I tHKY AINU JUJNI :> C^arwcU Ittitteraitg SIthtarg rs,fe.p.\^V»*tt,e Cornell University Library arW3853 A manual of practical pliysics, 3 1924 031 363 116 olin,anx I Cornell University 'j Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924031363116 A MANUAL OF PRACTICAL PHYSICS I A MANUAL OF PRACTICAL PHYSICS For Students of Science and Engineering VOLUME I. — Fundamental Measurements and Prop- erties of Matter. — Heat. By ERVIN S. FERRY and ARTHUR T. JONES VOLUME II. —Wave Motion, Sound, and Light. \In preparation VOLUME III. — Electrical Measurements. [/« preparation LONGMANS, GREEN, AND CO. NEW YORK, LONDON, BOMBAY, AND CALCUTTA A MANUAL OF PRACTICAL PHYSICS FOR STUDENTS OF SCIENCE AND ENGINEERING BY ERVIN SIDNEY FERRY PROFESSOR OP PHYSICS, PURDUE UNIVERSITY AND ARTHUR TABER JONES ASSISTANT PROFESSOR OF PHYSICS, PURDUE UNIVERSITY VOL. I FUNDAMENTAL MEASUREMENTS AND PROPERTIES OF MATTER HEAT LONGMANS, GREEN, AND CO. 91 AND 93 FIFTH AVENUE,' NEW YORK LONDON, BOMBAY, AND CALCUTTA 1908 Copyright, 1908, By LONGMANS, GREEN, AND CO. All rights reserved. ' ^ I NorfatfotF T^tti% J. 8. Oushing Co. — Berwlpk & Smith Co. Norwood, Muss., tr.S.A. PREFACE The aim of the present work is to furnish the student of pure or applied science with a self-contained manual of the theory and manipulation of those measurements in physics which bear most directly upon his subsequent work in other departments of study and upon his future professional career. Only those experimental methods have been included that are strictly scientific and that can be depended upon to give good results in the hands of the average student. Although several pieces of apparatus, experimental methods, and deriva- tions of formulae that possess some novelty appear, our fixed purpose has been to use the standard forms except in cases where an extended trial in large classes has demonstrated the superiority of the proposed innovation. It has been assumed that the experiment is rare that should be performed before the student understands the theory in- volved and the derivation of the formula required. Conse- quently the theory of each experiment is given in detail and the required formula developed at length. The more important sources of error are pointed out, and means are indicated by which these errors may be minimized or accounted for. The book is designed to be commenced during the second college year. It presupposes a working knowledge of trigo- nometry and college algebra, but does not require analytic geometry nor calculus. vi PREFACE Most of the experiments here given were printed privately some years ago and have since been in constant use, under our direction, by classes of from one to two hundred students each semester. They have all been carefully revised for the pur- poses of this volume. We are indebted to Mr. G. G. Becknell, Instructor in Physics in Purdue University, for the method we have adopted for solving the equation for the coefficient of expansion of a gas. E. S. F. A. T. J. CONTENTS PART I FUNDAMENTAL MEASUREMENTS AND PROPERTIES OF MATTER CHAPTER I General Notions Regarding Physical Measurement ABT. PAGE 1. Introductory 1 2. Errors 2 3. Methods of expressing Results 8 4. Notation 15 CHAPTER II Methods and Apparatus for the Measurement of Fundamental Quantities 1. Measurement of Distance 16 2. Measurement of Mass 25 3. Measurement of Time 32 CHAPTER III Length, Area, Angle EXP. 1. The Thickness of a Thin Plate by Means of a Spherometer and an Optical Lever 42 2. The Radius of Curvature of a Spherical Surface .... 45 3. The Radius of Curvature and Sensitiveness of a Spirit Level . . 49 4. Verification of a Barometer Scale 53 5. The Correction Factor of a Planimeter 54 6. The Correction for Eccentricity in the Mounting of a Divided Circle 61 vli vm - CONTENTS CHAPTER IV Velocity and Acceleration EXP. PAGE 7. The Change in Speed of a Flywheel during a Revolution . . 63 8. The Speed of a Projectile by the Ballistic Pendulum ... 64 9. The Acceleration due to Gravity by Means of a Simple Pendulum 66 10. The Acceleration due to Gravity by Means of a Compound Pen- dulum 69 CHAPTER V Friction Introduction 75 11. The Coefficient of Friction between Two Plane Surfaces . . 75 12. The Friction of a Belt on a Pulley 77 13. The Coefficient of Friction between a Lubricated Journal and its Bearings — Golden's Method 80 14. The Coefficient of Friction between a Lubricated Journal and its Bearings — Thurston's Method 82 CHAPTER VI Mass, Density, Specific Gravity 15. The Calibration of a Set of Standard Masses 86 Density and Specific Gravity 88 16. The Density of a Solid by Measurement and Weighing ... 89 17. The Density and Specific Gravity of a Liquid with a Pyknometer 90 18. The Density and Specific Gravity of a Solid with a Pyknometer . 92 19. The Density and Specific Gravity of a Solid by Immersion . . 95 20. The Density and Specific Gravity of a Solid or Liquid with Jolly's Spring Balance 97 21. The Density and Specific Gravity of a Liquid with the Mohr- Westphal Balance 99 22. The Calibration of an Hydrometer of Variable Immersion . . 102 23. The Relative Densities of Gases with Bunsen's EfEusiometer . . 107 CHAPTER VII Moment of Inertia Introduction , . HO 24. The Moment of Inertia of a Rigid Body 115 CONTENTS IX CHAPTER VIII Elasticity EXP. PAGE Introduction 118 25. The Elastic Limit, Tenacity, and Brittleness of a Wire . . . 119 26. The Tensile Coefficient of Elasticity, or Young's Modulus- — by Stretching 120 27. A Study of the Flexure of Rectangular Rods 123 28. The Tensile Coefficient of Elasticity, or Young's Modulus — by Bending 128 29. Simple Rigidity — Vibration Method 134 30. Simple Rigidity— Static Method 137 31. The Modulus of Elastic Resilience of a Rod ..... 140 CHAPTER IX Viscosity Inti'oduction 143 32. The Absolute Coefficient of Viscosity of a Liquid — Poiseuille's Method 143 33. The Specific Viscosities of Liquids — Coulomb's Method . . 148 PART II HEAT CHAPTER X Temperature Introduction 155 The Beckmann Thermometer 159 34. The Calibration of a Mercury-in-Glass Thermometer . . . 160 35. The Calibration of a Resistance Thermometer .... 166 36. The Flash Test, Fire Test, and Cold Test of an Oil . . . 170 37. The Relation between the Boiling Point and the Concentration of a Solution 173 CHAPTER XI Expansion Introduction 176 Reduction of Barometric Readings . . .... 176 CONTENTS 38. The Coefficient of Linear Expansion of a Solid .... 179 39. Tlie Absolute Coefficient of Expansion of a Liquid by the Method of Balancing Columns 183 40. The Coefficient of Cubical Expansion of Glass .... 186 41. The Coefficient of Expansion of a Gas by Means of an Air Ther- mometer . ' 188 CHAPTER XII Vapors 42. The Maximum Vapor Pressure of a Liquid at Temperatures below 100° C. — Static Method 194 43. The Maximum Vapor Pressure of a Liquid at Various Tempera- tures — Dynamic Method 196 44. The Density of an Unsaturated Vapor by Victor Meyer's Method . 198 CHAPTER XIII Hygrometey Introduction 202 45. Relative Humidity with Daniell's Dew Point Hygrometer . . 203 46. Relative Humidity with the Wet and Dry Bulb Hygrometer . 205 CHAPTER XIV Calorimetry Introduction 207 The Correction for Radiation — 1. Regnault's Method 209 2. Rowland's Method 212 3. Rumford's Method 214 47. The Emissivities and Absorbing Powers of Different Surfaces . 215 48. The Specific Heat of a Liquid — Method of Cooling . . . 219 49. The Specific Heat of a Solid — Method of Mixtures . . .221 50. The Specific Heat of a Solid — Method of Stationary Temperature 226 51. The Specific Heat of a Solid — Joly's Method . . . .229 52. The Heat Equivalent of Fusion of Ice 231 53. The Heat Equivalent of Vaporization of Water .... 235 54. The Heat Value of a Solid with the Combustion Bomb Calorimeter 237 55. The Heat Value of a Gas with Junker's Calorimeter . . . 243 CONTENTS XI CHAPTER XV Thermodynamics Exp. page Introduction 247 56. The Mechanical Equivalent of Heat by Rowland's Method . . 247 57. The Mechanical Equivalent of Heat with Barnes's Constant Flow Current Calorimeter 250 TABLES TABLE 1. Conversion Factors 254 2. Densities of Solids and Liquids 256 3. Specific Gravities of Water at Different Temperatures . . . 257 4. Specific Gravities of Aqueous Solutions of Alcohol . . . 257 5. Specific Gravities of Aqueous Solutions at 15° C 258 6. Reduction of Arbitrary Hydrometer Scales 258 7. Specific Gravities of Gases and Vapors 259 8. Coefficients of Friction 260 9. Elastic Constants of Solids 260 10. Viscosities of Liquids 261 11. Corrections for the Iniluence of Gravity on the Height of the Barometer ... 262 12. Boiling Point of Water under Different Barometric Pressures . 263 13. Pressure of Saturated Aqueous Vapor 264 14. Pressure of Saturated Mercury Vapor 264 15. The Wet and Dry Bulb Hygrometer 265 16. Coefficients of Linear Expansion of Solids 266 17. Coefficients of Cubical Expansion of Liquids 266 18. Heat Values of Various Fuels 266 19. Specific Heats of Solids and Liquids 267 20. Melting Points and Heat Equivalents of Fusion .... 267 21. Boiling Points and Heat Equivalents of Vaporization . . . 268 22. Thermal Emissivities of Different Surfaces 268 23. The Greek Alphabet 269 PRACTICAL PHYSIOS CHAPTER I GENERAL NOTIONS REGARDING PHYSICAL MEASUREMENT 1. Introductory Experimental work has one of two objects ; either to find out what hind of a result follows under given conditions, or to find out t\i& numerical relations between different quantities. The first class of experiments is called qualitative, the second quantitative. In the earlier days of any science qualitative experiments are numerous ; when the science is more mature, the majority of the experiments are quantitative. The determination of various quantitative relations is the object of physical measurement. In making a physical measurement, the magnitude of each quantity concerned has to be expressed in terms of some unit, and the process of measurement consists essentially in finding how many times this unit is contained in the given quantity. The distance between two points, for example, may be expressed in terms of the number of foot rules which could be laid end to end between those points. Some quantities can thus be measured directly, others can be measured only indirectly. Thus the Young's modulus of a brass wire cannot be experimentally determined by finding how many times the unit of Young's modulus is contained in the Young's modulus of the wire. The Young's modulus of the wire is usually determined by measuring a force and three lengths, and from them calculating the Young's modulus. The great majority of physical measurements are indirect measurements. 1 PRACTICAL PHYSICS 2. Errors Every measurement is subject to errors. In the simple case of measuring the distance between two points by means of a meter stick, a number of measurements usually give different results, especially if the distance is several meters and the measurements are made to small fractions of a millimeter. The errors introduced are due in part to — (1) Inaccuracy of setting at the starting point ; (2) Inaccuracy of setting at intermediate points when the distance exceeds one meter ; (3) Inaccuracy in estimating the fraction of a division at the end point; (4) Parallax in reading, i.e. the line from the eye to the divi- sion read not being perpendicular to the scale; (5) The meter stick not being straight ; (6) The temperature not being that for which the meter stick was graduated ; (7) Irregular spacing of divisions ; (8) Errors in the standard from which the division of the meter stick was copied. Besides the above there are doubtless other sources of error. It may be well here to note that blunders, such as mistakes due to mental confusion in putting down a wrong reading, or mistakes in making an addition, are not usually classed as errors. Of the above errors, (1), (2), and (3) can be very much decreased by having fine divisions on the scale and reading with microscopes ; (4) can be made small by bringing the scale on the meter stick close to the object to be measured ; (5) can be made very small by using a meter stick of special design, or, in rough work, by holding the meter stick against a straight edge ; (6) can be nearly eliminated by using the meter stick only at the proper temperature, or, if its temperature and coefficient of expansion are known, by calculating a correction to be applied ; (7) can be diminished only by a careful comparison of the lengths of the NOTIONS REGARDING PHYSICAL MEASUREMENT 3 different divisions ; and for (8) corrections can be applied only when something is known about the accuracy of the standard from which the meter stick was copied. But even with the most refined methods and the most careful application of cor- rections, different measurements of the same distance usually give different results. Errors due to (6), (7), and (8) may be determinate errors, i.e. errors for which more or less accurate corrections can be calcu- lated, whereas those due to (1), (2), and (3) are indeterminate errors, i.e. errors for which corrections cannot be calculated, Moreover, of those errors for which corrections are not applied, some, like those due to (1), (2), and (3), will be variable in amount and will tend to make the value obtained sometimes too large and sometimes too small; while others, like those due to (7) and (8) when corrections for them are not applied, will be constant and will tend to make the value obtained always too large or always too small. Since the average value of those variable errors which tend to make a result too large will after a considerable number of measurements be about the same as the average value of those variable errors which tend to make the result too small, the mean of a large number of measurements is usually nearly free from variable errors. In order as nearly as possible to do away with constant errors, the same quantity should be measured by as many different methods as possible. The results by the differ- ent methods will usually differ somewhat, but from them all a value can be calculated which is probably nearer the true value than any one of the separate results. The magnitude of an error may be defined as the amount by which the value obtained exceeds the true value. That is, if the true value — which is not usually known — is denoted by T, the value obtained by 0, and the error by E, I!=0-T. (1) The magnitude of the correction which ought to be applied may be defined as the amount which would have to be added to 4 PRACTICAL PHYSICS the value obtained in order to get the true value. That is, if Q denotes the required correction, Q^T-0. (2) From (1) and (2) it will be seen that the error in a measure- ment and the correction which ought to be applied to it are equal in magnitude and opposite in sign. This does not mean that the error is exactly equal in magnitude to a correction which actually is applied, because for the correction itself only an approximate value is usually known. Tkustwoethy Figures. — Since all measurements are sub- ject to errors, it is important to be able to determine how many figures of a result can be trusted. In direct measurements it is usually possible to make a fairly accurate estimate of the extent to which a reading can be trusted. Thus in reading by the unaided eye the position of a fine line which crosses a meter stick, the reading will not be in error by so much as a millimeter but pretty surely will be in error by more than a thousandth of a millimeter. So the extent to which the reading can be trusted will lie between these limits. A person who is accustomed to estimating fractions of a small division will be rather sure of not making an error so great as the tenth of a millimeter, and he can often trust his reading to a twentieth of a millimeter. It is convenient always to put down all the figures that can be trusted, even if some of them are ciphers. Thus the state- ment that a distance is 50 cm. implies that there is reason for supposing that .the distance really lies between 45 cm. and 55 cm., whereas the statement that the distance is 50.00 cm. implies that there is reason for supposing that the distance really lies' between 49.95 cm. and 50.05 cm. When the dis- tance is said to be 50 cm. the second figure is the last in which any confidence can be placed ; when the distance is said to be 50.00 cm., the fourth figure is the last in which any confidence can be placed. If a distance is about 50,000 Km. and the third figure is the last in which any confidence can be placed, NOTIONS REGARDING PHYSICAL MEASUREMENT 5 this fact may be indicated by saying that the distance is 50.0 • 103 Km. In indirect measurements the result is usually calculated by some formula. To find out how many figures should be kept in the result consider the following two cases: — 1. If the result is the algebraic sum of several quantities, such as 314.428, 32.6, and 7.063, it is seen that in the sum, 354.091, no figure beyond that in the first decimal place can be trusted, because in the quantity which has the fewest trustworthy deci- mal places, viz. 32.6, no figure beyond the 6 can be trusted — otherwise it would have been expressed. So the sum will not be written 354.091, but 354.1. This suggests the following rule : — Rule I. — In sums and differences no more decimal places should be retained than can be trusted in the quantity having fewest trustworthy decimal places. 2. If the result is the product of two quantities, such as 314.428 and 32.6, then the product cannot be trusted to more figures than appear in the quantity having fewest trustworthy figures, irrespective of the decimal place. To make this clear consider the following products : — 314.428 X 32.4 = 10187.4672 314.428 X 32.6 = 10250.3528 314.428 X 32.8 = 10313.2384 314 X 32.6 = 10236.4 It is seen that if the quantity which is supposed to be 32. 6 is really 32.4 or 32.8, then after the first three figures the true value of the product differs materially from the value obtained. The second and fourth of the above products show that if more than three figures cannot be trusted in one of two quantities which are to be multiplied, it is not worth while to use more than three — or at most four — figures of the other. These facts suggest the following rule : — Rule II. — In products and quotients no more figures should be kept than can be trusted in the quantity having fewest trust- worthy figures. 6 PRACTICAL PHYSICS Until the final result is reached, it is often worth while to keep one more figure than the above rules indicate. For logarithms a safe rule is the following : — Rule III. — When any of the quantities which are to be multiplied or divided can be trusted no closer than 0.01 % use a five-place table, when any of them can be trusted no closer than 0.1% use a four-place table, and when any of them can be trusted no closer than 1 % use a slide rule. Required Accueacy op Measurement. — From Rule I. it will be seen that if a small quantity is to be added to a large one, the percentage accuracy of the measurement of the small quantity need not be so great as that of the large one. Thus if JI= a-\-h, and if a is about 100 cm. and h about 1 cm., a 1 % error in a will produce in ^no greater effect than a 100 % error in 6. When quantities are to be added or subtracted, they should be measured to the same number of decimal places. From Rule II. it will be seen that if a small quantity and a large one are to be multiplied the percentage accuracy of the meas- urement of the small quantity should be at least as great as that of the large one. Thus if iZ'= a5, a 1 % error in a will produce in JI the same effect as a 1 % error in h. So that if a is about 100 cm. and h about 1 cm., and if h cannot be trusted closer than 0.01 cm., there is no gain in accuracy by measuring a much closer than to within 1 cm. When quantities are to be multiplied or divided, they should be measured to within the same fraction of themselves, e.g. all of them within 1 % and none of them much closer, or all of them within 0.01 % and none of them much closer. The last statement needs modification in the case of a power. If the value found for a quantity a is 1 % too large, i.e. is 1.01 a, then the value that will be obtained for a^ ig 1.0201 a, which is about 2 % too large, and the value obtained for o^ is 1.030301 a, which is about 3 % too large. In general, if the value found for a\s,h% too large, the value that will be obtained for a" will be nlc% too large. So that a quantity which is to be squared, cubed, or raised to some higher power should be measured with more care than if it entered the formula to the first power. NOTIONS REGARDING PHYSICAL MEASUREMENT 7 ERRORS INTRODUCED BY COMMON APPROXIMATIONS Num- ber Tritb Value Appkox. Value When Ap- plicable ! How Obtained APPEOX. Ekeoe Introduced BY THE Approximation 1 1 + a + a2 1 + a a small Neglect a^ -a2 a error e.^f. 0.1 -1% [0.01-0.01% 2 (l+a)(l + &) 1 + a+b a and 6 small Neglect ab -ab 3 (1 + «)» 1 + ma a small Expand by binomial theorem. Neglect second and higher powers of a TO(ra-l) ^2 2 4 (1 + ay l + 2a a small Apply (3) -a^ 5 1 1 + « 1-a a small ('^ 1 .n-'i-i -a2 Apply (3) 6 1 + ia o small + ia' Vl-f a=(l + a)2 Apply (3) VI + 7 8 Va6 Ka+&) 6 nearly equal to a Let b = a + e. Then Ab-ay 8a Vab=Va-^ + ae = a-\^l+- Apply (6) sin a a* a small Neglect third and higher powers + |a^ 9 cos a 1 a small Neglect second and higher powers -f U2 10 tana a* a small tana sin a . 12. -ia^ COS a 1 ^ 1 . Apply (5) 11 tano sin a a small Like (8) and (10) -ia^ * Expressed, of course, fti radians. 8 PRACTICAL PHYSICS Approximate Formula. — Beside the errors of observa- tion, errors may be introduced into indirectly measured quanti- ties by the use of formulae which are only approximate. Thus, the sine and tangent of small angles are used as equal to the angles, the reciprocal of (1 + a) is written equal to (1 — a) when a is small, 3.14 is used for tt, a number of figures are dropped from the end of a product, etc. Whenever such an approximation suggests itself, the error introduced by using it should be investigated and the approximation not made unless the error thereby introduced is so small as not to affect any figure that could otherwise be trusted in the result. The preceding table of a few common approximations may prove useful. 3. Methods of expressing Results The object of a quantitative experiment is sometimes the measurement of some quantity, and sometimes the determina- tion of the relation between various quantities. "When the relation between several quantities is sought, the usual method is, keeping all but two of the quantities constant, to vary by known amounts one of these two and then determine the changes produced in the other. Another pair of the quantities is then varied while the rest are kept constant, and so on until a sufficient number of pairs of quantities have been investi- gated. The various relations found to exist between the various pairs of quantities can then be combined to give the relation sought. When one quantity has been given various known values and the corresponding values of a second quantity have been determined, the relation between them can always be expressed graphically; it can also be expressed more or less accurately by means of an empirical formula; and when this formula is sufficiently simple, the relation can without difficulty be ex- pressed in words. To illustrate these methods, suppose that it is desired to determine the relation between the distance a body has fallen NOTIONS REGARDING PHYSICAL MEASUREMENT 9 from rest and the time it has been falling. Suppose that a number of determinations are made, in each of which a ball is allowed to fall a known distance, and the time required is ob- served, the values obtained being those in the following table : — . DtsTANOE Fallen Time Required 2.00 cm. 0.084 sec. 5.00 0.101 10.00 0.143 20.00 0.202 30.00 0.247 40.00 0.286 60.00 0.350 80.00 0.404 Plotting of Results on unifoemly divided Cookdi- NATE Paper. — These values may be plotted in the same way that curves are drawn in Analytic Geometry. The scales should be so chosen as to make the curve extend nearly across the sheeit in both directions, unless by so doing a unit in the last place that can be trusted is represented by a distance greater than one of the smallest divisions on the paper. If, for instance, times can be trusted only to 0.01 sec, then the scale for abscissas chosen in Fig. 1 is two or three times what it ought to be. If, however, the times can be trusted to 0.004 sec. or closer, then the scale is satisfactory. A curve should not always be drawn through all the points, but should be a smooth curve which fits the points as nearly as possible. If either scale has been so chosen that a unit in the last place that can be trusted is represented by one of the smallest divisions on the paper, a deviation of points from this curve usually indicates errors of observation. The curve in Fig. 1 shows at once that the distance fallen increases as the time increases; but since the curve is not a straight line, the distance fallen is not proportional to the time. Since the curve is convex toward the time axis, it follows that the distance increases at a continually increasing rate, i.e. that 10 PRACTICAL PHYSICS as the body falls it goes continually faster and faster. The curve also serves to find the distance fallen in any time not much exceeding 0.4 sec, or to find the time required to fall any distance not much greater than 80 cm. The next step is to find the equation which represents this curve. Let the time which has elapsed be represented by t, and the distance fallen by I. If I decreased when t increased, the equation might be of a form l = a+ , t t^ if etc. or or NOTIONS REGARDING PHYSICAL MEASUREMENT 11 In the case in hand, however, since I increases when t increases, the relation cannot be one of the above forms ; it may, perhaps, be of a form l = a-\-U, (3) or l = a + U+ ct\ (4) or l = a + U + ct'^ + d1?, (5) etc. If the relation were of the form in (3), two points would suffice to determine a and h. For if the coordinates of the two points were (t^, Zj) and (t^, l^, we should have and l^ = a + bt^, and from these two equations we could determine the two quantities a and b. Similarly, if the relation were of the form in (4), three points would suffice to determine a, b, and c. Thus, if only two points are determined, there can always be found a relation of the form in (3) that will be satisfied by both those points ; if three points are determined, a relation can always be found containing three constants which will be satisfied by all three points ; if n points are known, a relation can always be found containing n constants which will be satis- fied by all n points. But an equation containing many con- stants is cumbersome, and it is usually possible to find an equation with only three or four constants which is very nearly satisfied by a considerable number of points. A convenient method- of finding how many constants should be used in an equation like (3), (4), or (5) will be illustrated by considering the curve plotted in Fig. 1. The maximum abscissa is divided into some half dozen or more convenient equal parts, the ordinate at each division point is read, and the corresponding values of abscissas and ordinates are recorded in the first two columns of a table : — 12 PRACTICAL PHYSICS t I A,/ A,; 0.00 sec. 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.0 cm. 1.3 5.0 11.0 19.4 30.7 44.1 59.9 78.3 1.3 cm. 3.7 6.0 8.4 11.3 13.4 15.8 18.4 2.4 cm. 2.3 2.4 2.9 2.1 2.4 2.6 In the third column are the differences of the first order, i.e. the differences between the successive values of I ; in the fourth column are the differences of the second order, i.e. the differ- ence between the successive differences of the first order; in a fifth column would be the differences of the third order, etc. In the present case it is seen that in going down the columns the differences of the first order continually increase, whereas the differences of the second order, although varying somewhat, on the whole neither increase nor decrease to any great extent. By a simple application of the Differential Calculus it can be shown that if the differences of the wth order neither increase nor decrease, then (w+1) constants are enough to retain in a formula of the general form y= a+'bx + cx^+ do^ + • • • . For the case in hand, then, three constants should be retained, and the formula may be written l = a + ht+ ct^- The three constants are to be determined by choosing three points on the curve as far apart as convenient, and so obtaining three equations from which to solve for a, b, and c. If the points selected are (0, 0), (0.20, 19.5), and (0.40,78.2), the values found for a, b, and c are respectively 0, —0.5, and 490. Con- sequently the empirical equation is 1 = 0-0. 5t+i90tl NOTIONS REGARDING PHYSICAL MEASUREMENT 13 From theoretical considerations the formula for a body fall- ing from rest should be To compare the two equations compute by each of them the distance fallen in a given time. Thus if g is 980 cm. per sec. in a sec, the distance fallen in 0.3 sec, computed from the 100 so so 70 GO 50 40 3 i :::::::: t i 7 ■ :::;:. Z_ I / / / X / / ... —. = 3 :::::: — ;eeZ:::::;;— — ; /:::::::: — / A. / ' 1 / s + / £ " 7' % /___ t / = ^^ — EE?:E:::: y — -Tjin-e Clup wd- /- -:---d:----:::^zz- "KX::::-.-. =^ — „J — -> 1 Jt __ S T 8 10 Fig. 2. 40 so eO TO 80 90 10(1 theoretical equation, is 44.1 cm., while according to the empiri- cal equation the distance is 43.95 cm. Plotting of Results on logarithmically divided Co5rdinate Paper. — -In the graphical representation in Fig. 1, 0.2 sec is placed twice as far from the origin as 0.1 sec, 14 PRACTICAL PHYSICS 0.3 sec. three times as far as 0.1 sec., etc., and similarly for the distance fallen. Another method of plotting results is often adopted, viz. to plot along each axis a distance which, instead of being proportional to the value itself, is proportional to its logarithm. In order to save looking up logarithms coordinate paper having the rulings spaced logarithmically can be used. Fig. 2 represents a sheet of logarithmic coordinate paper with the values for times and distances fallen plotted upon it. The curve connecting these points is seen to be a straight line. If a straight line had been obtained when plotting on uni- formly divided coordinate paper, it would be known at once that the equation of the curve was l=a + bt, where a would denote the intercept on the Z-axis, and b the tangent of the angle between the curve and the i-axis. Since, instead of t and I, the quanti- ties which have been plotted in Fig. 2 are log t and log I, the equation of the straight line which is obtained is log 1= a+ h log t. But a is, of course, the logarithm of some number, and so the equation may be written log I = log A + b log t. Whence I = At^. Since b is the tangent of the angle made by the curve with the f-axis, its value can be found by dividing c, in Fig. 2, by d. On measuring these and dividing, the value found for b is 2.002. Since A is the number whose logarithm is the inter- cept on the Z-axis, the value of A may be read off directly. It will be noticed that the values of the times have been multiplied by one hundred before plotting. This does not alter either the shape of the curve or its slope, but merely throws it far enough to the right to get it on the paper. If it were moved to the left to its proper place, it is seen that it would cut the Z-axis some place between 100 and 1000, and since the triangle efg is equal to the triangle that would be formed by the Z-axis, the curve, and the 100 cm. line, it follows NOTIONS REGARDING PHYSICAL' MEASUREMENT 15 that the value of A is the same, aside from decimal point, as the intercept fg on the 10-sec. line. The point where the curve crosses the 10-sec. line is at 4.9 cm. Moving the decimal point so as to make the value lie between 100 and 1000, the value obtained by this method is about 490. The empirical equation obtained by this method is, then, Z = 490 «2 that can slide along the length of the scale. In using this instrument the jaw B is nearly closed upon the 22 PRACTICAL PHYSICS object to be measured, the screw E is tightened, and the final adjustment carefully made with the screw F. The zero read- ing should always be noted, and care should be taken that F is turned only until a slight pressure is felt. The Cathetometer is an instrument for measuring vertical distances in cases where a scale cannot be placed very close to the points whose distance apart is desired. It consists essen- tially of an accurately graduated scale, together with a hori- zontal telescope capable of being moved up and down a rigid vertical column. In one pattern of the instrument (Fig. 10) the scale is en- graved on the supporting column, while in another pattern the scale is independently supported parallel to the object being measured and close beside it. In the case of an instrument of the first type, the carriage can be clamped at any point along the length of the vertical column and its position read bj^ means of the scale on the column and a vernier ( V, Fig. 10), attached to the carriage. In the case of an instrument of the second type, the position of the carriage is obtained by observing through the telescope the point on the distant scale that appears to coincide with the cross hair of the telescope. Before taking a reading with a cathetometer three adjust- ments are necessary. The first adjustment is to make the axis AB vertical. To effect this, the telescope is set approximately parallel to the line connecting two of the three leveling screws in the base, and one or both of these two screws is turned until the bubble in L is near the middle of the vial. The telescope MEASUREMENT OF FUNDAMENTAL QUANTITIES 23 is then rotated about AB until it points in the opposite direc- tion. If the bubble is not still in the middle, it is brought back to the middle by turning one or both of the two screws, the number of turns being counted. Half of that number of turns is then made in the opposite direc- tion and the bubble brought back to the middle of the vial by means of the screw D. The telescope is then turned so as to be 90° from its original position and the third screw in the base adjusted until the bubble is in the middle. If the bubble does not now re- main in the middle of the vial, however the telescope may be turned about AB, the entire ad- justment is repeated. The second adjustment is to make the axis of the telescope horizontal. In doing this the telescope is taken from its wyes, turned end for end, and replaced. If the bubble does not come to rest at the middle of the vial, it is brought to the middle by the screw J), the numbers of turns required being counted. Half this number of turns is made in the opposite direction and the bubble then brought to the mid- dle by means of the screws at the ends of the vial. The tele- scope is again reversed in the wj^es, and if the bubble does not still come to rest in the middle, the above operations are repeated. The third adjustment is to focalize the telescope. The front tube containing the eyepiece is moved in and out until the cross Fig. 10. 24 PRACTICAL PHYSICS hairs appear as distinct as possible. Then, while sighting along the outside of the telescope, the latter is brought to about the right height and turned so as to point approximately at the object to be viewed. The eye is then placed at the eyepiece and the focalizing screw F turned until the image of the object does not move with reference to the cross hairs when the observer's head is moved slightly from side to side. If the scale is engraved on the column which carries the tele- scope, the latter is focalized first on one of the points and then on the other, the final setting being made in each case by the screw F. After each setting the height of a mark on the carriage is read by the vernier V. The difference between the two readings gives the desired distance. If the scale is independently supported, it is placed vertical, close to the object being measured, and so that the scale and object are at about the same distance from the telescope. The telescope may be focalized on one of the points and then rotated about a vertical, axis until the scale is in the field of view, the height of the cross hair being then read directly ; or a small mirror capable of rotation about a vertical axis may be attached to the telescope just beyond the objective, so that by rotating this mirror, an image of either object or scale can be seen without rotating the telescope. The height of the second point is then observed in the same manner. When the scale is independently supported, the error introduced by lack of verticality of the scale may be easily found as follows: 'Let AB (Fig. 11) be a vertical line drawn through the point B of the scale CB. Then in place of the real height AB, we read GB, and the error is CB — AB = CB - CB cos d=CB (1- cos 61). For a given inclination of the scale, this error will evidently be greatest when CB is greatest. If, then, the scale is 100 cm. long and readings are to be trusted within 0.01 cm., the scale should be so placed that its departure Fio. 11. MEASUREMENT OF FUNDAMENTAL QUANTITIES 25 from verticality is not greater than that given by 0.01 = 100 (1 — cos ^). From this equation 6 is found to be somewhat more than 0.01 radian, which means that for the given degree of accuracy the scale is nearly enough vertical when a plumb line dropped from its top would fall within 1 cm. of its bottom. 2. Measurement of Mass One of the most common as well as the most accurate methods for the comparison of masses is afforded by the beam balance (Fig. 12). The beam BB' can rotate about a knife edge K^ which rests upon an agate plate. Suspended from knife edges at K^ and K^ are the scale pans^j and p^. A handle H operates an arrestment consisting of a horizontal rod and three cams Cj, C^, and 0^, by means of which the knife edges 26 PRACTICAL PHYSICS may be relieved of the weight of the beam and pans when the balance is not in use and when the masses in the pans are being changed. Fastened to the beam is a long pointer / which swings in front of a graduated scale S. Whether the divisions on this scale are numbered or not, it is convenient to assume that the middle division is numbered 10, and that the divisions are numbered from left to right. Projecting from the side of the case is a rod R by means of which a bent aluminium wire called a rider can be placed at any point along the beam. This rider is used in place of standard masses smaller than 10 mg. The top of the beam is often divided into twenty equal parts, the 0-line being over the central knife edge, and the 10-lines over the other knife edges. If the mass of the rider is 10 mg., and it is placed on one of the 10-lines, it produces the same effect as if a 10 mg. mass were in the corresponding pan ; but if it is placed at division 3, it has a turning moment only three tenths as great, and so produces the same effect as would a 3 mg. mass placed in the pan. Occasionally a rider of some other mass is used and the beam divided accord- ingly. The Method of Vibrations is usually employed in making ac- curate weighings. When using this method, the case is at first left closed and the arrestment released. If the pointer does not begin to swing, the case is opened, the hand waved lightly over one pan, and the case again closed. With the pointer swinging in front of the scale, but not beyond it, the zero point of the balance is 'determined ; i.e. the point at which the pointer would finally come to rest, either with no load on the pans, or with equal loads on the two pans. This is done by observing an odd number of successive turn- ing points of the pointer. As the pointer swings, the distance between any two successive turning points on the same side of the scale gradually decreases, but in a few swings the decrease is slight. The zero point is about halfway from b (Fig. 18) to a point midway between a and c. It is also about halfway from c to a point midway between b and d, about halfway from MEASUREMENT OF FUNDAMENTAL QUANTITIES 27 d to a, point mid^vay between e and e, etc. Since the distance from a to c is about the same as that from c to e, the average of a, e, and e is nearly the same as c. The zero point, then, is very near the point found by taking the average of a, e, and e, and averaging with it the average of b and d. Suppose, for instance, that five successive turning points are observed to be : — ^•^ 11 9 li 11-8 8.7 FiQ, 13, Then the average of the turning points at the left is 8.53 and of the turning points at the right is 11.85. Consequently the zero point is in the neighborhood of [J(8.53+ 11.85) = ]10.2. Five successive turning points are usually enough to observe, but any odd number of successive turning points may be used in the same way, viz. by averaging the left turning points and averaging the right turning points and then finding the average of the two results. It should be noted that this method of finding the zero point is most accurate when the pointer swings with a small amplitude. Since the zero point varies from day to day, and even from hour to hour, it should be determined for each experiment. For very accurate work it sliould be determined both at the beginning and at the end of a weigh- ing, and the average value used. After the zero point has been determined and while the arrestment is elevated so as to lift the beam off the knife edge, the object is placed on one pan and standard masses on the other. Right-handed persons find it most convenient to place the object on the left pan so that the mass pan is in front of the hand that makes the adjustment of the standards. Each time that a new mass is placed on the pan the arrestment is lowered just enough to see in which direction the pointer would swing, but no masses are ever put on or taken off while the pointer is free to swing. When the masses are so nearly adjusted that 28 PRACTICAL PHYSICS when the arrestment is entirely released the pointer swings back and forth near the zero point, the position at which the pointer would finally come tp rest is determined from several successive turning points in the same way that the zero point had been. The rider is then moved so as to alter the effective mass on tlie mass pan by one or two milligrams, and the new position of rest determined. From these observations the mass which would be required to make the point of rest coincide with the zero point can be calculated without taking the time to effect the balance experimentally. Suppose, for example, that the zero point of the balance is 10.2 scale divisions, and that with the object on the left pan and a mass of 24.166 g. on the right pan, the point of rest is found to be 11.6 scale divisions. Since this point of rest is to the right of the zero point, the mass on the right pan is too small. Suppose that by means of the rider the effective mass on the right pan is increased by 2 mg., and that the new point of rest, determined as before, is found to be 7.4 scale divisions. Then the addition of 2 mg. has moved the point of rest through [11.6 — 7.4 = J 4.2 scale divisions, andl mg. would have moved it 2.1 scale divisions. It follows that the mass which would have to be added in order to move the point of rest through the [11.6 — 10.2= ] 1.4 scale divisions to the zero point of the balance is [1.4-=- 2.1 =j 0.7 mg. Consequently the apparent* mass of tlie object is [24.166 + 0.0007 =] 24.1667 g. The sensibility of a balance is defined as the number of scale divisions through which the point of rest is moved by the ad- dition of one milligram to the load on one of the pans. In the above example the sensibility was 2.1 scale divisions per milli- gram. The sensibility, however, depends upon the load and should therefore be determined for each weighing. The fact that it depends upon the load may be shown as follows : — Let Kj^, K^, JTg (Fig. 14), denote the three knife edges of the balance, and M the center of mass of the beam. Let * See below, Errors in Weighing. MEASUREMENT OF FUNDAMENTAL QUANTITIES 29 IM^+P^'lg Pi and P2 be the respective masses of the left and right pans, and M^ the mass of the beam. Suppose that with a mass M^ on the left pan and a mass M^ on the right pan the beam comes to rest in the position indi- cated. Then, since the balance is in equilibrium, the sum of the moments of CM, + p,)g, (M^ +P2)ff, and i% (Af,+p^te taken about IC^ must equal zero, i.e. CM^+p^-)gxl^ sin {6^ + ^-) — M^g X r sin ;8 = 0, (iHfj + pj)l,Csin 0, cos yS — cos 0^^ sin /3) — (ilfj +i»2)^2(®i'^ ^2 ^^^ ^ + ^os ^2 ^i^ /^) - iffgr sin /3 = 0. (10) Since in the actual case yS is always small, we may replace sin /3 by /3, and cos /3 by 1. Then if l^ = l,= I, if ^^ = ^1 = ^, and if ^2 =Pi=Pi we have (ifeTj + ^)Z(sin (9 - /3 cos 6") - (ifg + p)Z(sin (9 + /3 cos 61) Whence, /8 Fig. 14. (lfl+Pl)5'XZiSin(^l-^) or ? sin iUfi - ilfj ^ (^1 +-^2 + 2p)^ cos e + M/ (11) If ilfi - ifg = 0, then /3 = ; and if iUfi - ilfj = 1 ™g- ' then the left member of (11) denotes the movement of the pointer for 1 mg. change in the load. That is, each member of this equa- tion is proportional to the sensibility of the balance. If 6 is 90°, cos 9 is zero, and whatever the value of the load, iffj -I- ii^ , the right member of (11) is unaltered ; that is, when 30 ■ PRACTICAL PHYSICS 6 is 90°, the sensibility is independent of the load. If 6 is less than 90°, cos 6 is positive, and as the load, M-^ + M^, increases, tlie right member of (11) decreases ; that is, when 6 is less than 90°, the sensibility decreases as the load increases. If 6 is larger than 90°, cos 9 is negative. It follows that as M-^^ + M^ increases, the denominator of the right member of (11) de- creases, and the sensibility therefore increases ; that is, when 6 is larger than 90°, the sensibility increases as the load increases. Since different loads necessarily bend the beam different amounts, it follows that the sensibility is different for different loads. The maker usually arranges to have the three knife edges in line when the balance has about half its maximum load. Errors in Weighing. — The errors to which a weighing is espe- cially liable are due to (1) the buoyant effect of the air, (2) errors in the standard masses, (3) difference in the lengths of the bal- ance arms, and (4) difference in the masses of the scale pans. (1) The buoyant effect of the air will be different upon the bodies on the two scale pans unless their volumes are equal. The true mass may be found as follows : liCt M, B, and V de- note respectively the mass, density, and volume of the body the mass of which is desired, and m, d, and v, the mass, density, and volume of the standard masses which just balance it in air of density p. Then the difference between the weight of the body in vacuum and its weight in air is equal to the weight of the air displaced pVg, and the weight of the body in air is consequently Mg—pVg. In the same manner, the standard masses when in air weigh mg — pvg. Since the weight of the body in air equals the weight of the standard masses in air, Mg-pVg = mg~pvg, or Mg-p—g=mg-p—g. 1-1 Whence M=m ■ (12) 1-1 M= MEASUREMENT OF FUNDAMENTAL QUANTITIES 31 For ordinary temperatures and pressures p is about 0.0012 g. per cc, so that if any solid or liquid is being weighed, p is very small compared with i), and we may apply approximation (5), p. 7, obtaining from (12) or, employing approximation (2), U. Jf^„[l+,(l-l)J, (IS) or, since for the brass standards ordinarily used in weighing, d is about 8.4 g. per cc, 1+0.0012 ('1-0.12)1. (14) It will be noticed tliat a considerable error in 2) can produce in the value for M only a small error, so that a fairl}"- rough value for D can be used in (14). The error introduced by the approximations employed in obtaining (13) will almost always be negligible, but for accurate wofk the values of p and d should be determined and not assumed. (2) Errors in the standard masses may be corrected as ex- plained under Experiment 15. (3) and (4) Errors due to difference in the lengths of the balance arms and to difference in the masses of the scale pans can be nearly eliminated by weighing the body first in one pan and then in the other. Let Zj and l^ denote the respective lengths of the left and right arms of the balance, and p-^ and p^ the respective masses of the left and right pans. If an object of mass M'\8 balanced by standard masses m^ when the object is in the right pan, and by standard masses m^ when the object is in the left pan, then in Fig. 14, /S = 0, and if ^2=^i' and iPi+M)\=--{p^+m^^l^. (16) 32 PRACTICAL PHYSICS If the pointer swings near the middle of the scale with no load on the pans, we have also i'iZi=/'2^2' ^° ^^^^ (1^) ^^^ 0-^^ ^^°°°^^ m,l, = Ml, and Ml^ = 'm^l^. Whence M^-^:^^. (17) In case of a balance in ordinary adjustment m^ will so nearly equal «ij that we may use approximation (7), p. 7, and in place of (17) write M=lim, + m,). (18) Precautions in the use of a balance, 1. Do not place on the pans anything wet, any mercury, nor anything that might injure the pans. 2. Never change the masses on the pans nor move the rider when the beam is free to swing. 