series is the sum of their separate resistances. The* current strength may be determined by Ohm’s law, the total resistance in- cluding the internal resistance of the cell and the external resistance of the circuit, so that on short circuit (zero external resistance) the current strength is the same as that in a single cell. When several cells are connected in parallel, all the positive terminals are joined together and to one end of 210 Manual of Experiments the circuit, and all the negative terminals together and to the other end of the circuit. The potential difference from one set of terminals to the other is evidently the same as the potential drop through a single cell if they are all of the same kind, and the combined e.m.f. of the cells in parallel is the e.m.f. of a single cell. But the current passes through all of them simultaneously and though the length of the liquid path is the same for all of them as for one cell, the cross section for n of them in paralled is n times as large as that for a single cell, and since the resistance of a conductor varies inversely as its cross section, the resistance in this case will be 1/n times that of a single cell. As a result the current would be n times as great as from a single cell on short circuit It is sometimes advantageous to combine both methods of connecting cells. For instance if we have 6 cells, we may connect them in 2 rows, each row having 3 cells in series. The 2 rows may then be joined in parallel by connecting their free positive terminals together at one end of the circuit and their free negative terminals together at the other end. Or we may connect them in 3 rows each having 2 cells in series and then join these 3 rows in parallel. It may be shown that in order to obtain the maximum current from a combination of cells in a given circuit, they should be arranged in such a way that the internal resistance of the cells is equal to the external resistance of the circuit. In the general case let n be the number of cells of the same kind to be connected, s the number in series in each row, p the number of rows joined in parallel, r the resistance and e the e.m.f. of each cell, and R the external resistance. Then the total number of cells will be n = ps. The e.m.f. of each row will be se and its resistance sr . The e.m.f. of p rows in parallel will be se also, but their combined In General Physics 211 resistance will be sr/p. The current in the external circuit is therefore se pse ne 1 = = — = R-\-sr/p pR-\-sr pR-\-sr This current will evidently be a maximum when the de- nominator pR+sr is a minimum. Suppose pR and sr to be the sides of a rectangle, their product being its area. This product psRr is a constant in any given case, for the number of cells (ps) is fixed, the external resistance is fixed, and for a given current the internal resistance of the cells will be constant. But it is known that of all rectangles of a constant area that one has the least peri- meter which has its sides equal, (the square.) There- fore the sum of two sides pR-fsr will be a minimum when pR = sr , or when R=sr/p. In other words, in a series-parallel combination of cells , the current will have its maximum value when the external and total internal re- sistance are equal. When two cells have two like terminals joined to- gether, and two to the external circuit, their e.m.f.'s are opposed to each other and their combined e.m.f. is the dif- ference of their separate e.m.f.'s. Directions. — Connect the apparatus as shown in the diagram, being careful to have the cells to be tested opposed to the storage cells, and the latter in series with each other. The Daniel cell to be used as a standard is to be tested first. By means of the sliding key, contact should be made with one wire after the other until two adjacent wires are found for w T hich the galvanometer gives deflections in opposite direc- tions. The key is then moved along between the contact points on these two wires and contact made at different points until the position s' is found for which there is no gal- 212 Manual of Experiments variometer deflection. The length of wire AS' should be noted. The key must not be held in contact with the wire while it is being moved, and should not be pressed hard at any time, as this may injure the wire. Each of the gravity cells is then to be put in place of the standard and the balance positions found, the lengths AS being noted in each case. This is repeated with both gravity cells in parallel, then in series, and then in opposition. In the latter case the cell having the higher e.m.f. should be connected so that it has the same terminals (+ or — ) joined to A and D as the storage cells have. Finally the dry cell and the salam- moniac cell are each connected to the potentiometer and balanced. From the data obtained the e.m.f. of each cell and of each combination is to be determined by comparison with that of the Daniel cell. The e.m.f. of each combination should then be compared with the separate e.m.f. 's of the two gravity cells. Questions. — 1. — What theoretical arrangement of y pressing gently between sheets of filter paper without rubbing. The copper will not adhere to the plates unless they are perfectly free from oil or grease. Those parts of the plates which are to be immersed, mud therefore not be touched by the hand nor laid on a dusty table or balance pan. When quite dry the kathode should be carefully weighed on the fine balance and the mass recorded. Replace the plates in the cell and close the circuit, noting the exact 228 Manual of Experiments time when the current starts to flow. Read the deflections of both ends of the galvanometer pointer and at the end of five minutes reverse the current. While throwing over the reversing switch the current will be interrupted. It is there- fore necessary to reverse as rapidly as possible. Repeat the procedure, reversing the current and reading the galvano- meter deflections at the end of every five minutes until the current has been flowing for one hour. Then open the switch^ remove the kathode, and wash, dry and weigh it as before, and record the mass. The copper sulphate solution should be returned to the supply jar. While the copper is being deposited, the number of turns of wire on the galvanometer coil, and the mean radius of a turn should be recorded, and from these the galvanometer constant G calculated. Determine the mean current strength from the change in mass of the kathode. As the copper plates dissolve in the electrolyte at a definite rate per second per unit area of electrode surface, it is usually necessary to allow for this in calculating the amount of copper deposited but for the small plates used and the short time of deposit, this correction is negligible. Note the tangent of the mean angle of deflection and calculate K , the.reduction factor of the galvanometer. The horizontal component of the earth ’s field is calculated from the galvanometer constant and the reduction factor. Questions. — 1. — What is the advantage of the tangent galvanometer over the D’Arsonval for this experiment? ( b ) — Why is the switch reversed every five minutes? 