3. Never touch any standard masses with the fingers — use forceps. 4. Keep all standard masses in the proper compartments in the box when not actually in use upon the balance pan. 6. Never raise nor lower the arrestment so quickly as to cause any jerk. 6. When not actually altering masses keep the case closed. 7. Before leaving the balance bring the arrestment into play so that the beam is not free to swing, set the rider at the zero mark, dust off the pans and the floor of the case with a camel's- hair brush, and close the case^ 3. Measurement of Time Instruments. — In nearly all apparatus for measuring time use is made of the principle that the period of vibration of a body oscillating with harmonic motion is constant. The most commonly used vibrating bodies are the pendulum, the balance wheel, and the tuning fork. Any one of them may be kept going indefinitely by a slight impulse given it each tim6 it passes through its position of equilibrium. MEASUREMENT OF FUNDAMENTAL QUANTITIES 33 In order to give this impulse to a pendulum or a balance wheel, and also to count its vibrations, there is usually attached to it a mechanism called clock work. A pendulum of such a length as to make in each second one beat, i.e. half a complete vibration, is called a seconds pendulum. Such a pendulum is used in standard clocks. Where accuracy must be sacrificed to portability, the balance wheel is employed, as in the watch and the chronometer. A stop watch is a watch provided with a starting and stopping device so that the interval between two events can be easily determined. The impulse to keep the tuning fork going is usually given by an electro-magnet which is periodically actuated by a cur- rent made and broken by the motion of the tuning fork itself. Attached to one prong of the tuning fork is a sharp point which rests lightly against a sheet of smoked paper. The paper is wrapped round a metal drum which rotates and at the same time moves slowly in the direction of its axis, so that the trace made by the vibrating tuning fork is a wavy helix. The instants at which two events occur may be marked by minute holes made in the blackened surface of the paper by electric sparks which are caused to pass from the tracing point to the metal drum. If the period of the tun- ing fork is known, the number of waves and fraction of a wave in the part of the line between the two small holes shows the interval of time between the two events. A tuning fork and drum arranged in this manner constitute a tuning-fork chronograph. The Astronomical Chronograph (Fig. 15) differs from the tuning-fork chronograph principally in that the drum runs Fig. 15. 34 PRACTICAL PHYSICS more slowly, the paper is usually not smoked, and in place of a tuning fork there are one or more pens, A and 5, which, by means of electro-magnets, can be slightly displaced parallel to the axis of the drum. One electro-magnet is included in a circuit which is so connected to a clock pendulum that every second a notch is made in the line its pen is drawing. In the circuit containing the other electro-magnet a telegraph key can be so placed that an observer can produce a series of notches corresponding to a series of observed events, or the circuit may contain some device whereby the successive events may auto- matically close the circuit for an instant. A clock is seldom read closer than to seconds ; a stop watch is usually graduated in fifths of a second ; an ordinary watch, due to eccentricity in the mounting of the second hand, can usually not be trusted within three or four tenths of a second ; an astronomical chronograph can often be trusted to a hun- dredth of a second ; a tuning-fork chronograph may without difficulty be made trustworthy within a thousandth of a second. Methods op Measuring Time. — The measurement of a short interval of time between two separate events is usually made with a stop watch or chronograph. But for determining the period of a regularly recurring event, like the swing of a pendulum, there are several methods of procedure, the choice between which depends upon the magnitude of the period and the accuracy required. The movement of a vibrating point from one end of its path to the other is called an oscillation. The complete to-and-f ro movement from the instant when the vibrating point leaves any given position to the instant when it next passes through the same position in the same direction is called a vibration. The interval of time between two successive passages of the vibrat- ing point in the same direction through a given position is called the period of the vibration. The period of an oscillation is half the period of a vibration. The most useful methods of determining the period of a regularly recurring event will now be considered. 1. The Direct Method consists in noting by means of a clock or stop watch the interval of time between two recurrences of MEASUREMENT OP FUNDAMENTAL QUANTITIES 35 the event and dividing this interval by the number of recur- rences. The accuracy of this method depends upon the accu- racy of determining the times of beginning and ending the count, and upon the time that is allowed to elapse. 2. The Method of Omitted Transits. — The preceding method may be slightly modified so as to increase somewhat the accu- racy without materially increasing the time or labor required. Suppose, for example, that a heavy horizontal disk is suspended by a vertical wire, about the axis of which it can vibrate, and that the instant at which a mark on the edge of the disk passes through the middle point of its path is noted for sixty-one con- secutive swings. Then the difference between the fifty-first time of passing and the first time of passing gives the time of fifty swings ; the difference between the fifty -second time of pass- ing and the second time of passing gives an independent deter- mination of the time of fifty swings ; and so on. Thus after counting sixty swings ten independent determinations of the time of fifty swings are obtained, and their average is more trust- worthy than a single determination. There is no need of noting the times of the transits between the tenth and the fiftieth. 3. By the Eye and Ear Method an experienced observer can readily estimate times of transits to a tenth of a second. For a concrete case consider again the vibrating disk that was used as an example for the method of omitted transits. After focalizing a telescope on the mark on the edge of the disk, the latter is set into vibration and the time of transit of the mark past the cross hair of the telescope is obtained as follows: Looking at the clock, the time in hours, minutes, and seconds is observed ; then counting seconds, and while continuing the count, the hour and minute are recorded. Without interrupt- ing the count, the eye is placed at the telescope and the time of a transit is noted. This time can, with practice, if the mark passes rapidly, be estimated to within a tenth of a second. Without interrupting the count this reading is recorded. Con- tinuing the count, the eye is again placed at the telescope ready for the next transit, and the time of the transit observed and 36 PRACTICAL PHYSICS recorded as before. After a little practice this method can be used with ease and confidence for the observation of the times of any number of transits. During the count one should occasion- ally glance at the clock to confirm the correctness of the count. 4. The Flash and Stop-watch Method. — On a stand directly in front of the disk B (Fig. 16) is placed a stop watch W, and a few inches above the watch a mirror M is adjusted to reflect an image of the watch into a telescope T. On the disk Fig. 16. a small bit of mirror, m, is so arranged that just at the equi- librium position of the disk the field of the telescope is brightly illuminated by light reflected from the lamp L. On placing the eye at the telescope, there is seen an image of the stop watch which is illuminated by a flash of light every time the disk passes through its equilibrium position. With the disk vibrat- ing, the motion of the second hand of the stop watch is atten- tively followed through the telescope, and when the flash occurs the watch is read to tenths of a second. 5. In the Method of Passages the value which would be found for the period if a large number of vibrations were counted is obtained from the actual observation of a much smaller number. To fix the ideas, consider the case of the MEASUREMENT OF FUNDAMENTAL QUANTITIES 37 vibrating disk referred to in the preceding paragraph. Sup- pose that the instant is observed at wliich a mark on the edge of the disk swings through its position of rest, once at the be- ginning and once at the end of twenty complete vibrations. From these observations an approximate value for the period can be calculated. Suppose that after a time the observations are resumed and the instant noted at which the mark is again passing through its position of rest — in the same direction as when the two preceding passages were noted. Then the time that elapsed between the first and last observations divided by the approximate period already found gives approximately the number of vibrations that occurred between those two obser- vations. If, for example, this approximate number of vibra- tions comes out 44.1, and if this value can be trusted within 0.3 of a vibration, then, since the number of vibrations that really occurred was a whole number, the actual number of vibrations was 44. Since the time that elapsed between the first and last observations was longer than that between the first and second, the time that elapsed can be trusted farther, and it is, therefore, possible to obtain a closer approximation to the period. After this more accurate value has been found, a considerably longer time could be allowed to elapse and another observation of a time of passage would give a still more accurate value for the period. This method may be made clearer by the following example: — No. OF Obs. Time of Passage Ih IG" 7.3" 1 17 5.9 1 18 50.6 1 23 36.6 1 45 51.8 Interval (1.2) 58.6 s. (1.3) 163.3 s. (1.4) 449,3 s. (1.5) 1784.5 s. Approx. No. of ViBS. 10 [counted] 163.3 5.86 449.3 _ 5.832" 1784.5 -=27.86 5.835 : 77.04 : 305.83 Period 58.6 10 163.3 28 449. 3 77 1784:5 306 = 5.86 s. = 5.832 s. = 5.835 s. 5.8317E 38 PRACTICAL PHYSICS In using this method it is essential that the time between observations of passages be so chosen that the approximate number of vibrations can be trusted within three or four tenths of a vibration. It will be seen that the successive values ob- tained for the period are more and more trustworthy, so that, instead of finding the mean of these values, the last one of them is to be used. 6. The Method of Middle Elongations is another method by which it is possible, without counting the number of swings that occur, to obtain for a period of vibration a value of con- siderable accuracy. The accuracy attainable is somewhat greater than by the method of passages, but the method of middle elongations is applicable only when the period is long enough to allow of recording two readings during each vibration. The point of its path where a vibrating body changes the direction of its motion is called its position of maximum elongation. The mean of the times at wliich any two succes- sive passages through the position of equilibrium take place gives the tinae at which the elongation between tliem occurred. If ten successive passages are observed, the mean of the times of the fifth and sixth passages, or of the fourth and seventh, or of the third and eighth, or of the second and ninth, or of the first and tenth, gives the time at which the middle elonga- tion of the series occurred. For a concrete case consider a disk suspended at the end of a wire about the axis of which it rotates. Suppose that a mark on the edge of the disk is observed to pass through its position of equilibrium at the times indicated in the table on the follow- ing page. Suppose that the first transits in the two series occurred when the mark on the disk was moving in the same direction. Then the elongations considered were on the same side of the position of equilibrium, and during the 790.31 sec. between these elongations a whole number of vibrations occurred. This number of vibrations is not counted, but by subtracting the time of the first transit from that of the ninth and dividing the MEASUREMENT OF FUNDAMENTAL QUANTITIES 39 First Series Second Series Transit Time Middle Elong-atioii Transit Time Middle Elongation 1 r20'" 3.7' 1 1- 33" 14.0" 2 12.0 2 22.5 3 20.4 3 30.5 4 28.4 4 38.6 5 36.7 5 47.0 6 45.0 (5, 6)1" 20° 40.85" 6 55.4 (5, 6)1" 33" 51.20" 7 53.2 (4, 7) 40.80 7 1 34 3.5 (4, 7) 51.05 8 1 21 1.8 (3,8) 41.10 8 12.0 (3,8) 51.25 9 0.9 (2,9) 40.95 9 20.1 (2, 9) 51.30 10 18.0 (1,10) 40.85 10 28.6 (1,10 51.30 Mean 1" 20° 40.91- Mean 1" 33° 51.22- difference by four, the period is found to be approximately 16.55 sec, and the number of vibrations that occurred between the two given elongations is, therefore, about [790.31 -i- 16.55 = J 47.75. If this number can be trusted within 0.3 vibration, the number of vibrations which actually occurred must be 48. The period is, therefore, [790.31 -- 48 =] 16.4648 sec. If a still more trustworthy value were desired, a third series of readings could be taken after a considerable time had elapsed, and from this third series and either of the others an exceedingly close approximation to the true period could be obtained. The calculations would be carried out as above except that in finding the approximate number of vibrations use would be made of the value for the period that was obtained as the result of the observations in the first two series. If the times of transit cannot be trusted closer than 0.3 sec, the time which elapses between the middle elongations of the two series should not be greater than about 3 T^, where T de- notes the approximate period of the motion; if the times of transit can be trusted to 0.2 sec, the time between the middle elongations may be allowed to be about 4 T^ ; and if the times of transit can be trusted to 0.1 sec, the time between the mid- dle elongations may be allowed to become 8 T^ ov 9 T^. 7. The Method of Ooinoidences is a very accurate method 40 PRACTICAL PHYSICS for the comparison of two. nearly equal periods of vibration. Suppose the period of oscillation of a simple pendulum is to be compared with that of a clock pendulum beating seconds. If the simple pendulum swings slightly faster than the clock pendulum, a moment will occur when both are at their lowest points at the same time. But since the simple pendulum is all the time gaining on the clock pendulum, after a certain inter- val it will have gained a whole oscillation, and then both pen- dulums will again be at their lowest points. If between two such coincidences the clock pendulum has made n swings, then the simple pendulum has made n + 1 swiiigs, and its time of sec. Similarly, if the simple pendulum oscillation is n + 1 were going slower than the clock pendulum, the time of oscil- lation of the simple pendulum would be sec. n— 1 One method of determining the instant of coincidence employs an electric circuit containing the two pendulums, a battery, and a telegraph sounder, all in series as shown in Fig. 17. When the two pendulums are in coincidence, they pass through the mercury contacts A and B at the same instant, and at this instant the sounder clicks. It is to be kept in mind that the n in the above expressions denotes the number of swings made by the clock pendulum — not by the simple pendulum. Since one pendulum gains only slightly on the other, and since the passage of the pendulums through the mercury cups at A and 5 is not instantaneous, there are often clicks for several successive swings. The mean time of the first and last of these suc- cessive clicks is used as the instant of coincidence. The actual instant of coincidence, i.e. the instant when each pendulum is distant from its position of rest by the same frac- FiG. 17. MEASUREMENT OF FUNDAMENTAL QUANTITIES 41 tion of a vibration that the other is, may occur when both pen- dulums are in some position other than at their lowest points, but it can never be more than half a swing from the lowest point. If there are only a few successive clicks, it will be safe to assume that in taking the mean of several successive clicks, the time of coincidence is not in error by so much as one swing. If the simple pendulum is swinging faster than the clock pen- dulum, the error introduced into the value for the period by getting for n one swing too few is the difference between the period found, , and the true period, , viz. n n+1 n— 1 n 1 n n + 1 n(n + 1) If n is large compared with unity, the error is almost -. Thus if w = 70, the error introduced into the period by an error of 1 in the number of seconds between coincidence is about — 0.0002 sec. If n is small, the accuracy may be in- creased by counting the number of seconds to some later coin- cidence instead of to the second. In this case one pendulum will have gained on the other more than one swing, and the above formulas must be modified accordingly. CHAPTER III LENGTH, AREA, ANGLE Exp. 1. Determination of the Thickness of a Thin Plate by Means of a Spherometer and an Optical Lever Object and Theory op Experiment. — The object of this experiment is to measure the thickness of a microscope cover glass by two methods and to compare the precision of measure- ment obtained by the two methods. The spherometer has already been described. The theory of the optical lever will now be developed. The optical lever to be used in this experiment consists of a piece of sheet brass about 3 cm. long and 1 cm. wide mounted upon four pointed legs, one at each end and the other two midway between the end legs and in a line normal to the line joining the latter. Fas- tened on the upper side of the optical lever, with its reflect- ing surface in the plane of the two middle legs, is a small mirror. The length of the four legs may be such that when the optical lever rests upon a piece of plate glass all four legs are iu contact with the glass, or the end legs may be slightly shortened so that the optical lever can be tilted forward and backward about the ends of the middle legs. From the differ- ence in the angle through which the optical lever can be tilted when the middle legs rest directly upon a large plane surface, and the angle through which it can be tilted when a thin plate is interposed between the middle legs and the plane surface, the thickness of the thin plate can be determined. Let mna (Fig. 18) be the optical lever with its mirror approximately normal to the base mn, T a telescope, and 0' 0" 42 LENGTH, AREA, ANGLE 43 a vertical scale about a meter from the optical lever. First assume that the ends of the feet of the lever are all in one plane. Imagine the thin plate x placed under the middle feet of the optical lever. W hen the lever is tilted forward an observer at the telescope sees the point of the scale at 0' reflected in the mir- ror, and when the mirror is tilted backward the re- flected image of the scale at 0" comes to the cross hair of the telescope. Meantime the optical lever has been tilted through the angle S. Consequently the angle between the normals to the mirror in its two positions is also Q. And since the angle of reflection equals the angle of incidence, the angle between 0' a 0' 0" and 0"a' equals 2 6. When 6 is small, —^ — =26 radians, and , mm! . n also = Q. Fig. 18 mc Consequently 2 mm' Oa mc O'O' Oa Let the thickness of the thin plate be denoted by h, the dis- tance mn between the two end feet by 2Z, the distance Oa between the scale and mirror by L, and the difference between the scale readings at 0' and 0" by S. Since e is midway be- tween m and n, the distance mm' is twice the thickness h of the plate. On substituting these letters in the above equation, we get *=^- (19) This is for the case of an optical lever having the lower ends of all four feet in one plane. But if the end feet are shortened so that the lever is capable of being tilted enough to produce 44 PRACTICAL PHYSICS a deflection S' when placed upon a plane surface, the thickness of the thin plate is given by h = (^-^')K (20) 4 _Z> The development of this equation is left as an exercise for the student. Manipulation and Computation. — In using the optical lever, the telescope and scale must first be adjusted ; that is, the telescope, the scale, and the mirror of the optical lever must be placed in such relative positions that on looking through the telescope a reflected image of the scale is seen. To make this adjustment, place the scale vertical, facing the mirror, and about a meter from it; standing behind the scale and looking at the mirror, move the eye about until a reflected image of the scale is seen; keeping the image of the scale in view, move the scale and the eye toward each other until the telescope, the eye, and the mirror are in the same vertical plane ; and then, still keeping the image of the scale in view, move the eye and telescope up or down toward each other until they come to the same level. By sighting along the outside of the tele- scope see that it is pointing at the mirror and then focalize — first on the cross hairs and then on the scale — as is described on p. 23. If the focalizing screw is turned so that the mirror is clearly seen, the scale will not be visible. In order to bring the scale into view the focalizing screw must be turned so as to shorten the telescope tube somewhat. With the optical lever on a piece of plate glass, adjust the telescope and scale as above directed, and observe the scale reading in the telescope when the optical lever is tilted forward and when it is tilted back. The difference between these two readings is S' . Now place under the middle legs of the optical lever the plate the thickness of which is to be measured and take two similar readings. The difference between these read- ings is S. L may be measured with a meter stick, and I is best obtained from the measurement of the distance between prick- LENGTH, AREA, ANGLE 45 marks made by pressing the feet of the optical lever on a sheet of paper. In using the spJierow-eter, determine first the zero point by placing the instrument on a piece of plate glass and noting the readings on the two scales when the point of the screw just touches the glass. Raise the screw, place under it the thin plate whose thickness is to be measured, lower the screw until it just touches the thin plate, and note the readings on the two scales. The difference between the readings with and without the plate gives the thickness of the latter. Make five determinations by each method and compare the two methods as to accuracy. Exp. 2. Determination of the Radius of Curvature of a Spherical Surface Object and Theory of Experiment. — There are three principal methods for determining the radius of curvature of a spherical surface. They are by means of (a) the spherometer, (6) the optical lever, (c) the reflection of light. The last method is applicable only to highly polished surfaces, and its consideration will be delayed until the subject of light is taken up. The object of this experiment is to determine the radius of curvature of a spherical surface by means of the spherome- ter and by means of the optical lever, and to compare the two methods as to accuracy. The theory of the two methods will now be considered. (a) By means of the spherometer. — The curva- ture of the surface is determined from the dimen- sions of the spherometer and the distance through which the point of the screw must be moved from ^ Yia. 19. the plane of the ends of the three legs in order that all four points may be brought into contact with the spherical surface. Let XYZ (Fig. 19) be the positions of the three fixed feet, and F the position of the point of the screw, when all four are 46 PRACTICAL PHYSICS where in one plane. Let the distances XF, YZ, and ZX be denoted by a, b, and c. A proposition in Trigonometry states that the radius of the circle circumscribing the triangle XYZ is r = ^^" (21) 4 Vs (s - a) (s — 5} (s - e) Now consider a plane passing through one of the feet of the spherometer Y (Fig. 20), the point of the screw H, and the center of curvature 0, of the surface whose radius is required. Then if R is the required radius, and h, when all four points are in contact with the spherical surface, the height of the point of the screw above the plane of the ends of the three feet, we have, from Fig. 20, Whence Jfi + r-^ R 2A On substituting in this equation the value tov r obtained from (21), we have (22) 7? =^ I . 2 32As(s- 2J2c2 a) (s— J)(s — e) It should be noticed that in deriving this equation it has been assumed that the axis of the screw is perpendicular to the plane of the three feet of the spherometer, and also that when the point of the screw and the three feet are in the same plane the point of the screw is at the center of the circle circumscribing the triangle formed by the three feet. Errors due to these causes are, however, almost always so small as to be negligible. If the distances between adjacent feet of the spherometer are equal, that is, if we set 1= a = i = c, (22) becomes ^=2 + 61- (23) LENGTH, AREA, ANGLE 47 In practice a, b, and o will not be exactly equal, but often they will be so nearly so that instead of using (22) it will be permissible to substitute for I in (23) the mean of a, b, and c. (6) By means of the optical lever. — Figs. 21 and 22, which are views at right angles to one another, show the optical lever rest- ing on the curved surface. The end points of the lever touch the spherical surface at F and Z), and the middle points at £ and H. Let H represent the required radius of cur- vature of the spherical surface, 2 1 the dis- tance between the end points, and 2 b the distance between the two middle points. From Fig. 21, whence 2 BCAO) = QA Of -h ( CBf = iABf. Similarly from Fig. 22, 2 i2(^^) = iARf. If i_AB) is small compared with (AD), and (A^) small com- pared with (^EE'), then (A-D) and (-4-H) will be approximately equal respectively to I and b, and the above equations may be ^^■itten 2 R^AB + BO) = P (24) and 2RiAE) = b\ (25) From the theory of the optical lever (p. 43) it has been seen SQi^i'^-'^y. (20') But AB in Fig. 21 equals AU in Fig. 22. On setting AB in place of AB in (25) and then eliminating AB and BO from (24), (25), and (20'), we obtain _^^ . (l + b)0-b) .26) ■S-S' I ^ ^ where S and S' denote the respective deflections observed on the scale of a telescope distant L from the optical lever (1) when 48 PRACTICAL PHYSICS the lever is rocked back and forth on the object being measured, and (2) when it is rooked on a plane. Manipulation and Computation. — (a) When the sphe- rometer is used, run the center point up out of the plane of the other three, press the three outer points on a piece of bristol board, and either by means of a glass scale laid face down on the bristol board, or by means of a pair of sharp-pointed dividers and a millimeter scale, measure the three sides of the triangle. If the three sides are nearly equal, use the average for I. Determine the zero point of the spherometer by placing the instrument on a sheet of plate glass and noting the readings on the two scales when the point of the screw just touches the glass. Place the spherometer on the spherical surface whose curvature is to be determined, and turn the screw down until its point just touches the surface. Again read the two scales of the instrument. The difference between this read- ing and the zero reading is h. Substitute these values of I and h in (23) and solve for M. Obtain the mean of five values of It determined in this way from five sets of obser- vations. (/)) When the optical lever is used, press the end points of the lever on a piece of bristol board and measure 2 1 either by means of a glass scale laid face down on the bristol board, or by means of a pair of sharp-pointed dividers and a millimeter scale. In the same way measure 2 b. Place the lever on the curved surface, and, exactly as described in the preceding experiment, adjust a telescope and scale, measure L, and take readings to deter- mine S and /S". Make five different sets of observations, each time having the telescope and scale a few centimeters farther from the optical lever. In each case find R by (26). From the results obtained, compare the two methods as to accuracy. The spherometer is especially useful for finding the radius of curvature of a surface of considerable extent, while the optical lever is available for surfaces of limited extent and small curvature. LENGTH, AREA, ANGLE 49 Exp. 3. Radius of Curvature and Sensitiveness of a Spirit Level Object and Theory op Experiment. — In many measure- ments in which a spirit level is used in connection with other phys- ical apparatus it is necessary that the sensitiveness of the level be at least as great as that of the other apparatus. An example is the case of the telescope and level of an engineer's transit. When used in leveling or in measuring vertical angles, the least vertical motion of the telescope which can be detected by means of the cross hair should also make itself evident by a displacement of the level bubble. A test of the suitability of a level for a particular use includes the determination of the uniformity of the run of the bubble in the vial and the sensi- tiveness of the spirit level. The sensitiveness of a spirit level may be defined as the distance the bubble moves for an inclina- tion of the level of one minute. Since the sensitiveness can be Fig. 23. proven to be directly proportional to the radius of curvature of the vial, it is often designated by the radius of curvature. The object of this experiment is to make a test of a spirit level. In the laboratory a spirit level is usually tested by means of a Level Trier consisting of a base plate upon which rests a ^-shaped casting supported by two projecting steel points H and F (Fig. 23) at the end of the arms of the T and a mi- crometer screw M at the foot of the T. The pitch of the micrometer screw must be measured and also the perpendicular distance from the micrometer screw to the line connecting the points U and F. The level to be tested, L, is placed on the T and the position of the bubble in the vial is noted by means of a scale engraved upon the glass or by a scale S attached to the level trier. In case it is inconvenient to separate the level 50 PRACTICAL PHYSICS from a piece of apparatus of which it forms a part, the entire apparatus, e.g. a telescope or theodolite, may be mounted in the grooves ABC or BEF. After the level tube is in place, the micrometer reading is noted. The T is now tilted through a small angle by turning the micrometer screw, and readings are again taken of the micrometer screw and the position of the bubble. Suppose that by means of the micrometer screw the T of the level trier is moved from the position FJ (Fig. 24) to the position FJ' , the middle of the bubble moving meantime from Cr to H. If a vertical line GiK were drawn through Gi before the micrometer screw was turned, and if this line were to move with the level, it would after the movement be in a position Gr'P, such that the angle through which it moved would equal the angle JFJ' through which the level moved. A vertical line through the middle of the bubble's position of rest has the direction of a radius of the -pia. a. bubble vial. If, then, MP is drawn vertically through JT, both IFF and Cr'F are radii of the vial. But since ITP and GrK are parallel, the angle Cr' PH equals the angle between GK and Gr'P, which latter has just been shown to equal JFJ' . It follows that ^GJPH=^JFJ'. Let &'P be denoted by R, Q'R by d, FJ' by x, and JJ' by y. Then A B = 6 radians, (27) LENGTH, AREA, ANGLE 51 and, since the screw is always perpendicular to the T, ^ = tan^. X Since is always very small, tan 6 = 6, and we have almost exactly ^.^y_ R • X Whence R=^. (29) ■ y Eliminating R from (27) and (29), d ^xd ~6^T Since one minute differs from the 3438th part of a radian by less than 0.01 %, ^ can be reduced to minutes by multiplying it by 3438. The sensitiveness is, therefore, from its definition ^^^""^ ^^ d xd ~ 3438 0~ 3438 «/■ ^ ^ Manipulation and Computation. — Place the ^-shaped casting upon a piece of bristol board, and by means of slight pressure obtain an impression of the three supporting points. Measure the perpendicular distance from the impression made by the end of the micrometer screw to the line connecting the impressions of the other two supporting points. The pitch of the micrometer screw may be obtained in the following manner : After placing the spirit level on the trier, adjust the micrometer screw until one end of the bubble is directly under a scale division near the middle of the vial; then insert under the micrometer screw a small piece of plate glass whose thickness has been already measured with a sphe- rometer or micrometer caliper, and again adjust the micrometer screw until the bubble rests at the same point as before. The thickness of the glass plate divided by the necessary number of turns of the micrometer screw gives the pitch of the latter. 52 PRACTICAL PHYSICS Again adjust the micrometer screw until one end of the bubble is directly under a scale division near one end of the vial. Observe the micrometer screw reading and the scale readings at both ends of the bubble; rotate the micrometer screw through a convenient number of spaces and take read- ings as before. Continue this operation until the bubble has been removed in some half dozen steps to the other end of its run, and then return step by step in the same manner. Repeat this series of readings three times. A series of such readings may be conveniently tabulated in the following form : — N"LrMBER OF Obsekvation MiOEOMETEE Reading Eeadingb DF Bubble Displacements Length of Bubble Left End Eight End Left End Eight End 1 3.700 mm. 1.8 mm. 10.2 mm. 8.9 mm. 2 3.800 6.1 14.9 4.8 mm. 4.7 mm. 8.8 3 3.900 11.1 19.8 5.0 4.9 8.7 4 4.000 16.2 25.1 5.1 5.3 8.9 5 4.100 21.1 30.2 4.9 5.1 9.1 6 4.000 16.1 25.1 5.0 5.1 9.0 7 3.900 11.1 19.9 5.0- 5.2 8.8 8 3.800 6.2 15.0 4.9 4.9 8.8 9 3.700 1.3 10.2 4.9 4.8 8.9 Mean 4.95 5.00 8.88 The values in columns 2, 3, and 4 are read, and those in 5, 6, and 7 are calculated from these readings. The values in columns 5 and 6 show the uniformity of the run of the bubble, or the variation in sensitiveness when the bubble is at different positions in the vial. The average radius of curvature and sen- sitiveness of the vial are obtained by substituting for d in (29) and (30) the mean displacement obtained from columns 5 and 6. Care must be taken to keep the entire vial at the same tem- perature. It must not be touched by the fingers nor breathed upon, as when unequally heated the bubble tends to move toward the point of highest temperature. LENGTH, AREA, ANGLE 53 Exp. 4. Verification of a Barometer Scale Object and THEORy of Experiment. — In the ordinary form of Fortin's barometer, the lower end of the tube dips into a mercury reservoir wliich can be raised or lowered by a screw. By this means, before taking an observation, the surface of the mercury in the reservoir is always brought to the level of the point of an ivory pin extending downward from the cover of the reservoir. The barometric height is the length of the mercury column from the bottom of this pin to the top of the meniscus at the upper end of the col- umn. A brass scale attached to the metal case inclosing the barometer tube is supposed to be adjusted so that its divisions indicate distances measured from the point of the ivory pin. The object of this experiment is to measure the barometric height by a cathetometer and to compare with this height the reading by the brass scale. Manipulation and Computation. — Set the cathetometer on a stand about a meter distant from the ba- rometer. After the cathetometer has been adjusted as described on p. 23 raise the telescope until the horizontal cross hair in the eyepiece is tangent to the meniscus at the upper end of the barometer column, and take the cathe- tometer reading by means of the scale and vernier. Lower the telescope until the horizontal cross hair coincides with the level of the mercury in the reser- voir and again take the cathetometer reading. The difference between these two readings is the barometric height. Now, by means of the screw T (Fig. 25), at the bottom of Figs. 25 and 26. 54 PRACTICAL PHYSICS the barometer, bring the surface of the mercury in the reser- voir to the level of the ivory point P, and read the barometric height by means of the scale and vernier F(Fig. 26) attached to the case. Attached to the sliding vernier there is a similar piece of metal directly back of the barometer tube. These two slides move together. In order to avoid parallax, the vernier is moved up and down tntil the position is found where the lower edge of the vernier, the upper surface of the meniscus, and the lower edge of the rear slide are in line. Find the error of adjustment by taking the difference be- tween the mean of five determinations of the barometric height by means of the barometer scale and the mean of five with the cathetometer. Exp. 5. Determination of the Correction Factor of a Planimeter Object and Theory of Experiment. — A planimeter is a direct-reading instrument which is used to determine the areas of irregular figures on drawings. The correction factor of a planimeter is that number — usually near unity — by which the area read from the instru- ment must be multiplied in order to get the true area. The object of this experiment is to determine the correction factor of a given planimeter. Amsler's polar planimeter (Fig. 2J) consists of two arms, a tracer arm AC, and a pole arm ^0, jointed at C. The point U is fixed and the point A is carried around the boundary of the figure in the direction of the hands of a watch. Attached to the tracer arm is a small roller D, the axis of which is parallel to the line AO. This roller and the LENGTH, AREA, ANGLE 55 points A and E are the only parts of the planimeter that touch the paper. As the point A passes over the boundary of the figure, the roller rotates unless the motion A is entirely in the direction of AC, — in which case the roller slides. It will now be proved that when the tracing point circumscribes any closed plane figure which does not contain the pole point, the cir- cumference of the roller rotates a distance proportional to the area circumscribed by the tracing point. This proof will be in four parts. First, consider two concentric circular arcs AA" and A' A'" (Fig. 28) cut by radii AE and A"E. Let the pole point of the planimeter be fixed at the center of these arcs, and let the tracing point be moved along the radius AE from Ato A' . Then the roller will move from D to D' while a point in its circum- ference will rotate through the distance BH. Again, let the tracing point be moved along the radius A" E from A" to A'", causing the roller to move from D" to D'" while a point on its circumference rotates through the distance I)"H'. Since the shape and size of the figure EB'HDA'A are the same whether the tracing point has moved along the radius AE or along some other radius A" E, it follows that I)"H' equals BE. Therefore, while the tracing point passes over the portions of any radii intercepted between the same two circles having the pole point E as center, the rolling components of the motion of the roller are equal. Second, let ECA (Fig. 29) represent the planimeter in one position, and EC A' the planimeter in another position. Draw EB and JB normal to AQ, and HB' normal to JB; also draw EA, EB, and EB' . For brevity let 8^ denote the distance Fig. 28. 56 PRACTICAL PHYSICS through which a point in tlie circumference of the roller moves with reference to AO when A describes any line x. Let the instrument start from the position EGA, and, keeping the angle EGA constant, rotate about E through a small angle A@ into the new position EC A'. AA! is, then, the arc of a circle described about E as center. The roller, meantime, moves through a small distance DI>' , sliding through a distance SI)\ and rolling through a distance DH. Whence S^^. = I)S= BD' ■ cos HDD' = ED A© • cos HDD'. (31) Since HD is by construction normal to AQ, and the very small arc DD' is normal to the radius ED, the angle HDD' equals the angle BDE. And since BE is by construction normal to^D, ,cos HDD'l= cos BDE^ = ^■ ED Equation (31) becomes, therefore, S^^- = ^i> . A@ • ^ = A® • BD. (32) ED Now BD = BO- DO = EC- cos AOE- DC. (33) And since in the triangle A OE, {AEy = (A oy + QEoy ~ 2 ac-eo- cos ace, (33) may be written BD = EC iAGy±(ECy-iAE^_j)Q_ 2 AC -EC Equation (32) becomes, therefore, B^^, = Ae[ ^^<^y+(f^y-(^ -^-DO-]. (34) For this particular case, then, where the tracing point moves over the very small arc of a circle described about the j)ole LENGTH, AREA, ANGLE 57 point, S^^. is expressed in terms of the radius of this circle, the dimensions of tlie instrument, and the very small angle sub- tended by the given arc. Third, let any figure KLMN (Fig. 30), not inclosing the pole point, be cut into a large number of very narrow strips by a series of circles having E for center. Let these strips be cut into very small areas by radii drawn from E. Thus the entire figure is divided into a great uumbeT of areas, each as small as we choose. If the tracing point cir- cumscribes in the clockwise direction one of these small areas, ah' , we have, from the first division of this proof. And since S^j is described in the opposite direction from that assumed in the second division of this proof, the entire value of ^aib'a'a is equal to h^.a, — ^ab- Whence, by (34), _ ^p,[ UO)^+(Ecy-QaEy L ^AO DO ^ AQ r 2AC\_ (aEy {a'Ey~\- - A@(aE} 2 = 1. Ae CaE){aE) = l Qah-yiaE), But and this last expression measures the area of the circular sector ohE. In the same way i A© (a'E) ^ measures the area of the circular sector a'b'E. So that ^ahb'a'a area (obE^ — area {a'b'E) __ area (aS') .gg. AC AO In Fig. 29 the angle BBE was drawn acute. If this angle is drawn obtuse, it will be seen that the roller then ro- tates in the opposite direction. Calling rotation in this oppo- 58 PRACTICAL PHYSICS site direction negative, and making the changes in sign involved in the new figure, we find that (35) holds whether BBE is acute or obtuse. That is, when the elementary area ah' is cir- cumscribed by the tracing point, that area is given by the prod- uct of the length of the tracer arm and the small distance through which a point in the circumference of the roller has rotated. Fourth, let the tracing point move over the whole figure KLMN (Fig. 30) in such a way as to traverse the boundary once in a clockwise direction, and each of the radial lines and circular arcs twice, once in each direction. By taking lines in the proper order, this can be done without lifting the tracing point from the paper. Describing these lines in the manner indicated amounts to the same thing as going once in the clock- wise direction around the whole figure ; it also amounts to the same thing as going once in the clockwise direction around each of the small areas into which the figure is divided. The total value of S^ will then be ;; _>rAarea (a6') _ area (KLMJV} /■or\ Oklmnk- 2^ -j^ -^ (.00) This equation shows that when an area which does not contain the pole point is circumscribed hy the tracing point, the area is measured hy the product of the length of the tracer arm and the distance through ivhich a point in the circumference of the roller has rotated with reference to the tracer arm. The dimensions of the planimeter are usually so selected that the product of the length of the tracer arm by the circumfer- ence of the roller is equal to ten square inches or a hundred square centimeters. That is, they are so selected that if the tracing point circumscribes an area of ten square inches or a hundred square centimeters, as the case may be, the roller rotates once. The circumference of the roller is then divided into a hundred equal parts, and these by means of a vernier (F", Fig. 27) can be read to tenths. The counting wheel B indicates the whole number of revolutions of the roller. LENGTH, AREA, ANGLE 59 In the practical use of a planimeter, the figure the area of which is desired may be so large that it cannot be circumscribed without placing the pole point inside it. In this case the area may be determined as follows : — If the angle ADE (Fig. 29) is a right angle, then BD is zero, and, therefore, from (32), S^^, equals zero. That is, as A moves about E in the circular arc AA' , tlie — — ». roller slides, without rolling at all. The l circle generated by the tracing point A about the pole point E as center when the / roller does not rotate, and so makes no ; record, is called the "zero" or "datum" ^ circle. In Fig. 31 let RSW be this datum cir- ^^^- ^'^• cle. Then if the tracing point were to circumscribe the area TS, (85) shows that ;; _ area (2W) "TUSRT— ~T-p ' (3') and if now the tracing point were to circumscribe the rest of the shaded area, then ;; _ shaded area ( UVT} "TRwsavT xTy (38) If these two paths were to be described successively, then, hj adding (37) and (38), we find that r, total shaded area 'TUSRTRWSnVT'- AO In tracing this whole path, the lines US and RT have each been described twice, once in each direction, so tliat the result- ant motion of the roller produced by tracing them is zero, and, since RSW is the datum circle, the roller did not rotate while it was traced. It follows that if the tracing point had simply described the perimeter TUVT, the roller would in the end have turned from its first position just as much as it did while the more complicated outline was being traced. That is, if the tracing point were to describe the entire perimeter of the 60 PRACTICAL PHYSICS figure, the area indicated by the roller would be the area of that part of the figure outside of the datum circle. If the trac- ing point were ever to cross the boundary of the datum circle, the roller would move in opposite directions before and after crossing. From this it may be shown, if proper attention be paid to the sign of the roller reading, that whenever the pole point is inside of the figure circumscribed hy the tracing point, the area actually circumscribed is greater than that indicated by the roller, the difference being the area of the datum circle. To sum up, if the tracing point circumscribe in the clockwise direction any area, the difference between the final and initial readings of the roller gives the area when the pole point lies out- side the figure ; when the pole point lies inside the figure, the area is obtained by adding to this difference the area of the datum circle. Equation (36) suggests at once a method of determining the correction factor of a planimeter. If d denotes the diame- ter of the roller, and I the length of the tracer arm AO, then the area which can just be circumsci'ibed by the tracing point while the roller rotates once is, by (36), equal to Trdl. If the roller is so graduated that the area indicated for one rotation is /, the correction factor K is given by K='^. (39) Manipulation and Computation. — With a steel scale and a sharp pencil lay off a rectangular area of not less than 150 sq. cm. Make five careful readings of the length and breadth of the rectangle. If the tracer arm is adjustable in length, note the reading on its scale. Place the pole point outside the rectangle, bring the tracing point to one corner, and read the planimeter. Using the steel scale as a straightedge to guide the tracing point, circumscribe the rectangle in the clockwise direction, and again read the planimeter. In this manner measure the area at least ten times. The product of the average length and average breadth of the figure divided LENGTH, AREA, ANGLE 61 by the average difference between the final and initial readings of the planimeter gives the correction factor. With a micrometer caliper determine the diameter of the roller. With the steel scale make five readings of the length of the tracer arm. From these calculate the correction factor by (39). Compare the results obtained by the tv^o methods. Exp. 6. Correction for Eccentricity in the Mounting of a Divided Circle Object and Theory of Experiment. — Angles are often measured by means of "a divided circle and an index or vernier attached to an arm capable of rotation about an axis passing through the center of the circle. This method is subject to a source of error due to the mechanical difficulty of mount- ing the arm carrying the vernier so that its axis of rotation accurately coincides with the normal axis of the divided circle. The object of this experiment is to construct a correction curve for a divided circle having an eccentrically mounted vernier. Let Q (Fig. 32) be the center of the divided circle, A and B the zero points of the two verniers carried on arms capable of rotation about the point D. If the line AB passes through D, and I) coincides with C, there is no eccentricity in the mounting, and correct angular readings are obtained by means of a single vernier. But in the general case where neither of these conditions is fulfilled, correct angular readings can be obtained only from simultaneous readings of the two verniers A and B. Let A° and B° be the observed readings. Draw A^B-^ through G parallel to AB. If there were no eccentricity in the mounting, and if A and B were diametrically opposite, the readings would be A^ and B^. In other words, A^ and B^ 62 PRACTICAL PHYSICS are the true readings corresponding to the observed readings A° and B°. Through O draw the lines BU and AF. Since A^Bj^ is parallel to AB, and ^(7 equals BO, HOAi = Z OBA = ZBAC=ZAOAy Therefore Z XOA^ = ^ (^ ^C'^ + ^XOA-), or 4i° = K-^° + ^°)- If the division lines on the circle are numbered as shown in the figure, E° = B° — 180°. Consequently the corrected reading of the vernier A is A^° = 1 (4° +i?° - 180°). (40) This is the corrected reading for the verhier giving the smaller reading. In precisely the same manner, since B-^ = ^(^B° + F°^ and since F° = 180° + A°, the corrected reading of the vernier B is 5i° = i(4° + -B°+180°). (41) This is the corrected reading for the vernier giving the larger reading. In this manner, by means of two verniers, is obtained the reading of either vernier corrected for eccentricity of mounting. Manipulation and Computation. — Starting with one vernier near the zero point of the circle, read both verniers. Then move the verniers about thirty degrees and again read them both. Repeat at intervals of about thirty degrees until the entire circumference is traversed. The corrections for the observed vernier readings are found by subtracting the observed readings from the corrected readings. On cross-section paper lay off the observed readings of one vernier on the axis of abcissas and the corresponding correc- tions on the axis of ordinates. The curve drawn through the points thus obtained is the correction curve for this vernier. From the form of this curve decide whether C and D are coin- cident, and whether AB passes through B. CHAPTER IV VELOCITY AND ACCELERATION Exp. 7. Determination of the Change of Speed of a Flywheel during a Revolution Object and Theory of Experiment. — For many purposes, as when used to drive high-speed machinery, it is important that throughout a revolution the angular speed of a flywheel shall be nearly constant. The object of this experiment is to deter- mine the angular speed and the acceleration of a flywheel at different points in its revolution. Attached to the shaft of the flywheel is a brass disk in the edge of which thirty-six slots of equal width have been cut and then filled to the edge with pieces of hard rubber. If one terminal of an electric circuit including a chronograph be pressed against. the edge of the disk, and the other against the revolving shaft, then during each revolution of the flywheel the circuit through the chronograph will be made and broken thirty-six times. That is, at every 10° rotation of the flywheel a break will be made in the record line on the chronograph drum. If the flywheel revolves through equal angles in equal times, the distances between notches in the record line will be equal. Any irregularity of motion will thus be rendered apparent. Manipulation and Computation. — Plot a curve with in- tervals of time between any selected notch and the succeeding notches as abscissas, and the corresponding angles of rotation as ordinates. If the angular speed of the flywheel is uniform, this curve will be a straight line. If a straight line be drawn tangent to this curve at a point corresponding to any particular time, the speed of the flywheel 63 64 PRACTICAL PHYSICS at that instant equals the tangent of the angle between this tangent line and the axis of abscissas. In this manner compute the speed of the flywheel at points 20° apart throughout an entire revolution. Construct a second curve by plotting times as abscissas and speeds as ordinates. If a straight line be drawn tangent to this second curve at a point corresponding to any particular time, the acceleration of the flywheel at that instant equals the tangent of the angle between this tangent line and the axis of abscissas. Construct a third curve by plotting times as abscissas and accelerations as ordinates. Carefully interpret each curve. Exp. 8. Determination of the Speed of a Projectile by the Ballistic Pendulum Object and Theory op Experiment. — The object of this experiment is to determine the speed of a bullet from a rifle. Newton proved that if two bodies are moving along the same straight line, the speed of the first with respect to the second after a collision between the two is directly proportional to the speed before the collision, the proportionality factor depending upon the elasticity of the two bodies and being called the coef- ficient of restitution of the given bodies. He also proved that if no external forces act upon a system of bodies, the total momentum of the system is constant. Imagine that a projectile of mass m and speed u strikes a body of mass M and speed U, and that after the impact the speeds are u' and U' respectively. Then before impact the speed of the projectile with respect to the other body is (u — V"), and after impact it is (m' — CT"'). It follows, then, from the state- ments in the preceding paragraph, that u' - U' = e(iu- U) (42) and mu' + MU' =mu + MU, i (43) where e is the coefficient of restitution of the bodies. If the bodies are perfectly elastic, e = 1, and if they are perfectly in- VELOCITY AND ACCELERATION 65 elastic, e = 0. If the experiment is so arranged that the initial speed of the large mass is zero, and so that after the impact the two masses move together, thus acting like inelastic bodies, then ^7=0 and e = 0. On making these substitutions in (42) and (43) and then eliminating u' between them, we get \ m J (44) The conditions necessary to fulfill the requirements of this equation are met by the use of the ballistic pendulum. This consists (Fig. 33) of a block of wood so suspended that it can swing freely about C as an axis. When a bullet strikes the pendulum bob, the whole impulse may be used in giving to the bob a motion of translation in the direc- tion in which the bullet was moving, or part of the impulse may be used in pro- ducing torques which tend to set up wob- bling motions that are not taken into account in the above equations. If the bullet strikes at a point called the center of percussion, these torques are not pro- duced. The center of percussion is at a distance from the axis of rotation equal to the length of the equivalent simple pendulum, and when the masses of the supporting cords are small compared with that of the bob, the lower end of this equivalent simple pendulum is verj'- near the center of mass of the bob. If the angle through which the pendulum is deflected by the impact of the bullet is denoted by d, the height through which the center of mass of the pendulum is elevated by A, and the distance from the axis of rotation to the center of percussion by I, then h = l (1 — cos^). By the time the bullet has ceased to move through the pendulum bob they both have a speed U' , and consequently kinetic energy equal to J (m + M) U''^- When the end of the swing is reached, this kinetic energy has all been Fig. 33. 66 PRACTICAL PHYSICS used ill lifting them through the distance h ; i.e. in doing work equal to (m+M)gh. Consequently ^(m+M} U''^ = (m+M)gh. Whence U' = ^2gh=^ 2gl(l-cose'). On substituting in (44) this value for V, we obtain ^^m+M^2gl(l-cose^. (45) m Manipulation and Computation. — In setting up the apparatus see that the line of flight of the bullet is horizontal, that it is perpendicular to the axis of rotation of the pendulum, and that it passes through the center of percussion of the pendulum. Weigh the wooden plug in the center of the pen- dulum bob both before and after the bullet is fired into it. Weigh the rest of the bob, measure I, and observe 0. Exp. g. The Acceleration Due to Gravity by Means of a Simple Pendulum Object and Theory of Experiment. — The object of this experiment is, from measurements of the length and time of oscillation of a simple pendulum, to find the value of the ac- celeration due to gravity. In elementary text-books on General Physics it is shown that the period of a complete to-and-fro vibration of a simple pendulum of length I vibrating through a small arc at a place where the acceleration due to gravity is g, is very nearly Whence g = (^Y^- ' (46) If the length of the pendulum is about 100 cm. and the amplitude of vibration about 3 cm., the value that (46) gives for g is about 0.01 % too small. This error is so slight that in VELOCITY AND ACCELERATION 67 the above equations the approximation sign is omitted. More- over, (46) is deduced on the assumption that the pendulum has its mass concentrated at a point on the end of a perfectly flexible suspension. An increase either in the size of the bob or in the mass of the suspending wire increases the error intro- duced by using the above equation. If the length of the pendulum is taken as the distance from the supporting knife-edge to the cen- ter of mass of the bob, and if this distance is about 100 cm., and the diameter of the bob about 3 cm., the value found for g is about 0.01 % too small. With the same length of pendulum, if the mass of the supporting wire is about 0.3 g. and the mass of the bob about 75 g., the value found for g is about 0.07 % too large. Manipulation and Computa- tion, — In finding the period of oscillation of the experimental pen- dulum by the method of coinci- dences, the time of coincidence can be observed by the electric method described on p. 40, or by the opti- cal method used by Borda. In the apparatus for Borda's method (Fig. 34), the experimen- tal pendulum is suspended directly in front of a clock pendulum. Between the two pendulums is a screen C containing a vertical slit. Attached to the bob of the clock pendulum is a small mirror which produces an image of a portion of the filament of an incandescent lamp placed at one side of the two pendulums. This image is viewed with a Fig. 34. 68 PRACTICAL PHYSICS telescope placed a meter or more from the clock. When the axis of the telescope, the points of support of the two pendu- lums, and the slit in the screen Care all in a plane perpendicular to that in which the clock pendulum swings, a flash will be seen in the telescope — if the lamp and mirror are properly ad- justed — every time the clock pendulum passes its lowest point except when the two pendulums are in coincidence. To make this adjustment, set the experimental pendulum swinging, and while looking through the slit in the direction perpendicular to the screen, move the incandescent lamp until a bright line of light fills the slit every time the clock pendulum passes. Then place the telescope a meter or more from the slit and in such a position that when the experimental pendulum is deflected a bright flash is seen every time the clock pendulum passes the slit, but when the experimental pendulum hangs at rest no flash is seen. If the slit in the screen C is too narrow — especially when the periods of the two pendulums are almost the same — the eclipse will last for several seconds. In this event, the time of coincidence is the mean of the time of the beginning and the time of the end of the eclipse. If the slit is too wide, no eclipse will be seen, but only a dimming of the flash. Unless the alignment of the two pendulums is better than is usually attained, two eclipses will be observed within a few seconds of each other, one on even-numbered seconds and the other on odd- numbered seconds. The average time of these two eclipses is very close to the true time at which the coincidence occurred. Since the coincidences which occur when the two pendulums are moving in opposite directions are more sharply marked than those which occur when the two pendulums are moving in the same direction, and since these two types of coincidence alternate with each other, it is usually better to observe only the coincidences when the pendulums are moving in opposite directions, and pay no attention to the others. When the apparatus is in adjustment, note the times at which a series of coincidences occur. When the pendulums are swing- VELOCITY AND ACCELERATION 69 ing in opposite directions, and it is seen that a coincidence will soon occur, note the hour and minute and begin counting seconds. Keeping both eyes open, put one eye at the telescope and by the flashes of light that are seen keep on counting seconds. Record the hour, minute, and second every time that a flash does not appear. Repeat about five times. The calculation of the period is explained on p. 40. In mak- ing this calculation it is necessary to know which pendulum goes faster. To determine this, watch both pendulums for a few moments immediately following a coincidence. It will soon be evident that one of them reaches the end of its path before the other, and is, therefore, the one that goes faster. To determine the length of the pendulum, mount a cathe- tometer in front of the experimental pendulilm, make the ad- justment described on p. 23, and read the positions of the knife edge, the top of the bob, and the bottom of the bob. For the length of the pendulum use the distance from the knife edge to the center of the bob. Make at least two determi- nations — each time readjusting the cathetometer — and take the mean. Exp. 10. Determination of the Acceleration Due to Gravity with a Compound Pendulum Object and Theory op Experiment. — The method described in the preceding experiment for determining the acceleration due to gravity requires a clock for each person who is to perform the experiment. If several students are to make determinations at the same time, this would involve ex- cessive cost of apparatus. In the following method only one clock is needed, and, instead of flashes of light appearing at every swing except when the pendulums are nearly in coinci- dence, the flash appears only when the pendulums are nearly in coincidence. The object of this experiment is to determine the acceleration due to gravity by means of a compound pendulum. 70 PRACTICAL PHYSICS Fig. 35. Let A be an axis about whicb a body £ is free to swing, and be the center of mass of B. If £ is swinging back and forth about A, at some instant the line A makes with its equilibrium position an angle 0, and if p denotes the distance from A to (7, and M denotes the mass of JB, then the torque tend- ing to restore B to its equilibrium position is L = - Mgp sin 0, the negative sign being used because the torque and the displacement are in opposite directions. If JST denotes the moment of inertia of B about the axis ^, and a denotes the angular acceleration with which B swings through the indicated position, then we know that It follows that Kr = — Mpg sin 0, or, if is small (see p. 7), KsL=- Mpg0. (47) Since a is proportional to 0, the motion is simple harmonic. If T denotes the period of a complete to-and-fro vibration of a body which is vibrating with simple harmonic motion, it is shown in elementary dynamics that a From (47) it follows, then, that T=2' 'Mpg (48) Now the moment of inertia of B is the sum of the moments of inertia Ki of the n different parts of B. That is, «=1 ... n And if the masses of the different parts are Mj, if the centers of mass of these various parts are at distances pj from the axis of VELOCITY AND ACCELERATION 71 rotation, and if the lines pj are all parallel to p, then from the definition of center of mass we know that Mp=^M^p^. 3 = 1 ■■• n (50) On substituting in (48) the values of K and Mp from (49) and (50), and writing in place of T its value 2 1, where t denotes the time taken by a single oscillation, we get =l-n "Whence, ^=7 X^i X^iPi (51) The pendulum to be used consists (Fig. 36) of a stout piece of steel shafting P, which car- ries at its upper end an adjustable collar to which are fixed the knife edges on which the pendulum swings. Project- , ing below the bottom of the steel rod is a light vertical plate O, in the middle of which is a vertical slit. This pendulum swings in front of an incandescent lamp D, the light from which can be seen only through a vertical slit in a sheet-iron jacket about it. Electrically connected with the clock pendulum is a telegraph sounder which i-s mounted with its arma- ture vertical. Connected with the armature is a light shutter B, in which is a vertical 72 PRACTICAL PHYSICS slit. When the sounder is actuated, this shutter moves just far enough to bring its slit in line with another slit in a screen A, mounted close to it on the base of the sounder. When the pendulum is at rest, and the slits and the filament of the lamp are all in line, a person looking through the screen on the sounder sees a flash each second. When the pendulum is swinging, no flash is seen unless the pendulum passes through its position of equilibrium at the same instant that the current from the clock passes through the sounder. Since a seconds pendulum of the type described above would be about a meter and a half long — that is, too long to deter- mine its length conveniently with a cathetometer — the pendu- lum used is a half-seconds pendulum. During the interval between coincidences, then, the experimental pendulum makes one swing more or less than twice the number made by the seconds pendulum. That is, if the number of clicks between coincidences is w, the number of oscillations made in the same time by the experimental pendulum is 2 w ± 1, and its time of oscillation is, therefore, sec. In general, if n clicks occur 2w±l during the interval from any coincidence to the /th following coincidence, the experimental pendulum makes j swings more or less than twice the number made by the seconds pendu- lum, and the time of oscillation of the experimental pendulum is n : sec. 2w ±^ When the clicks of the sounder are counted, it will often be observed that there is a series of flashes on odd-numbered clicks, and then very soon a series on even-numbered clicks, after a time a series on odd clicks, and then very soon a series on even clicks, and so on. If the apparatus were more accu- rately adjusted, the odd and even sets of clicks would come together instead of one after the other. The interval between coincidences is, therefore, the interval from the middle of one series of odd clicks to the middle of the following series of odd clicks, or from the middle of one series of even clicks to the VELOCITY AND ACCELERATION 73 middle of the following series of even clicks, or from the middle of the time between a series of odd and a series of even clicks to the middle of the time between the following two series. Manipulation and Computation. — Take the diameter of the pendulum rod once with a micrometer caliper. Then see that the pendulum is at rest with room enough to swing freely, and with the knife edges perpendicular to the wall and the plate parallel to the wall. With a cathetometer (p. 23) determine the heights of the top of the rod, the knife edge, the bottom of the rod, and the bottom of the plate. Make two sets of determinations. Adjust the position of the lamp and its jacket until the glow- ing filament in the lamp is seen through the pendulum slit when the eye is directly in front of the pendulum. Place the sounder some 25 cm. or 30 cm. in front of the pendulum, and in such a position that when the armature is held in its position nearest the magnet, the filament in the lamp and the slits in the jacket, pendulum, and sounder diaphragms are all in line. Connect the sounder with the clock, and with the eye close to the shutter watch to see if a flash occurs every time the sounder clicks. Then set the pendulum . swinging with an amplitude not much exceeding 1 cm., and with the eye again close to the shutter watch for the flashes that indicate coincidences. When a flash occurs, begin counting the clicks of the sounder and record the number of each click on which a flash is seen; or, if this is too difficult, count the alternate clicks of the sounder, record the number of each counted click on which a flash is seen, and pay no attention to flashes which occur on clicks that are not counted. In one of these ways make at least two sets of observations, each set running through the times of four or eight series of flashes. If all clicks are counted, the interval between coincidences can be obtained by finding the interval from the middle of the first series of flashes to the middle of the third series, or the interval from the middle of the second series to the middle of the fourth, or, better, the average of these two intervals. If only alternate 74 PRACTICAL PHYSICS clicks are counted, the second and fourtli series of flashes are not recorded, but the interval from the middle of the first series to the middle of the third series that was observed gives twice the time between coincidences, and the interval from the middle of the second observed series to the middle of the fourth observed series also gives twice the time between coincidences. It is to be noted that, if only the alternate clicks are counted, the value for the interval should be doubled to reduce it to seconds. Whether the pendulum is swinging faster or slower than a lialf-seconds pendulum may be determined by watching it for a few moments at about the time when the clicks of the sounder occur when the pendulum is at one end of its path. The apparatus can be so designed that the moment of inertia and mass of the collar may be neglected in comparison with those of the rod, and, further, so that the moment of inertia of the plate about an axis through its center of mass is neglible in comparison with the moment of inertia of the rod ; that is, so that the radius of gyration of the plate may be assumed to be the distance from its center of mass to the axis of rotation. The masses and moments of inertia to be taken into account are, then, those of the rod and the plate. The masses will be given by an instructor, and the moments of inertia are to be calculated by the use of (6) and (2) on p. 111. These values will complete the data necessary for the calculation of g by means of (51). CHAPTER V FRICTION If a body resting on a plane surface is acted upon by a force parallel to the surface, the body does not start to move until this force has reached a certain definite value. Moreover, the force Fp which is necessary to start the body is directly propor- tional to the force F^ which presses the two surfaces together. That is, Fp = jj, #„, in which the constant jjl is called the coefficient of static friction. When the body does start to move, the force which is required to keep it moving uniformly is somewhat less than the force that is needed to start it. And this force FJ which is necessary to keep the body moving uniformly is also directly proportional to the force F^ which presses the two sur- faces together. That is, FJ = hF„, in which the constant b is called the coefficient of kinetic friction. Since Fp is greater than Fp', /J- is greater than b. Exp. 11. Determination of the Coefficient of Friction between Two Plane Surfaces Object aistd Theory op Experiment. — The object of this experiment is to determine the coefficient of friction between two plane surfaces. The apparatus consists of a horizontal plate having a small pulley fastened at one end, and a block that can be drawn along the length of the plate by means of a cord passing over the pulley. Since the pulley possesses friction, the weights on the cord do not accurately represent the force required to overcome the friction between the plate and the block. On this account the 75 76 PRACTICAL PHYSICS force #1 (Fig. 37) is greater than the force F^. The differ- ence between these two forces (^^ — jP^) is a force /^ which has to be applied along the circumference of the pulley in order to start it. But this /^ is proportional to the force /2, — the resultant of the two forces, F-^ and F^,, — which presses the pulley against its bearings. That is, /p =fi^f^, where fi^^^ although not the coefficient of static friction between the pulley and its bear- ings, is a quantity proportional to that coefficient. From Fig. 37, F^^+F^^ = fi Whence, ]• (52) Since Fj, is less than F-^, the numerator of the quantity in square brackets is less than the denominator. Since /ig cannot ^ F" 1 \ f2 w X Fig. 37. W+X Fig. 38. be negative, this means that the negative sign before the radical is to be chosen. /Lt2 may be determined by passing the cord over the pulley as in Fig. 38, applying to one end a weight W-, and finding what weight, X, is required at the other end to start the pulley. Then _ X-W _A ^^ X+W B' where A is written for X— TFand B for X+W. tuting this value for fi^ in (52), it becomes On substi- F,= F, &-A^2B^-A^ &-A^ (53) FRICTION 77 Since the quantity in square brackets is the same for all values of Fp and ^j, it can be denoted by the single letter k and (53) written in the abbreviated form F^ = kF^. The coefificient of static friction between the two surfaces is, then, '^ = S=X'" ^^^^ When the coefficient of kinetic friction is to be determined, (53) must be modified by using in place of Fp, F^, A, and B, the quantities FJ, F^', A', and B', where the primed symbols have the same meanings as the unprimed, except that the primed are taken when the bodies are moving uniformly instead of when they are just starting. Manipulation and Computation. — After cleaning the block and the surface of the plate and making the plate hori- zontal with the aid of a spirit level, place the block near one end and add weights to the pan until the block on being started keeps in uniform motion. Make not less than five determina- tions with different weights on the block. Carefully clean the plate and block before each observation. For each case calcu- late the coefficient of kinetic friction. Having thus determined the actual forces Fp necessary to keep the block in uniform motion when it is pressed against the plate with various forces F„, plot a curve showing, the relation between the two. This curve should be very nearly a straight line, and, if the normal forces, F„, are plotted as abscissas, (54) shows that the tangent of the slope gives the coefficient of friction. Determine the tangent of the slope and see how the result checks with the mean of the previous results. Exp. 12. The Friction of a Belt on a Pulley Object and Theory of Experiment. — The object of this experiment is to determine the coefficients of static and kinetic friction between a belt and a pulley. 78 PRACTICAL PHYSICS Let U CrllJ repvesbnt the portion of the belt in contact with the pulley whose center is C. On account of the friction be- tween the two surfaces, the tension of the belt will vary all along the length in contact with the pulley. When the belt is just on the point of slipping, let the tensions at the ends of the arc GS subtending the indefinitely small angle A@ be de- noted by / and /'- Let F and F' represent the tensions of the belt where it leaves the pulley. By compounding the forces/and /' — which are approximately equal because A® is very small — it is found that, the normal force against the pulley due to the element of the belt GS is equal to ^ n, 180° -A® „ ., . A© . .,.^ 2/ cos^ = 2/' sin— - =/'A©. Therefore when the belt and pulley are in equilibrium, and the belt is just on the point of slipping, the coefficient of static friction is f —f ^ ■ f'AB Whence, Z. = i _ ^A0, or log,/- log,/'= log, (1 - /xA@). Expanding into a series the right side of this equation, log,/- log,/ = - /.A© - J- (yuA@)2 - 1 (/.A@)3 - etc. But when A® is chosen very small, its second and higher pow- ers become negligible in comparison with its first power, and we may write , j., i j. . » ^^ log,/'-log,/=M@- If we write expressions like the above for all the elementary arcs that the belt touches, and then write the sum of the left members equal to the sum of the right members, we get log,#'-log,J?'=/.0. FRICTION 79 Whence, ,. = log, F' - log, F ^ ^55^ in which the angle @ is measured in radians. The reason for not using the approximation sign in the last two equations is this: Approximations have been made in two places, and in each one the approximate value approaches the true value when A@ approaches zero. That is, in the limit, when A® = 0, these last two equations hold exactly. In precisely the same manner is obtained the coefficient of kinetic friction j^^log,F"-log,F^ (56) @ • where F and F" are the tensions of the belt where it leaves the pulley, when the belt is slipping at a uniform rate. Manipulation and Computation. — Stretch the belt over a pulley that can be rotated by means of a crank. To one end of the belt apply a 10 lb. weight and to the other end a verti- cally hanging spring balance whose lower end is fastened to the floor. Now turn the crank so as to carry the belt away from the spring balance until the belt is just on the point of slipping. The spring balance reading plus the weight of the balance is now F', F=10 lbs. weight, and @=180° = 7r ra- dians. Consequently a value of fi can be computed. Repeat, making F successively equal to 20, 30, etc., pounds weight, until the limit of the spring balance is reached. The mean of the values of fj, thus obtained is to be taken as the coefficient of static friction between the belt and the pulley. Determine this coefficient for both the flesh side and hair side of the belt. When the pulley is rotated until the belt slips and then the speed of rotation is kept constant, the spring balance reading is F", while, as before, F equals the weight acting on the other end of the belt, and © equals ir radians. From these values b is computed. 80 PRACTICAL PHYSICS Exp. 13. Determination of the Coefficient of Friction between a Lubricated Journal and its Bearings (GOLDEN'S METHOD) Object and Theory of Experiment. — The object of this experiment is to determine the coefficient of kinetic friction between a cylindrical jour- nal and its bearings for dif- ferent loads, speeds, and temperatures. The Golden Bearing and Oil Testing Dynamometer consists of a spindle passing through a bearing B (Fig. 40), form- ing part of a yoke Q. The spindle can be rotated at various speeds by means of a motor, and the yoke can be weighted by means of adjustable masses M' and M" . As the spindle is ro- tated the friction between it and the bearing tends to rotate the yoke also. This tendency to turn is measured by the spring dynamometer D, which is essentially a spring balance. For the testing of oils at different temperatures, the collar A is cored in such a way that a stream of water or steam can be passed through it and the temperature of the bearing determined by means of a thermometer T. If r be the radius of the shaft and F the total force of fric- tion tangential to the surface of the shaft, the turning moment resulting from the friction of the shaft and bearing is Fr. If / represents the force, having a lever arm I, required to keep the yoke from turning (Fig. 41), the resisting torque isfl. If the center of mass of the yoke with its appendages is vertically Fig. 40. FRICTION 81 below the axis of rotation of the shaft, then when the shaft is rotating and the yoke is held steady, Fr=fl. If the total weight on the bearing surface due to the yoke and its accessories together with the masses M' and M" be denoted by P, then the coefficient of kinetic friction P Pr If the surface of contact between the journal and bearing be projected upon a horizontal plane, and if the area of this projection be denoted by A, then the pressure on the bearing is P It follows that Fig. 41. pAr (57) Manipulation and Computation. — Measure (I + r) and 2r with calipers and scale. Find the area A of the projection, on a horizontal plane, of the surface of contact between the journal and bearing. After cleaning the journals and bearing with benzine, lubri- cate with the assigned oil and apply small and nearly equal loads to the ends of the arms of the yoke. The difference be- tween these two loads should be sufficient to develop a turn- ing moment due to gravity slightly greater than that due to friction. Start the motor and by means of the spring dynamometer D measure the tendency of the yoke to turn. Reverse the direction of rotations and take another dyna- mometer reading. By this operation the pull developed by the friction between the shaft and bearing is first added to the pull on the dynamometer due to the excess weight on one 82 PRACTICAL PHYSICS end of the yoke, and then subtracted from it. The difference between the two dynamometer readings is 2/. The data are now at hand for computing the coefficient oi kinetic friction between the given surfaces lubricated by the assigned oil, for the particular speed, temperature, and pressure per unit area of bearing surface, used in this determination. Proceeding as above described and keeping the temperature constant, determine, for a fixed speed of rotation s, the values of b corresponding to a series of values of p. With these values plot a curve coordinating b and p for the given speed and tem- perature. Keeping the temperature fixed, now change the speed of rotation and determine a new series of values of b and p. Plot a curve coordinating these values on the same sheet with the other. Proceeding thus, plot on the same sheet about five curves for the same temperature but different speeds. In the same manner obtain data for and construct on an- other sheet of coordinate paper a series of curves showing the relation between b and p for a given constant speed when the temperature t is changed. From the first set of curves construct another set coordinat- ing b and s for different fixed values of p when the temperature is constant. And from the second set of curves construct another set coordinating b and t for different fixed values of p when the speed is constant. Care should be exercised that the direction of rotation of the journal is frequently reversed, especially when the bearing is heavily loaded, so as to avoid error due to inequality of the wearing of the bearing. Exp. 14. Determination of the Coefficient of Friction between a Lubricated Journal and its Bearings (THURSTON'S METHOD) Object and Theoky op Experiment. — The object of this experiment is to compare the lubricating properties of different FRICTION 83 oils from their relative effect in reducing the friction between a journal and its bearings. The Thurston Oil Testing Machine to be used in this experiment consists of a heavy pendulum having at one end a bearing through which passes a horizontal shaft capable of rotation. The bearings can be caused to exert any given pressure on the journal by means of a heavy coiled spring and adjusting screw, forming part of the pendulum. When the shaft is rotated, the pendulum is deflected through an angle determined by the moment of the tangential effort at the Fig. 42. Fig. 43. circumference of the journal and the moment of the weight of the pendulum. Let W represent the weight of the pendulum ; S, the expan- sive force of the spring; J, the mean normal force between journal and bearings; H, the distance from the axis of the journal to the center of mass of the pendulum ; F, the tangen- tial effort at the circumference of the journal — numerically equal to the force of friction; r, the radius of the journal; I, the length of the journal; and b, the coefficient of kinetic fric- tion between the journal and its bearings. If the pendulum is in equilibrium when deflected through an 84 PRACTICAL PHYSICS angle 6, the couple due to the forces FF at the circumference of the journal equals the moment of the weight W. That is, 2Fr=WRsm 0. Since the upper bearing exerts on the journal a force equal to the sum of the weight of the pendulum and the expansive force of the spring, while the lower bearing exerts a force due only to the spring, the mean force between journal and bearing is j_ CW+S) + S ^ 2S+W 2 2 ■ Consequently the coefficient of kinetic friction between the journal and bearings is J ^^ -sin 0, (58) r{2S+W) or b = k sin 0, (59) where k represents the constant coefficient of sinO in (58). This constant can be determined from a single series of care- fully made measurements and used in any computation of b, so long as the force exerted by the spring is unchanged. Manipulation and Computation. — Measure the diameter 2 r of the journal with a pair of calipers. Obtain the weight Woi the pendulum. Observe the angle on the divided arc attached to the apparatus. Place the coiled spring in a testing machine and measure the forces required to produce given compressions. Plot a curve coordinating forces and resulting compressions. From this curve may be read off directly the force S corresponding to any compression measured by a scale and vernier attached to the side of the pendulum. The distance M from the axis of the journal to the center of mass of the pendulum can be determined as follows : While the pendulum is still suspended from the shaft, support the free end on a knife edge resting on the platform of a balance. (See Fig. 43.) The product of the weight observed and the horizontal FRICTION 85 distance L between the supporting knife edge and the center of the shaft equals WR. Compute the pressure on the journal. , Since the projection of the journal surface equals 2(2 r)^, the pressure is After cleaning the journal and bearings with benzine, lubricate with one of the oils to be tested, apply a given force S to the spring, set the shaft into motion, and observe the deflection 6 of the pendulum. With the speed of rotation kept constant observe the deflec- tions produced for several values of the force S. Repeat the same series of observations when the other oils are used. Care- fully clean the journal and bearings after each sample is examined. Plot for each specimen a curve coordinating the coefficient of kinetic friction h and the force per unit area of bearing surface p. The laws brought out by these curves should be carefully discussed. CHAPTER VI MASS, DENSITY, SPECIFIC GRAVITY Exp. 15. Calibration of a Set of Standard Masses Object and Theory op Experiment. — In making a set of standard masses it is impossible to get the mass of each piece exactly right. Moreover, the handling of a set of masses, even when carefully done with forceps, necessarily wears them a trifle. In addition, some dust settles upon them, there may be chemical action with vapors in the air, etc. It is consequently necessary in accurate work to compare the masses with each other ; and where absolute weighings are to be made, the masses in the set must be more or less indirectly compared with the ultimate stand- ard. The object of this experiment is to compare the masses of the various members of a set and to construct a table of corrections. The method employed is by means of a sensitive balance to find the differences between masses or groups of masses supposed to be equal, from these results to form as many separate equa- tions as there have been weighings performed, and from these equations to find the masses of the different pieces in terms of some convenient unit. In this experiment the unit of compari- son will be the mass of one of the standards in the set being calibrated. Consider a set consisting of a 10-mg. rider, eight aluminium or platinum masses ranging from 10 mg. to 500 mg., and nine brass masses ranging from 1 g. to 100 g. Call the mass of the rider /•, the masses of the fractional gram pieces respectively lOj, 20i, 2O2, SOp lOOj, 200^, 2OO2, SOOj, and the masses of the brass pieces respectively ij, 2^, 2^, 5^, lOj, 20^, 2O2, 50^, 100^. 