2. — A current of a( = 1) ampere will decompose how much b(= water) in *( = 1) hour if the atomic weight of d{ = hydrogen) is *( = 1.008)? — A voltameter containing a solution of a salt of a( = nickel) with a resistance of &( — M) ohm between its ter- In General Physics 229 minals, has a constant potential difference of c( = 2) volts maintained between them. How much of a will be deposited in d{ = 15) minutes? 4. — A copper voltameter is in series with a tangent galvanometer. A current passing through it deposits a ( = 0.912) grams of copper on the kathode in b( = 20) minutes, and the galvanometer deflection is c( = 30)°. What is the reduction factor of the galvanometer? 43 . ELECTROMAGNETIC INDUCTION To study electromagnetic induction qualitatively (D. 499-505; G. 957-963; K. 710-726; W. 516-519.) Apparatus. — A primary and secondary coil of wire ballistic D’Arsonval galvanometer, reversing switch, horse shoe magnet, iron core, 110 volt circuit with lamp resistance Theory and Definitions. — Whenever the number o lines of force threading through a closed circuit is altered, a current of electricity is induced in the circuit. This phenome- non is called electromagnetic induction. Whenever the in- duction results from relative motion between the circuit and a source of magnetic lines of force, Lenz’s law states that the induced current is always in such a direction that it opposes by its electromagnetic action , the motion which produced it. The change in the number of lines of force may be due to (a) Relative motion between the closed circuit and a magnet; ( b ) Relative motion between the closed circuit and a second closed circuit through which a steady current is flowing; 230 Manual of Experiments (c) A change in the strength of *the current flowing through the second closed circuit when both circuits are at rest. In cases (b) and (c) that circuit in which the current is induced is called the secondary circuit, the other is the pri- mary circuit. The current lasts only during the time that the change in the number of lines is taking place, and a ballistic galvanometer must be used to indicate the total flow of current. It is sometimes more convenient to think of the current as induced in a conductor by cutting across lines of force. In this case the direction of the induced current is given by Fleming’s rule. Place the thumb, forefinger and middle finger of the right hand mutually perpendicular to each other. Then if the thumb be pointed in the direction of motion and the forefinger in the direction of the lines of force, the middle finger will point in the direction of the induced current. The permeability of iron is greater than that of air; that is, for the same field strength, there will be more lines of force in a given cross section of iron than in the same cross section of air at the same point in the field. Hence if we place an iron core through one or both of the circuits, the change in the number of lines of force in the secondary will be greater than before. Directions. — Connect the apparatus as shown in the diagram, Fig. 59. The secondary coil B is fixed, and con- nected to the galvanometer G. The primary coil A is mov- In General* Physics 231 able and may be turned about a vertical axis or may be moved to different distances from B. The coil A is connected to the source of current B through the reversing switch K. (1) Relative motion between a magnet and a closed cir- cuit . — Bring the horse-shoe magnet rapidly astraddle of the coil B and note the direction of the deflection of the gal- vanometer. When the galvanometer coil has returned to its position of rest, rapidly withdraw the magnet from the coil and note the deflection. Repeat this with the magnet poles reversed. Relative motion between two circuits. (2) Primary and secondary circuits approaching or re- ceding from each other . — Place the primary coil A parallel to the secondary coil at the opposite ends of the supporting board, and close the switch so that the current flows through A. Move A suddenly up to B and note the direction of deflection. When the galvanometer is again at rest, with- draw A and note the deflection. Reverse the current and repeat. (3) Rotation of primary circuit . — Insert the brass pin through the holes in the base of the coil and the supporting board and place A parallel to B. Rotate A suddenly through 90° and note the amount and direction of deflection. When the galvanometer is again at rest, rotate A through a further 90° in the same direction and at approximately the same speed, and note the direction and amount of deflec- tion. Compare the two deflections. Then bring A back to its original position and when the galvanometer is at rest, rotate it in the same direction as before and with approxi- mately the same speed, but through 180°. Allow the gal- vanometer to come to rest and rotate A through another 180° as before. Compare the two deflections in this case in direction and amount. 232 Manual of Experiments (4) Variable current in the primary . — Bring A close up to and parallel to B , and when the galvanometer is at rest, suddenly close the primary circuit and note the deflection. After the galvanometer is again at rest, suddenly open the switch. Repeat with the switch reversed. Compare the four deflections. (5) Primary at different angles. — Place A on the brass pin and note the deflections on breaking the circuit when the plane of coil A makes the following angles with the plane of B : — 0°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 180°. Plot a curve having angles as abscissae and deflections as ordinates. (6) Effect of iron core. — With A and B parallel and about 3 cm. apart, note the deflection when the current is broken. Put an iron core through the center of both coils and repeat. Compare the deflections. Questions. — 1 . — With the data obtained in section five of the above experiment plot a curve having angles between the planes of primary and secondary coils as abscissae and corresponding deflections of the galvanometer as ordinates. 2. — Suppose a circular metal ring to rotate clockwise about an axis through its center, perpendicular to the paper but parallel to its own plane, in front of the north pole of a permanent magnet lying in the plane of the paper. Draw four diagrams showing the ring in its four quadrants, and indicate on each the direction of the induced current, and that of the lines of force which it sets up in the ring. Note. — Draw a section through the center of the coil parallel to the plane of the paper, representing the cut ends of the coil by circles. A dot or dash in one of these circles represents an approaching current (point of an arrow); a cross represents a receding current (tail of the arrow.) In General Physics 233 3. — Explain how Lenz's law applies in each case (by the aid of the north and south faces of the ring,) or how Fleming 's rule applies. 