86 MASS, DENSITY, SPECIFIC GRAVITY 87 First the position of rest is determined (a) when the balance is unloaded and (6) when r is at the 1-mark on the beam. These observations give with sufficient accuracy the sensitiveness of the balance for the first few small loads. 10^ is now placed on one pan and r either on the other pan or at the 10-mark on the other side of the beam. From the position of rest deter- mined under these conditions, the position of rest of the unloaded balance, and the sensitiveness of the balance the value of lOj in terms of r can be calculated : — lOj = ?• + a^r = r(l + a{), where the a^ is a small number which may be either positive or negative. 20^ is now placed on one pan, 10^ on the other, and r either on the pan with lOj or at the 10-mark on the beam. As in the previous case, the value of 20j can be determined in terms of r : 20i = lOj H- r -h a^r, or, substituting for lOj its value from the preceding equation. In the same way is obtained 202 = 10i-i-r -f-flgj- = r(2+ a^ +a3). Throughout the rest of the calibration the rider is kept on the beam, and the sensitiveness is determined for each load. With 50j in one pan and 20^, 2O2, and 10^ in the other, the po- sition of rest and the sensitiveness are determined. This gives the value of 50^ : — 50i = 2O1 -I- 2O2 + lOj -I- a^r, or substituting for lOj, 20j, and 2O2 their values from the above equations, 50j = ?*(5 + Sai + a2+ a^+ a^"). In the same way lOOi = 50i -f- 2O1 + 2O2 -I- lOj + a^r = r(10 + 6aj^ + 2a^ + 2a^+ a^ + a^). 88 PRACTICAL PHYSICS 200i = lOOj + 50^ + 20i + 2O2 + 10^ + a^r = r(20 + terms involving fljj . . . ttg). 2OO2 = lOOi + 50i + 2O1 + 2O2 + 10^ + a^r = r(2Q + terms involving aj . . . a^}. 500^ = 2OO1 + 2OO2 + 100^ + a^r = r(50 + terms involving a^ . . . a^). 1^ = 5OO1 + 200^ + 2OO2 + lOOj + UgV = r (100 + terms involving flSj . . . ag). 2i = 1 1 + 500i + 2OO1 + 2OO2 + lOOj + aio»- = ?'(200 + terms involving «! . . . ajg). 22 = li + 500i + 2OO1 + 2OO2 + lOOi + a^jT = r(200 + terms involving a^ . . . a^). 5^ = 2i + 22 + li + a^^r etc. In the above equation the a's are all experimentally observed, so that if the mass of any one of the pieces in the set is known in terms of the ultimate standard, then from the equation in- volving the mass of that piece can be calculated the mass of the rider. When the mass of the rider is known, the masses of all the pieces in the set can be calculated from the respective equations. Manipulation and Computation. — Perform the opera- tions indicated above. Make all weighings by the method of vibrations, and with the brass pieces use the method of double \eighing. Assuming that the 100 g. mass is correct, determine he masses of all the other pieces in terms of it. Record t le results in a three-column table, putting in the first column the symbol used to denote the particular mass considered, in the second the value obtained for this mass, and in the third the correction for the mass as obtained from (2). DENSITY AND SPECIFIC GRAVITY If a body of mass m occupies a volume v, then the average density of the body is given by i> = -• (61) MASS, DENSITY, SPECIFIC GRAVITY 89 From this expression it is seen that the number which ex- presses a density depends upon the units in terms of which the mass and volume are measured. For example, at 4°C. the density of lead is about 708 pounds per cubic foot, or 2868 grains per cubic inch, or 11.34 grams per cubic centimeter. Since density is a concrete quantity, the units in terms of which the mass and volume of the body are measured must always be stated. ' Since most bodies change their volume somewhat with changes of temperature, the density of a substance depends upon its temperature ; and so in accurate work the tempera- ture at which a determination is made should always be stated. The specific gravity or relative density of a substance is the ratio of its density to the density of some standard substance. In other words, the specific gravity of a body is the ratio of its mass to that of an equal volume of a standard substance. Specific gravity is thus a numerical ratio, an abstract number which is independent of the units employed. For solids and liquids, water at the temperature of its maximum density (4° C. or 39° F.) is arbitrarily taken as the standard substance. Since in the C.G.S. system of units the unit of mass is the mass of a unit volume of water at the temperature of its maxi- mum density, it follows that the density of a body in grams per cubic centimeter is numerically equal to its specific gravity. Exp. 16. Determination of the Density of a Solid by Measure- ment and Weighing Object and Theory of Experiment. — From (61) it will be seen that the density of any solid could readily be determined if a specimen of it could be obtained in a shape such that its volume could easily be computed. Manipulation and Computation. — The specimen to be used is a cylinder. Measure its diameter with a micrometer cali- per and its length with a vernier caliper (see pp. 17, 21) and cal- culate the volume. Determine the mass by weighing, using the method of vibrations (pp. 26-28). In order to get a very 90 PRACTICAL PHYSICS accurate value for the density it will be necessary to correct the weighing by allowing for the buoyancy of the air. First, without making this correction, divide the apparent mass by the volume and so get an approximate value for the density. Use this value in (14) to get the true mass of the cylinder. Calculate the density by (61). Exp. 17. Determination of the Density and Specific Gravity of a Liquid with a Pyknometer Object and Theory of Experiment. — The pyknometer is essentially a small glass vessel of definite volume. Various Fig. 44. Fio. 45. Fig. 46. Fig. 47. forms suitable for determining the densities of liquid are given in Figs. 44-48. The pyknometers in Figs. 45 and 48 can be used only for liquids, while the others can be used for either liquids or solids. The most common form, that shown in Fig. 46, consists of a MASS, DENSITY, SPECIFIC GRAVITY 91 small bottle fitted -with a perforated glass stopper that always comes accurately to a seat at the same point, so that the volume of the bottle is definite when the stopper is in place. This form is often called a specific gravity bottle. The volume of the pyknometer is obtained from two weighings, first when empty, and second when filled with a liquid of known density, e.g. water. If the mass of water contained in the filled pyknometer is denoted by My, and its density by /o„, then the vol- ume is M,„ v = Fig. 48. Now let the water be replaced by the speci- men. If the mass of this second liquid filling the pyknometer be denoted by M^ and its density by p^, then _M,_M,p, M„ (62) Denoting the maximum density of water by S, we have for the specific gravity of the specimen. Sp. Gr. =^ M,p„ (63) In the preceding equations no account has been taken of the buoyant effect of the atmosphere on the liquids being weighed and on the standard masses used in the weighing. In precise determinations this source of error cannot be neglected. The true weight of an object equals its apparent weight plus the weight of air displaced by it. And when the balance is in equilibrium, the apparent weight of the body equals the appar- ent weight of the standard masses. So that when the specimen is weighed in air, its true weight minus its loss of weight due to the buoyancy of the air equals the true weight of the standard masses minus their loss in weight. If the density of air is de- noted by Pa and the density of the standard masses by /)j, this 92 PRACTICAL PHYSICS last statement says that when the pyknometer was filled with the first liquid, meter was fille^ '*>Ps — vpa=]^B- Pb and when the pyknometer was filled with the second liquid, M,Pa _ Pb On eliminating v from these equations, we obtain p^=mh^+p^. (64) Manipulation and Computation. — Weighing by the method of vibrations, determine first the mass M^ of the empty pyknometer ; second, the mass (^Mp + My, ) of the pyknometer filled with recently distilled water ; and, third, the mass (iH^ + ilif^) of the pyknometer filled with the liquid in question. Take the values of /}„ and p^ from tables. Each time before filling the pyknometer, clean it by rinsing successively with nitric acid, distilled water, and alcohol, and then dry it by putting into it the end of a tube connected to an exhaust pump. Be sure that there are no air bubbles in the pyknometer, that the outside is dry, that the stopper is in place, and that the liquid fills the capillary tube in the stopper. In order to avoid changes in volume due to changes in tempera- ture, avoid touching the filled bottle with the bare hand. Exp. 18. Determination of the Density and Specific Gravity of a Solid with a Pyknometer Object and Theory of Experiment. — The object of this experiment is to determine the density and the specific gravity of a solid in small pieces. Three suitable forms of pyknometer have already been illus- trated (Figs. 44, 46, 47). To determine the volume of a solid by means of a pyknometer, four weighings are made : first, when the pyknometer is empty ; second, after the specimen MASS, DENSITY, SPECIFIC GRAVITY 93 has been introduced ; third, after the rest of the space in the pyknometer has been filled with water or other liquid; and, fourth, after the pyknometer has been emptied and then filled with the same liquid used in the third weighing. Let the mass of the pyknometer be denoted by TUfp, that of the specimen by M^, that of the water which fills the pyknometer by My,, and that of the water which was used with the speci- men by m„. Then if the mass found in the wth weighing is denoted by iff,„ ilfj = M^, (65) M^ = Mj, + M,, (66) Ms = M^ + M,+m„, (67) M, = Mp + M^. (68) The mass of the water which displaces the specimen is (il!f^ — m„), and from (67) and (68), M^-m^= [M, - M^-] - [Ms - (ilf^ + iff,)] . Substituting in this equation the values of Mp and (^Mp + Jf,) from (65) and (66), ^« - m^ = (^4 - ^i) - (M% - ^2) = (ilf2+ilf4)-(ilfi + ilf3). If py, denotes the density of the water used, the volume of the water which displaces the specimen, and therefore the volume of the specimen, is ^ AM.. + M,-)-CM,^M,^ Pw Now the density of the specimen is M, M„ - M, V V On eliminating v from the last two equations, we get P' CM, + M,-)-(M^ + Ms-) "^ ^ 94 PRACTICAL PHYSICS If 8 denotes the maximum density of water, it follows that the specific gravity of the specimen is This method is capable of very accurate results. In precise measurements account must be taken of the buoyant effect of the air on the specimen and on the standard masses used in the weighing. The true weight of an object equals its appar- ent weight plus the weight of the air displaced by it. When the balance is in equilibrium, the apparent weight of the body equals the apparent weight of the standard masses. Conse- quently, when the specimen is weighed in air, its true weight diminished by its loss in weight due to the buoyancy of the air, equals the true weight of the standard masses diminished by their loss in weight. If the density of air is denoted by p^ and the density of the standard masses by /a^, this last statement says that VPs - vpa = if, - Pi> ■ In the same way considering the water which displaced the specimen, vp. - vp, = (Jf„ - m„) _ (^«>-^»>a . Pb On eliminating v from the last two equations, we get or, substituting for M, and (^M^—m^) their values in terms of the masses actually observed. This gives the density at f, the temperature at which the fexperiment was performed. If 7 denotes the coefficient of MASS, DENSITY, SPECIJFIC. GRAVITY 95 cubical expansion of the specimen, then its density at 0° is given by {M.-M.Xp.-Pa^ + P J (1 + 70 ■ (72) .(lf2 + iJf4)-(ifi + if3) n The development of (72) from (71) is left as an exercise for the student. Manipulation and Computation. — Make all weighings hy the method of vibrations. Observe the precautions that are suggested in the last paragraph under Experiment 17. Exp. 19. Determination of the Density and Specific Gravity of a Solid by Immersion Object and Theory op Experiment. — The object of this experiment is to determine the density and specific gravity, of a solid of irregular form. Since a solid body immersed in a liquid is acted upon by an up- thrust equal to the weight of the liquid displaced by the body, it follows that if this upthrust is measured, the weight of the displaced liquid is known, and if the weight of a unit volume of the liquid is also known, then the volume of the liquid dis- placed — and, therefore, the volume of the body — ■ can be cal- culated. If the weight of the body in air is denoted by Ba, and its weight when immersed in the liquid by Bi, then the up- thrust of the liquid — and, consequently, the weight of the liquid displaced — is Ba — -B;. So that, if w denotes the weight of a unit volume of the liquid at the temperature of the experi- ment, the volume of the liquid displaced — and, consequently, the volume of the body — is z> = ^"~-^' - (73) w It follows, if m denotes the mass of the specimen, that the den- sitjr of the specimen is m _ mw _ B„pi .y .. f'- v-B^-B.-JSa-B: ^ ^ the last equation in (71) being true because m = Ba/ff and w = p,g. 96 PRACTICAL PHYSICS Since specific gravity is defined as the ratio of the density of the substance in question to the maximum density, B, of water, Sp. Gr. .^aPl (75) When the body is lighter than the liquid in which it is to be immersed, a sinker is attached. Weighings are made to deter- mine : first, the weight of the body in air, B^, ; second, the weight of the sinker immersed in the liquid, S, ; and third, the weight of the two together when immersed, (5 + S)/- The weight of the body alone when immersed in the liquid is nega- tive, but its value, sign included, is and this value can be substituted in (74) and (75), giving ^'"^,-(^ + ;S'), + ^, *^^^^ Aft (77) ' B Manipulation and Computation. — The liquid in which the body is immersed must be one which will not dissolve the body, act upon it chemically, nor cause it to change its volume. Whenever possible, use is made of water which has been freed of dissolved gases by boiling. If the liquid contains dissolved gases, bubbles will collect on the immersed body, causing an increased upward thrust, and therefore an error in the result. Water should be boiled for about half an hour and then cooled to the temperature at which the experiment is performed. As water slowly dissolves air, it must be boiled on the day it is used. The motion of the balance beam is so much damped by the immersion of the load in a liquid that it is useless to weigh by the method of vibrations. The values of pi and B are to be taken from tables. MASS, DENSITY, SPECIFIC GRAVITY 97 Exp. 20. Determination of the Density of a Solid or Liquid with Jolly's Spring Balance Object and Theory of Experiment. — The Jolly spring balance is especially suited to the determination of the densities of liquids and of solid bodies of small mass. The essential part of the instrument is a spiral spring which hangs vertically and carries at its lower end a weight pan. If the limit of elasticity is not passed, any increase in the length of this spring is proportional to the force which is applied to it. In Lin charger's form of the instrument (Fig. 49) the spring is supported by two vertical telescoping tubes. The inner tube can be adjusted up or down by turning the milled head D. To the lower end of the spring is attached an index J, consisting of a double cross of aluminium, half of which is painted black. This index hangs inside a short length of glass tubing which is whitened on the back, and which carries on its inside surface a horizontal black hair line. This line serves as a zero, to which the line separating the black- ened from the unpainted part of the index may be brought^ To the lower end of the index is at- tached a thin wire supporting two pans W and 0. If, after the index has been brought to the zero mark, a body be placed on one of these pans, the index can again be brought to the zero mark by adjusting the height of the inner supporting tube A. On this tube is engraved a millimeter scale, which, by means of a vernier at Fj can be read to tenths of millimeters. The difference between the reading at Fwhen one of the scale pans is loaded and when not loaded gives the elongation of the spring due to the weight of the body- When the body does not weigh \B w/ Pig. 49. 98 PRACTICAL PHYSICS more than about 5 g., accurate results are possible with this method. In the case of a solid that will sink in a given liquid of known density, the lower pan is submerged, and, after the in- dex has been brought to the zero line and the reading at V noted, the specimen is placed in the upper pan and the elonga- tion of the spring, b^, necessary to bring the index back to the zero line, is determined. The specimen is then placed on the submerged pan and the new elongation, J^, is found. Since 6„ is proportional to the weight of the specimen in air, and bi is proportional to the weighls of the specimen when sub- merged in the liquid, (74) and (75) can be put in the forms and Sp.Gr.[=|] = ^^, (79) where Pi is the density of the given liquid and S is the maximum density of water. .In the case of a solid that floats in the given liquid, a sinker must be attached. With the apparatus arranged as before, let the elongation of the spring when the speci- men is in the upper pan be represented by ba, the elonga- tion when the sinker alone is in the submerged pan be represented by s„ and the elongation when the specimen and sinker are tied together and are in the submerged pan by (b + s')i. Since these elongations are proportional to the forces which produce them, (76) and (77) can be put in the forms p r = ^ 1 = ^-^ (80) and Sp. Gr. [=^'1=77 77^^^^^ r^- (81) In determining the specific gravity of a liquid by this method a sinker that is unaffected by water and by the given liquid is weighed in air, in water, and in the liquid whose specific gravity MASS, DENSITY, SPECIFIC GRAVITY 99 is required. If the elongations of the spring when the sinker is in the air, in water at 4° C, and in the given liquid, are de- noted respectively by s„, s„, and S;, it is easily shown that the specific gravity of the liquid is given by Sp. Gr.=?5LZiL. (82) If the temperature of the water is t° instead of 4°, the right member of (82) is to be multiplied by the specific gravity of water at t°. Manipulation and Computation. — In determining each elongation it is necessary to make a reading of the scale at V before the spring is extended, as well as afterward. The suspended system must hang free without touching either the glass tube around the index or the beaker containing the liquid, and the submerged part of the suspended system must be kept free from air bubbles and always submerged to the same depth. The upper pan and its contents must be kept dry. After im- mersion in any liquid, the sinker, specimen, and scale pan should be carefully dried with filter paper. Exp. 21. Determination of the Specific Gravity of a Liquid with the Mohr-Westphal Balance Object and Theory op Experiment. — The object of this experiment is to determine the specific gravity of an aqueous solution by means of a Mohr-Westphal balance. From Archimedes' principle it follows that if a body of constant volume be immersed in various liquids, the corre- sponding losses of weight sustained by the body will represent the weights of equal volumes of the various liquids. Whence, if a body of volume v, when immersed in succession in two liquids, of densities p^ and p^, sustain the respective losses of weight Wj and w^i then '^^^M^Pl. (83) Wa '"P29 Pi 100 PRACTICAL PHYSICS If the second liquid be water at the temperature of its maximum density, then the ratio of w^ to w^ gives the specific gravity of the first liquid. If, therefore, a means be devised for measuring the loss of weight of a given body when immersed in any liquid, and also for determining what loss the same body would suffer if it were immersed in water at 4° C, the specific gravity of the liquid could be computed by means of the above equation. A convenient instru- ment designed for the' purpose is the Mohr- Westphal balance. This device (Fig. 50) consists of a decimally divided balance beam at one end of which is suspended a glass sinker for immersion. The other end of the beam is so counterbal- anced that the beam is held in equilibrium when the sinker is surrounded by air. The instrument is also provided with five riders which are ordi- narily equal in mass to 1.0, 1.0, 0.1, 0.01, and 0.001 of the mass of water displaced by the sinker. Thus, if the sinker be immersed in water, one unit rider placed at the end of the beam would be required to compen- sate for the loss sustained by the sinker and to bring the beam back to a horizontal position. Again, if with the sinker immersed in a certain liquid the beam is brought into a hori- zontal position when a unit rider is hung on the hook A^ the Fig. 50. MASS, DENSITY, SPECIFIC GRAVITY 101 teaths rider on the second notch C, and the hundredths rider on the third notch B, the theory of moments of forces show- that the upthrust on the sinker is 1.023 times as great as in the preceding case. Consequently the specific gravity of the given liquid is 1.023. If, as the temperature rose, tlie sinker were to expand at the same rate that water does, the temperature at which the Mohr-Westphal balance is used would make no difference, for the sinker would always displace the same mass of water. But, as a matter of fact, at ordinary room temperatures water expands more rapidly than glass, so that when the temperature is a little above 20° C. the Mohr-Westphal balance reads 0.1 % lower than it would at 15°. Moreover, the temperature in a laboratory is usually not so low as 4° C, and so the riders are usually adjusted to read specific gravities with reference to water at 15° — about the temperature at which European laboratories are usually kept. In order to use the balance in a laboratory at about 20° and to get specific gravities with reference to water at 4° it will then be necessary to apply a correction. To find what this correction is, let h-^^ and l^ denote the re- spective readings of the balance when the sinker is immersed, (1) in water of density pu at 15°, and (2) in the liquid whose density pt at f is desired, and let w^j and v^ denote the respec- tive volumes of the sinker. Then the weights of liquid dis- placed by the sinker in the two cases are respectively pi^v-^^g and pfVtff- Since the readings of the balance are proportional to these weights, -K'*i5 = Pui^u9 = Pi&^o9 (1 + 7 • 15) (84) and Kbt = p^Viff = ptV^g (1 -f yt), (85) where v„ denotes the volume of the sinker at 0° and 7 its coeffi- cient of expansion. On dividing (85) by (84), we obtain ht _, ft(l + 7t) hs Pi5(l + 7 ■ 15)" 102 PRACTICAL PHYSICS Whence, since the balance is so adjusted that Sjg = 1, or, employing approximation (5), p. 7, or, employing approximation (2), p. 7, pt=Pi,h\y-^(t~lb-)-]. (86) If the specific gravity of the liquid is desired, we have at once, if S denotes the maximum density of water, Sp. Gr. [= |]= ^[1- 7(« - 15)] . (87) Since 7 is small and pjg differs only slightly from S, it will be seen that if only fairly accurate values are desired, (86) and (87) give very nearly Pt = ^it (88) and Sp. Gr.=6,. (89) Manipulation and Computation. — With the sinker in air and no rider on the beam, the instrument is first leveled until the pointer attached to the beam indicates zero. The sinker is then immersed in the liquid whose specific gravity is to be determined, and riders are placed in the notches on the beam until the pointer again indicates zero. Exp. 22. Calibration of an Hydrometer of Variable Immersion Object and Theory op Experiment. — In the measure- ment of the specific gravity of liquids for technical purposes where great accuracy is unnecessary, some form of hydrometer of variable immersion is usually employed. The hydrometer (Fig. 51) consists of a closed graduated glass tube of uniform cross section with a weighted bulb on the lower end. The mass and volume of the instrument are so chosen that when it is placed in the liquid whose specific gravity is to be deter- mined it will float upright. The specific gravity of the liquid MASS, DENSITY, SPECIFIC GRAVITY 103 I » I,. is shown by the depth to which the hydrometer sinks. If the graduations on the stem are so spaced and numbered as to give directly the density of the liquid, the instrument is called a densimeter. Often, however, the gradu- ations are equidistant and are referred to some arbitrary scale. Thus we have the scales of Baume, Beck, Cartier, and Twaddell. The specific gravities corresponding to readings on these vari- ous scales are given in Table 6. Not infrequently the stem of the hydrometer contains two or more scales. When graduated with especial reference to use with some particular class of liquids, the hy- drometer is called the alcoholimeter, salinimeter, etc. A calibration curve for any instrument is a curve in which the actual readings of the instrument are plotted against the readings that the instrument ought to give. The object of this exercise is to calibrate an hydrometer. (a~) Scale with divisions of equal length. If an liydrom- eter of mass m sinks to scale division d-^ when placed in a liquid of density /j^, and to division d^ when placed in a liquid of density p^, then by Archimedes' principle the volume of the first liquid displaced is — and of the second is m Pi —. If u denotes the volume of that part of the stem which is P2 included between two consecutive scale divisions, then Fig. 61. mm .-, — = uQdj^- P2 Pi ■C?2). Whence u=- or P2 = m(p,. - Pi) PiP2i<^i - '^%) ' m — Pi^'id^ — d^ (90) (91) From (90), if p■^, p^, and m are known, the value of u can be found, and from (91), if u, p^, and m are known, p^ can be found. 104 PRACTICAL PHYSICS If the maximum density of water is denoted by S, the spe- cific gravity of the second liquid is Sp. Gr. S mPx Im-pjuidi-d^-)}^ ' (92) (6) Scale in which the successive divisions express equal dif- ferences in density. Consider a wooden rod of mass m, of uniform cross section q, and so loaded at one end that it will float upright. When the rod floats, the weight of liquid dis- placed is by Archimedes' principle equal to the weight of the rod. That is, if the rod sinks a distance l^ in a liquid of density pj, PihW = ing- r- Whence ?i = — . (93) <■ Pii K- r \J Fig. 52. I Similarly, if the rod sinks a distance l^ in a liquid I of density p^i i h=~- (94) ! Pa? f Dividing (93) by (94), I ^ = £2. (95) I h Pi That is, the distances to which this hydrometer of uniform cross section sinks in various liquids are inversely proportional to the densities of those liquids. Consider now an hydrometer of the usual form, which is not of uniform cross section throughout, but which is of uniform cross section above some point JT (Fig. 52). For this hydrom- eter there is at some unknown distance x below IC a point to which the hydrometer would extend if it had still the same mass and volume which it really has, but if, instead of the varying cross section which it really has, it continued through- out with the same cross section which it has above JBT. Sup- pose that in one liquid this hydrometer sinks to a point distant MASS, DENSITY, SPECIFIC GRAVITY 105 hi above K, and in another liquid to a point distant h^ above K. Then from (95), L^2 J K + - Pi or V2 - ^iPi , (96) (97) Sp. Gr. (98) (99) P1-P2 If the subscript 2 is dropped, (96) gives If the maximum density of water is denoted by B the specific gravity of the liquid is, then, _P~]_ Pi(^i+a:) Sj S(A+a;) Thus if we determine to what distance above K the hydrom- eter sinks in each of two liquids of known densities, we can by (97) determine x. And if we know to what distance above ^the hydrometer sinks in one liquid of known density, and know also x, then if we determine to what distance above K the hydrometer sinks in any other liquid, we can by (99) determine the specific gravity of that liquid. Uniformity of cross section of the hydrometer may be tested by reading diameters at various points with a micrometer cali- per. If the cross section is not uniform above IC, the above method of calibration is not applicable. In this dozen or twenty solutions having densities varying somewhat uniformly within the range of the hydrometer should be made up, the density of each determined, and the reading of the hydrometer in each taken. This method of calibration is, of course, accurate, but is more tedious than the other. Manipulation and Computation. — The surface of the liquid about an hydrometer is usually of a shape similar to that in Fig. 53. AB is the stem of the hydrometer and CD is a tall case some Fig. 53. 106 PRACTICAL PHYSICS narrow jar in which the liquid is placed. First be sure that the hydrometer is floating freely, and then place the eye below the level of the liquid surface and raise it until it is sighting the hydrometer along the dotted line. The point of the scale crossed by this line is the required reading. The temperature of the liquid should be noted at the time of each observation. When changing from one liquid to another, the jar, hydrome- ter, and thermometer are to be thoroughly washed and dried. Determine the densities of two liquids either with a pyknometer or with a Mohr-Westphal balance. Observe the scale readings on the hydrometer when it is floated in turn in the two liquids. (a) If the hydrometer has a scale with equal divisions, weigh the instrument, place it in succession in two liquids of known densities, and then bymeans of (90) calculate the value of u. By means of (92) calculate the specific gravity corresponding to each of the numbered scale divisions on the stem of the hydrome- ter. Plot a curve with these calculated specific gravities as abscissas and the corresponding scale readings as ordinates. This is the calibration curve of the instrument. The calibra- tion curve should be checked by comparing two or three values obtained by means of the hydrometer in connection with the curve, with values obtained by means of a pyknometer or a Mohr-Westphal balance. (5) In the case of the densimeter or direct-reading hydrome- ter, lay a steel scale along the stem of the hydrometer and read the steel scale at each numbered division on the hydrometer. In addition, read the steel scale at the points to which the hydrometer sank when floated in the two liquids whose densi- ties were previously determined. From these last two readings and the densities already determined, and taking K as any con- venient point, calculate x by (97). Knowing x and the distances from K to the various hydrometer divisions, use (99) to deter- mine what the hydrometer readings ought to be at the various points along its scale. The quantity which has to be added to a reading in order to obtain the corrected reading is called the correction for that MASS, DENSITY, SPECIFIC GRAVITY 107 reading. Plot both a correction curve, coordinating the read- ings of the hydrometer and the corrections to be applied, and a calibration curve, coordinating the actual readings with the corrected readings. The observations and results should be arranged in a table, somewhat as follows : — Hydrometer Reading Steel Scale Reading Distance a BOTE K h + x Specific Gravity .g_ Pi(^'i + a') a(A + X) Correction Exp. 23. Determination of the Relative Densities of Gases with Bunsen's Effusiometer Object and Theory of Experiment. — The object of this experiment is to determine the ratio of the density of a gas to the density of air or hydrogen. The density of a gas might be determined by weighing a large bulb of known volume, first when quite empty, and then filled with the gas under investiga- tion. But on account of the difficulty in completely evacuating the bulb before the first weighing, and in obtaining an accurate value of the mass of gas contained in the bulb at the time of the second weighing, this method requires unusual care and many precautions. Consider a gas of density p^ inclosed in a vessel at a pressure of p dynes per sq. cm. above that of the surrounding atmosphere. If there be a small opening of area a in the vessel, then the gas will escape into the atmosphere at some speed s^ cm. per sec. That is, in one second there will issue from the opening a column of gas of length gj cm. and cross section a sq. cm. Conse- quently the mass of gas that escapes per second through the 108 PRACTICAL PHYSICS opening is pias^ grams, and the kinetic energy of this mass is Again, since the gas in the vessel is under a pressure exceeding that of the surrounding atmosphere by jo dynes per sq. cm., it follows that the force producing the flow is pa. Consequently the work done on the escaping gas in one second is pas^^. This is the loss of potential energy of the gas in the vessel. Since the loss of potential energy equals the gain in kinetic, it follows that Therefore the speed of efflux of the escaping gas is Pi (100) Similarly, if a second gas of density p^ is allowed to escape through the same opening under the same difference of pressure, its speed of efflux is F2 Dividing (101) by (100), F2 ~ «i' ~ ^2' (101) (102) Fig. 54. where t-^ and fg ^-re the times required for equal volumes of the two gases to effuse through the same opening. That is, when under the same conditions as to pressure, the densities of two gases are inversely ^ proportional to the squares of their speeds of effusion, and are directly proportional to the squares of the times required for equal volumes to effuse through the same orifice. This is the principle of Bunsen's Eff usiometer. The apparatus (Fig. 54) consists of a glass tube open at the bottom and sur- mounted by an enlargement containing a diaphragm D pierced with a small opening about 0.01 mm. in diameter. This tube MASS, DENSITY, SPECIFIC GRAVITY 109 is inserted in a larger vessel containing mercury. The gas under investigation is inclosed in the tube and the time noted that is required for a certain volume of gas to effuse through the diaphragm. is a three-way cock by means of which the gas holder can be put into direct communication with the atmosphere, or with the orifice in the diaphragm, or can be closed entirelj'^. ^ is a float for indicating the change in vol- ume of the gas, and »S' is a stopper. Manipulation and Computation. — If the ratio of the density of a gas to the density of air is to be determined, put the gas holder into direct connection with the atmosphere by means of the three-way cock (7, and then, by raising the gas holder, fill it with air. Close the stopcock, depress the gas holder, and clamp it into position. Next remove the stopper and turn the three-way cock so as to connect the gas holder with the diaphragm D. As the gas effuses through the dia- phragm, observe the interval of time between the instant when the upper point P of the float arrives in the plane of the upper surface of the mercury in the well, and the instant when the mark at M on the float reaches the same level. Now empty the gas holder and fill it with the other gas. De- press the gas holder as far as possible while it is in direct communication with the atmosphere. This will expel most of the air. Connect with the gas being examined and elevate the gas holder. This operation will fill the gas holder. By repeatedly refilling and emptying the gas holder, it will become practically freed of air and filled with a specimen of the gas whose density is sought. Proceeding as in the case of air, find the interval of time between the instant when the apex of the float appears above the surface of the mercury in the well and the instant when the mark at M appears. The times required for equal volume of the two gases under the same pressure to effuse through the same opening have now been obtained. Their relative density can, therefore, be calculated by (102). CHAPTER VII MOMENT OF INERTIA That property of matter in virtue of which a force from out- side must act upon a body in order that either the speed of the body or the direction in which it is moving may be changed is called inertia. Similarly, that property of matter in virtue of which a torque from outside must act upon a body in order that either the angular speed of the body or the axis about which it is rotating may be changed is called moment of inertia. The inertia of a body is numerically equal to the sum of the masses of its component particles. The moment of inertia of a body can be shown to be numerically equal to the sum of the products of the masses of the particles composing the body and the squares of their respective distances from the axis of rotation, i.e.., K=^mr^. (103) When a resultant torque is applied to a body there is produced an angular acceleration a numerically equal to the ratio of the applied torque L to the moment of inertia K of the body, i.e., a=^. (104) The moment of inertia of a body of simple geometric form can be computed, but the moment of inertia of an irregularly shaped body may often be determined most easily by experi- ment. The experimental determination is usually made by comparison with a body whose moment of inertia can be com- puted. The computations for a few simple cases are effected as indicated in the following table : — • 110 MOMENT OF INERTIA 111 No. Shape Axis ABOUT WUIOH EoTATioN Occurs Moment of Inertia Meaning of Symbols K-c, Moment of inertia 1 Plane lamina Any axis normal about any line XX' of any shape to it ^a:+ Ky (see Fig. 55) in the plane of the lamina and intersecting the axis. 2 Any shape Any axis Kc + Mf Ky, Moment of inertia about another line YY' in the plane of 3 Solid circu- lar cylinder Geometric axis \ M

,W+dr) = \MW + d,^y (110) Exp. 24. Determination of the Moment of Inertia of a Rigid Body Object and Theory of Experiment. — The object of this experiment is to determine a moment of inertia. In any case where a body can be set into torsional vibration about the axis about which the moment of inertia is required, it is a simple matter to determine experimentally the moment of inertia of the body. From (141) it follows that if a body is suspended so that it can vibrate torsionally, its moment of inertia is proportional to the square of its period of vibration. That is, if the proportionality factor is called k, Ky = kT^\ (111) If a mass of known moment of inertia K^ be added to the body above considered, we have K, + K, = ]cT^^ (112) where 2^12 is the new period of vibration of the system. Eliminating k between (111) and (112), Fig. 57. K, m 2 -^12 ■ rp2 (113). Manipulation and Computation. — A convenient form of apparatus for this experiment consists (Fig. 57) of two 116 PRACTICAL PHYSICS horizontal disks connected by three thin vertical rods. From the center of the upper disk rises a short spindle for attachment to the supporting torsion wire. The body whose moment of inertia is required can be placed on the lower disk in such a position that the line about which its moment of inertia is to be determined coincides with the axis of the sup- porting wire. The positions of the masses MM are then ad- justed until the axis of vibration of the system passes through the center of the two disks. Below the vibrating system is a device by means of which the apparatus can be set into tor- sional vibration with very little swinging motion. Find the period of vibration, 2j, of the apparatus ; then add a body of known moment of inertia, K^, and find the new period of vibration, T-^^. From (113) the moment of inertia of the apparatus is Now substitute for the body of known moment of inertia the body whose moment of inertia is required, and find the period of vibration as before. If this period be denoted by ^^g, then the moment of inertia K^ of the body under investigation is by (113) . or, substituting the value of K^ from (114), -^3 =-^2 In finding the various periods of vibration, first with the apparatus at rest set the pointer P directly in front of one of the three vertical rods. Then set the apparatus into torsional vibration with an amplitude of perhaps 90°. At some instant when the vertical rod passes the pointer start a stop watch. Count some ten or fifteen complete vibrations and stop the watch. After recording the time that has elapsed, again at the instant of a passage start the watch. After some ten minutes. MOMENT OF INERTIA 117 during which time no attention has been paid to the vibrating system, stop tlie watch at an instant of passage. Calculate the period by the Method of Passages, given on pp. 36-38. Take all the required linear dimensions with a vernier caliper and make all weighings with a balance of moderate sensibility. Calculate K^ as indicated on p. 111. Determine K^ both by (115) and as indicated on p. Ill, and see how the values check. CHAPTER VIII ELASTICITY When a body is perfectly elastic, a given deforming force keeps it distorted to the same extent no matter for how long a time the force is applied. This means that the distortion calls into play a restoring force which, so long as the body is at rest, is exactly equal and opposite to the deforming force. It fol- lows that, when the deforming force is removed, this restoring force causes the body completely to recover its original shape and size. When a body is imperfectly elastic, a given deform- ing force produces a gradual yielding so that the restoring force which the distortion calls into play is in this case not quite equal to the deforming force. It follows that when the deforming force is removed from a body which is imperfectly elastic, the body does not completely recover its original shape and size. It is said to have received a permanent set, or to have been deformed beyond its elastic limit. So long as any body is not deformed beyond its elastic limit it is perfectly elastic. The ratio of a force to the area on which it acts is called a stress. The ratio of a deformation to the original value of the length, volume, or whatever has been deformed, is called a strain. When a body has not passed its elastic limit, the ratio of the restoring stress to the strain which produced it is constant and is called a coefficient of elasticity. Since forces applied to a body in different ways produce different types of deformation, there are various coefficients of elasticity. If a wire is stretched or a pillar shortened by a load applied 118 ELASTICITY 119 to it, the strain is the change of length divided by the original length. In this case the ratio of the stress to the strain is called the tensile coefficient of elasticity or Young'' s modulus. If a toy balloon were fastened under water and then pressure applied to the water, the balloon would decrease in volume with- out changing its shape. In this case the strain is the change in volume divided by the original volume, and the corresponding coefficient of elasticity is called the hulk modulus. ' If a rectangular parallelopiped of rubber ac (Fig. 58) has two opposite faces glued to two boards, and if one of these boards is pushed sideways in its own plane, there is no change in the volume of the block but its shape is changed to fgcd. In this case the strain is the ratio of af to ad, and is called a shear or a shearing strain. If F is the force applied, and A the area of the face ah, then F divided by A is called a shearing -^^^ gg stress. If the block of rubber is very thin in a direction normal to the paper, and if it is bent around until ad coincides with he, it is seen that a shear is the kind of strain involved in the twisting of a wire about its geometric axis. The ratio of a shearing stress to the shearing strain which it produces is called the simple rigidity or the slide modulus of the material sheared. Exp. 26. Determination of the Elastic Limit, Tenacity, and Brittleness of a Wire Object akd Theory of Experiment. — The elastic limit of a material is the stress beyond which the material cannot go without becoming permanently set. Since it is found that the curve showing the relation of a stress to the strain which it produces is a straight line until the elastic limit is reached, the elastic limit is the stress corresponding to the point on the stress-strain diagram where the curve departs from being a straight line. The tenacity or tensile strength of a material is 12D PRACTICAL PHYSICS the greatest longitudinal stress it can bear without rupture. The brittleness of a material is the ratio of its elastic limit to its tenacity ; in other words, it is the ratio of the force just sufficient to produce permanent set to the force just sufficient to produce rupture. The object of this experiment is to plot a curve showing the relation between the longitudinal stress and strain of a wire, to determine from this curve the elastic limit of the . material composing the wire, and also to determine its tenacity and brittleness. Manipulation and Computation. — Arrange a wire verti- cally so that it cannot twist, with one end fastened to a rigid bracket and the other end attached to a scale pan. Place on the supporting bracket, directly above the wire, a number of iron masses whose aggregate weight exceeds the breaking strength of the wire. Focus the cross hairs of the telescope of a cathetometer, or the cross hairs of a microscope containing an eyepiece micrometer, on a well-defined mark on the lower end of the wire, and take the reading. Take a weight off the supporting bracket, place it on the scale pan, and take a new reading of the position of the fiducial mark. Continue chang- ing weights from the supporting bracket to the scale pan and taking the corresponding readings until the wire breaks. On coordinate paper plot the stresses applied to the wire as abscissas, and the strains produced as ordinates. The sti-ess corresponding to the point where the curve departs from being a right line and bends toward the vertical axis is the elastic limit. The tenacity is the .breaking weight divided by the area of cross section of the wire, and the brittleness is the elastic limit divided by the tenacity. Exp. 26. Determination of the Tensile Coefficient of Elasticity, or Young's Modulus (FIRST METHOD, BY STBETCHING) Object and Theory of Experiment. — From the definition of Young's modulus (p. 119), it follows that if L denotes the ELASTICITY 121 length of a wire, d its diameter, and e the elongation produced by a force F, then the Young's modulus of the material compos- ing the wire is E 4:F e T 4FL ird^e (116) If the force is measured in dynes and the* other quantities in centimeters, the value of E will be in dynes per sq. cm. The object of this experiment is to determine the value of Young's modulus for a metal in the form of a wire. Of the quantities which have to be meas- ured, the only one that it is difficult to get with moderate accuracy is the value of the elongation e. One means of finding this is by an optical lever. The upper end of the wire is securely clamped to a rigid support (Fig. 59), and to the lower end of the wire is fastened a rectangular piece of metal ^S* terminating in a hook for the attachment of a weight pan H. This rectangular piece of metal is kept from twisting or swinging by being let through a loosely fitting rectangular hole in a second bracket fastened to the wall. One leg of the optical lever is supported in the axis of the wire by the rectangular hook^ while the other two legs are supported by the bracket. In Fig. 60 mnb is the optical lever with its mirror ch vertical, T i& a, horizontal telescope, and oo' is a vertical scale divided into centimeters and millimeters. If the wire be stretched by a small amount, the optical lever will assume the position m'nh' making an angle 6 with its previous position. When light is reflected from a mirror, the angle of reflection equals the angle of incidence. Whence o'a'i=oa'i=0. Consequently oa'o' = 2 ^. Pig. 59. 122 PRACTICAL PHYSICS And since the small distance aa' is negligible in comparison with ao, tan 2 = ^. ao If 6 is small, approximation (10), p. 7, may be employed, giving 2 6=°-^. Fig. 60. The elongation is the vertical distance through which the point m moves in passing to the position m' . So that e = m'n sin 6 = mn sin or, employing approximation (8), 0, e =f mn • 6 = mn ■ 00' 2 CIO On putting this value of e in (116) it becomes 8FL E = ao ■jT oP ■ mn go' (117) Manipulation and Computation. — See that the wire is straight and carefully suspended. Place three or four kilo- grams on the supporting bracket directly over the clamp hold- ing the upper end of the wire, and one kilogram on the pan below. Put the optical lever in place and the telescope and scale a meter or so from it, clamp the scale vertical, and adjust the height of the telescope until it is at about the same level as the optical lever. Move the head to such a position that the image of the telescope is seen in the middle of the mirror of the optical lever. If the eyes are not now at the level of the tele- ELASTICITY 123 scope, turn the thumb screw beneath the front legs of the optical lever until the image is seen when the eyes are at the same level as the telescope. This makes the mirror vertical. Focal- ize the telescope as directed on p. 23. Read the telescope, move the masses from the supporting bracket down to the weight pan, read the telescope, move the masses back to the supporting bracket, and read the telescope again. If the elastic limit has not been exceeded, the last reading should be about the same as the first. Repeat two or three times. Make about five determinations, each one after moving the telescope and scale a few centimeters farther from the optical lever. Measure the diameter of the wire in some half dozen places with a micrometer caliper. Determine the length mn of the optical lever by pressing the three feet upon a piece of card- board, connecting the prick points made by the two front feet by a fine line, and then measuring the normal distance between the remaining prick point and this line by means of a milli- meter scale. Determine the length of the wire with a meter stick, and the loads added to the weight pan with a platform balance weighing to grams. For each distance ao find the average deflection oo' and cal- culate ^- Find the average of all the values for — , and by oo' oo' (117) calculate U. Give the result in dynes per sq. cm., in Kg. wt. per sq. mm., and in lb. wt. per sq. in. Exp. 27. Study of the Flexure of Rectangular Rods* Object and Theory op Experiment. — Even before a given phenomenon is sufficiently understood to permit the deri- vation by purely analytic methods of a formula that will show the relation between the various factors entering into the phe- nomenon, it is often possible to construct from purely experi- * This experiment is taken with slight modification from Reed and Guthe's " Manual of Physical Measurement." 124 PRACTICAL PHYSICS mental data an equation that will give the law connecting the various related quantities. An equation obtained from experi- mental data is called an empirical equation. One of the methods used in the construction of empirical equations is illus- trated in the present exercise. If a number of rods of any material,' differing iii length, breadth, and thickness be supported on a pair of knife-edges and loaded in the middle, it would be expected that the flexure produced, that is, the displacement I of the middle point of any rod, would be a function of the load F, the distance L between the supports, the breadth of the rod £, and its depth D. The law of flexure of rectangular rods of a given material might, perhaps, be expressed by an equation of the form I = kF^L^BW, (118) where Jc, «, /S, 7, and e are constants to be determined by experiment. The object of this experiment is to ascertain whether the facts warrant the acceptance of the above tenta- tively assumed equation; and, if they do, to determine the values of the five constants. The constants a, /3, 7, e, can be most easily obtained by varying the independent variables one at a time and noting the change of the dependent variable I. When, in this way, these four constants have been determined, the value of k is obtained by solution. - First, let the load F be varied while the other independent variables remain constant. This will give a separate equation for each value of F used. Thus lp^ = hF^'L^B''D\ Z^3 = hF^lFB'B', lp^ = hF^^L^B^B'. Dividing the first of the above equations by the third and the second by the fourth, \n = EiL and ^ = :^. 7 35"* 7 7^ "> tj?3 JJg Ip^ J! ^ ELASTICITY 125 Putting these equations into the logarithmic form, log Ifi - log Ip^ = «i3(log F^ - log F^) (119) and \oglp^-\ogljr^ = a^llogF^-\ogF^, (120) in which Wjg denotes the value of a derived from the equation expressing the ratio of Ip^ to Ip^ If the values of Wjg and a^^ obtained by solving (119) and (120) are nearly the same, their average is to be taken as the value for a in (118). Second, the other independent variables remaining constant, let the length L be varied, the flexure I being observed when the force F is applied. By the process described above we get log ?ii - log Zi3 = ;ei3(log ij - log ig) (121) and logZi2-logZi4 = /324(logi2-logi/4). (122) If the values of /Sjg and ^^ obtained by solving (121) and (122) are nearly the same, their average is to be taken as the value for yS in (118). Third, let the breadth B be varied and the other independent variables remain constant. This will give log h^ - log Z,,3 = 7i3(log B, - log ^3) (123) and . log ?^2 - log l^ = y^i^og B^ - log B^. (124) From these equations a value for 7 will be found. . Fourth, let the depth I> be varied. This will give log Ijy^ - log ?J3 = £i3(log i>i - log 2)3) (125) and logZj2-log?j4 = e24(log2>2-logZ>4). (126) From these equations a value for e will be found. If the values found for « are nearly the same, for ^ nearly the same, for 7 nearly the same, and for e nearly the same, this justifies the form assumed for the desired relation. If the values experimentally obtained for any one of the quantities a, )8, 7, e are not nearly the same, this means that the form as- sumed is not the form of the relation which actually exists, and some other form must be tried. The equation obtained by substituting for a, /S, 7, e their values thus experimentally determined is called an empirical 126 PRACTICAL PHYSICS formula. The statement of the facts expressed by this for- mula constitutes the law of bending. The values obtained for these four constants should be very nearly « = 1, ;S = 3, 7 = — 1, 6 = — 3. If exactly these values are obtained, the law of bend- ing will be expressed analytically by the equation Whatever the form of the empirical equation that is actually found, the value of Te is determined by substituting in this equation a set of corresponding values for Z, F, L, B, and D. Several such sets of valuers should be substituted and the average value for k used. If similar series of measurements are made upon rods of different materials, the values of a, /3, 7, 6 are found to be very nearly the same for the different materials, but the values of k are different. This means that k depends upon the material of the bar and not upon its dimen- sions, whereas the other constants depend only upon the dimensions of the bar. Manipulation and Computation. — The rods to be ex- perimented upon should be 70 or 80 cm. long and their trans- verse dimensions so selected that the same bars can be formed into two series, one in which the bars have constant depth and variable width, and another in which they have constant width and variable depth. The variable length is secured by adjust- ing the distance between the supporting knife edges. After a bar is placed on the knife-edges a weight pan is suspended from the bar about halfway between the knife edges and sufficient weight applied to insure good contact between the bar and its supports. On the addition of a known load the flexure of the bar, that is, the depression of the middle point, is measured. This measurement may conveniently be made by means of a microscope furnished with a micrometer eyepiece, or by means of a micrometer screw fastened to an adjacent sup- port directly above the middle of the bar. In the latter case the instant when the micrometer screw comes into contact with ELASTICITY 127 the bar can be determined either by means of a telephone receiver in a battery circuit including the bar and micrometer screw, or by observing the image of some fixed object in a small mirror, one end of which rests upon the rod and the other end upon some adjacent fixed support. In order to be certain not to load the rods beyond their elastic limits, the student should ask an instructor what loads may safely be applied. Following the division of the experiment as outlined above, make a series of observations on a single rod by noting the flexures produced by different loads on the pan. Add, say, 500 g. and observe the flexure, add 500 g. more and observe the flexure, and so on until six equal increments of load have been added. Then reverse the process, removing 500 g. at a time and taking an observation for the flexure after each change of load. Combine the six values of load and cor- responding flexure as in (119) and (120) so as to get three values for a, viz., a^^, a<^^ and agg. Second, by moving the knife edges a few centimeters, obtain six lengths of a single rod, and for each length determine the flexure produced by the same load of, say, 2 Kg. Combine the values of L and I as in (121) and (122) so as to obtain three values for /3. Third, with the distance between the knife edges constant, find the flexure produced by a constant load of, say, 2 Kg. act- ing on each of four or six bars of the same material and depth but different breadth. Measure the breadth of the bars with a micrometer caliper. Proceed as directed in the preceding paragraph, using equations like (123) and (124) to find 7. Fourth, with the distance between the knife edges constant, find the flexure produced in each of four or six bars of the same material and breadth but different depth. Measure the depth of the bars with a micrometer caliper. Proceed as di- rected in the preceding paragraphs, using equations like (125) and (126) to find e. Insert the final values of «, /8, 7, e in (118), and formulate in words a statement of the facts expressed by the resulting em- pirical equation. 128 PRACTICAL PHYSICS Fig. 61. Exp. 28. Determination of the Tensile Coefficient of Elas- ticity, or Young's Modulus (SECOND METHOD, BY BENDING) Object and Thboky of Experiment. — Consider a rec- tangular rod of length L, breadth B, and depth D, fixed at one end and weighted at the other. The rod will become bent as in the figure. The upper portion of the rod is extended and the lower por- tion compressed. Since the rod is strained by a longi- tudinal stress, and since Young's modulus is defined as the ratio of the longitudi- nal stress to the longitudinal strain, Young's modulus may be determined from an observa- tion of the amount of bending which a given force produces in the rod. The object of this experiment is, by the method of bending, to determine the Young's modulus of the material composing a rectangular rod. Imagine the unstrained rod to be cut up into m laminae, each of width w, by a series of planes normal to its length. Then let the rod be bent slightly by the force F applied downward at the end of the rod, and let the yth lamina from the free end be thereby so distorted that its sides ac and Id make with each other a small angle 6^. The restoring stress in this lamina produces a couple which tends to bring the rod back to its undistorted position, and is prevented from doing so only by the distorting force F. The first step in the development of the formula for deter- mining the Young's modulus of the rod is to find an expression for the restoring couple due to the stress in this yth lamina. Halfway between the upper and the lower surfaces of the rod ELASTICITY 129 is a neutral surface gJi which is neither extended nor com- pressed. Through the point e, where ao cuts gh, draw a'c' parallel to hd. Then the original length of any line vz in the yth lamina is that part of it included between a'c' and hd, and the increase in its length is the part of it between ac and a'c' . Imagine the upper half of the yth lamina to be made up of n layers, each of breadth B equal to that of the rod and of depth t. Then, counting upward from ef, the top of the «th layer is stretched it6j and the bottom of it is stretched (J, — V)tdj. The effective elongation, e^, of this ith layer lies, therefore, between these limits. That is (i-l)te^ The difference between the third and first members of (129) is i t. Since -r is the radius of curvature of the neutral iw Oj surface in the/th lamina, and since this radius is never very small, it follows that, by choosing t sufficiently small, the dif- ference between the third and first members of (129) can be made less than any assigned quantity. It follows that the value of L is th« common limit approached by the first and third members of (129) when t approaches zero. That is, ,^EBejB^ 2iw ' Now the resultant moment of the restoring forces below the neutral surface equals the moment of those above. It follows that the whole torque due to the strain in the/th lamina is 2L. Since the bar is in equilibrium, this restoring couple equals the distorting moment of E about e. If the rod is bent only slightly, the moment of E about e is so little smaller than Ejw that we may write EBe.BK J.. .-tons ——2—= Ejw. (130) 12 w The next step is to find the depression of the end of the rod. The entire depression I may be regarded as made up of parts, ?j, ?2i • • • 5 4> due to the bending in the different laminae. At e and / draw es and fk tangent to the neutral surface and equal in length respectively to the arcs eh and fh. Then the angle between these lines equals the angle 6j between ac and hd, and this angle is so small that sk is practically the arc of a circle two of whose radii extend in the directions es and fk. ELASTICITY 131 It follows, since /A;, if prolonged, would cut e« between e and/, that fk-ejii - lyCAry ^ p, ^ 2 ■7riJi,(j>i\Ary I ' / It follows that if the torque which acts on the ring is denoted by Li, 2 7riJ,Ci-iy(Ary _^^ 2777i^i^(A?;}^ I ''^ I On summing the torques which act on all the rings and letting L denote the resultant torque, we obtain ^^/^■^(AO^ X (,-_l)3-)3Ar + (»iAr)2(Ar)2]. Since re is the number of rings of width Ar between the center and the circumference of the wire nAr = r, and the above ex- pression may be rewritten ^ [r* - 2 r^Ar + r\Ary] < L < ^ [r* + 2 r^Ar + rXAry] . The difference between the two outer expressions in the above inequality is ^^•4?'^Ar, a quantity which, by choosing Ar sufficiently small, may be made less than any assigned quantity. It follows that the value of L is the common limit which both of the above expressions approach when Ar approaches zero. That is, if d denotes the diameter of the wire, 2Z 32 ? ^^'^^■> This is the torque that must be applied at the end of the wire to keep it twisted. If a massive bodj'^ B is suspended from the lower end of the wire and twisted about the axis of the wire through an angle (/), the wire and B react upon each other — 5 exerting upon the wire this torque L that keeps the wire twisted, and the wire exert- ing upon B an equal and opposite torque L' that tends to swing B back to a position such that the wire is not twisted. That is, L'=-L=-^?^^*. (139) The second negative sign in (139) means that the torque L' and displacement ^ are in opposite directions. If- B is twisted about the axis of the wire and then released, the torque L' will swing it back towards its original position with an acceleration which by (104) and (139) is where ^denotes the moment of inertia of B. ELASTICITY 137 Since the quantities within the parenthesis in (140) are all constants, it is seen that the angular acceleration'is proportional to the angular displacement, and that the acceleration and the displacement are in opposite directions. From this it follows that the motion of B is simple harmonic. Its period of com- plete vibration is, therefore, T\=21^^}-^=2^^fi^. (141) Whence, the simple rigidity of the material composing the wire is 128^rZ^ (142) Manipulation and Computation. — Suspend from the lower end of the wire a massive body of such a shape that its moment of inertia can easily be computed, a solid iron cylinder, for instance, with its axis coincident with that of the wire. Find the period of vibration of the suspended system by one of the methods outlined in Chapter II. Take the diameter of the wire with a micrometer caliper. Since the diameter enters the equation to the fourth power, it must be determined with considerable care. Measure it in not less than ten places distributed about equally along the length of the wire and take the mean. Measure the length of the wire with a meter stick or steel tape, and take the necessary dimen- sions of the suspended body. The mass of the suspended body should be determined within 0.1%. Express the value for the simple rigidity in dynes per sq. cm.. Kg. wt. per sq. mm., and lb. wt. per sq. in. Exp. 30. Determination of Simple Rigidity (STATIC METHOD) Object and Theory op Experiment. — The method given in the preceding experiment is applicable only to wires of radius so small that if a heavy body is suspended by the 138 PRACTICAL PHYSICS wire, the period of torsioBal vibration will be large enough to admit of accurate determination. The present method is ap- plicable to heavier rods. In the method here employed there is fastened to the lower end of the rod a massive disk which has its upper face grad- uated in degrees, and has around its edge a series of pins placed 20° apart. In front of the disk and in back of it are two horizontal scales. The twisting couple is applied to the disk by hori- zoiital forces acting tangentially at its circumference. Masses m^ and m^ are suspended by cords which pass in front of the two horizontal Scales. Tied to each supporting cord at about the level of the pins in the disk is another short cord which has at its other end a loop that can be slipped over one of the pins, thus twisting the graduated disk through an angle which can be read by means of a pair of pointers fixed above it. Let the forces in the horizontal cords be denoted by -fj and F^. Then from the diagram (Fig. 6-i) h tan w ■■ (143) Fig. 63. system of concurrent forces in equilibrium. and since F^ and m-^g are perpendicular to each other, and #j, m^cf, and the tension in the supporting cord are a ^1 = tan w. (144) From (143) and (144) it follows that jr, _ m.,gx ^'- h ■ ELASTICITY 139 This is the force with which the horizontal cord pulls on the point where the three cords join. The pull F^ which the cord exerts on the disk is equal to this, but in the opposite direction. That is, -F, 1 --^i---T- (145) The second negative sign in (145) denotes that F-^ and x have opposite directions. If the two masses m^ and m^ are equal and their supporting threads looped over diametrically opposite pins, and if the points from which the upright cords hang are equidistant from the plane of the wire and supporting bracket, J'j = F^. If we drop the subscripts and de- note by D the diameter of the disk increased by twice the radius of the horizontal cords, the moment of the couple that tends to turn the disk farther from its equilibrium position is F'D = mgxD (146) On pp. 134-136 it has been shown that if a cylinder of length I, diameter d, and made of a material of simple rigidity /tt is twisted through an angle of (j> radians, the torque which this twisted wire exerts on the body that holds it twisted is L'=- riJ,^ '^ = A^-d = As ^^^'^ Exp. 32. Determination of the Absolute Coefficient of Viscosity of a Liquid (POISEUILLE'S METHOD) Object and Theory of Experiment. — Consider a column of liquid flowing through a tube of length I, and with a radius, 143 144 PRACTICAL PHYSICS r, so small that there will be no eddies in the liquid column. Imagine this column to be made up of a large number n of concentric hollow cylinders of very small thickness Ar. Sup- pose that all of these hollow cylinders but one could be made solid, so that there would be a solid rod sun-ounded by a thin layer of the fluid, and this again surrounded by a solid tube. While the rod was moving, two forces would be acting on it — one due to the viscous resistance in the tube that was still liquid, tending to retard the motion of the rod, and the other due to the difference between the pressures at the two ends of the rod, tending to accelerate it. If the radius of the rod were a;Ar, the viscous resistance in the liquid tube surround- ing it would, by (150), be ^'^^r ■' and if^ denotes the difference between the pressures at the two ends of the rod, the force to which this difference in pressure would give rise would be Fj, = p ■ -TrCxAry. If the rod were moving uniformly, F^ would equal Fp, i.e. 77 • 2 TTxArl ■ s /- A NO ■1 = p . Trfx^rY. Ar Whence ^^ p(xAr)Ar ^^5^^ This equation shows that the difference in speed between the outside of an axial cylinder of the liquid of any radius and the outside of the adjacent layer is small near the axis where xAr is small, and increases in direct proportion with the radius of the cylinder. If it is imagined that these concentric layers of liquid are congealed without interfering with their ability to slip past one anotlier, then on account of their difference in speed, at any instant after the flow has begun, the end of the inmost cylinder will protrude beyond the end of the adjacent layer, this second VISCOSITY 145 layer will protrude beyond the end of the third layer, and so on. Let Sj represent the speed of the inmost cylinder relative to the second layer ; Sj, the speed of the second layer relative to the third, etc. Also let v^ represent the volume of the portion of the inmost cylinder that protrudes beyond the end of the second layer ; v^, the volume of the second solid cylinder that protrudes beyond the third layer, etc. Then the entire volume discharged by the capillary in time t is V=vi + v^ + v^+ ... +V (152) Now Vj = 7r(A/*)2sj^, v^ = 7r(2 Ar^s^t, v^ = 7r(3 Arys^t, etc. Putting these values in (152), we have r=7r«[Si(Aj-)^ + 82(2 Ar)2 + 83(3 Ar)2+ ... + s„(wAj-)^], and on substituting for Sj, s^, Sg, etc., their values from (151), we obtain P^^ 7rp^(AO-t .-^3 ^3 33 3. 2r}l "- -■ _ TrptjAry 2r,l ' n\n + l) n 4 J = ^[(nAO* + 2(nA?-)3(Ar) + (wAr)2(Ar)2] . But nAr = r. Therefore in the limit, when Ar = 0, r=2^. (153) If the pressure is due to a column of liquid of height h and density p, then p = pgh. On putting this value in (153), and solving for ^, we have j^^ ^ "= sT-F ^^^^^ It should be noticed that in deriving (154) it has been tacitly assumed (a) that the viscous resistance to the flow of the liquid is uniform throughout the entire length of the tube, (J) that the lines of flow of liquid in the tube are parallel to the axis of the tube throughout its length, (c) that no part of the energy supplied to the liquid in the tube appears as energy of motion, (c?) that there is no effect at the outlet due to surface tension. 146 PRACTICAL PHYSICS The conditions demanded by (a), (6), and (c) can be realized to a sufficient degree of approximation by using a tube that is both long and of narrow bore and having the liquid flow through at a uniform rate. Condition (^d) is met by immersing the dis- charge orifice in a portion of the liquid having a considerable free surface. 1 Manipulation and Computation. — A vis- cometer that fulfills the above conditions is illus- trated in Fig. 65. The vertical tubes AB and CD are of uniform bore and are graduated in millimeters throughout their length. The capillary tube BE is straight and of uniform circular bore. In order that the temperature of the liquid being investigated shall be constant and definite, the viscometer is supported in a suitable water jacket supplied with a thermometer. The length I of the capillary tube is measured with a meter stick. The mean radius of the bore is determined by measuring the length of a known mass of mercury at different positions along the length of the tube. An amount of mercury sufficient to make a thread about four centimeters long is drawn into the tube by suction applied at the opposite end, and this thread is measured in length at different equally spaced positions along the length of the tube by means of a dividing engine. Knowing the mass of the mercury thread and the average length, the average radius of the bore of the tube is determined. A tube with a bore departing very much from uniformity must be rejected in determining the absolute coefficient of viscosity. In order to determine the V in (154) it is necessary to cali- brate the lower part of the tube CD. This may be done by putting a solid stopper at E, removing the one just above 0, and dropping into GB known volumes of water from a burette. Pig. 65. VISCOSITY 147 After each small volume of water is dropped in, a reading is made of the top of each water column — the one in QB and the one in the burette. From these readings a curve is to be plotted coordinating the volume of water in CD with the read- ing of its surface on the CD scale. After thoroughly cleaning and drying the parts of the vis- cometer, it is assembled, a quantity of the liquid under investiga- tion is introduced, and this liquid column run back and forth until it is free of air bubbles and the tubes are coated with a thin film of the liquid. The quantity of liquid introduced should be such that it will form a column extending from a point near the upper end of the tube AB to a point near the lower end of QB. With all the rubber stoppers tight, and the stopcock S open, run the liquid into the position mentioned above, close the stop- cock, and place the viscometer in the water bath. After the temperature has become constant and of the desired value, with stop watch in hand open the cock «S', and when the meniscus in AB reaches some previously selected scale division X, start the watch ; when the meniscus reaches some second selected scale division X.\ stop the watch. This gives the value for t in (154). When the upper meniscus was at X. the lower meniscus was at some point y, and when the upper meniscus had fallen to x' the lower meniscus had risen to some point y' . The posi- tions of y and «/' can be obtained by opening S and again run- ning the liquid into AB to the points x and a/. The mean of the vertical distances between x and «/, and d and «/', is the value for h in (154). These distances can be obtained by the scales engraved on AB and QB. p can be obtained by means of a balance and a 5cc. pipette. Fis obtained by finding from the curve already plotted the volume of water that would be held between the marks y and y' . At least five sets of observations should be taken and the average value for -= used in (154) to get ?; at the temperature of the experiment. 148 PRACTICAL PHYSICS Exp. 33. Determination of the Specific Viscosities of Liquids (COULOMB'S METHOD) Object and Theory of Experiment. — On account of such experimental difficulties as that of obtaining a capillary tube of uniform bore and circular cross section, of accurately measuring its diameter, and of keeping the capillary free from minute air bubbles and particles of foreign substances, the determination of a coefficient of viscosity by the method given in the pi'eceding experiment is very troublesome. Coulomb's Method is especially suited to the determination of the relative viscosities of those liquids used in engineering and the arts which are liable to contain particles of solid substances in suspension, e.g. lubricating oils. By specific or relative vis- cosity is meant the ratio of the viscosity of the liquid to the viscosity of water. The object of this experiment is to deter- mine the specific viscosities of a series of liquids. If a massive disk suspended axially by a thin vertical wire be immersed in a liquid and set into torsional vibration, it will be shown in the chapter on Damped Angular Vibration, Vol. II, that the ratio of the lengths of any two successive swings from one end of the path to the other is a known function of the " damping constant," a, which is proportional to the viscos- ity of the liquid surrounding the moving body. Let o-j represent the ratio of the lengths of any two successive oscillations of the disk when immersed in the first liquid ; ?\', the period of vibration of the disk when immersed in the first liquid ; and aj, the damping constant for the first liquid. Let o-g, T^., and a^ represent the corresponding quantities for the second liquid. Then, from (31), Vol. II, (7, = e *ii . (155) where e is the base of the natural logarithms and K is the VISCOSITY 149 moment of inertia of tlie suspended system, by (156) and putting the resulting equation into tlie logarithmic form, we obtain log a-^ ^ a^Tj' log 0-2 a^T^'' Whence the relative viscosity of the two liquids, z, is (157) Dividing (155) „_ai_ T^ log 0-1 Fig. 66. "2 ^l' log 0"2 If the second liquid is water, z is the specific viscosity of the first liquid. Manipulation and Computation. — In the apparatus here employed (Fig. 66), one end of a thin piano wire is fastened to a rigid support while the other end is attached to a vertical rod carrying a divided circle and the massive disk which is to be immersed in the various liquids. The disk has a thin stem by which it is fastened to the rod carry- ing the divided circle. The vessel containing the liquid being studied is surrounded by an oil bath heated by means of a Bunsen burner. As the viscosity of many liquids is very different at different temperatures, it is always necessary to make the determination at the temperature at which the liquid is to be used. For in- stance, a test of cylinder oil should be made at about 150° to 175° C, while most machine oils should be tested at about 50° C. Since the relative viscosities of many pairs of specimens of oil are even reversed with a change of temperature of less than 100° C, it is impossible to judge the relative lubricating values of oils from their relative viscosities determined at a tempera- ture much different from the temperature at which they are to be used. After cleaning and assembling the apparatus and allowing the temperature of the specimen to attain the required value, 150 PRACTICAL PHYSICS twist the disk through about 180° by rotating the rod above the divided circle. With a stop watch observe the time of ten complete vibrations. One tenth of this time in seconds is the period T{. By means of the pointers P and P' make a series of readings of the turning points of successive swings to the right and to the left. The number of scale divisions through which the disk turns in rotating from one end of its path to the other is the magnitude of thait oscillation. Calling the magnitudes of these successive oscillations f^, ^2' ^s' ^^c, we have Ii = l2 = l3= ... =0-,. ?2 U U ' Whence f g = ^^a^, f 1 = ^i'^i = h'^i^ and, in general, ?„ = ^^o-j'"-''. If, say, twenty oscillations were observed, we have then etc., see fc and by finding the average of |^, 1^, l^,--- f^i and taking fell 612 513 ?20 the tenth root of this average, o-j is found. In the same manner find T^ and cr^. These values of ?\', T^, o-j, and cr^ substituted in (157) will give the relative vis- cosity of the two liquids. With liquids having viscosities not very different, the value of 2\' will be so nearly equal to T^ that their ratio may approximate unity ; but it is never allow- able to assume their ratio to be unity without experimental verification. Instead of reckoning viscosity in absolute units or with ref- erence to water at some standard temperature, the viscosity of VISCOSITY 151 a liquid is sometimes rated in comparison with the viscosity of an aqueous solution of sugar having a definite concentration and temperature. For example, the viscosity of a certain oil at 50° C. may be specified as being equal to the viscosity of an 18% aqueous solution of pure sugar at 20° C. Values for the viscosities of aqueous sugar solutions of various con- centrations, referred to water, are given in Table 10. PAET 11. HEAT PART II. HEAT CHAPTER X TEMPERATURE The comparison of temperatures involves several arbitrary- conventions. Temperatures cannot be directly measured, — they can be compared only in terms of some other phenomenon which depends upon temperature. Of the various phenomena which are used for the comparison of temperatures, the follow- ing are the most important : (a) change of the volume of a gas or liquid kept at constant pressure, (S) change of the pressure of a gas kept at constant volume, (c) change of the electric re- sistance of a metal wire, (c?) production of an electromotive force at the junction of two dissimilar metals, (e) quantity of energy radiated by the hot body, (/) luminous intensity of the radiation of a particular color radiated by the hot body. In cases (a), (5), (c), and (c?) it is necessary to select a particular thermometric substance. In all cases it is necessary to adopt two particular temperatures as standard or fixed points of a thermometric scale, and to divide the interval between these points into a definite number of spaces or degrees. The scale of temperatures that has been adopted as standard is based on the change of pressure which a change of tempera- ture produces in a fixed mass of hydrogen kept at constant vol- ume. By means of a gas thermometer temperatures as high as 1700° C. can be compared. However, as the standard gas ther- mometer is both bulky and fragile, it is seldom used except in scientific work and for the purpose of standardizing other ther- mometers. All other thermometers are calibrated in terms of the gas thermometer. 155 156 PRACTICAL PHYSICS On account of the comparatively simple technique necessary in its use, the mercury-in-glass thermometer is employed when- ever the conditions of the measurement permit. By using a very hard glass, and filling the space above the mercury with an inert gas at a pressure sufficient to prevent boiling of the mercury, a mercury-in-glass thermometer can be used up to about 650° C. (1000° F.). Mercury -in-quartz thermometers having the space above the mercury filled with gas at 60 atmos- pheres pressure can be used up to 700° C. Thermometers available for temperatures above 500° C. are often called pyrometers. The electric resistance and the ther- moelectric instruments are available for temperatures up to about 1500° C. For temperatures above this, radiation pyrome- ters are available. Measurements of temperature, even by means of the mercury- in-glass thermometer, are subject to so many sources of error that an accurate determination of temperature is a task of some diificulty. Nevertheless, the thermometric methods have be- come so highly developed that if proper precautions are taken and proper corrections made, a measurement of temperature made with a mercury-in-glass thermometer between 0° C. and 100° C. can be trusted to 0°.005. The methods described in the following pages correspond to an accuracy of about 0°.05. The principal sources of error in the use of a mercury-in- glass thermometer are : — 1. Errors in reading the thermometer due to parallax. Usu- ally the scale of a thermometer is at some distance in front of the capillary, so that, unless the line of sight is normal to the length of the tube, the reading is too high or too low. The two principal methods employed for keeping the line of sight normal to the length of the thermometer tube are, (a) to hold a small mirror against the back of the thermometer, and to place the eye in such a position that the top of the mercury thread is in line with the image of the eye seen in the mirror; and (J) to observe the thermometer at a distance by means of a telescope containing a cross hair in the eyepiece, the telescope being fas- TEMPERATURE 157 tened normal to a rod placed parallel to the thermometer tube. The telescope must be arranged so that when it is moved along the supporting rod to observe the end of the moving column at different heights, it will always remain normal to the support- ing rod. A cathometer is usually most convenient for this pur- pose, but a short open tube without lenses, having crosshairs at the two ends, and sliding either on the thermometer itself or on a parallel rod, serves the purpose very well. 2. Errors due to the changes in the volume of the bulb lagging behind the changes in temperature. A rising thermometer in- dicates too low, and a falling thermometer too high, a tempera- ture. This lag is due to the viscosity of the glass of which the thermometer is made. If a thermometer be kept for some days at a uniform temperature of, say, 20° C, and then plunged into a bath of melting ice and the temperature observed ; and if it then be heated to a temperatvire of 100° C, and again plunged into the bath of melting ice, the tem-perature now observed will be lower than the one previously obtained. The increase in the volume of the bulb due to the high temperature does not at once disappear, and the zero point may be depressed as much as half a degree for some kinds of glass. This depression of the zero point is greater when the temperature to which the thermome- ter has been raised is greater, and when the time is greater that the thermometer is kept at the higher temperature. The de- pression persists .for weeks and even months before the normal volume of the bulb is regained. It follows that while the ther- mometer is being used at various temperatures the zero point is constantly changing. This makes no temperature determinate unless the value of the zero point at this particular time is known. The value of the zero point can be obtained by cool- ing the thermometer down to the temperature of melting ice immediately after the desired temperature reading has been made. Then, if no other errors affect the observation, the true temperature is the difference between the observed temperature and the value of the depressed zero. This is called the "de- pressed zero method" of measuring temperature, and is the 158 PRACTICAL PHYSICS only method capable of yielding the most accurate results attainable. 