4. — At a place where the vertical component of the earth 's magnetic field is a( = 0.50), a horizontal coil of wire of &( = 10) turns, with diameter c( = 3) meters, and resistance d{ = 2 )ohms, is suddenly pulled out into a straight line loop in e( = y 2 ) second. Find a) the induced e.m.f.; b ) the average current in amperes; c) th^ total flow of electricity in coulombs; and d) the energy used up. g 4 / 44. JOULE'S LAW AND THE ELECTRO-CALORIMETER To determine Joule's equivalent by the electro-calori- meter. (D. 458-461; G. 403, 858-861; K 654-657; W. 493, 494.) Apparatus. — Calorimeter with high resistance coil, tangent galvanometer, reversing switch, source of current, thermometer, coarse balance and weights, P. 0. box bridge, dry cell, connecting wires and watch or clock. Definitions and Theory. — Whenever a current of electricity flows through a conductor it generates heat. If the difference of potential between the terminals of the con- ductor be V the energy expended when unit quantity of electricity flows through it will be V units by definition of difference of potential. If in the time T, q units flow through the conductor, the work done will be W = qV But if i be the current strength, q = iT and if r be the re- 234 Manual of Experiments sistance of the conductor, V = ir, by Ohm ’s law, when ce W= i 2 rT When i, r and t are measured in C. G. S. absolute units, the work done will be expressed in ergs, but if C be the current in amperes and R the resistance in ohms, since one ampere equals lO- 1 C. G. S. units of current and one ohm equals IQ 9 C. G. S. units of resistance, the work done will be W= 10 7 G 2 RTergs = C 2 RT joules the joule being defined as a unit of energy equal to 10 7 ergs. Then if one ampere flows through a resistance of one ohm for one second the work done will be one joule. If we wish to express the energy expended as heat in calories we must divide by the mechanical equivalent of heat, /, in ergs per calorie, whence 10 7 C 2 R T II = calories. */ This is known as Joule’s law. By passing a current of electricity through a coil of wire immersed in a water calorimeter and measuring the strength of current, resistance of coil, time of flow and heat generated, we may determine Joule’s equivalent, /, by the above for- mula. If however, we assume the value of /, we may de- termine the resistance of the wire or the strength of the current. Directions. — Connect the coil of calorimeter and tangent galvanometer in series with the source of current and arrange the switch so that the current may be reversed. Before starting, measure the resistance of the coil and weigh the calorimeter and stirrer. Cool some water to about 10 degrees below the room temperature and fill the calorimeter In General Physics 235 about three-fourths full. Determine the weight of water added. Note the temperature of water, and at the instant of closing the circuit note the exact time and observe the gal- vanometer deflection. At the end of every five minutes reverse the current and observe the deflection. Keep the water well stirred. Watch the thermometer and when the temperature has risen to about 10 degrees above that of the room, open the circuit, carefully observing the time and temperature at the instant of opening. Redetermine the resistance of the coil, and find the mean of these two deter- minations. Calculate the water equivalent of the calori- meter cup and stirrer. Determine the heat generated from the change in temperature. Average the galvanometer de- flections and determine the current strength. With this data determine the value of /. Questions. — 1. — Given two conductors of the same material and length but of different cross sections. In which will the energy loss per unit time by heating be the greater a) when they are in series? b) When they are in parallel? Give reasons for your answers. (Notice that C 2 R = E 2 /R.) 2. — If coal costing r, v < r, for u = r, v = r, for u < r, v > r, for u = /, v = infinity, for u < f, v = negative. Positive values of p and of / correspond to real images, negative values to virtual images, hence the image formed by a concave mirror is real , except for values of u less than f, in which case it is virtual. We see that as the source moves up from an infinite distance to the principal focus, the image moves off from the focus to an infinite distance, the two points meeting at the center of curvature. In the case of the convex mirror, the value of v is always less than / and negative for positive values of u. If however the rays of light incident on the mirror converge toward a point — u behind the mirror, v may become positive, and the image will be real. Since the object and its image subtend equal angles at 'the mirror, it is evident that the relative sizes of both are in the ratio of their respective distances from the mirror or if 0 and / represent their corresponding dimensions, 0 u The apparent position of an image is sometimes obtained by the parallax method. By parallax we mean the apparent displacement of an object {or image) relatively to another object , due to a real displacement of the observer. If the two objects are coincident or are equally distant, their relative parallax vanishes. When waves from a point source fall upon a spherical mirror obliquely, they do not form a point image, but there 246 Manual of Experiments will be formed two line images at different distances and at right angles to each other. The nearer the source is to the axis of the mirror, the more nearly these focal lines will coincide, and will be reduced to form a point image on the axis. When the source of light is not a point, but an extended object, each point of this object will have its own point image formed, and all of these images together make up the image of the source. But those points which lie off of the axis of the mirror will have their images formed at different distances from the pole than that of the point on the axis. It will therefore be impossible to find a position of the screen at which all parts of the projected image are equally in focus. 46. MIRRORS. To show that the angle of incidence is equal to the angle of reflection, and to find the position, size and shape of the image formed by a plane mirror; to show the relation between image and object distance and focal length of a concave spherical mirror, and to measure its focal length. Apparatus. — Optical bench, incandescent lamp with cover and slit, plane and concave spherical mirrors and hold- ers, sliding blocks, right triangle, protractor, pins, screen with slit and cross wires, and sheet of paper. Description. — The optical bench is a slotted channel iron bench about one and a half meter long, with a mm. scale attached to one side. (See Fig. 63.) The mirrors, lenses and screens to be experimented with are carried by metal blocks which are arranged to slide along the slot. In General Physics 247 Their positions on the scale may be read by means of an indicator mark on the side of each block. One block carries a screen with one side enameled white so that images pro- jected on it may be sharply focussed. It has also a rectangu- lar opening in the center, across which cross wires are stretched. These may be used as the object to be focussed on the screen when a strong light is placed behind the open- ing. An incandescent lamp is covered with a copper screen, in one side of which is a rectangular slit. This generally serves as the object to be projected on the white screen, and it is this opening, not the incandescent filament of the lamp, which must be sharply focussed. To aid in focussing, a sharp spur is cut on one side of the slit. The mirrors and lenses are mounted so that they may be