3. Errors due to the exposed column of the thermometer being at a temperature different from that of the bulb. Let T denote the true temperature of the bulb ; t, the temperature indicated by the thermometer ; s, the temperature of the exposed part of the stem ; and e, the reading where the stem emerges from the bath. Then the length of the exposed column is (i — e) degrees, and the difference between its temperature and that of the bulb is (y — s). Since the coefficient of apparent expansion of mercury in glass is about 0.000156 per degree C, this exposed part of the column, if it were to be raised in temperature {T — s) degrees, would increase in length 0.000156 (t — e) {T — s) degrees. That is, T=t + 0.000156 (t - e)(T- s). Whence T = t - 0.000156 s ^t - e^ 1-0.000156 Ct- e) or, employing approximation (5), p. 7, T=lt- 0.000156 s (t - e)] ■ [1 + 0.000156 (t - e)], or, neglecting the term which involves the square of 0.000156, r=< + 0.000156 («- s)0-e). (158) 4. Errors due to inequalities in the bore of the tube. These errors are corrected by calibrating the tube as described in Experiment 34. 5. Error in the graduation of the stem; that is, although the divisions are of equal length, their length is not such as to make just a hundred divisions between the boiling point and the freezing point of water. Let T^ denote the true temperature of the vapor above boiling water as determined by reading the barometer (see pp. 176-178, and Table 12), t^ the temperature indicated by the thermometer when it is immersed in the vapor above boiling water, and t„ the depressed zero reading taken TEMPERATURE 159 immediately after t^ was observed. Then the number of de- grees that ought to be between the point where the thermome- ter reads t„ and that where it reads t^ is T^, and the number of degrees that really are between those points is (^« — *<,). It follows that any temperature difference read from the ther- mometer is to be multiplied by a factor 6. Errors due to changes in the pressure to which the bulb is subjected. Any change of pressure will cause a change of the height of the mercury column independent of any change of temperature. Usually the experimental method can be arranged so as to eliminate this source of error. 7. Error due to capillarity. In a thermometer of very small bore the mercury does not move smoothly but moves in little jumps. This error is much greater when the temperature is falling than when rising. In fact, the capillary action makes it impossible to measure accur- ately a falling temperature by means of a mercury-in- glass thermometer. The Beckmann Theemometer. — A thermometer designed to estimate temperatures to thousandths of a degree requires such a long space for each degree of scale, that, if constructed on the ordinary plan, the range of the instrument would be limited to a few degrees. This would require the use of a number of instruments to cover the range of ordinary laboratory work. When it is not required to determine definite temperatures but only small temperature differences, the thermometer devised by Beckmann can be used at any temperature for which a mercury-in-glass ther- mometer is available. The peculiarity of this ther- mometer is a reservoir R (Fig. 67) at the upper end Fig. 67. of the tube, by means of which the quantity of mercury in the bulb can be increased or diminished. The scale is usually about 160 PRACTICAL PHYSICS five centigrade degrees in length and is divided into hundredths of a degree. In setting the instrument, a sufficient amount of mercury must be left in the bulb and stem to give readings between the required temperatures. First invert the thermometer and tap the tube so that the mercury in the reservoir will lodg-e in the bend B at the end of the stem. Now heat the bulb until the mercury in the stem joins the mercury in the reservoir. (See Fig. 67.) Place in a bath one or two degrees above the upper limit of temperatures to be measured. If now the upper end of the tube be flipped with the finger, the mercury suspended in the upper part of the reservoir will be jarred down, thus separating it from the thread at the bend B. The thermometer is now set for readings between the required temperatures. Exp. 34. Calibration of a Mercury-in-Glass Thermometer Object and Theory of Experiment. — If the bore of a thermometer is not uniform in cross section, the length of the tube corresponding to a degree difference in temperature will not be the same at different parts of the tube. And as it is impossible to get a perfectly uniform capillar}^, it is necessary to determine the correction to be applied to any particular reading to ta-ke account of the irregularity in the bore of a thermometer. Again, if the fixed points are incorrectly placed on the stem, this will introduce an error throughout the scale. The object of this experiment is to construct for a given thermometer a curve by which to correct errors due either to the irregularity of the bore or to the location of the fixed points. The experiment consists of two parts. First, the length of a short thread of mercury is measured at different parts of the tube, and from these lengths points are found throughout the whole length of the tube that separate equal volumes. Second, the position of the fixed points is determined by placing the ther- mometer in the vapor of boiling water and also in melting ice. TEMPERATURE 161 Manipulation and Computation. — The length of the thread to be broken off depends upon the thermometer. If the thread is too long, local irregularities of bore are not evident ; if the thread is too short, its changes in length are minute. If a dividing engine is available, a thread not more than a centi- meter long is advisable. If no magnification is to be used and the thermometer is an ordinary Centigrade thermometer grad- uated from 0° to 100° in degrees, a thread some fifteen degrees long is perhaps most satisfactory. The separation of the calibrating thread requires som^ dex- terity. In blowing the bulb on a thermometer tube, a slight constriction is usually left where the bulb and tube join. If such a thermometer is inverted and then given a sudden jar, the thread is likely to separate at this point. If there be no such constriction, the thread may be separated by laying the thermometer on a table and striking the upper end of the tube with a small block of wood. If this is not carefully done, how- ever, cracks may be produced inside the stem near the bulb. If the bore has an enlargement at the upper end, the column of mercury that has been broken off is allowed to run into this enlargement and to remain there while the tube is being cali- brated. The bulb is then slightly warmed until a thread of mercury of the proper length runs into the tube, and this, in turn, is separated from the mercury in the bulb. This is the thread that is used in the calibration. If the capillary has no enlargement at the upper end in which to store part of the mercury, it may be necessary to use two mercury threads to calibrate the two ends of the tube. When this is the case, the bulb is cooled, with a mixture of ice and salt if necessary, until all the mercury has run into the bulb except the length that is to be broken off. This thread is separated and run to the farther end of the tube. In order to make measurements in the lower end of the tube, this part of the thermometer must be freed of mercury and another thread separated as before. When a thread has been broken off, it is to be brought nearly to one end of the tube and the position of both ends carefully 162 PRACTICAL PHYSICS read, then moved along through a quarter or a third of its length and the positions of both ends again read, this process being repeated until the thread has been moved to the other end of the tube. Suppose that when this is done — a mirror being used as suggested on p. 156, and readings being made to twentieths of a degree — the readings are those in the first, second, fourth, and fifth columns of the following table : — Lower end of Upper end of Thread Lower end of Upper end of Thread Length in Degrees Thread at • Thread at Degrees TlIKBAD AT Thread at - 12°.30 + 3°.60 15°.90 39°.95 55°.65 15°.70 - 7 .25 8 .65 15 .90 45 .30 60 .95 15 .65 - 2 .05 13 .85 15 .90 50 .00 65 .65 15 .65 + 3 .00 18 .85 15 .85 55 .20 70 .80 15 .60 8 .15 24 .00 15 .85 60 .15 75 .75 15 .60 14 .30 30 .15 15 .85 64 .90 80 .45 15 .55 20 .00 35 .80 15 .80 70 .10 85 .60 15 .50 24 .55 40 .35 15 .80 74 .85 90 .35 15 .50 30 .05 45 .80 15 .75 80 .05 95 .50 15 .45 34 .90 50 .65 15 .73 83 .30 98 .75 15 .45 The quantities in the third and sixth columns are calculated from the observed quantities. The quantities in the first and third columns give the following curve, which shows the length of the thread when at different points of the capillary. From this curve the lengths of equal volume portions of the tube can be determined as follows : — The curve shows that if the bottom of the mercury thread were at 0° its length would be 15°. 87, so that the top of the TEMPERATURE 163 thread would be at 15°. 87; it also shows that if the bottom of the thread were at 15°. 87 its length would be 15.°81, so that the top of it would be at 31°.68 ; if the bottom of the thread were at 31°. 68 its length would be 15°. 73; etc. These values are recorded in the following table : — Points on Scale be- tween WHICH Vol- umes OF Bore are Equal Lengths of Thread BETWEEN Equal Vol- ume Points Positions of Equal Volume Points if Bore had ^been Uniform Corrections for Points in First Column 0.00 15.87 31.68 47.41 63.07 78.64 94.12 15.87 15.81 15.73 15.66 15.57 15.48 Av. 15.687 0.00 15.69 31.37 47.06 62.75 78.44 94.12 ±,0.00 -0.18 -0.31 -0.35 -0.32 -0.20 ±0.00 The quantities in the third column are found by multiplying by 1, 2, 3, etc., the average length of thread between equal volume points. The quantities in the fourth column are found by subtracting the quantites in the first column from those in the third. The quantities in the first and fourth columns give the upper curve in Fig. 69. This curve gives the corrections that must be applied to readings at different points along the scale on account of irregularities in the bore. The corrections for the displacement of the fixed points will now be considered. By definition, the lower fixed point (0° C. or 32 F.) is the temperature of melting ice. The upper fixed 164 PRACTICAL PHYSICS point (100° C. or 212° F.) is defined as the temperature of the steam produced by water boiling at sea level and latitude 45° under a barometric pressure of 76 cm. of mercury when the barometer is at the temperature 0° C. Observe the barometric height, noting the temperature of the barometer by means of the thermometer attached to the instru- ment. Ascertain from the laboratory instructor the latitude and altitude of the Jaboratory. From these data compute, in the manner explained on pp. 176-178, the corrected baromet- ric pressure H reduced to standard conditions. From a con- sideration of the number of figures that can be trusted in the uncorrected readings determine which of the corrections on pp. 177, 178 it is worth while to make. Suspend the thermometer in the vapor of boiling water. It must not be immersed in the water itself nor be so near the surface that the bulb will be spattered by drops of water, because the temperature of boiling water is influenced by the nature of the surface composing the vessel and by the presence of slight quantities of dis- solved impurities. But the temperature of the vapor depends only upon the pressure. Re- gnault's hypsometer consists of a reservoir R (Fig. 70) in which the water is boiled, sur- mounted by a tube in which the thermometer is suspended. After passing through this tube the steam passes through the jacket J and escapes into the air at U- M is 2k water manometer which serves to measure any difference of pressure between the steam inside and the air outside. If the manometer indicates a pressure of d mm. of water, i.e. ^„ „ mm. of mercury, then the total pressure on the surface of the boiling water \s H + mm. Call the observed boiling point T^. Draw the thermometer up until the upper twenty degrees or so Fig. 70. TEMPERATURE 165 of the stem is exposed. After about five minutes note the read- ing and also the reading at the top of the stopper, draw the thermometer up some twenty degrees farther, and, after about five minutes more, read again at both points. Repeat until the zero point is at the stopper. The difference between the read- ing at the top of the mercury when the thermometer was wholly immersed and the reading in any of the other cases is the stem exposure correction for that particular case. Note ap- proximately the temperature of the air in the neighborhood of the hypsometer. From (158) and (2) calculate the stem ex- posure correction for each case. Plot on the same sheet two curves, one coordinating the thermometer reading at the stopper and the observed stem exposure correction, and the other coor- dinating the thermometer reading at the stopper and the calcu- lated stem exposure correction. Remove the thermometer from the hypsometer, allow it to cool in the air to about 40° C, and then immerse it in a vessel filled with snow or shaved ice which contains enough water to fill the interstices. This gives the depressed zero point. By reference to Table 12, obtain the temperature of the vapor of water boiling at a pressure oi JI+ ..., „ ■ Call this true temperature T^ Then (y„ — Tt} is the error of the upper fixe'd point, and (^^ — T„') is the correction to be applied to the reading. Suppose that in the above example the error of the boiling point is found to be + 0°.6, and the error of the freezing point + 0°.8. Then the correction for the boiling point is — 0°.6, and for the freezing point — 0°.8. If now on the same sheet of coordinate paper on which the correction curve for irregularities of bore was plotted, the freezing point correction be entered along the axis of ordinates opposite the zero of ab- scissas, and the boiling point correction be entered opposite the observed boiling point, and these points be connected by a straight line, as shown by the dotted line in Fig. 69, this line gives the corrections for all intermediate points of the scale due to the displacement of the fixed points. 166 PRACTICAL PHYSICS By adding the ordinates of the correction curve for the irreg- ularities of bore — the upper curve — to the corresponding ordinates of this correction curve for displacement of the fixed points — the dotted line — ^the lower curve in Fig. 69 is obtained. This is called the Calibration Curve of the thermometer. If this calibration is done with two mercury threads instead of one, the calibration should extend from each end to a dis- tance past the middle of the tube. The curve analogous to that in Fig. 68 will be a continuous line, but along the region where data were taken with both mercury threads, one branch of the curve will be above the other. In this region find the ratio of the ordinates of the two curves for three or four posi- tions on the thermometer scale. This ratio must be really the same for all points on the thermometer scale. By multiplying any ordinate of one curve by the averages of the values found for the ratio, the corresponding ordinate of the other curve will be obtained. Proceeding in this manner, a continuous curve is obtained, just as though all of the calibration had been per- formed with a single mercury thread. Exp. 35. Calibration of a Resistance Thermometer Object and Theory op Experiment. — The mercury-in- glass thermometer is unavailable for the measurement of tem- peratures much below — 30° C, or above -|- 300° C. Although the gas thermometer can be used for any temperature for which a suitable material to construct the bulb can be found, it is such a large awkward instrument, and the difficulties of the manipu- lation are so considerable, that it is suitable only for stand- ardizing more convenient types of thermometer. Since the electrical resistance of metals varies continuously with the tem- perature according to definite laws, and since the accurate measurement of resistance is attended with no considerable difficulty, thermometers depending upon this change of resist- ance are in common use for measuring high and low tempera- TEMPERATURE ' 167 tures. Platinum is the material usually employed, both because its resistance at any given temperature does not change with time, and because the law connecting the temperature and resistance of a wire made of it is expressible by a simple for- mula throughout a very wide range of temperatures. It has been shown by experiment that if iJo represents the resistance of a piece of metal at 0° C, then throughout a more or less definite range of temperatures the resistance ij^ at t° C. is expressible by the equation Rt =R^\l + at + U^^, (160) where i^o, a, and h are constants. In order that a resistance thermometer may be used, these three constants must be known. The object of this experiment is to determine the values of these constants for a given resistance thermometer. If the resistance of the wire at three different temperatures be known, three equations of the form of (160) are obtained, and from these equations the values of the three constants can be calculated. The freezing point and boiling point of water are two convenient temperatures for the experiment. The re- maining temperature can be the boiling point of any convenient substance, e.g. sulphur, which boils at 444°. 5 C. But from measurements of the resistance of wires at very low tempera- tures, Dewar and E'leming have shown that at the absolute zero of temperature it is highly probable that the resistance of all pure metals is zero. Assuming this relation, the resistance of the thermometer wire need be measured at but two tempera- tures. This simplified process gives the values of the constants in (160) with sufficient accuracy for most purposes. Repre- senting the resistance of the thermometer wire at the tempera- tures t-^, ^2' ^^^ ^^^ absolute zero by the symbols Rt^, Rt,^, and -^-273' ^^ have Ri^ = ii!o [1 + « The Wheatstone bridge is to be used to determine the re- 3^((gj^ sistances. In texts on General c/o ^1 1 Physics it is shown that, if the ^° \ g;^Galv. ^ ( ^ r symbols have the meanings indi- _!■/" ^ -1 cated in the figure, ^'" ■ ^ = h (162) If the resistance ^g ^^^ ^^® lengths Zg and l^ are known, the other resistance i2j can at once be calculated. Manipulation and Computation. — The resistance ther- mometer consists of fine platinum wire wound on a mica frame enclosed in a wrought-iron capsule. In order to diminish errors due to a change in the temperature of the leads which run down into the capsule, a second pair of leads- precisely like the first is placed side by side with them but short circuited at Fig. 72. the bottom. By measuring at each temperature the resistance between the terminals of the coil and also the resistance between the terminals of the dummy leads the change in the resistance of the coil alone can be obtained, so that any change in the resistance of the leads does not need to be taken into account. The particular form of Wheatstone bridge called the "slide wire " or " meter " bridge will be used in this experiment. This apparatus is illustrated in Fig. 71. A uniform wire AO is stretched over a divided scale. The ends of this wire are con- TEMPERATURE 169 iiected to a parallel copper rod in which are two gaps. In one of these gaps is inserted the resistance to be measured M^, and in the other gap a resistance box M^. The current enters at one end of the bridge and leaves at the other. One side of the galvanometer is connected to the binding post B and the other to a key K^ which can slide back and forth and make contact at any point on the wire AC Whenever the keys are closed, K^^ is closed first and then jfiTg. Until the bridge is nearly balanced K^ is closed for as short a time as possible, and as soon as IC^ is open IC^^ is opened. If a mirror galvanometer with its telescope and scale are used, the telescope and scale are to be adjusted as described on p. 44. With the resistance thermometer packed in a bath of melting ice, make the connections indicated in Fig. 71. With no re- sistance in the resistance box and K^ about halfway from A to C, close the circuits just long enough to see in which direction the pointer of the galvanometer swings. Put a large resistance in the box, and again see in which direction the pointer swings. If the direction of swing is the same as before, something is wrong with the connections or else a larger resistance is needed in the box. If the pointer swings out in the opposite direction, the resistance needed in the box lies between zero and the re- sistance now in the box. Try half the resistance now in the box and note the direction of swing. Proceeding in this way, a value of the resistance can soon be found for which there is not much movement of the pointer. The remaining adjustment is to be made by means of K^, finding two positions of K^ such that the deflections for the two are in opposite directions, and then closing down upon a point where there is no deflec- tion. When no deflection is obtained, note the resistance in the box and the reading on the bridge scale. Note also how far the key can be moved in each direction without producing any observable deflection. Then, from (162), the value of the resistance being measured is B, = rJ^. (163) 170 PRACTICAL PHYSICS In the same manner find the resistance of the dummy leads. If the temperature of the resistance thermometer when in the bath of melting ice is represented by ij, then the difference between the two resistances just found is the value of Rf in (161). Proceeding in the same manner, find the resistance of the platinum coil when immersed in a steam bath. If this tem- perature be denoted by t^, the resistance will be the value of B,^ in (161). The values of the three constants can now be determined from (161). On substituting their values in (160), an equa- tion is obtained which gives the relation between the tempera- ture of the coil and its resistance. Such an equation, containing experimentally determined constants, is called an empirical formula. Substitute for t in this empirical formula the values — 200°, — 100°, 0°, 100°, 200°, and compute the corresponding values of Rf With these values plot a curve coordinating R and t. The accuracy of the preceding work should be tested by meas- uring the resistance of the thermometer coil at two or three known temperatures and comparing these observed values with the corresponding values given by the curve. Exp. 36. The Flash Test, Fire Test, and Cold Test of an Oil Object and Theory of Experiment. — If an inflam- mable gas is mixed with air in proper proportion, the mixture will explode on ignition. The air above a volatile oil is satu- rated with the oil vapor. If the temperature of the oil is slowly raised, the proportion of oil vapor in the air will in- crease until, at a certain temperature, the saturated air will become an explosive mixture. This temperature is called the flashpoint of the oil. If the temperature of the oil is still farther increased, a point will be reached at which the oil will evolve vapor so rapidly that, when ignited, it will burn con- tinuously. This is called the fire test of the oil. The cold TEMPERATUKE 171 test of an oil is the lowest, temperature at which the oil will flow. The object of this experiment is to make a flash test, fire test, and cold test of a sample of oil. The general method of determining the flash point is to heat the specimen gradually in a covered cup and at frequent inter- vals pass a small flame near the surface of the oil. In making a fire test, the specimen is heated in an open cup and the tem- perature is noted at which the vapor will burn continuously when ignited. The flash point depends upon (a) the rate of heating, (6) the depth and diameter of the cup, (c) whether the cup is closed or open, {d) the quantity of oil used, (e) the size of the testing flame and its distance from the surface of the oil. Consequently, the size and design of the testing apparatus and the method of carrying out a determination are explicitly described in the legislative enact- ments of the various states. Manipulation and Computation. — The form of apparatus most commonly used in this country for the flash point is the " New York State Board of Health Tester." This consists (Fig. 73) of a seamless copper cup C covered by a glass plate perforated with two holes — one for the insertion of the ther- mometer and another for the testing flame. This cup is heated in a water or air bath B by means of an alcohol lamp or small Bunsen burner. The whole apparatus should be placed in a sheet-iron pan filled with sand. In using this apparatus to test illuminating oils, the New York State Board of Health publish* the following regulations : — "Remove the oil cup and fill the water bath with cold water up to the mark on the inside. Replace the oil cup and pour in enough oil to fill it to within one eighth FiQ. 73. * Report of N. Y. State Board of Health, 1882. 172 PRACTICAL PHYSICS of an inch of the flange joining the. cup and the vapor chamber above. Care must be taken that the oil does not flow over the flange. Remove all air bubbles with a piece of dry paper. Place the glass cover on the oil cup, and so adjust the ther- mometer that its bulb shall be just covered with oil. " If an alcohol lamp be employed for heating the water bath, the wick should be carefully trimmed and adjusted to a small- flame. A small Bunsen burner may be used in place of the lamp. The rate of heating should be about two degrees per minute, and in no case exceed three degrees.* " As a flash torch, a small gas jet one quarter of an inch in length should be employed. When gas is not at hand, employ a piece of waxed linen twine. The flame in this case, however, should be small. "When the temperature of the oil has reached 85° F., the testing should commence. To this end insert the torch into the opening in the cover, passing it in at such an angle as to well clear the cover, and to a distance about halfway between the oil and the cover. The motion should be steady and uni- form, rapid and without a pause. This should be repeated at every two degrees' rise of the thermometer until the thermom- eter has reached 95°, when the lamp should be removed and the testings should be made for each degree of temperature until 100° is reached. After this the lamp may be replaced if neces- sary and the testings continued for each two degrees. " The appearance of a slight bluish flame shows that the flashing point has been reached. " In every case note the temperature of the oil before intro- ducing the torch. The flame of the torch must not come in contact with the oil. " The -water-bath should be filled with cold water for each separate test, and the oil from a previous test carefully wiped from the oil cup." Make five determinations of the flash point and take the mean. * This refers to degrees Fahrenheit. TEMPERATURE 173 After each determination, remove the cover from the oil cup and blow the burnt gases out of the cup. After the flash point has been determined, remove the cover from the oil cup and continue to heat the oil at the rate of two degrees per minute. About every half minute test the oil with the small flame as above described. The lowest temperature at which the vapor of oil will burn continuously is the fire test. Remove the thermometer and smother the flame by placing on top of the oil cup a piece of asbestos board. Such a damper should always be at hand for emergencies. In the case of lubricating oils the method of finding the flash point and the fire test is exactly as above described except that the rate of heating should be 15° F. per minute and the testing flame should be applied first when the oil is about 200° F. In making the cold test, a glass vial or boiling tube of about 100 CO. capacity is one fourth filled with the oil under investi- gation, and then placed in a freezing mixture of ice and salt. When all of the oil has congealed, it is removed from the freez- ing mixture and thoroughly stirred with a thermometer until it is sufficiently softened to flow from one end of the tube to the other. The temperature at which this occurs is the cold test of the oil. Exp. 37. Relation between Boiling Point and Concentration of a Solution. Object and Theory of Experiment. — The object of this experiment is to find the relation between the boiling point and the concentration of a solution of common salt. The boiling point of a solution of a non-volatile substance is higher than the boiling point of the pure solvent. If a current of steam be passed into an aqueous solution below its boiling point, steam will be condensed in the solution until the heat thereby liberated raises the temperature of the solution to its boiling point. Consequently steam that passes through a solu- tion will leave it at the boiling point of the solution and not at 174 PRACTICAL PHYSICS that of pure water. However, as the steam eBcapes into the space above the solution it cools somewhat by expansion, and wherever it comes into contact with the walls of the vessel, with the thermometer, or with any body that can gradually conduct away the heat given up by the condensation of the steam, this cooling continues until the steam becomes saturated, that is, until its temperature falls to the boiling point of pure > water. Consequently, in determining the boiling point of the pure solvent the thermometer is suspended in the space above the liquid, while in determining the boiling point of a solution the thermometer bulb must be immersed in the solution. Buchanan has recently utilized the principle stated in the preceding paragraph for finding the boiling point of a saturated solution. A quantity of the pure solute is placed in the bottom of a tall test tube containing a thermometer. A current of steam is sent through a glass tube extending to the bottom of the test tube until a saturated aqueous solution of the given solute is obtained. As long as any of the solute remains un- dissolved and the current of steam is uninterrupted, the tem- perature of this saturated solution remains at its boiling point. Manipulation and Computation. — The apparatus used in determining the boiling point of a dilute solution consists of a flasli provided with a cork fitted with a thermometer and con- denser. Without the condenser the solution would gradually increase in concentration through the loss of steam. To prevent "bump- ing," a handful of clean dry pebbles or pieces of broken glass is placed in the flask. With the flask about one third filled with a solution of some 50 g. of sodium chlorid to each liter of water, insert the thermometer until its bulb is 5 cm. or more above the surface of the liquid. With the condenser in place, heat the solution until it boils fairly rapidly. Read the thermometer and also the laboratory Fig. 74. TEMPERATURE 175 barometer. Push the thermometer down until its bulb is in the boiling solution, and when it has become steady read it again. From the corrected barometer reading (see pp. 176- 178 and Table 12) the true temperature of the vapor of boiling water can be found. This value minus the reading of the ther- mometer when in the vapor is the correction to be applied to the given thermometer in the neighborhood of 100°. This cor- rection added to the reading of the thermometer when in the boiling solution gives the boiling point of this solution. In the same manner find the boiling points of solutions which contain respectively three and five times as large a proportion of salt. To find the boiling point of a saturated solution, fasten a large boiling tube in a vertical position in a retort stand, and fill the tube to a depth of one or two centimeters with salt. Suspend in the tube the same thermometer used before so that the bulb just touches the layer of salt and then push halfway through the layer of salt the end of a glass tube in which flows a current of steam. When the bulb of the thermometer is sub- merged in the solution formed by the salt and condensed steam, observe the temperature. When the correction determined in the first part of the experiment is applied to this reading, it gives the required boiling point. Plot a curve showing the relation between concentration and corrected boiling point. The concentrations may be expressed in grams of salt per liter of water, and the concentration of a saturated solution of sodium chlorid may be taken as 396.5 g. per liter. The temperature of the vapor above the boiling solutions is the boiling point of a solution of zero concentration. By the method outlined on pp. 10-12 find an empirical equa- tion connecting the concentration and boiling point of a sodium chlorid solution. CHAPTER XI EXPANSION If a body has at 0° a length If, and -at t° a length l^ it is found that the relation between length and temperature is usually ex- pressed very nearly by the equation I, = lo(l + «0. (164) where « is a constant for any one substance and is called the coefficient of linear expansion of that substance. Similarly, if a body has at 0° a volume Wq and at t° a volume Vf, it is found that the relation between volume and tempera- ture is usually expressed very nearly by the equation ^t = ^0 (1 + 70' (165) where 7 is a constant for any one substance and is called the coefficient of cubical expansion of that substance. In the case of gases it is shown in texts on General Physics that if p, V, m, and T denote respectively the pressure, volume, mass, and absolute temperature of the gas, pv=RmT, (166) where i2 is a constant which depends only upon the units chosen and not at all upon the nature of the gas nor any other condition. (166) is obtained by combining Boyle's and Charles's laws and is known as the Fundamental Law of Gases. REDUCTION or BAROMETRIC READINGS If the Torricellian vacuum above a barometric column were devoid of matter and if there were no capillary force between the mercury and the tube, the weight per unit cross section of 176 EXPANSION 177 a barometric column would equal the pressure of the atmos- phere at the place where the barometer is situated. The space above the mercury column is, however, filled with mercury vapor which exerts a small pressure depressing the mercury, and the capillary action between the mercury and the glass tube also diminishes the height to which the mercury rises. Again, even if the pressure of the atmosphere does not change, the actual height of the mercury column may be altered in two ways : first, by a change of temperature which not only alters the density of the mercury in the barometer and consequently its height, but which also alters the length of the scale used to measure the height ; second, by a change in the force of gravity acting on the mercury as it is moved to different parts of the earth's surface. Consequently, in order that barometric read- ings taken at different temperatures and at different parts of the earth's surface may be compared with one another, they must be reduced to the heights that would have been observed if the barometer had been at some standard temperature and at some standard position on the earth's surface. The standard condi- tions arbitrarily selected are the temperature of melting ice and the altitude of the sea level at latitude 45°. In precise work a barometric reading must be adjusted in the above four particulars, of which two are corrections and two are reductions to standard conditions. The method of making these corrections and reductions to standard conditions will now be considered. 1. Temperature. Let A and p represent the observed height and the density of the mercury at t°, and let v represent the volume of mass m of mercury at this temperature. Let Aqi Pai and ^0 represent the corresponding quantities at 0° C. Then pgh = p^g\ and m = vp = v^p^. (167) If /3 denotes the coefficient of cubical expansion of mercury, v^v.il + ^f). . Whence ^ = — ^. (168) V 1+ ^t 178 PRACTICAL PHYSICS And since, from (167), -£ = ^ and £- = ^, it follows that ^ = — ^. (169) On using approximation (•^),p. 7, the above equation reduces to Ao % A (1 - /30- (170) But the brass scale used to measure the height A is ruled so as to He correct at 0° C. That is, a space on the scale having at 0° unit length has at t° a length (1 + at), where a is the co- efficient of linear expansion of the brass scale. Whence, a dis- tance which at f is presumably A units long is really h (1 + a^) units long. Consequently, the barometric height that would be observed if the barometer were cooled to 0° C. is A„ % A (1 + «0 (1 - /SO. or, employing approximation (2), p. 7, Ao = A[l + (a-^)«]. (171) Since /8 = 0.000182 and « = 0.000018 per degree centigrade, (171) becomes A^ =A(l-0. 00016 (172) if the temperatures are taken in centigrade degrees. If, how- ever, the temperatures are taken in Fahrenheit degrees A32 % A [1 - 0.00009 (t - 32)]. (173) 2. Depression due to Capillarity. This depends upon the diameter of the bore of the glass tube. Its magnitude may be taken from the following table : — Bore of tube in mm. 2 4 6 8 10 Depression in mm. 2.18 0.70 0.25 0.10 0.04 3. Reduction to Sea Level at Latitude 45°. This is most easily effected by means of Table 11. 4. Depression due to Pressure of Mercury Vapor. This may be neglected except in the most refined work. The values of EXPANSION 179 the vapor pressure of mercury at different temperatures are given in Table 14. Exp. 38. Determination of the Coefficient of Linear Expansion of a Solid Object and Theory op Experiment. — The object of this experiment is to determine the coefficient of linear expansion of a metal. If Zj denotes the length of the body at tempera- ture to the knife edge HF with a dividing engine. After assembling the apparatus set up the telescope and scale about a meter from the optical lever and see that the telescope is about at the level of the mirror. After adjusting EXPANSION 183 the telescope and scale as directed on p. 44 set the optical lever in such a position that when the eye is at the level of the telescope and close to it, the image of the telescope in the mirror can be seen. This makes the optical lever vertical. Measure l^, the distance between the wyes on which the specimen rests, and L, the distance from the mirror to the vertical scale. Note the readings of both thermometers, take the scale reading in the telescope, send a current of steam for some minutes through the jacket surrounding the specimen, and then take the new scale reading and again read both thermometers. By (158) correct the thermometer readings for steam exposure. Calcu- late a by (178). In the case of the roller method, measure the diameter of the roller with a micrometer caliper. Assemble the apparatus, being careful that the roller carrying the pointer is normal to the length of the bedplate, and also that it is at the center of the divided circle. It is well to start with the pointer about as far to one side of the vertical as it will come to be on the other side of the vertical. It can be set in this way after a preliminary experiment in which the angle through which it will turn is determined roughly. The carriage should be so placed that the wye is directly above the roller that carries the pointer. Measure Zj, the distance between the edges of the two wyes on which the specimen rests, and note the readings of both thermometers and of the pointer. Send a current of steam for some minutes through the jacket surrounding the specimen and then observe the*new position of the pointer and again read both thermometers. By (158) correct the thermometer readings for steam exposure. Calculate a by (179). Exp. 39. Determination of the Absolute Coefficient of Expan- sion of a Liquid by the Method of Balancing Columns Object and Theory op Experiment. — The object of this experiment is to determine the coeflficient of expansion of mer- cury by a method which is independent of the change in 184 PRACTICAL PHYSICS volume of the containing vessel. The method employed in this experiment is to determine the coefficient of expansion of the liquid from the ratio of its densities at different tempera- tures. The apparatus used by Regnault is illustrated in Figs. 79 and 80. Consider a W-shaped tube ABCDEF containing mercury, having the branch A kept at a high temperature by means of a steam jacket, and the remainder of the apparatus Fig. 79. at the temperature of the room by means of water jackets. The mercury columns A and F are connected at the top by the tube (7, so that the pressures of the mercury in both columns are the same at this level. At the bottom the two columns A and F are kept separated by means of compressed air in the tube CD. Let ITj, H^, h^, and h^ represent the differences in level indicated in the figure. Let the temperature and density of the mercury in the hot part of the apparatus be denoted bj' EXPANSION 185 <2 and p^ respectively, and the temperature and density of the remainder of the mercury by t^ and py Let P denote the atmospheric pressure and P' the pressure of the air in P. Then the pressure at the bottom of the ^-column is (P + p^gH^^ and the pressure at the bottom of the i)- column is (P' + Pighj). Now these are pressures at the same level in a fluid at rest and are therefore equal. That is, P + p,gff, = P'+p,gh,. (180) In the same w&j for the ^-column and the (7-column, P + p^H^ = P' + p,gh^. (181) On subtracting (181) from (180) and solving for ^i P2 ^1 - «! + «2 From the figure ^^ + 6 = Ag + «. Whence, Aj.— }i,^ = a—h. •■•^i-(^i-^2) = -^i- (,a-h~) = H^-a. So that (182) becomes Pi= ^i . (183) Pi -02 ~ * Now the density of a given mass of mercury is inversely proportional to its volume. So that, if yS denotes the coefficient of expansion of the mercury, &)=!!l = l + /3«i, (184) Pi ^0 and Psi = h = l + ^t^. (185) Pi "o On dividing (185) by (184) a value is obtained for ^. If this P2 value is equated to that in (183), and the resulting equation solved for /3, we obtain 186 PRACTICAL PHYSICS Manipulation and Computation. — An instructor will pump air into the reservoir R so that the mercury rises in the tubes A and F until its surface stands about at the axis of the tube Gi. Water should flow through the water jacket only slowly and its temperature, ij, should be near that of the room. This temperature is to be taken by a thermometer, and the tem- perature of the hot jacket calculated from a reading of the barometer. (See pp. 176-178, and Table 12.) Measure -Hg with a meter stick and a with a cathetometer. Make at least five independent determinations of a, readjusting the cathetometer before each one, and use the mean. Exp. 40. Determination of the Coefficient of Cubical Expansion of Glass Object and Theory op Experiment. — When a vessel filled with liquid is raised in temperature, the apparent expan- sion of the liquid depends upon the coeiiicients of cubical ex- pansion of the liquid and the containing vessel. If the apparent expansion is observed and the coefficient of expansion of either the liquid or the containing vessel is knoAvn, then the coefficient of expansion of the other can be determined. The object of this experiment is to determine the coefficient of cubical expan- sion of a glass bulb from an observation of the apparent expan- sion of mercury contained in it. The absolute coefficient of expansion of mercury is supposed to be known from the pre- ceding experiment. Let the bulb be filled with mercury at the temperature t-^ and then be heated to t^. This will cause some of the mercury to be expelled. A bulb used in this manner is called a weight dilatometer. Let itfj and pj represent the mass and the density of the mercury contained in the bulb at t-^; M^ and p^, the mass and the density of the mercury contained in the bulb at t^; EXPANSION 187 f 1 and v^, the respective volumes of the mercury in the bulb at t^° and t^"; v^-, the volume at t^ of the mercury which at t-^ filled the bulb ; 7, the mean coefficient of cubical expansion of glass be- tween ij° and t^; y8, the mean coefficient of cubical expansion of mercury between t-^ and t'^. From (61) we have ^1 = ^ v^ = —\ «2 =-^- (187) Px Pi Pi Since yS and 7 are both small, it may be shown from (165) in the same way that (176) was developed from (164), that ^= V.~''\ a"d 7 = ''^"''1 • (188) ■^1^2-^1) '^1(^2-^) On substituting in (188) the values of v^, v^, and Vg' from (187), and eliminating Q from the resulting equations, we get or, writing mj for the mass of the mercury expelled when the bulb is heated from t-^ to t^, Manipulation and Computation. — A convenient form of weight dilatometer for this experiment is a specific gravity bottle having a perforated glass stopper. After weighing the bottle, nearly fill it with mercury, and, without inserting the stopper, heat very carefully until all observable air bubbles have been removed. An iron wire will greatly facilitate the drawing out of these bubbles. Have under the bottle a vessel to catch the mercury in case the heating should be done too rapidly and the bottle thereby be broken. 188 PRACTICAL PHYSICS After the bottle has cooled enough to be held comfortably in the bare hand, place it on a cork stool in a beaker and pack it to a little below the opening with shaved ice. After -five or ten minutes insert the stopper, being sure that there is enough mer- cury in the bottle to fill the capillary and leave a tiny globule above it. If this globule does not decrease in size in a few moments, brush it off, and then begin slowly heating the beaker. As fast as mercury comes out of the capillary, brush it off into a piece of paper bent into a cup without any hole in the bottom. When the mercury has stopped coming out, remove the flame and allow the beaker to cool. Meantime read the barometer in order to determine the tem- perature of the hot mercury (cf . pp. 176-178, and Table 12), and then fold the paper containing the extruded mercury so that the latter will not run out and by the method of vibrations weigh paper and mercury. Then weigh the paper alone, and after the specific gravity bottle has cooled to the temperature of the room, weigh the bottle and the mercury which it still con- tains. The difference between the weight of the paper with the mercury in it and the weight of the paper alone is m^ ; the difference between the weight of the bottle with the mercury left in it and the weight of the bottle alone is M^ ; t^ is zero, and ^2 is the temperature of boiling water. The value of /3 may be taken as 0.000182 per degree C. Exp. 41. Determination of the Coeflacient of Expansion of a Gas by Means of an Air Thermometer Object and Theory of Experiment. — If a given mass of any substance is heated from 0° to 1°, and the pressure upon it kept constant, the ratio of the increase of volume to the initial volume is called the coefficient of expansion of the substance. If it is heated from 0° to 1°, and its volume kept constant, the ratio of the increase of pressure to the initial pressure is called the temperature coefficient of pressure of the substance. In the succeeding paragraph it will be shown that for a perfect gas EXPANSION 189 these two coefficients are equal. Since it is easier to measure the pressure of a gas under constant volume than to measure the volume under constant pressure, the coefficient of expansion will be determined in the present experiment from observations of the changes produced in the pressure of a gas when its mass and volume remain nearly constant and its temperature changes. Consider a given mass of gas at temperature 0° C. or Tq absolute, pressure pg, and volume Vf^. When it is heated to t° C, i.e. (^Tf^ + ty absolute, let the pressure be represented by pt and the volume by Vi. From the fundamental law of gases, (166), ^=i2m=^. (190) If the volume be kept constant, which is denoted below by the subscript v outside of the parenthesis, the Vf in (190) becomes equal to the Vq, and (190) solved for — gives ^JPlUlo),' (191) To V Pot A or, if the pressure be kept constant, 1 _fvt-v^ (192) But the coefficient of expansion of a gas, j3, is, from its definition, given by It follows that h'^"^)- '-'''' That is, the coefficient of expansion of a gas is equal to the reciprocal of its absolute temperature and is also equal to its pressure coefficient. An apparatus well suited for determining the pressure coeffi- cient is some form of Jolly's Air Thermometer. This consists 190 PRACTICAL PHYSICS (Fig. 81) of a glass bulb B filled with air or other gas, connected to an open manometer tube M filled with mer- cury. Immediately below the bulb is a tube containing an index finger F made of colored enamel. The volume of the gas is made defi- nite by adjusting the plunger P until the mer- cury surface is brought to the point F. The pressure of the gas in the bulb is measured by the difference between the levels of the mer- cury .surface at F and the mercury surface in tlie manometer tube. The bulb is inclosed by a vessel in which can be placed water or ice. i 2) is a drying tube used in filling the bulb. On account of the temperature of the small amount of gas in the exposed part of the bulb being different from that in the jacketed part of the bulb, and also on account of the change in the volume of the bulb when its temperature is changed, (193) cannot be used in its present Fig. 81. simple form. The corresponding equation in which these facts are taken into account will now be derived. Let Pq denote pressure in bulb when at 0° C. ( Ti^° absolute) ; Pi, pressure in bulb when at t° C; v^, volume of jacketed part of bulb at 0° C. ; Vf, volume of jacketed part of bulb at t° C. ; tHq, mass of gas in jacketed part of bulb when at 0° C; »M(, mass of gas in jacketed part of bulb when at ^° C. ; Vf^', volume of exposed part of bulb when jacketed part is at 0° C. ; «/, volume of exposed part of bulb when jacketed part is a.tt° C; wig', mass of gas in exposed part of bulb when jacketed part is at 0° C. ; w?/, mass of gas in exposed part of bulb when jacketed part is at t° C. EXPANSION 191 Without great error, the temperature of the exposed part of the bulb may be assumed to be constant and equal to that of the room. Let this temperature be denoted by t'° C. It follows that v/ = v^ . Applying (166) to (a) the jacketed part of the bulb when at 0°, (hj the jacketed part of the bulb when at t°, (c) the exposed part of the bulb when the jacketed part is at 0°, and (c?) the exposed part of the bulb when the jacketed part is at f°, the following four equations are obtained : — PtVi = Bmt(^TQ + t'), p,v,' = Bm:(^To + t'}. Since the mass of gas in the apparatus remains constant^ we have also Wq + Wq = ?W( + m/ . Eliminating from these five equations the four unknown masses, we get Representing by 7 the coefficient of cubical expansion of glass between the temperatures 0° and t° C, we have Substituting this value in (194), remembering that, from (193), 1 V ' Tq = — , and denoting the constant ratio -2- by k, we obtain ?.(i+TT5?)=^'(rTl+rw> ^"«> When (195) is solved for /3 the resulting formula is very long. This procedure is, therefore, seldom adopted. One of the methods which may instead be employed to find /3 is the following : It will be seen that as long as the room temperature 192 PRACTICAL PHYSICS remains the same the left member of (195) does not change. It follows that the right member is also constant. That is, if t had some value t^, this right member would have the same value as if t had some other value t^. That is, ^\1 + ^/ 1+/3W ^^l+/S^/l + y8«7 ^ ^ If ^2 is chosen as the room temperature t', if the subscript 1 is dropped, and if p' is used to denote the pressure in the bulb when all the gas in it is at the temperature t' , (196) becomes .1 + /3«' ^ 1 + y8«7 ^\\ + ^ri + ^t' Whence, /3 = Aj + 7^^ + /fc) -^0 +7^ + ^) , (jgg^ It will be seen tlrat if h and 7 were both zero and if the tem- perature t were 0°, (198) would reduce to (193), as it should. Manipulation and Computation. — The air in the bulb has been dried once for all and the upper opening of the bulb permanently sealed. Hang a thermometer in the air just out- side of the bulb and adjust the plunger until the mercury touches the index F. After a few minutes, when the tempera- ture seems steady and the top of the mercury stays at F, set the slider S at the top of the column in M and read both the position of the slider and the temperature in the jacket. Then fill the jacket with shaved ice. Notice the index frequently for several minutes, readjusting whenever the mercury is not just touching it. When no more adjustment seems necessar}', set the slider S at the top of the mercury in M and read its position. Read also the laboratory barometer. For the ice substitute water at a temperature of about 40°' C, and after allowing a few moments for the bulb to aquire the tem- perature of the water, begin slowly heating the water by pass- ing steam into it. While the* water is heating, read on the manometer scale the height of F. This can be done best with EXPANSION 193 a cathetometer, but may be effected by a straightedge held horizontal with the aid of a level. As the water warms adjust the plunger occasionally, and when the steam is bubbling rapidly through the water in the jacket and no further adjustment seems necessary, set the" slider and read it again. Read also a thermometer which is pretty well immersed in the jacket. When through with the apparatus, draw off the water in tlie jacket and leave the mercury at about the same height in both tubes. The barometric height plus or minus the difference between the heights of S and I' when the bulb was at the room tempera- ture gives p'. The corresponding quantity when ice was in the jacket gives one value for pf, the t in this case being 0°. k will be given by an instructor, and 7 may be taken as 0.000027 per degree C. Use (198) to get one value for /3. Find a second value for yS by using for pt the pressure found when the bulb was surrounded by the hot water and for t the tempera- ture of that water., CHAPTER XII VAPORS Exp. 42. Determination of the Maximum Vapor Pressure of a Liquid at Temperatures below ioo°C. (STATIC METHOD) Object and Theory of Experiment. — When a liquid evaporates in a closed space, the vapor formed produces on the surface of the liquid and on the inclosing walls a pressure which increases with the mass of vapor and with the tempera- ture. For a given temperature the vapor pressure* reaches a maximum value when the space is saturated. The object of this experiment is to determine the pressure of saturated aqueous vapor at temperatures from about 50° C. to 100° C. In the method to be used in this experiment, the vapor pres- sure is determined from the decrease in the height of a barome- ter column produced by a small quantity of the specimen which is introduced into the vacuous space above the mercury. The apparatus (Fig. 82) consists of a barometer tube, the upper end of which is enlarged into a narrow bulb and its lower end joined to an open manometer tube, M. Opening into the hori- zontal tube joining the barometer and manometer is an iron cylinder filled with mercury. The height of mercury in the two tubes can be varied by means of a plunger, P, in this cylin- der. A small enamel finger, F, in the bulb of the barometer tube serves as a convenient fixed point from which heights can * The expression vapor tension is sometimes used instead of vapor pressure to denote elastic stress exerted by a vapor. Careful writers, however, use the word pressure to denote a push, and tension to denote a pull. Since vapors and gases cannot exert a pull, the term vapor tension is a misnomer. 194 VAPORS 195 be measured. The vapor being studied can be brought to the desired temperature by means of a water bath surrounding the bulb. The pressure in the tube M at the level of the mercury surface at S is the atmospheric pressure. The pressure of the vapor in the bulb is less than this by the pressure due to the column of liquid between the levels of aS* and F. Manipulation and Computation. — The bulb has been freed from air, a specimen of air- free water introduced, aud the upper end of the bulb permanently sealed. Observe the atmospheric pressure from the laboratory standard barometer. Fill the water jacket with water and pass steam into it until it reaches a temperature of about 45° C. On account of danger of cracking the glass, the current of steam should not be directed against the bulb nor against its jacket. Hold the , temperature as nearly steady as possible for a few minutes, and by means of the plunger adjust the height of the mercury in the barometer tube until it is brought just into contact with the tip of the index finger. Stir the water in the jacket, observe its temperature, readjust the plunger if necessary, move the slider *S' until its index line is tangent to the meniscus in the manometer tube, and read the position of this index line. Determine to the nearest millime- ter the height of the water column above the mercury. Di- vide this height by 13.6, the specific gravity of mercury, and add the result to the difference between the levels of the mer- cury in the two tubes. Subtract this result from the barometric pressure as given by the standard barometer. The result is the vapor pressure of water at the temperature of the experiment. Take similar readings every ten degrees up to about 95° C. When through the experiment, draw off the water in the jacket, Fig. 82. 196 PRACTICAL PHYSICS and adjust the mercury to about the same level in both arms. Plot a curve with vapor pressures as ordinates and corre- sponding temperatures as abscissas. On the same coordinate axes plot another curve from the values given in Table 13. This method is liable to several errors. The surface tension of the dry mercury in the manometer tube is different from that of the wet mercury in the barometer tube. This will cause a rise of the column having the wet surface of 0.10 to 0.15 mm. The fact that the lower part of the barometer tube is at a lower temperature than the upper causes the final result to be too low. This error will be of the order of 0.15 mm. If the position of the end of the index finger is read through the water jacket, the refraction of the glass and water will intro- duce an uncertainty that may amount to 0.5 mm. This error is obviated by carefully measuring the distance from the end of the index to a fine scratch on the tube below the water jacket before the apparatus is assembled. In order to use this scratch as the fiducial line from which heights are measured, the posi- tion of the line is read on the meter stick by means of a cathe- tometer (see p. 23). The greatest limitation to the use of this method, however, is due to the large error introduced in the depression of the barometer column by an impurity of the specimen. Exp. 43. Determination of the Maximum Vapor Pressure of a Liquid at Various Temperatures (DYNAMIC METHOD) Object and Theory of Experiment. — The object of this experiment is to determine the maximum vapor pressure of water at various temperatures from about 50° C. to about 120° C. The dynamic method to be employed in this experi- ment is based upon the following two laws of vapors: first, a liquid boils when the pressure of its vapor equals the external pressure ; second, if the pressure does not change, the tempera- VAPORS 197 ture of the boiling liquid remains constant as long as there is any liquid to vaporize. In Regnault's apparatus (Fig. 83) the water is inclosed in a boiler B, from the top of which runs a tube through the condenser C to a large metal reservoir inclosed in a water bath kept at con- stant temperature. The reservoir is filled with air the pressure of which is varied by means of a pump connected to P. The air in the reservoir serves to equalize any sudden changes of pres- sure due to irregularities in boiling. If it were not for the condenser, most of the steam formed in the boiler would not be condensed, but instead would increase the pressure in the boiler and reservoir and thus prevent much boiling. That is, when the burner was lighted both temperature and pressure would gradually rise. The pressure of the vapor is measured by means of the open manometer at the right. The tempera- ture of the vapor in the boiler is ob- tained from the four thermometers T, placed in tubulures which project into the boiler. Two of these tubulures are long, projecting into the water, and two are short, projecting into the vapor only. Fig. 83. The bottoms of all of them are filled with mercury, so that the bulbs of the thermometers quickly acquire the temperatures of the water and vapor in the boiler. Manipulation and Computation. — The boiler already contains sufficient water. Start a stream of cold water flowing through the condenser, and then light the burner under the boUer. Pump air out of the reservoir until the pressure is reduced to about 10 cm. of mercury, i.e. until the difference be- 198 PRACTICAL PHYSICS tween the heights of the mercury in the two arms of the manom- eter is about 65 cm. Then close the stopcock in the tube con- necting the pump and the reservoir. When the thermometers have become steady, record the reading of each thermometer and of each of the manometer tubes. Note also the temperature of the manometer and the barometric height. The corrected barometric height diminished by the corrected difference of level between the manometer columns gives the pressure of the vapor at the temperature indicated by the thermometers in the boiler. Assuming that the manometer scale is correct at 20° C, reduce the pressure to 0° C, i.e. so correct it as to make it the pres- sure that would have been observed if barometer and manom- eter had been at 0° C. This can be effected for the barometer by (172) and for the manometer by (170). Allow air to enter the reservoir until the difference in the heights of the mercury columns is about 20 cm. less than before. This increase of pressure requires that a higher temperature be attained before the water will boil. When the temperature has reached the new boiling point, a second set of observations is to be made. In the same manner, the boiling points correspond- ing to about eight different pressures are to be determined, the difference in pressure in passing from each case to the next being about 20 cm. of mercury. Plot a curve with pressures as ordinates, and temperatures as abscissas. On the same coordinate axes plot for the same range another curve with values from Table 13. This curve, showing the way in which the pressure of saturated water vapor changes with temperature, is called the steam line. Exp. 44. Determination of the Density of an Unsaturated Vapor by Victor Meyer's Method. Object and Theory op Experiment. — Probably the most accurate method for determining the density of an unsat- urated vapor is to allow a known mass of the liquid whose vapor density is to be determined to vaporize in the Torricellian vacuum VAPORS 199 of a barometer, and then to observe the volume occupied by the vapor. The ratio of the mass of the liquid vaporized to the volume occupied by the vapor is the density of the vapor at the temperature and pressure of the experiment. But if the result need not be trusted more closely than to within some three to five per cent, a method due to Victor Meyer ■will be found much more convenient. The apparatus used in this method is shown in Fig. 84. It comprises a gas measuring tube ^, and a vapor chamber con- sisting of a long glass tube ter- minating in a bulb 5, surrounded by a bath containing a liquid of higher boiling point than the substance under examination. The specimen is contained in a small bulb whicli can be sup- ported in the upper cooler part of the vapor chamber by means of a rod H capable of a back and forth motion in a side tube. When the bath has attained a constant temperature, high enough to vaporize the speci- men, the rod H is drawn back so as to allow the little bulb containing the specimen to fall to the bottom of the chamber. Here it either breaks by con- cussion with the bottom, or bursts due to the expansion of the contained liquid. When the contained liquid vaporizes, it pushes out of the vapor chamber a volume of air equal to its own volume, and the vol- t\ -- -J Fig. 84. 200 PRACTICAL PHYSICS uiiie of this expelled air is determined by means of the measur- ing tube E. In bubbling up through the water in the measuring tube the air expelled from the vapor chamber becomes cooled and so con- tracts. Since all gases have nearly the same coefficient of ex- pansion, the vapor, if it could be cooled to the temperature of the air in the measuring tube without becoming saturated, would after this cooling occupy the same volume that the air does. Therefore the density of the vapor at the temperature and pressure of the air in the measuring tube equals the mass of substance vaporized divided by the volume of air thereby forced into the measuring tube. The temperature of the bath sur- rounding the vapor chamber must remain constant during the vaporization of the specimen, but its value need not be known. Since the densities of gases and vapors vary greatly with changes of pressure and temperature, it is customary to reduce the values to what they would be at some standard pressure and temperature. The pressure usually adopted as standard is the pressure of 760 mm. of mercury, and the temperature adopted as standard is 0° C. Let m be the mass of substance vaporized, and v^ and Vq the volumes of the vapor when at the respective pressures, tempera- tures, and densities pt, p^, t, 0, pi, and p^. From the funda- mental law of gases, (166), p^v^ = RmT^ (199) and ptVt =Bm. Down to the temperature of saturation, both water vapor and air obey approximately the fundamental law of gases. There- fore, from (166), p"v" = Bm"T (203) and pv" = RmT. (204) As long as p" does not change, (203) shows that -^ does not change, and, therefore, from (204), p does not change. That is, whatever change there may be in the temperature of a part of the atmosphere, if the pressure of the atmosphere as a whole is not altered, then the pressure of the water vapor in it is not altered. Consequently, the pressure of the water vapor in any portion of air can be determined by cooling the air down to the dew point and looking up in the proper table the pressure of saturated water vapor corresponding to this temperature. From (202) and the definition of relative humidity it follows that the relative humidity of any portion of air is given by Sh^r (205) 204 PRACTICAL PHYSICS where p and p' represent respectively the pressures that satu- rated water vapor would exert at the dew point and at the actual temperature of the air. Manipulation and Computation. — Daniell's hygrometer consists of two glass bulbs connected by a bent tube as shown in Fig. 85. The lower bulb contains ether and a thermometer. The upper bulb is wrapped witli a piece of muslin. In determining the dew point with this apparatus all of the contained ether is passed into the lower bulb and then the upper bulb is moistened with ether. The evaporation of the ether poured on the upper bulb causes the bulb to cool and a part of the vapor in the apparatus to condense. This condensation in the upper bulb decreases the pressure which the ether vapor exerts on the surface of the liquid ether in the lower bulb. Under the de- creased pressure some of the ether in the "lower bulb evaporates and so cools the lower bulb. Thus the temperature of the lower bulb gradually falls until dew is deposited on its surface. The beginning of a dew deposit is usually detected by watching to see if any effect is produced on drawing a fine brush lightly across a gilded surface on the bulb. The temperature of the lower bulb is then read. After this the apparatus is allowed to remain until equilibrium is restored and the temperature begins to rise. The temperature at which the deposit of dew disappears is noted. The mean of the temperatures when the deposit appears and when it disappears is taken as the dew point. The temperature of the surrounding air is noted by the thermometer attached to the wooden stand supporting the hygrometer. From Table 13 find the pressure of saturated aqueous vapor at the dew point and at the temperature of the room. Make at least five determinations and take the average. Fig. 85. HYGROMETRY 205 Exp. 46. Determination of Relative Humidity with the Wet and Dry Bulb Hygrometer Object and Theory of Experiment. — If two exactly similar thermometers, the bulb of one naked and the bulb of the other covered with a wet wick, are placed near each other in a current of air, the thermometer with the naked bulb will indicate the temperature of the air while the other will indicate a lower temperature. The difference between the indications of the two thermometers is due to evaporation at the surface of the wet bulb and depends upon the degree of saturation of the air. The relation between the relative humidity of the air and the indications of the thermometers has never been obtained in an entirely satisfactory manner from purely theoretical con- siderations. But by comparing the indications of this hygrom- eter with the indications of hygrometers of other types, tables have been constructed by means of which the relative humidity of the air can readily be determined from a single pair of si- multaneous readings of the wet and dry bulb, thermometers. For several years simultaneous readings of the Daniell and of the wet and dry bulb hygrometers were taken, and from a comparison of these readings the numbers in Table 15 were obtained. As an example, on placing these two instruments near one another, tlie following simultaneous readings were made : — Temperature of the air, 21° C, Temperature of the wet bulb, 19° C, Dew point, 18° C. In Table 13 the pressure of saturated aqueous vapor at 18° is given as 15.33 mm. It follows that corresponding to an atmos- pheric temperature of 21° and a wet bulb temperature 2° lower, the pressure which the aqueous vapor in the atmosphere would exert at the dew point — and, consequently, does exert at the given temperature — is 15.33 mm. of mercury. In Table 15 this number in placed in the line numbered 21° C. and in the column numbered 2°. 206 PRACTICAL PHYSICS Manipulation and Computation. — The wet and dry- bulb hygrometer, sometimes called August's psychrometer, con- sists of two similar thermometers, one with a naked bulb and one with the bulb covered by an envelope of wet muslin. A current of air is caused to blow over the two bulbs with a fan or some other means. In one convenient arrangement for this purpose (Fig. 86) the two thermometers are supported by a frame that can be rotated by hand. With the muslin envelope dry take simultaneous readings of both thermometers for several minutes. Then see that the muslin envelope about the bulb „ „ of the wet thermometer is kept Fig. 86. , , , . n . , ,. thoroughly moist, and with a fan or the rotating device shown in the figure change rapidly the air about the instrument. When the wet bulb has reached a stationary temperature, read each thermometer again. From the corrected difference in the readings of the two thermometers find in Table 15 the pressure p of the aqueous vapor present in the atmosphere. In the 0° difference column find the pressure p which the aqueous vapor would exert if there were enough of it present to be a saturated vapor. Calculate the relative humidity by (205). Make not fewer than five determinations and take the average. Before each determination be certain that the muslin envelope is thoroughly moist. ^ CHAPTER XIV CALORIMETKY Caloeimetry is the theory and art of measuring quantities of heat. Unfortunately there is no single quantity of heat that is universally adopted as the unit. A common unit in scientific work is the amount of heat required to raise the temperature of one gram of water from 15° C. to 16° C. This unit is called the 15° calorie or simply the calorie or the gram-degree-centi- grade thermal unit. In the British system the unit adopted is the amount of heat required to raise the temperature of one pound of water from 60° F. to 61° F. This is called the British thermal unit or the pound-degree-jFahrenheit thermal unit. Throughout this book the calorie will be used exclusively. The number of thermal units required, to raise the tempera- ture of unit mass of a substance from t° to (t + 1)° is called its specific heat at t°. The specific heat of bodies is slightly differ- ent at different temperatures, but the difference is so minute that except in the most refined measurements it need not be considered. The average specific heat of a body between any two temperatures is the number of heat units required to raise a unit mass of it from one of those temperatures to the other, divided by the difference in the two" temperatures. That is, the quantity of heat 5" required to raise from t^ to t^" the tem- perature of m grams of a substance of average specific heat s is n=mB(t^°-t{). (206) Throughout the above paragraph it is assumed that between the temperatures considered the body does not melt, solidify, vaporize, nor condense. When a body does melt, solidify, vaporize, or condense, with- 207 208 PRACTICAL PHYSICS out changing at all in temperature, the amount of heat required or giA'-en out is proportional to the mass of the substance that changes state and depends upon what that substance is. That is, H=mf, (207) where / is a constant called the heat equivalent* of fusion, solidification, vaporization, or condensation, as the case may be. The mass of water which requires the same amount of heat as a given body in order to change its temperature by the same amount is called the water equivalent of the body. Thus, if e represents the water equivalent of a bod}% and o the mean specific heat of water between t^ and t^, the quantity of heat required to raise from fj° to t^ the temperature either of the body or of e grams of water is JI=ec{t^°-ty°). (208) Dividing (208) by (206), ^ e=— • (209) Ordinarily, the specific heat of water may be taken as constant and equal to unity. In this case e = ms. (210) That is, the water equivalent of a body equals the product of its mass arid its specific heat. In determining the water equivalent of a thermometer, only that part of it which changes in temperature is to be consid- * From the fact that the heatr absorbed by a body during fusion or vaporiza- tion does not change the temperature of the body, it used to be supposed, when heat was considered to be a fonm of matter, that the heat absorbed during fusion and vaporization existed in the melted or vaporized body in a latent, i.e. a hidden, form. This heat was then called the latent heat of fusion or vapori- zation. Now that it is known that heat is a form of energy, viz. that form which changes the temperature of bodies, we prefer to say that the heat absorbed by a body during fusion or vaporization does not exist in the melted body as heat but as some other form of energy. Consequently, the expression latent heat is now obsolescent and is giving place to the term heat equivalent. CALORIMETRY 209 ered. This may be taken as somewhat more than the part immersed. Fortunately, the product of the density of mercury by its specific heat is nearly the same as the corresponding product for glass. That is, the water equivalent of a given volume of mercury is about the same as that of the same vol- ume of glass. Since the value of this product is about 0.5 g. per c.c, the water equivalent of a thermometer in grams may be taken as somewhat more than half the volume of the im- mersed part in cubic centimeters. Although simple in theory, calorimetric experiments require great care and many precautions. One of the most important sources of error is radiation, i.e. there is a gain or loss of heat because neighboring bodies are at temperatures different from that of the body being studied. The principal methods of diminishing this error are (a) to compute the amount of heat actually gained or lost by radiation ; (S) to determine the temperature which the body would have attained if there had been no radiation; (c) to employ a method in which the tem- perature is kept the same as that of the surroundings. THE CORRECTION FOR RADIATION 1. Regnault's method is based on Newton's Law of Cooling. This law may be stated as follows: The rate at which a body cools is proportional to the difference between its temperature and the temperature of its surroundings. If, then, A t denotes the fall of temperature due to the radiation which occurs in the short time A T, and if the temperatures of the body and its sur- roundings are denoted respectively by tf, and t^, Newton's law of cooling may also be stated by the equation where k' is the proportionality factor. If both members of this equation are multiplied by the water equivalent, e', of the cool- ing body, then since e'At is, from (210) and (206), the heat, A ff, 210 PRACTICAL PHYSICS lost while the temperature falls At°, the equation becomes Aff=rAT(:t,-Q, (211) where r is written in place of the product e'k'. This r is a con- stant which depends only on the nature and area of the radiating surface, and is called the radiation constant of the body. New- ton's law is now known to be only a rough approximation to the true law of cooling; but it is simple, and, if the difference in temperature between the body and the surroundings is not greater than 15° or 20°, it holds fairly well. Let CD and UF in Fig. 87 represent the respective changes in temperature of the body and its surroundings while the body cools by radiation. Since (^j— ^j) is at any instant the vertical distance from UF to OD and A y is the horizontal distance be- tween two of these vertical lines which are near together, it fol- lows that the product AT Ob — ts} is represented very nearly by the shaded area A A. That is, from (211), AH = rJcAA, where A; is a constant which depends upon the scales chosen in plotting. If the temperature of the body, instead of falling the small amount Ai, falls from *2 to *i' t^6 entire fall being due to radiation, the total heat lost H^ involves the sum of the elementary areas A A, that is, the area OF. If this area is denoted by Acf: we have ff^rJcAcF. (212) Now suppose that the body has heat given to it in such a way that its rise of temperature can be represented by the curve ABO in Fig. 88. The maximum temperature is reached when the body ceases to receive heat from the source faster Fig. 87. CALORIMETEY 211 than it radiates heat to the cooler surroundings. After this point is reached, the body falls in temperature in a manner that can be represented by the line CD. While the body is be low the temperature of its surroundings it absorbs heat from them, and while it is above the temperature of its surroundings it loses heat to them. The radiation correction, now to be found, is the difference between the amount of heat lost by the body through radiation and the amount gained by absorption while the body was rising from its original to its maximum tempera- ture. r:::::: >- ;- . I- . -X _- -3 - -.-.-.-/.Zl - '";:::;;;- y An ja. --'.'.'.'. jS :::: J £c = iE = "*E 8 9 10 11 5 6 7 Fig. 88. Curve showing rate of change of temperature of heated body. 12 Let EJ denote the heat lost by radiation during the time T' that the body is above the temperature of its surroundings, and H" the heat gained by absorption during the time T" that the body is below the temperature of its surroundings. Then from (212) and Fig. 88, H'^rltA', and E" = rkA", where A' denotes the area BOU, and A" the area AJB. Since, in addition, the radiation correction B is, from its definition. we have B=H'-H", B = rk(iA' -A"). (213) 212 PRACTICAL PHYSICS If the radiation constant r were known, (213) could be used to determine the radiation correction R. The purpose of allow- ing the body to cool for a time by radiation after the other heat changes have taken place is to make possible the determi- nation of this r. From the part QB of the curve we have from (212), if ^denotes the heat lost by radiation while the body falls in temperature from *' to f", S= rkA, and from (206) and (210) we have also 5"= e' («'-«"), where e' denotes the water equivalent of the body. It follows that e' (t> - t") = rkA. (214) On substituting in (213) the value of r from (214), we have R = e'(t' -t"^- /■ . (215) 2. Instead of finding the number of heat units lost by the body due to radiation while the temperature of the body is rising to its maximum value, the effect of radiation can be accounted for if the temperature is determined which the body would have attained if there had been no radiation. In the following modification of a method due to Rowland this tem- perature can be obtained to a close approximation by a simple graphical construction. Suppose that a body at a temperature below that of its sur- roundings is given a quantity of heat 5" such that its tempera- ture rises to a value above that of the surroundings. While the temperature of the body is lower than that of the surround- ings the body absorbs heat, and while the temperature of the body is above that of the surroundings, the body loses heat. The way in which the temperature changes before the heat H is added is represented by the line AB in Fig. 89. The line BB shows how the temperature changes while the body is absorbing the heat H. From B to C the body is, in addition, CALORIMETRY 213 receiving heat from the surroundings, and from O to D is losing heat to the surroundings. The line DU shows how the temperature of the body changes due to radiation alone. Through B and O draw vertical lines. Prolong JD^ backward until it cuts the vertical line through in /. Through / draw a line fg parallel to AB until it cuts the vertical line through B in g. The temperature indicated by ff is the desired tempera- ture. To see that the above method of finding the desired temperature is reasonable, consider the following. If the heat H had not been given to the body, it would have con- tinued to rise in temperature iu the same way that.it was rising from A to B, so that by the time it really attained the tempera- ture indicated by it would have reached the temperature indicated by h. That is, while the body really rose in tempera- ture from B to O, the rise in temperature from B to h was due to heat from the surroundings and the rise from h to was due to a part of the heat H. Again, if the body had not been given the heat IT, but if it had been at first at such a tempera- ture that as it cooled it reached the temperature indicated by 2) at the same instant that it really reached that temperature — and thereafter cooled as shown by B^ — it would have been at a temperature / at the instant when it really was at the temperature 0. That is, while the body really rose in tempera- ture from to B, the fall in temperature due to radiation was the fall from / to 2>, so that if there had been no loss of heat by radiation, the rise of temperature during this time would have been from to /. If, then, there had been no gain nor 214 PRACTICAL PHYSICS loss of heat by radiation, the body would have risen in tem- perature the amount indicated by the distance from h to /. But the temperature when it began to receive the heat H was that indicated by B. So that the temperature which would have been reached if there had been no radiation is a tempera- ture as far above B as / is above h — that is, the temperature indicated by g. While the temperature of the body rose from C to _Z) it was really at a lower temperature than if it had been cooling from / to B, and so did not really lose as much heat by radiation as has above been supposed. That is, the point / is higher than it ought to be. For a similar reason A is also somewhat higher than it ought to be. If the time from ^ to C is about the same as that from G to B, these two errors will nearly balance each other. 3. Another method should be referred to, although it is con- siderably less accurate than the two already discussed. In this method, first suggested by Rumford, the initial and final tem- peratures of the body are so arranged that the difference be- tween the temperature of the surroundings and the initial temperature of the body equals the difference between the tem- perature of the surroundings and the final temperature of the body. The idea is that, by this arrangement, the heat absorbed from the room while the body is colder than the surroundings equals the heat lost to the room wliile the temperature of the body is higher than that of the surroundings. That this, how- ever, may be only a rough approximation can be shown as follows : — When a body is heated and then immersed in cold water, the temperature of the water rises in a manner very like that repre- sented by the curve RA in Fig. 90. During the first part of the time, the temperature rises rapidly because the body is at a temperature considerably higher than that of the water, whereas when the temperatures become more nearly the same, the temperature of the water rises more slowly. This means that the first half of the temperature rise is accomplished in CALORIMETRY 215 less time than the second. And this, in turn, if the tempera- ture of tlie surroundings is halfway from the lowest to the highest temperature of the water, means that less heat is gained by absorption during the first half of the temperature rise than is lost by radiation during the second half. In fact, from (212) it follows that in the case represented in Fig. 90 the ratio of the heat lost by radiation to the heat 4o| gained by absorption equals the ratio of the areas FAQ- and HEF. The radiation in would compensate the radiation out if the temperature of the surroundings were raised to BD, so that the areas CAD and EBOv^ere equal. That is, in the given case, the rise in temperature before reach- ing the temperature of the surroundings should be about two and a half times that after passing the temperature of the surroundings. 20 10 ifiiiliiiiil lililli iiiJ! !• «!:!::■!: - .\ ■aa ■■"■i ■! :::::::::: : ::i:: 'Jn EiiiiiiJtfi: jjjjjj i-~ :■■:;::::::::; :::::: :::: :;■:::■::::■:: :::::!„ '^■i :: :::::t^: :: : ::::■ a imniis-Kiiiiii 1 i ■ .■ •: : :: ;: :: ::: ::: : ::: :: : :: :: :■ ::: ::: : ::: il ::■■: ■■ :::::• :::::::: 0.5 1.0 1.5 Fig. 90. Exp. 47. Determination of the Emissivities and Absorbing Powers of Different Surfaces Object and Theory of Experiment. — The emissivity or radiating power of a surface is defined as the number of heat units lost by radiation at atmospheric pressure, per second, per unit area, per degree excess of temperature of the cooling body above the temperature of the surrounding air. Similarly, the absorbing power of a surface is defined as the number of heat units absorbed at atmospheric pressure, per second, per unit area, per degree excess of temperature of the surrounding air above the temperature of the absorbing body. The object of this experiment is to determine the emissivities and the ab- sorbing powers of different surfaces for various temperature differences between the surfaces and the surrounding air, and also to compare the emissivity and absorbing power of a given surface under similar conditions, 216 PRACTICAL PHYSICS Consider a mass M of water filling a closed vessel, the water equivalent of which is e and its external surface area A. If during a short time AT the vessel and its contents cool Ai°, their mean temperature during the time being t^, and the tem- perature of the surrounding air being t,, then from the above definition the emissivity of the surface is LTA(t,-t,y (216) Similarly, if the vessel and its contents rise in temperature A^°, due to absorption of heat from the surrounding air, the absorbing power of the surface is (M+ e) At ATA(t,-t,y (217) Manipulation and Computation. — For this experiment there are provided two or more metallic cylinders (Fig. 91), exactly alike except for the nature of their external surfaces. One cylinder, for example, may be highly polished, one may have a tarnished surface, and one may be coated with lampblack. In finding the emis- sivities of these different surfaces, the cylin- ders are in succession filled with warm water and suspended inside of an inclosure formed by two concentric cans J, IC, the space out- side ^and inside >T being filled with water at the temperature of the room. The temper- atures of the water in the central vessel and of the water in the jacket are observed every two minutes for at least half an hour. The water in the central vessel and in the jacket is kept thoroughly stirred throughout the whole experiment. From these observations are plotted two curves coordinating temper- one for the radiating body and one for the Fig. 91. ature and time water jacket. CALORIMETRY 217 Suppose that in a particular experiment the curve shown in Fig. 92 was obtained and the following data found: — 3o r"""!": "":::": :■:: :":"::::: 11 = N Qrt 5" .s ._ """" ^ ..