turned about both a vertical and a horizontal axis, in order to place them in any desired position, and the indicator mark on the carrying block is in all positions under the pole of the mirror or lens. On the front of these blocks are a pair of clamps for holding a small plane mirror for use in the parallax method of fo- cussing. Directions. — Relation between angles of incidence and reflection . — The plane mirror is fastened to one face of a supporting block. Draw a straight line across the sheet of paper and place the silvered (reflecting) side of the mirror along this line. Stick a black headed pin about 10 cm. in front of the mirror and a second ordinary pin to one side of the former and about 2 cm. in front of the mirror. Move the eye on a level with the mirror until a position is found where the images of the two pins coincide. Place a third pin c in line with the coincident images and the eye. Then ■ draw the line through a and b to the reflecting surface and a perpendicular to this surface at the point of intersection o. Also join c and o. See Fig. 61. If the glass of the mirror 248 Manual of Experiments is thin the deviation of the rays ao and co, by refraction, may be neglected. Measure the angle of incidence and the angle of reflection and compare their values. Place the pins a and b in different positions and repeat, using large angles for the sake of accuracy. Fig. 61 Position , size and shape of the image in a plane mirror . — Draw a line on the other side of the paper, through its center and place the mirror as before. Lay the triangle in front of the mirror, trace its outline with a sharp pencil, and mark the angles a , b and c. Place a short pin at the vertex a and a large pin a' behind the mirror, a' is to be placed at the image of a by the method of parallax, hence a' must be moved to different positions until such a position is reached that upon turning the head from side to side or looking at the mirror from different positions, the image of a in the mirror coincides with the pin a' as seen above the mirror, for all angles of incidence. So long as a' is not at the image of a there will appear to be relative motion between these two as the head is turned from side to side. Locate the images V and c' of b and c respectively in the same way. Join the points a, b and c to their corresponding images. Measure and compare the perpendicular distance of each vertex from the mirror, with that of its image. Join a\ b' and c' to form the image of the triangle. Measure and com- pare the lengths of its sides with that of the corresponding sides of the triangle. In General Physics 249 Focal length of a concave spherical mirror. — Place the mirror on the optical bench facing the screen; carry them to a shaded window, and turn bench and mirror toward some distant object, such as a church tower, or a tree. Move the screen on the bench until a sharply outlined image of this object is obtained. Have the angles of incidence and re- flection as small as possible. This image is formed at the principal focus. Measure the focal length between screen and pole of mirror. Relation between distances of object and of real images from a spherical mirror. — Return the apparatus to the place as- signed for performing the experiment. Arrange the lamp close to the bench so that the light passing through the slit may be reflected from the mirror to the screen, the slit being on the same level with the mirror and screen. To get sharper, though less brilliant images, the mirror may be covered by a diaphram made of paper or sheet metal, having the outline of the mirrox and a circular hole at the center one to one and a half inches in diameter. The rectangular slit is the object whose image is to be formed and to aid in getting the image sharp a fine spur is cut so that it projects at one side of the slit. The distances u and v, of the slit and of the screen respectively from the center of the mirror may be measured on the scale. It should be remembered that for oblique incidence (which is unavoidable here) two images are formed at slightly different distances and a mean position must be found in each case. Place the slit close to and in the same plane with the screen and move the mirror until a sharp image of the slit (not the lamp filament) appears on the screen. The dis- tance between mirror and screen is now the radius of curva- ture of the latter. Measure this distance r and compare it with the focal length /. 250 Manual of Experiments This distance may also be determined by removing the disk with the slit from the lamp cover, placing the lamp di- rectly behind the white screen, at one end of the bench and adjusting the position of the. mirror until a sharp image of the cross wires is projected on the screen just beside the slit. The object and image are then both at the distance r from the mirror. With the lamp cover and slit once more at the side of the bench, place the mirror at a distance between / and r, from the slit and move the screen until a sharp image of the slit is formed. Measure the distances u and v on the scale, and the length of the slit and of its image with the aid of dividers. Repeat with the mirror at a distance greater than r from the slit. Compare the sum of the reciprocals of u and v in the two cases with each other and with the re- ciprocal of the focal length /. Also compare the ratio of the object and image lengths in each case with the corre- sponding ratio of u and v. Virtual image . — Remove the diaphram and fasten a short pin in the sliding block at a distance from the mirror less than the focal length, f. With a large pin behind the mirror locate the image of the short pin by the method of parallax, as described for a plane mirror. Measure the distances u and v and, remembering the adopted convention concerning signs, compare the sum of the reciprocals of u and v with the reciprocal of f. Owing to the spherical aberration, the image of the pin at the edges of the mirror curves from side to side as the eye is moved, and makes it difficult to determine these conjugate foci. When this curvature is objectionable the following method should be used: Place a small plane mirror M in front of the lower half of the spherical mirror, C. (See Fig. 62.) Place a black headed pin 0 at a distance less^fchan f In General Physics 251 in front of C , and a white headed pin P in front of M. By looking down at an angle at both' mirrors, the image of 0 in C, (/), and of P in Af, (0, may be seen simultaneously without interference from the pins themselves. P must now be shifted in position until there is no parallax between the images / and <2, when the eye of the observer moves from side to side. I and Q then occupy the same position, and Q is as far behind the reflecting surface of M as P is in front of it. Calling this distance “ui” and the distance between reflecting surfaces of C and M “ d,” we see that the distance of I from C is v = Ui~d Note on the scale the distances u, d and calculate v, and compare the reciprocals of f with the sum of the reciprocals of u and v. Problems. — 1 . — Draw large clear diagrams showing the formation of a real, and of a virtual image of an arrow, by 252 Manual of Experiments means of a concave mirror, when the arrow is perpendicular to and symmetrical to its axis. Trace the path of two rays from each end of the arrow; one parallel to the axis of the mirror, or one through its center of curvature, and one through its principal focus. Mark center C, focus F, object (9, image /, and the direction of the rays with arrow heads. Use full lines for actual rays, dotted lines for construction and prolongation of rays. 2. — In the same way draw diagram of image formation by a convex spherical mirror. 3. - — a ) A concave spherical mirror has a radius of curvature of #( = 30) cm. Find the position and size of the image formed when a candle flame with a height b( = 2) cm. is placed r( = 20)cm. in front of the mirror, on its axis. b) Find the position and size of the image when the candle is moved a distance J( = 8) cm. toward the mirror. 4. — A spherical mirror forms an image of an object having a height #( = 5) inches, and placed b( = 20) inches in front of the mirror, a) If the image is at a distance c( = 12) inches behind the mirror, find the focal length of the latter, b) Is the mirror concave or convex? c) What is the height of the image? REFRACTION THROUGH LENSES. Theory and Definitions. — (D. 620, 656, 666-667; G. 552, 562-572; K. 834-836, 849-858; W. 348, 349.) A lens is a portion of a refracting medium bounded by two curved surfaces, which are usually spherical. A line through the centers of curvature of the two surfaces is termed the prin- cipal axis of the lens. The points where this axi£\quts the two surfaces are termed the poles of the surfaces. In the In General Physics 253 following discussion the lenses are supposed to be thin; i. e., the two poles are so close together that distances may be measured from the center of the lens. For definitions of aperture, real and virtual images, see previous theory of reflection. Lenses are of two general kinds. Those which are thicker at the poles than at the periphery usually cause the incident light from a point source to converge after refraction to a point beyond the lens, when surrounded by a less refractive medium, and are called convergent. Those which are thinner at the poles than at the periphery cause the incident light to diverge after refraction from a point in front of the lens, when surrounded by a less refractive medium, and are called divergent lenses. When the incident light is parallel to the principal axis the lens will cause it to converge to, or diverge from a point called the principal focus. The distance from the lens to the principal focus is called the focal length of the lens. The focal length remains unaltered when the lens is reversed. We shall adopt the same conventions for signs as in the case of reflection; namely: — (1) All distances are to be measured from the poles along the axis. (2) Distances measured from the poles in the direction in which the incident light travels are negative , those measured in the opposite direction are positive. According to this convention the focal length of a con- vergent lens is negative, that of a divergent lens positive. Let u be the distance of an object from the lens center, v the distance of its image ? and / the focal length. Then for lenses having a small aperture , the following relation holds for both kinds of lenses: — 254 Manual of Experiments 1 i i U V f It should be remembered that besides the signs of the fractions in this equation each symbol (u, v, /,) has its own sign depending upon the conventions given above. This equation gives us the following relations: — Convergent lenses: For u — infinity, For oo > u > 2/, For 2/ > u > /, For / > u > o, v = —f. —f>v> -2 f - 2/ > v > - o°. oo > v > o. In the last case the image is virtual. The other images are all real. Divergent lenses . — For u = 00 , v — f. For °o > u > o, + f > v > o. The image formed by the divergent lens is always virtual and always nearer to the lens than the object. Since the object and image subtend equal angles at the center of the lens, their sizes will be in the same ratio as their respective distances from the lens. Hence, if 0 and I represent their corresponding dimensions 0/I = u/v. If a lens forms an image of an object on a screen when it is half way between them, it may be shown that the dis- tance between them is four times its focal length, f. If the fixed distance between object and screen is greater than 4/, there will be two positions of the lens between them, at which In General Physics 255 it forms a distinct image of the object on the screen. This fact suggests a method of determining the focal length of the lens. For if s be the distance between the object and the screen (greater than 4/), and / be the distance between the two positions of the lens, at which it projects a sharp image on the screen, it may be shown that the focal length is s 2 -l 2 47. LENSES. To study the relation between focal length, image and object distance, and image and object size in the case of convex and concave lenses, and to measure their focal lengths. Apparatus. — Optical bench, double convex and double concave lens and stand, incandescent lamp and leads, cover and slit, sliding screen, two sliding blocks meter stick and pins. (Apparatus described in Exp. 46.) Directions. — Convex lens . — Take the optical bench to a shaded window. Place the convex lens in its holder at one end of the bench with its axis pointed toward some distant object, such as a chimney or church steeple. Move the screen along the bench until a sharp image of the object is projected on it. The rays from the object are practically parallel and the distance from lens to screen is the focal length of the former. Measure this length on the attached scale. Place the lamp with its slit vertically over zero of the scale and place the lens at a distance between / and 2 f from the slit. Fig. 63. Then move the screen until there is a sharp image of the slit (not of the lamp filament) upon it. 256 Manual of Experiments Adjust for sharp definition by observing the spur on the side of the slit. Measure the distances u and v and also the lengths 0 and / of the object and image. Repeat with the lens at a distance equal to 2/ and then at a greater distance from the slit. Compare — l/u + l/v with 1 If in both cases, remembering the adopted conventions regarding signs. Also compare O/I with u/v. Virtual image . — Place a block with a small pin behind the lens at less than the focal distance f. Locate the virtual image of this pin by the method of parallax; i. that the wavelets at D and B being in the same phase, form a new wave front. When this is brought to a focus at R it produces an image of the slit called the first order spectrum. Similarly for some point on the screen beyond R a new wave front is formed by a wavelet just starting from B and one from A which has passed over two wave lengths. This produces on the screen the second order spectrum; and so on for higher orders. An exactly similar set of images is formed in the same way on the other side of P. For all points between these images there is interference between the wavelets from A and i?, and more especially when the path difference from A and B is an odd multiple of a half wave length, the waves are in opposite phase, and complete interference results in a dark “image” or space on the screen. The same explanation applies in the case of more than two ruled lines, but the spectra in this case are much brighter because of the greater number of lines from which-The light comes. If in place of monochromatic light we use white light the central image will be white, because all wave lengths travel over equal dis- tances to this point. But the distance AD will differ with the wave lengths, consequently the first order images will not all be formed' at i?, but at some point nearby. As a In General Physics 279 result a continuous spectrum of all the colors will be formed in the neighborhood of R , with the violet (due to the shortest wave lengths) nearest to the central image. The same is true of the spectra of higher orders. Let the distance AB between adjacent rulings be < 2 , the difference of path AD be X, and the angle ABD be 0. Then from triangle ABD we have \ = a sin 9 (1) or if the angle of the nth order spectrum be n\= a sin <£. (2) As the triangles ABD etc., are not exactly right triangles these equations are not quite true, but for the lower orders 280 Manual of Experiments of spectra they may be used without making an appreciable error. If the grating is placed on a spectrometer table, the angle d may be measured directly. The number of lines per centimeter is usually marked on the grating so that a is easily calculated. From these we calculate the value of X. If a spectrometer is not available, we may fix a scale in front of the grating, and move a telescope along it to measure the distance PR between images. The distance RC (or PC, between grating and scale, since these are pactically equal) is readily measured and the sine of 6 will be PR/RC . Directions. — Focus the eyepiece of the telescope on the cross hairs and, having focussed the telescope for parallel light, focus the slit of the collimator so that there is no parallax between the image of the slit and the cross hairs of the telescope. (See Exp. 51). Place the grating in the holder on the platform so that the ruled side of the grating is turned away from the collimator. See that the grating lines are vertical and the plane of the grating is at right angles to the axis of the collimator. Make the slit of the collimator very narrow and illuminate it with a sodium flame. Place the cloth over the spectrometer to cut off external light. Move the telescope so as to be in line with the collimator and, having the cross hairs on the image of the slit, observe the readings of the vernier. All parts of the spectrometer, exce pt the telescope, should be claipped. Move the telescope a few degrees to one side and adjust it till the cross hairs are on the image of the slit as seen in the first order spectrum. The slit must be vertical and parallel to the rulings on the grating for best definition. If the line is not readily found, an incandescent light may be placed in front of the slit which will produce a continuous spectrum In General Physics 281 that is readily located and then the sodium light substituted for it. Note the readings of the verniers in degrees, minutes, and seconds of arc, when the cross hairs are on the first order spectrum, and find the mean of the differences be- tween corresponding vernier readings for the first order spectrum and the central image. This is one value of 6. Move the telescope to the other side of the central image and make a second determination of the angle 6 by adjust- ing the telescope so that the image of the first order spectrum on that side is in coincidence with the cross hairs. Repeat each determination two times and find the mean of the six values of 6 (three on one side and three on the other side,) the angle through which the telescope must be turned from its position of coincidence with the central image to its po- sition of coincidence with the first order spectra. Note the number of lines to the centimeter as recorded on the grating and determine the distance a between two adjacent lines. Substitute the values of a and 6 in equation (1) and calcu- late X the wave length of sodium light. In a similar manner make three determinations of the angle for coincidence of the cross hairs with the image of the second order spectra on each side of the central image, and find the mean of the six values of 4> thus observed. The second order spectrum will be at an angle about twice the angle for the first order. It will be rather indistinct, and consequently care must be taken to have the slit well illu- minated and all external light cut off the grating as much as possible. Substitute the value of <£ found for the second order spectra together with the value of a in equation (2) where n has the value 2, and calculate the wave length X of sodium light. By careful adjustments it will be possible to see that the sodium line is composed of two lines, and especially so in 282 Manual of Experiments the second order spectra. Replace the sodium light by the incandescent lamp and observe the order of the colors in the spectra of both the first and second orders on each side of the central image. Observe which spectra are the most distinct and which spread out the most. Make a sketch in your report showing the order of the colors in diffraction grating spectra. Questions. — 1 . — How does the order of* the colors in a grating spectrum differ from the order in a prism spectrum? 2. — What is the frequency of the vibration which pro- duces a yellow sodium line? EXPERIMENTS IN SOUND. Theory and Definitions. — Sound is transmitted by means of longitudinal waves, the direction of vibration of the particles of the transmitting medium being parallel to the direction of propagation of the wave. The wave is transmitted as a series of alternate condensations and rare- factions of the medium. The length of the wave is the dis- tance from any one vibrating particle to the next succeeding (or preceding) one which has the same phase. It is there- fore the distance between two successive points of maximum condensation or of maximum rarefaction. The velocity of the wave transmission is the wave length times the frequency of the vibration producing it. The pitch of the sound depends upon the frequency, the intensity upon the amplitude, and the timbre upon the shape of the wave. When waves impinge upon an elastic body which has the same natural period or the same frequency as the wave, the In General Physics 283 body is set in sympathetic vibrations and there is said to be resonance between the two. When two bodies vibrating near each other have very nearly the same natural period, pulsations of sound or beats, are produced due to the alternate reinforcement and inter- ference of the waves. Two bodies may be brought to the same pitch by altering the period of one of them until the beats disappear and a sound of uniform intensity is heard. When a column of air in a tube closed at one end, is set in vibration by resonance, there is a condensation at the closed end or a node is formed there and a rarefaction at the open end producing an antinode. The shortest closed tube in which resonance could be produced by a given wave would have a length equal to one-quarter of the wave length. The next length in which resonance would occur would be three- quarters of the wave length, etc. In an open tube resonance would occur for lengths equal to multiples of a half wave length. When a stretched string is set in vibration, the wave length of the sound produced cannot be greater than twice the length of the string since there must be a node at each end. It may be shown that the frequency of vibration or the pitch of the sound produced is inversely proportional to the length of string and to the square root of its mass per unit length, but directly proportional to the square root of its tension. If T represents tension, /, length of string, and m, mass per unit length, the frequency is 1 /T n = — / — 2/ V m 284 Manual of Experiments 53. GRAPHICAL DETERMINATION OF PITCH OF TUNING FORKS. To determine the pitch of two tuning forks graphically. (D. 592, 609; G. 245; K. 304; W. 267, 288.) Apparatus. — Two tuning forks to be tested, a long board slide-way with slides and clamps for pendulum and tuning fork, strip of plate glass, rubber hammer for fork, steel dividers, paste of whiting or chalk dust and dauber. Description and Method. — The slide-way is a smooth board about 5 feet long and 6 inches wide, provided with side guard strips. The wooden slide — about 2 feet long — slips easily in the slide-way. A tuning fork and a small pendulum are supported above the slide-way so that they may vibrate across the path of the slider. The clamps allow an adjustment of the pendulum and the fork. The strip of glass may be attached to the top of the slide by appropriate fasteners. The method used is to get traces of the vibrating fork and swinging pendulum in the same line on the glass and to count the number of waves of the fork in one vibration or wave length of the pendulum; then by timing the pendulum vibration, to determine the number of vibrations of the fork in one second, i. m (r 2 +ri 2 ) Circ. cylinder transverse through center m(r 2 /4+l 2 /12) Thin rod transverse through end ml 2 /3 Thin rod transverse through middle ml 2 /12 Rectangular/ block / through center perpendicular to face with sides 1 and b m(l 2 +b 2 )/12 Sphere through center 2mr 2 /5 Hollow sphere through center 2 (r 5 — r+) — m 5 (r 3 — r-, 3 ) V. BAROMETRIC CORRECTIONS. Capillary Correction— +0.031 cm. Temperature Corrections Temp. °F. Correction. Temp. °F. Correction. 60 0. 171 cm. 70 0.225 cm. 61 0 . 176 cm. 71 0.231 cm. 62 0 . 181 cm. 72 0.236 cm. 63 0. 185 cm. 73 0.241 cm. 64 0 . 192 cm. 74 0 . 246 cm. 65 0.198 cm. 75 0.251 cm. 66 0.203 cm. 76 0.256 cm. 67 0.209 cm. 77 0.261 cm. 68 0.214 cm. 78 0.266 cm. • 69 0.220 cm. 79 0.271 cm. Note — Temperature readings should be taken from the Fahrenheit thermometer on the side of the barometer. 294 VI. HEAT CONSTANTS. Solids Substance Coefficient of Linear Expansion Specific Heat Aluminum .0000255 .219 Brass .... .0000189 - .090 Copper . . . .0000169 .0936 Glass (crown) .0000078 .161 Glass (flint) . . : .000009 .117 German Silver .000018 .095 Ic'e .0000507 .502 Iron (cast) .0000102 .11 Iron (wrought) .0000119 .11 Lead .0000276 .031 Nickel ..... .0000128 .109 Platinum . . . . .0000089 .032 Quartz (fused) .00000039 .174 Silver .0000188 .056 Steel (tempered) . .0000132 .12 Tin .0000214 .055 Zinc .0000263 .094 Liquids, Vapors and Gases (at constant pressure). Substance Coefficient of Cubical Expansion Specific Heat Air .003671 .237 Alcohol (ethyl) .00110 .548 Alcohol Vapor .453 Benzine .... .00124 .40 Benzine Vapor .299 Ether (ethyl) . .00163 .54 Hydrogen .... .003661 3.409 Kerosene .... .0012 .39 Mercury .... .0001818 .0333 Sulphuric Acid .00059 .34 Turpentine .00094 .420 Water (15° — 100°) .000372 1 . Water Vanor .442 295 VII. STEAM TEMPERATURE. At Different Barometric Pressures. h.cm. 70 71 72 73 74 75 76 77 .0 97 72 98 11 98 50 98 88 99 26 99 63 100. 00 100 36 .1 97 76 98 15 98 54 98 92 99 30 99 67 100 04 100 40 .2 97 80 98 19 98 58 98 96 99 33 99 71 100 07 100 44 .3 97 84 98 23 98 61 98 99 99 37 99 74 100. 11 100 47 .4 97 88 98 27 98 65 99 03 99 41 99 78 100 15 100 51 .5 97 92 98 31 98 69 99 07 99 44 99 82 100 18 100 55 .6 97 96 98 34 98 73 99 11 99 48 99 85 100. 22 100 58 .7 98 00 98 38 98 77 99 14 99 52 99 89 100. 26 100 62 .8 98 03 98 42 98 80 99 18 99 56 99 93 100. 29 100 65 .9 98 07 98 46 98 84 99 22 99 59 99 96 100 33 100 69 VIII. HYGROMETRY Pressure of aqueous vapor, p, and weight of water, m, contained in one cubic meter of air saturated at the temperature t°C. t P m t P m t P m —10 cm. .22 gm. 2.4 4 cm. .61 gm. 6.4 18 cm. 1.54 gm. 15.2 — 9 .23 2.5 5 .65 6.8 19 1.64 16.2 — 8 .25 2.7 6 .70 7.2 20 1.74 17.1 — 7 .27 2.9 7 .75 7.7 21 1.85 18.2 — 6 .29 3.1 8 .80 8.2 22 1.97 19.3 — 5 .32 3.4 9 .86 8.8 23 2.09 20.4 — 4 .34 3.7 10 .92 9.4 24 2.22 21.6 — 3 .37 4.0 11 .98 10.0 25 2.35 22.8 — 2 .40 4.2 12 1.05 10.6 26 2.50 24.1 — 1 .43 4.5 13 1.12 11.3 27 2.65 25.5 0 .46 4.9 14 1.19 12.0 28 2.81 27.0 1 .49 5.2 15 1.27 12.7 . 29 2.98 28.5 2 .53 5.6 16 1.36 -13 . 5 30 3.16 30.1 3 .57 6.0 17 1.45 14.4 31 3.34 31.7 296 IX. SPECIFIC RESISTANCE. (Ohms per cm. 3 at 0°C.) Aluminum 3. X10-6 Manganin 42. X10-6 Brass 8. Mercury 94. Copper .... 1.5 Nickel .... 10. German Silver . 20. Platinum 8.9 Gold . . . . . 2. Silver . 1.5 Iron ..... 10.5 Steel .... 50. Lead 20. Zinc 5.7 X. ELECTROCHEMICAL EQUIVALENTS. Element Atomic Weight V alency Chemical equivalent Electrochemical equivalent Chlorine 35.45 1 35.45 .0003672 Copper, (cupric) 63.6 ■ 2 31.8 .000329 Hydrogen 1.008 1 1.008 .00001044 Iron, (ferric) 55.9 2 27.95 . 000289 Iron, (ferrous) . 55.9 3 18.49 .000193 Oxygen . . 16.0 2 8.0 .00008283 Silver . . . 107. 93 1 107.93 .001118 Zinc 65.4 2 32.7 .000338 Nickel .... 58.6 2 29.3 .000305 XI. INDEX OF REFRACTION FOR D LINE (>1=0.00005893) Air 1 000294 Cedar oil .... 1.516 Alcohol, ethyl . 1 362 Ether 1.36 Alcohol, methyl 1 332 Glass (crown) 1.51-1.62 Benzine .... 1 .504 Glass (flint) . . . 1 . 57-1 . 96 Canada Balsam 1 .54 Quartz .... 1.54 Carbon bisulphide 1 .63 Water . . . ... 1.334 Notes on Tables XII and XIII Table XII — The functions of angles between 0 and 45° (left hand column) are marked at top of page, those from 45° to 90° (right hand column) at bottom. For decimal fractions of a degree, add pro- portional parts of the difference given in columns next to functions. Table XIII — Columns on right of page give numbers to be added to logarithms for the corresponding decimal fractions given at the head of these columns. 297 XII. NATURAL TRIGONOMETRICAL FUNCTIONS Degree Sine Tangent Cotangent Cosine Degree 0 0.0000 0.0000 0*0 1 . 0000 90 1 .0175 175 .0175 175 57.29 0.9998 02 l 89 2 .0349 174 .0349 174 28.64 .9994 04 88 3 .0523 174 .0524 175 19.08 .9986 08 87 4 .0698 175 174 .0699 175 176 14.30 .9976 10 14 86 5 0 . 0872 0 . 0875 11.43 0 . 9962 85 6 .1045 173 .1051 176 9.514 .9945 17 84 7 .1219 174 .1228 177 8.144 .9925 20 83 8 .1392 173 . 1405 177 7.115 .9903 22 82 9 . 1564 172 172 .1584 179 179 6.314 801 643 .9877 26 29 81 10 0.1736 0.1763 5.671 0.9848 80 11 .1908 172 .1944 181 5.145 526 .9816 32 79 12 .2079 171 .2126 182 4.705 440 .9781 35 78 13 .2250 171 .2309 183 4.331 3/4 .9744 37 77 14 .2419 169 .2493 184 4.011 320 .9703 41 76 169 186 279 44 15 0 . 2588 0 . 2679 3.732 0.9659 75 16 .2756 168 .2867 188 3.487 245 .9613 46 74 17 .2924 168 .3057 190 3.271 216 .9563 50 73 18 .3090 166 .3249 192 3.078 193 .9511 52 72 19 .3256 166 .3443 194 2.904 174 . 9455 56 71 164 197 157 58 20 0.3420 0 . 3640 2.747 0.9397 70 21 .3584 164 .3839 199 2.605 142 .9336 61 69 22 .3746 162 .4040 201 2.475 130 .9272 64 68 23 .3907 161 4245 205 2.356 119 .9205 67 67 24 .4067 160 .4452 ■""20 7 2.246 110 .9135 70 66 159 211 101 72 25 0.4226 0 . 4663 2.145 0.9063 65 26 .4384 158 .4877 214 2.050 95 .8988 75 64 27 . 4540 156 . 5095 218 1.963 87 .8910 78 63 28 .4695 155 .5317 222 1.881 82 .8829 81 62 29 .4848 153 . 5543 336 1.804 77 .8746 83 61 152 231 72 86 30 0.5000 0 . 5774 1.732 0.8660 60 31 . 5150 150 .6009 23 o 1.664 68 .8572 88 59 32 .5299 149 .6249 240 1.600 64 .8480 92 58 33 .5446 147 .6494 245 1.540 60 .8387 93 57 34 .5592 146 .6745 251 1.483 57 .8290 97 56 144 257 55 98 35 0.5736 0.7002 1.428 0.8192 55 36 .5878 142 .7265 263 1.376 52 .8090 102 54 37 .6018 140 .7536 271 1.327 49 .7986 104 53 38 .6157 139 .7813 277 1.280 47 .7880 106 52 39 .6293 136 .8098 285 1.235 45 .7771 109 51 135 293 43 111 40 0 . 6428 0.8391 1.192 0 . 7660 50 41 .6561 133 .8693 302 1.150 42 .7547 113 49 42 .6691 130 .9004 3 l 1 1.111 39 .7431 116 48 43 .6820 129 .9325 321 1.072 39 .7314 117 47 44 .6947 127 .9657 332 1.036 36 .7193 121 46 124 343 36 122 45 0.7071 1.0000 1.000 0.7071 45 Degree Cosine Cotangent Tangent Sine Degree 298 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 XIII. LOGARITHMS. 8 4 5 6 7 8 9 L523 L818 7007 7093 7177 7259 0128 0170 0212 0253 0294 0334 0374 4 8 12 17 1 21 25 29 33 '37 1 r 0531 0569 0607 0645 0682 0719 0755 4 8 11 15 19 23 26 30 34 0899 0934 0969 1004 1038 1072 1106 3 7 10 14 17 21 24 28 31 1239 1271 1303 1335 | 1367 1399 1430 3 6 10 13 16 19 23 26 1 29 1553 1584 1614 1644 1 1673 1703 1732 3 6 9 12 15 18 21 24 27 1847 1875 1903 1931 4959 1987 2014 3 6 8 11 14 17 20 22 25 2122 2148 2175 2201 2227 1 2253 2279 3 5 8 11 13 16 18 21 24 * 2380 2405 2430 2455 2480 2504 2529 2 5 7 10 12 15 17 20 22 2625 2648 2672 2695 2718 2742 2765 2 5 7 9 12 14 16 19 '21 2856 2878 2900 2923 2945 1 2967 2989 2 4 7 9 11 13 16 18 20 3075 3096 3118 3139 3160 3181 3201 t 2 4 6 8 11 13 15 17 19 3284 3304 ^324 3345 3365 '3385 '3404 2 4 6 8 10 12 14 16 18 3483 ,3502 3522 3541 3560 3579 3598 2 4 6 8 10 12 14 15 17 3674 3692 |3711 3729 3747 3766 ,3784 1 1 1 2 4 6 7 9 11 13 15 17 1 3856 3874 3892 3909 3927 3945 3962 2 4 5 7 9 11 12 14 16 4031 4048 4065 4082 [4099 4116 14133 2 3 5 7 9 10 12 14 15 4200 4216 1 4232 4249 4265 4281 4298 2 3 5 7 8 10 11 13 15 '4362 4378 ! 4393 4409 ! 4425 4440 4456 2 3 5 6 8 9 11 13 44 4518 4533 4548 4564 4579 4594 ,4609 2 3 5 6 8 9 11 12 14 4669 4683 4698 4713 4728 4742 4757 1 3 4 6 7 9 10 12 13 1 4814 4829 4843 1 4857 4871 4886 4900 1 3 4 6 7 9 ,10 11 13 4955 4969 '4983 4997 5011 5024 5038 1 3 4 6 7 8 10 11 12 5092 5105 5119 5132 5145 5159 5172 1 3 4 5 7 8 9 11 12 5224 5237 5250 5263 5276 5289 5302 1 1 1 3 4 5 6 8 9 10 1 12 '5353 5366 5378 5391 5403 5416 5428 1 3 4 5 6 8 9 10 11 5478 5490 5502 5514 5527 5539 5551 1 1 2 4 5 6 7 9 10 11 5599 5611 5623 1 5635 i5647 ,5658 5670 1 1 2 4 5 6 7 8 10 1 11 5717 5729 '5740 5752 5763 5775 5786 1 2 3 5 6 7 8 9 10 ! 5832 5843 15855 5866 5877 5888 5899 1 2 3 5 6 7 8 9 10 5944 5955 5966 5977 5988 5999 ^010 1 2 3 4 5 7 8 9 10 6053 6064 6075 6085 6096 6107 6117 1 2 3 4 5 6 8 9 10 '6160 6170 6180 6191 6201 6212 '6222 1 2 3 4 5 6 7 8 9 ,6263 6274 6284 6294 6304 6314 ,6325 1 2 3 4 5 6 7 8 9 6365 6375 6385 6395 6405 6415 6425 1 2 3 4 5 6 7 8 9 6464 6474 6484 6493 6503 6513 '6522 1 2 3 4 5 6 7 8 9 6561 6571 ,6580 6590 6599 6609 6618 1 2 3 4 5 6 7 8 9 16656 6665 6675 l 1 i ^684 6693 6702 ^712 1 2 3 4 5 6 7 7 8 6749 ‘6758 6767 6776 '6785 6794 6803 1 2 3 4 5 5 6 7 8 6839 6848 6857 6866 6875 6884 6893 1 2 3 4 4 5 6 7 8 6928 6937 6946 ,6955 6964 1 1 6972 6981 1 2 3 4 4 5 6 7 8 7016 7024 7033 7042 7050 7059 *7067 1 2 3 3 4 5 6 7 8 7101 7110 7118 7126 7135 7143 7152 1 2 3 3 4 5 6 7 8 7185 7193 7202 7210 ,7218 7226 7235 1 2 2 3 4 5 6 7 7 7267 7275 7284 7292 7300 7308 7316 1 1 2 2 3 4 5 6 6 7 17348 7356 7364 7372 7380 7388 7396 1 2 2 3 4 5 6 6 7 299 Logarithms — Con. 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 1 2 2 1 3 4 T * O 5 6 7 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 1 2 2 3 4 5 5 6 7 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 1 2 2 3 4 5 5 6 4 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 1 1 2 3 4 4 5 6 L! 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 1 2 3 4 4 ! 5 6 7 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 1 1 2 ' 3 4 4 5 6 6 61? 7853 7860 7868 7875 7882 ( 7889 7896 7903 7910 7917 1 1 2 3 4 4 5 6 6 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 1 1 2 3 3 4 5 6 6 63 7993 8000 8007 8014 8021' 8028 8035 8041 8048 8055 1 1 2 3 3 4 5 5 6 64 8062 8069 8075 8082 8089 8096 1 8102 8109 8116 8122 1 1 2 ! 3 3 4 _ 5 5 ^6 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 1 1 2 3 3 4 5 5 6 66 8195 8202 8209 8215 8222 1 8228 8235 8241 8248 8254 1 1 2 3 3 4 5 5 6 67 8261 8267 8274 8280 8287: 8293 8299 8306 8312 8219 1 1 2 3 3 4 5 5 6 68 8325 8331 8338 8344 1 8351 8357 8363 8370 8376 8382 1 1 2 3 3 4 4 5 6 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 1 1 2 3 3 4 4 5 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 1 2 2 3 4 4 5 6 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 1 1 2 2 3 4 4 5 ,5 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 1 1 2 2 3 4 4 5 5 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 1 1 2 2 3 4 4 5 *5 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 1 2 2 3 4 4 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 1 2 2 3 3 4 5 5 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 1 1 2 2 *s 3 4 5 5 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 1 1 2 2 3 3 4 4 5 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 l 1 8971 1 1 2 2 3 3 4 4 5 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 1 2 2 3 3 4 4 5 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 1 1 2 2 3 3 4 4 5 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 1 1 2 2 3 3 4 4 5 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 1 1 2 j 2 3 3 4 4 5 84 9243 9248 9235 9258 9263 9269 9274 9279 9284 9289 1 1 2 ! 2 3 3 4 4 5 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 1- 1 2 2 3 3 4 4 5 86 9345 9350 9355 9360 9365 9370 9375 938o' 9385 9390 1 | 1 2 2 3 3 4 4 5 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 0 1 1 2 2 3 3 I 4 4 88 9445 9450 9455 9460 9465 9469 9474 9479 | 9484 9489 0 1 1 2 2 3 3 I 4 4 89 9494 9499 9504 9509 9513 9518 9523 9528 ! 9533 9538 0 I 1 1 2 2 ■ 3 ; 3 4 4 90 9542 9547 9552 9557 9566 9566 9571; 9576 9581 9586 0 1 1 2 2 3 | 3 4 4 91 9590 9595 | 9600 9605 9609 9614 9619 9624 1 ’ 1 r 1 9628 9633 1 0 1 1 2 2 | 3 3 4 4 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 0 1 1 1 2 2 3 3 4 4 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 0 1 1 2 2 31 3 4 4 94 9731 9736 9741 1 9745 9750 9754 9759 9763 III! 9768 9773 0 1 1 2 2 I 3 ! 3 4 4 95 9777^ 9782 9786 9791 9795 9800 9805 9809 9814 9818 0 1 1 2 2 3! 1 3 4 4 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 0 1 1 2 2 3 3 4 4 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 0 1 1 2 2 3 3 4 4 98 9912 9917 9921 9926 9930 9934 9939 1 9943 9948 9952 0 1 1 2 2 3 3 4 4 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 0 1 1 1 2 2 3 1 3 3 4 300 / ,