___. Sj.. 20 ^:^==ii m -U-- 4 ±::. III ~"±i"^ y------MM ff- -■■ t---^ -i;¥ ■Uf ii ff'll"ll 5 1 ) IS 20 2C 30 Fig. 92. Curve showing the rate of change of temperature of the hot body and of the water jacket. Mass of copper radiating vessel and stirrer, 78.1 g. Mass of water contained in vessel, 126.3 g. Area of external surface, 153.1 sq. cm. Since the specific heat of copper is known to be 0.093, it 218 PRACTICAL PHYSICS follows from (210) that the water equivalent of the radiating vessel and stirrer is e = 78.1 -0.093 = 7.3 g. In computing the emissivity by means of (216), ?j, *,, and A^ may be taken from the curve. For a curve of this sort a con- venient value for ^is five minutes. For example, to find the emissivity of the surface of the radiating body when at 34° C, while the inclosure was at 20°. 23 C, proceed as follows: To the right and left of the point where the 34° line crosses the cooling curve, lay off distances corresponding to 2.5 minutes. At A and B, the ends of this line, erect perpendiculars until they intersect the cooling curve, and at the points of intersec- tion draw two lines parallel to the time axis. The distance be- tween the last two lines represents the At° through which the radiating body cooled during an interval of five minutes. In this particular case Ai=l°.84. Substituting in (216) the values thus obtained, we have _ (126.3 -f 7.3)1.34 _ n QQQogQ '"^ - 300x153.1(34.0-20.23) ~ ^-^^O^^^- In the same way are found the emissivities at other temperatures. Proceeding as described above, with each of the surfaces being studied, plot on a sheet of cross section paper an emissivity curve for each surface. Now fill with cold water the vessels heretofore used as radi- ating bodies and by means of (217) knd an experimental method similar to that used to find emissivity, determine the absorbing powers of the different surfaces for various temperature differ- ences between the absorbing surface and the inclosure. State in words the conclusion reached from a comparison of the emissivity and the absorbing power of the same surface. At the beginning of the radiation experiment the tempera- ture of the water in the radiating body should be about 15° C. above that in the water jacket, and in the absorption experiment the temperature of the water in the absorbing body may be about 15° C. lower than that in the water jacket. But in no CALORIMETRY 219 case should the temperature of the absorbing body be so low that dew will be deposited on its surface. Exp. 48. Determination of the Specific Heat of a Liquid (METHOD OF COOLING) Object and Theory of Expekiment. — Suppose that a mass rrii of a liquid of a specific heat S; is contained in a vessel which has a water equivalent e and a radiation constant r. If the temperature of the liquid is somewhat above that of the surroundings, and if the temperatures of both liquid and surroundings are observed for some little time, and then the temperatures are plotted against times, two curves like those in Fig. 87 will be obtained. While the liquid and vessel fall in temperature through a range of At°, the heat which they lose is, from (206), Hi=imi8i + e')Mi. (218) Since this heat is lost by radiation, (212) shows that it is also given by Ei^rkAi, (219) where r, k, and Ai have the same meanings as the r, h, and A^p in (212). From (218) and (219) we have at once (jniSi + e)Mi = rkAi. (220) In the same way, if the liquid in question is replaced by warm water, and if subscripts w mean that the symbols to which they are appended refer to this water, (m„ + e)Af„ = rJcA^. (221) If the scales chosen in plotting are the same for both pairs of curves, the k in (220) and (221) is the same ; and if the nature of the surface of the containing vessel remains the same, the r is the same. On dividing (220) by (221) and solving for Sj, we get ^^ ^ AiAt„(m^ + e) _ _£_ ^222) A^AtiMi mi 220 PRACTICAL PHYSICS Fig. 93. Manipulation and Computation. — The apparatus used in this experiment consists of a closed metal radiating vessel suspended in an inclosure surrounded by an ice jacket. The radiating vessel is provided with a stirrer for agitating its contents and a thermometer for reading temperatures. Weigh the radiating vessel and stirrer and, by multiplying their mass by the spe- cific heat* of the material of which they are composed, determine their water equiv- alent. Fill the radiating vessel just to the bottom of the neck with the liquid whose specific heat is to be determined and set it in water in a saucepan over a burner until its temperature is about 40° C. Then wipe the outside of the radiating vessel dry, sus- pend it in the inclosure inside the ice jac- ket, and while continually stirring, observe the temperature every minute for quarter of an hour or longer. Throughout this time keep the jacket full of ice. Then remove the radiating vessel from the jacket and weigh it, thus finding the mass of the liquid. Clean the vessel, rinse it out with water, and fill to the bottom of the neck with water. Then heat, dry, and suspend in the ice jacket as before, and again observe the temperature every minute for a quarter of an hour. Remove from the jacket and weigh, thus finding the mass of the water. Plot the cooling curves for both substances on the same sheet. Since the jacket is packed with shaved ice, the temperature of the surroundings in each case is zero. If, then, times are plotted as abscissas, A„ is the area bounded on the top by the water curve, on the bottom by the temperature axis, and on the sides by any two convenient ordinates, — one near the beginning and one near the end of the time employed, — and Ai is the * If the radiating vessel or stirrer is of unknown composition, the water equivalent can be obtained experimentally by the method of mixtures, p. 224. CALORIMETRY 221 same area except that its upper boundary is the other curve. These areas, may be obtained with a planimeter, determined by counting the millimeter squares, or, perhaps most easily, found by the method of average ordinates. At^ is the difference between the temperatures at the points where the water curve crosses the ordinates which bound the area A^, and Atf is the corresponding difference for the other curve. All of the data are now at hand for calculating the specific heat of the specimen by means of (222). Two or three cooling curves for each substance should be taken, and two or three values for the specific heat thus obtained. Exp. 49. Determination of the Specific Heat of a Solid (METHOD OF MIXTUEES) Object and Theory of Experiment. — The Metliod of Mixtures depends upon the principle that when a number of bodies of different temperatures are brought together, the amount of heat lost by the bodies that fall in temperature equals the amount of heat gained by the bodies that rise in temperature. Consider a body of mass m, specific heat s, and temperature t, to be placed in a mass wi, of liquid of specific heat S; and tem- perature ti contained in a vessel of mass m„ made of a material whose specific heat is s^. Let the final temperature of the mix- ture be ty. Then, if t is higher" than ti, the heat lost by the given body equals the sum of the heat gained by the vessel and its contents and that gained by the surrounding air. That is, from (206), if It denotes the radiation correction, msQt-t;) = {mtSi+m^s^Xtf- td+R- (223) The correction for radiation may be applied either by Regnault's method of calculating R, or graphically by the modification of Rowland's method given on pp. 212-214. When the specimen is in small pieces the rise of temperature is rapid and Rowland's method is perhaps to be preferred. 222 PRACTICAL PHYSICS If water is the liquid used, s,= l. For purposes of ab- breviation the water equivalent of the vessel, 711^8^1 will be denoted by the single letter e. Then if tj denotes the tempera- ture that the mixture would have reached if there had been no gain nor loss by radiation, (223) gives s = (m, + e}Ct;-t,-) (224) It should be noted that e represents the water equivalent of the vessel in which the mixing occurs, together with any ac- cessories it may contain, such as a stirrer or thermometer. Manipulation and Computation. — The special apparatus used in this experiment consists of a calorimeter and a heater. Fig. 94. Fig. 95. A calorimeter is any apparatus used to measure quantities of heat. The ordinary "water calorimeter" used in this experi- ment (Fig. 94) consists of a thin polished copper vessel held centrally within a jacket by means of non-conducting supports. The inner vessel contains a thermometer T' and stirrer S, while a second thermometer T is suspended in the air space between the two concentric vessels. A convenient form of heater, shown in Fig. 95, consists of a closed copper can in which water can be boiled. Extending through one side and projecting nearly through the boiler at an angle of 45° with the bottom, is a tube sealed at the lower end and having the upper end closed with a cork through which extends a thermometer. The specimen to CALORIMETRY 223 be heated is placed in this tube, and when the temperature indicated by the thermometer T has become steady, the specimen is dropped into the calorimeter. If the specimen is in small pieces, e.g. lead shot, it can be poured into the calorimeter by simply tilting the heater, if the specimen is in a single piece, it is drawn out of the heater with a thread and quickly lowered into the calorimeter. A very compact form of apparatus designed by Regnault is shown in Fig. 96. In this apparatus the calorimeter is on a little carriage that can easily be moved up to the heater and with- drawn. Tlie tube BE extends entirely through the heater. At its lower end is a shutter A by means of which the ^^"■=-^^'""^ tube can quickly be opened or closed. A thermom- eter extending through the stopper at B permits the observation of the tem- perature of the specimen. The specimen is held in the middle of the tube by a string extending out through B. If the speci- men is in small pieces it is contained in a small wire basket. When it is desired to drop the specimen into the calorimeter, the latter is moved under jE^, the shutter A is opened, and the string holding the specimen is released so as to allow the speci- men to fall quickly into the calorimeter. The water equivalent of the inner vessel of the calorimeter should first be determined. If the mass and specific heat be known for each part of the calorimeter that changes in tempera- ture, the water equivalent is most easily and most accurately obtained by taking the sum of the products of these masses and the corresponding specific heats. When this method can- not be applied, the method of mixtures can be employed. In this case, after weighing the inner vessel and stirrer, half fill Fig. 96. 224 PRACTICAL PHYSICS the inner vessel with water at a temperature some 10° or 15° below that of the room, and weigh again to determine the mass m^ of cold water. With one thermometer in the water in the calorimeter and another in water at a temperature some 15° or 20° above the temperature of the room, watch both thermome- ters for a few moments, and immediately after reading both t„ and t/^, the temperatures respectively of the cold water and of the hot water, pour rapidly into the inner vessel enough of the hot water nearly to fill it. Meantime stir briskly and watch the thermometer in the calorimeter. After noting the tempera- ture of the mixture t^, weigh again to determine the mass ?«/, of hot water added. Since the heat lost by the hot water equals that gained by the calorimeter and contents plus that lost to the surroundings, ^kik - U = (m, + e) (t^ - t,) + R', (225) where e represents the water equivalent of the calorimeter and M' the radiation correction. From the discussion on pp. 214-215, it follows that M' can be made very small by selecting proper values for the temperatures. Since e is usually small compared with wi,, and since in the only place where e is used in (224) it is added to to,, a smaU error introduced into e by failure entirely to eliminate H' would cause in s a very small error. If this very small error is neglected, (225) gives e = "^a(^;. - ^m) _ ^^_ (226) ^m ''c The satisfactory determination of a water equivalent by this method requires deft and rapid manipulation and careful deter- mination of temperature. Be sure that the specimen is dry, and place it in the tube in the heater until its temperature assumes a constant value t. While the specimen is heating weigh the inner vessel of the calorimeter, if this has not already been done. Then pour into this inner vessel water at a temperature three or four degrees below that of the room until the vessel is somewhat more than CALORIMETRY 225 half full and determine the mass of the water. Assemble the parts of the calorimeter, placing one thermometer in the water contained in the inner vessel and another thermometer against the inner surface of the jacket. The thermometer in the water should have its bulb entirely covered by the water, but should not be low enough to be touched by the specimen. The tem- perature of the water should now be observed at quarter or half minute intervals, and the temperature of the jacket every minute or two. For each reading the hour, minute, and second at which the reading is made should be recorded. The readings are taken continuously but belong to three successive periods. Before beginning the first period be sure that the thermometer in the heater is steady in the neighborhood of 100° and record its reading. First period. While stirring the water read times and corre- sponding temperatures for some three to five minutes before transferring the specimen to the calorimeter. Second period. At a given instant transfer the specimen rap- idly to the calorimeter. Continue to stir the water and to take temperature readings every quarter or half minute. While the heated specimen is giving up its heat, the water rises rapidly to' a maximum temperature ty. This period is frequently over in fifteen or twenty seconds. The maximum temperature is attained when the rate at which heat is radiated by the water to the air equals the rate at which the water receives heat from the specimen. The temperature may remain stationary at this value for an appreciable length of time. Thereafter, if the water has risen to a temperature above that of the jacket, the loss by radiation exceeds the gain of heat from the specimen. If the water does not rise above the temperature of the jacket inside of a minute after the specimen is dropped into the calo- rimeter, the specimen is to be dried and the experiment begun again,. If the temperature rises rapidly and then almost at once falls again somewhat, the specimen has come too close to the thermometer and the experiment should be begun again. Third period. Without interruption continue to stir the 226 PRACTICAL PHYSICS water and to take readings of temperature and time for at least five minutes during the cooling of the water in the calorimeter. After the third period weigh the inner vessel and contents, and so determine the mass of the specimen. With these readings, plot on the same sheet two curves — one coordinating temperature and time for the water in the calo- rimeter, and another coordinating temperature and time for the surroundings. A pair of such curves is shown in Fig. 89. From these curves the temperature which would have been reached if there had been no radiation can be determined by Rowland's method. The data are now at hand which when substituted in (224) give a value for the specific heat of the specimen. One or two preliminary experiments may be necessary in order to determine just how much water to use and at what initial temperature to have it. After a satisfactory set of read- ings is obtained, another set should be taken and two values found for the specific heat. Exp. 50. Determination of the Specific Heat of a Solid (METHOD OF STATIONARY TEMPERATURE) Object and Theory of Experiment. — The object of this experiment is to determine the specific heat of a solid by a modified form of the Method of Mixtures in which the water equivalent of the calorimeter is avoided and the radiation cor- rection is eliminated. This is accomplished by maintaining the temperature of the calorimeter throughout the experiment the same as that of the surroundings. Consider a body of mass m, specific heat s, and temperature t, dropped into a calorimeter containing TOj grams of water at the temperature of the surroundings ^j. Let cold water be added to the calorimeter at such a rate that the temperature of the calorimeter remains constant. If the mass and temperature of this cold water be represented by m^ and t^ respectively, then the heat emitted by the specimen is, by (206), ms(t — tj), and the CALORIMETRY 227 heat gained by the cold water added to the calorimeter equals ^cCh ~ O- Since the water originally in the calorimeter has not changed in temperature, the heat lost by the specimen equals that gained by the cold water. That is, Whence ms (t — ij) = m^(t^— *,.) . m(t— t^ (227) Manipulation and Computation. — The apparatus used in this experiment includes a calorimeter of special design Fig. 97. T 1 J E T =flt r^ L V *a o Fio. 98. (Fig. 97), together with a heater H and a water dropper 2), both capable of rotation about vertical axes. The water drop- per consists of a reservoir R (Fig. 98), having a valve Fj by means of which the flow of water through the orifice can be regulated. Surrounding this reservoir is an ice jacket J. By means of the thermometer T the temperature of the water at the moment it issues from the water dropper can be observed. The calorimeter (Fig. 99) is essentially the metallic bulb of an 228 PRACTICAL PHYSICS air thermometer into which projects a copper tube X for the reception of the water and specimen. Any change in the tem- perature of the calorimeter is indicated by an open manometer tube M. To prevent any effect due to changes in the tempera- ture of the surrounding air, the calorimeter is placed in a water bath yat the temperature of the room. After the apparatus has been assembled ready for use, the specimen is weighed and placed in the heater. The mixing tube of the calorimeter is unscrewed and weighed, first when empty, and then when filled with enough water at the temperature of the room to cover the specimen. The mixing tube is now replaced and the stopcock attached to the manometer is opened for an instant. By this means any difference of pressure between the inside of the air ther- mometer bulb and the outside air is equalized. When the thermometer in the heater indicates a steady temperature near 100°, the water dropper is made ready for use by allowing cold water to escape until the thermometer in the escaping steam indicates a stationary tempera- ture. The temperatures of the specimen in the heater, the water in the mixing tube of the calorimeter, and the cold water in the water dropper are now noted. The heater is rotated into position over the calorimeter and the specimen quickly lowered into the mixing tube. The heater is immediately rotated out of position and the water dropper rotated into place. By operating the valve V, cold water is now allowed to fall into the mixing tube at such a rate that the index in the manometer tube of the air thermometer remains stationary. The proper rate can be ascertained only by previous trials ; it depends largely upon the conductivity and fineness of division of the specimen. When no more cold water is needed, the mixing tube with its contents is again weighed. All of the data necessary for the computation of the specific heat of the speci- men by means of (227) are now at hand. Fig. 99. CALORIMETRY 229 Exp. 51. Determination of the Specific Heat of a Solid (JOLY'S METHOD) Object and Thboey of Experiment. — A cold body placed in an atmosphere of steam absorbs heat until its temperature is the same as that of the steam. A certain amount of steam is thereby condensed. If the steam is at the boiling point of water, the amount of heat lost equals the product of the mass condensed and the heat equivalent of condensation of steam. By "heat equivalent of condensation" of steam is meant the number of heat units given up by the condensation of unit mass of steam. This is nu- merically equal to the "heat equivalent of vaporization" of v^ater, i.e. the number of heat units required to vaporize unit mass of water. The object of this experiment is to determine the specific heat of a solid from a measurement of the mass of steam condensed on the body as it rises in tempera- ture to the boiling point of water. Joly's apparatus (Fig. 100) consists of a steam chamber inclosing one pan of a delicate balance. The pan is suspended from the balance beam by a fine wire passing through a small hole in the top of the steam chamber. Steam is first passed into the steam chamber and the mass of steam which condenses on the scale pan is weighed. The apparatus is now allowed to cool to the temperature of the room. The scale pan is dried and upon it is placed the specimen whose specific heat is required. Steam is again passed into the steam chamber and the mass of ■■^"""■l" ■—■■■■■I ^ 1 ^^^ , 1 1. 1 Fig. 100. 230 PRACTICAL PHYSICS steam which condenses on the specimen and on the scale pan is weighed. Let s denote the specific heat of the specimen ; e, the water equivalent of the scale pan and suspending wire ; ^^ and t^, the respective temperatures of the room and of the steam ; h, the heat equivalent of condensation of steam ; and mj, m^, m^, and m^, the respective masses required to balance (1) the empty scale pan, (2) the pan with the steam which condenses on it, (3) the pan and the specimen, and (4) the pan and specimen with the steam which condenses on them both: The amount of heat absorbed by the scale pan and suspending wire as they rise in temperature from ^j to t^ is e(^t^ — t^). This heat is supplied by the heat liberated in the condensation of the mass (m^ — w^x ) °^ steam. Therefore, e(t^-t.^) = (m^-m-^;)h. (228) Similarly, the amount of heat absorbed by the specimen, the mass of which is (m^ — jw^), together with the balance pan and suspending wire is (m^ — m^')s(t^ — t^')+e(t^—t-^). This heat is due to the condensation of the mass of steam (m^~m^). Consequently (wzg — m{)s(t^ — t{)+ e(t^ — ^j) = (m^ — m^')h. (229) Subtracting (228) from (229), (Wg — mj )s(<2 ~^i) = (^4 — JWg — W2 + m-^h. Whence ^^ (.m,-m,-m^ + m,-)h ^ 230) It will be noticed that (m^ — m^ — m^ + m-^) is the mass of steam condensed on the specimen, and that (m^ — wij) is the mass of the specimen. Manipulation and Computation. — A common source of error in this method is an uncertainty in weighing produced by steam condensing on the suspending wire where it emerges from the steam chamber. In tlie apparatus illustrated in the CALORIMETRY 231 figure, this trouble is diminished by having the suspending wire pass through a small tube surrounded by a steam jacket (Fig. 101). By passing the steam through this jacket before it enters the steam chamber, the neighbor- hood of the aperture is sufficiently heated to prevent a large amount of condensation on the suspending wire outside of the yiq. loi. chamber. Take care to have the suspending wire hang free. Then with standard masses m^ balance the lower scale pan. When water in a detached boiler is boiling vigorously, note the tem- perature ^j of the inside of the steam chamber and then connect the boiler to the steam chamber with a good-sized rubber tube. Steam will immediately be condensed on the object pan. After one or two minutes diminish the flow of steam to such an extent that the current will not disturb the object pan. With standard masses m^ again bring the balance into equilibrium. Disconnect the boiler from the steam chamber and allow the chamber to cool to the temperature of the room ty Dry the object pan, place upon it the specimen whose specific heat is required, and determine the mass m^ now required to bring the balance into equilibrium. Again connect the boiler to the steam chamber, and after four or five minutes, when the speci- men and object pan have acquired the temperature t^ of the steam, diminish the flow of steam and with standard masses m^ again bring the balance into equilibrium. Since h, the heat equivalent of vaporization of water, is known, all of the data are now at hand for computing the specific heat of the specimen by means of (230). Exp. 52. Determination of the Heat Equivalent of Fusion of Ice. Object and Theory oe Experiment. — The Heat Equiva- lent of Fusion of a substance is the number of heat units re- quired to melt unit mass of it without changing its temperature. 232 PRACTICAL PHYSICS Suppose that when OTj grams of ice at 0° C. are dropped into m^ grams of water at tj', the ice melts and the temperature of the mixture of the two becomes t^°. During this operation, the ice has absorbed the heat required to melt it and also after melting to raise its temperature from 0° to t^", while the calo- rimeter and its contents have lost heat. If there were no gain of heat from the surroundings nor loss to them, the heat gained by the ice in melting and then rising to the temperature t^ would equal the heat lost by the calorimeter and contained water. That is, if e denotes the water equivalent of the calorimeter, and / the number of heat units required to melt unit mass of ice, we should have, from (206) and (207), That is, the heat equivalent of fusion would be (m^ + e)(^„-i{^) -h- (231) In most cases, however, the error due to radiation is too great to be neglected. This error may either be computed by Re- gnault's method or determined graphically by the modification of Rowland's method given on pp. 212-214. If the latter method be selected, it is necessary to determine the tempera- ture that the mixture would have attained if there had been no radiation nor absorption. Denoting this corrected value by t^, we obtain the corrected equation ^^(m^ + e)(^^-V)_^^,^ (232) The simple theory given in this experiment applies only to a solid whose temperature is at its melting point at the moment it is introduced into the calorimeter. In the general case not only will the temperature of the specimen be below its melting point at the moment of its introduction to the hot water of the calorimeter, but in addition its specific heat will be different in the solid and the liquid states. Even though neither of these specific heats is known, by means of three experiments. CALORIMETRY 233 similar to the above, in which the masses of the specimen and the water, as well as the original temperature of the water, are different, the heat equivalent of fusion of a substance can be found. We have thus three simultaneous equations containing but three unknown qantities, viz. the required heat equivalent of fusion and the specific heats of the specimen in the solid and in the liquid states. By eliminating the specific heats, the heat equivalent of fusion can be determined. Manipulation and Computation. — Weigh the inner vessel of the calorimeter and the stirrer. The product of their mass and the specific heat of the material of which the vessel and stirrer is composed gives the water equivalent e. Fill their vessel somewhat over half full of water at about 60° C, weigh, and then assemble the calorimeter. Cut a piece of ice having a mass somewhat over a fourth that in the calorimeter. Keeping the water in the calorimeter well stirred, read the temperature of the water about every half minute and of the surroundings every minute or two. Record the hour, minute, and second at which each reading is made. At a given instant, after reading temperatures for four or five minutes, drop the carefully dried ice into the calorimeter and continue reading temperatures and stirring for seven or eight minutes longer. The ice must be kept submerged, and under no circumstances must the temperature of the inner vessel fall so low that dew forms on it. Now weigh the inner vessel with its contents. The data for deter- mining m„ and m^ are now at hand. The corrected temperature of the mixture can be determined graphically, as follows. On a single pair of coordinate axes plot two curves — one coordinating temperature and time for the water in the calorimeter, and the other for the air between the two vessels. Such a pair of curves is shown in Fig. 102. Through the point of intersection of the two curves draw a line parallel to the temperature axis. Produce the cooling curve AB until it intersects this line PS at some point X. Prolong DE backward until it intersects the line 234 PRACTICAL PHYSICS PS at some point H. Then in the manner given on p. 213 it may be shown that the point corresponding to the tempera- ture t^' is as far above ^ as w is above x. That is, to find t^' add to the temperature indicated by M the temperature differ- ence represented by wx. It may be necessary to make one or 45 40 35 SO 25 20 15 lO, l. ._ ■»,. _._. .-■ = --■'■' . -X iif:;]:--"" ----- 6 8 lO 12 Fig. 102. Curve sliowing rate of change of temperature of calorimeter. two preliminary experiments to determine -just how warm to have the water and just how much ice to use. After the ex- periment has been performed successfully it should be repeated once or twice, two or three values for the heat equivalent of fusion being thus obtained. CALORIMETRY 235 Exp. 53. Determination of the Heat Equivalent of Vaporization of Water Object and Theory op Experiment. — If heat be applied to aliquid, the liquid rises in temperature untilits maximum vapor pressure becomes a trifle greater than the external pressure on its surface. The vapor pressure is then great enough to make the bubbles in the liquid expand in spite of the pressure of the liquid outside of them. As the bubbles grow they rise to the surface and burst and the liquid is said to boil. Further addi- tion of heat does not raise the temperature, but simply makes the evaporation into these bubbles go on faster, i.e. produces more rapid boiling. The number of heat units required to vaporize unit mass of a liquid is called the heat equivalent of vaporization of the liquid. The object of this experiment is to determine the heat equivalent of vaporization of water. Let Wj grams of steam be condensed in m^ grams of water contained in a calorimeter of water equivalent e. Let 4 denote the temperature of the steam ; t„, the temperature of the calorim- eter and contents at the moment the steam began to enter; t^, the temperature of the two after they are mixed; and i;, the heat equivalent of vaporization of water. Then the heat giveii up by the steam in condensing and then cooling to the temperature t^ equals the heat taken up by the calorimeter and contents plus the heat lost by radiation. That is, from (206) and (207), m,v -\- m, (t, - fg) = (m„ -f e)(t^ - t„) + E, where It is the radiation correction. Whence, ^^(m„+ e) (t, -t„) + B_ .^^ _ ^. ^233) Manipulation and Compcttation. — The apparatus used in this experiment comprises a boiler in which the liquid is vaporized and a calorimeter containing a copper worm in which the vapor is condensed. The liquid in the. boiler A (Fig. 103) 236 PRACTICAL PHYSICS Fig. 103. is heated by means of an electric current passing through a coil of wire. The arm holding the boiler is attached to a vertical rod supported by the tubular column B. Below the clamp D there is a horizontal slit extending through an arc of about 90°, and from one end of this horizontal slit there is a vertical slit ex- tending about halfway down the tubular column. A pin in the vertical rod supporting the boiler extends through this slit. By means of this arrangement, the boiler can be rotated quickly into a definite plane and dropped in a vertical line so as to cause the outlet of the boiler to register with the end W of the copper worm contained in the calorimeter C. Weigh separately the condensing worm and the inner vessel of the calorimeter with the stirrer, and determine their total water equiva- lent e. Pour water into the inner vessel of the calorimeter until all the convolutions of the condensing worm are covered. The temperature of this water should be below that of the room, but not so low as to cause dew to be deposited on the calorimeter. Determine the mass m„ of this water. Assemble the apparatus and adjust the position of the calorim- eter until the outlet of the boiler will register accurately with the opening in the rubber stopper on the end of the condensing worm. Raise the boiler, thus disconnecting it from the calorim- eter, rotate it to one side, and pour into it enough distilled water to cover all the turns of the wire. Connect a 110-volt circuit to the terminals of the wire spiral and adjust the rheo- stat until the water boils rapidly but does not spatter over into the outlet tube. Now commence stirring the water in the calorimeiter, every half minute recording its temperature, and every minute or two recording the temperature of the air in the jacket. Record the hour, minute, and second at which each reading is made. After reading for two or three minutes rotate the boiler into CALORIMETRY 237 position, drop it into place, and, without interrupting the stir- ring and reading of temperatures, allow steam to flow into the condensing worm until the temperature of the water in the calorimeter rises to 45° or 50°. Disconnect the boiler from the calorimeter, rotate it to one side, throw off the current, and continue stirring and taking temperature readings at one min- ute intervals for about ten minutes. Remove the condensing worm from the calorimeter, carefully dry the outside, and weigh. The difference between this mass and the mass of the worm, already determined, is the mass m, of the condensed steam. Read the barometer, correct it as indicated on pp. 176-178, and by Table 12 find the temperature of the steam. Compute the value of the radiation correction R by Regnault's method in the manner given on pp. 209-212. In determining this correction notice that the e' in (215) is the water equivalent of everything that cooled along the curve CD (Fig. 88). In the present case e' is the water equivalent of the inner vessel of calorimeter, contained water, stirrer, thermometer, worm, and condensed steam. Exp. 54. Determination of the Heat Value of a Solid with the Combustion Bomb Calorimeter Object and Theory op Experiment. — The object of this experiment is to determine the amount of heat developed by the complete combustion of a unit mass of coal. The heat value of a solid or liquid is expressed either in British thermal units per pound or in calories per gram. The method to be employed in this experiment is to burn a known mass of the given substance in a strong steel bomb filled with oxygen under high pressure. During the combus- tion the bomb remains immersed in a water calorimeter and the heat developed is obtained by the ordinarj^ method of mix- tures. Thus 'suppose that by the combustion of m grams of the substance, the bomb together with the calorimeter, its acces- sories, and the contained water rise in temperature from t° to 238 PRACTICAL PHYSICS t^ C. If the mass of water in the calorimeter is m„ grams, the total water equivalent of calorimeter, bomb, thermometer, and stirrer is e grams, and the radiatioii correction is B, calories, then the heat value of the substance is E- (m, +e)(*2-«i)+-K calories per gram. (234) The superiority of this method is that since in it complete combustion is attained and all the products of the combustion remain in the apparatus, the quantity of heat developed is readily computed. Manipulation and Computation. — The apparatus used in this experiment consists of a water calorimeter, a combustion bomb, a press for molding the specimen into a small coherent pellet, and a retort for generating oxygen. Hempel's combustion bomb consists of a soft steel or cast-iron capsule D (Fig. 104), closed by a massive plug Q. The inside surface of the bomb is coated with enamel. The plug is pierced by two passages — one JH for filling the bomb with oxygen, and the other for the in- troduction of an insulated conductor KF. The gas passage is controlled by the compression valve A. The rod KF is insulated from the metal plug by the rubber packing ilf and asbes- tos packing N. (r is a metal rod screwed into the plug. A little basket E^ made of incombustible material, is suspended by means of heavy plati- num wires from the ends of the rods 6r and F. The ends of the rods Gr and F are connected by a thin platinum wire. CALORIMETRY 239 ^>?^ :\,. '^ I' ^?" " A >^ , ^ * X Fig. 105. In preparing a specimen of coal for a determination, the coal is first pulverized in a mortar and then molded into a compact coherent pellet by means of a screw press (Fig. 105). The mold of the press consists of a' block of steel x (Fig. lOG) bored out to the required size. The upper portion of this hole is cylindrical and is fitted with a cylindrical plug A. The lower portion of the hole is reamed out to a conical form and is fitted with a conical plug B. On the conical surface of the plug are two narrow channels which extend from one face to another. Loop a piece of thread ^^S*^ over the conical plug and lay its ends in these slots. Pack some 1.25 g. of pulverized coal about this loop of thread, put the cylinder A in place and the plunger P on top of it, and turn the screw down until the specimen has been compressed into a compact pellet. Raise the screw, slip the mold into the upper horizontal guides, and again depress the screw until the pellet is forced out through the bottom of the mold. With a sharp knife pare down the pellet until it weighs about one gram. Cut off one end of the thread close to the specimen. Remove any loose particles of coal by means of a small brush, place the pellet on a watch glass, and weigh. Do not touch the pellet with the fingers, but handle by means of the thread. Unscrew the plug C (Fig. 104) of the bomb, mount it in a retort stand, connect the terminals of an electric circuit to the binding posts ^and i, and carefully decrease the resistance in the circuit until the current will just bring the platinum wire connecting Gt and J' to a red glow. Without disturbing the resistance in the circuit, open the switch and disconnect the terminals from the binding posts ^and L. Place the specimen of coal in the basket E and tie the free end of the thread to 240 PRACTICAL PHYSICS the wire connecting Gr and F. Without disturbing the speci- men, remove the plug from its support and screw it tightly into the bomb. The bomb is now ready to be filled with oxygen. Into the gas generating retort i2 (Fig. 107) put a mixture of about two hundred grams of potassium chlorate and some fifty grams of manganese dioxid. Put a tightly wound roll of copper or brass wire gauze into the tube leading from the retort, and connect the retort and a pressure gage Gr to the combustion bomb in the manner shown in the figure. Before beginning to heat, shake the retort so as to spread the mixed potassium chlorate and manganese dioxid along its whole Fig. 107. length. The pressure gage and combustion bomb are immersed in a vessel of water for the purpose of detecting any leak in the bomb and also for the purpose of cooling the oxygen coming from the hot retort. Open the gas valve in the combustion bomb and apply a Bunsen flame near the farther end of the gas generating retort until the gage indicates a pressure of about 1 Kg. per sq. cm, (14 lb. per sq. in.). If the flame be now removed, the heat already given to the retort will generate enough oxygen to raise the pressure to about 5 Kg. per sq. cm. (70 lb. per sq. in.). Now loosen the flange coupling P so as to allow the mixture of oxygen and air contained in the apparatus to escape. By CALORIMETRY 241 tightening the coupling P and repeating this operation the entire apparatus can be freed of air. Now tighten the couplings and slowly heat the retort until the gas pressure rises to about 12 Kg. per sq. cm. (170 lb. per sq. in.). Close the gas valve on the combustion bomb and immediately afterwards discon- nect the bomb at the coupling H from the remainder of the . apparatus. Cool the bomb to about the temperature of the room and carefully dry it with a towel. Place the bomb in a water calorimeter Q (Fig. 107) containing iw„ grams of water at about the room temperature. Connect the terminals of the previously arranged electric circuit to the binding posts K and L, and see to it that K and L are not short-circuited by the cover of the calorimeter. Before closing the switch in the electric circuit take temperature readings of the continuously stirred water at half minute intervals for at least five minutes. At a given instant close the switch so that the electric current will ignite the specimen. The switch should be closed for a moment only or the heating effect of the current will need to be taken into account. Continue stirring the water and taking half minute temperature readings for at least ten minutes after ignition. Take the bomb out of the water, open the valve, unscrew the head, wash out the inside, and oil the screw threads. From a curve coordinating temperature and time find by the graphical method described on pp. 212-214 the highest tem- perature that would have been attained by the calorimeter if there had been no loss of heat by radiation. Let t^ represent this corrected temperature. Then instead of (234) we can write ^^ (ma + e)(%-t{) gg^iories per gram. (235) m in this equation the water equivalent e is still unknown. This can be determined in any of three ways : (a) By taking the sum of the products of the masses and the assumed specific heats of the various parts of the apparatus. In an apparatus like this, composed of so many different materials of uncertain 242 PRACTICAL PHYSICS composition, this method is unreliable. (6) Experimentally, by the method of mixtures. The large amount of water re- quired in this experiment and the difficulty of obtaining temperatures accurately make this method unsatisfactory for inexperienced observers, (c) By means of a supplementary experiment in which a definite amount of heat is developed in the apparatus by the combustion of a known mass of a sub- stance having a known heat value. There are a number of substances the heat values of which are accurately known and which can easily be obtained pure. The last method is the one that will be employed, in this experiment. Suppose that when using the same apparatus as before, the burning of m! grams of a substance of heat value H' raises the temperature of the apparatus and of m'^ gfams of water from ij° to t'^G. Let ^g be the temperature that the calorimeter would have attained if there had been no loss of heat by radia- tion. Then jj, _ CW+e)(V-^/) calories per gram. (236) Whence, on solving for e, (237) t ' — f ' (,3 tj Naphthalin is a suitable substance to use in this supplemen- tary experiment. Make a pellet of somewhat smaller mass than that of the coal already used and proceed exactly as in the experiment with the coal. Use (237) to find e, and then (235) to find H. Before putting away the apparatus dig the remaining solid substance out of the gas retort, rinse out the combustion bomb with water, and carefully oil the threads of the bomb and all parts of the press. Be certain that no water or oil is left inside of either the retort or the bomb. If oil or any other organic substance is heated in the retort with the oxygen producing mixture an explosion is liable to occur. CALORIMETRY 243 Exp, 56. Determination of the Heat Value of a Gas with Junker's Calorimeter Object and Theory oe Experiment. — The object of this experiment is to determine tlie number of heat units devel- oped by the combustion of unit volume of a given sample of gas. In Junker's method the heat developed by a steady flame is determined by measuring the heat absorbed by a steady stream of water inclosing the flame. Fig. 108. The apparatus consists of an accurate gas meter if (Fig. 108), a gas pressure regulator R, and a calorimeter (7, of special 244 PRACTICAL PHYSICS design. The calorimeter consists of a combustion chamber A (Fig. 109), inclosed by a water jacket B, traversed by a large number of tubes for the pas- sage of the products of com- bustion. The water jacket is surrounded by a closed space L filled with air. After trav- ersing the meter and pressure regulator the gas is burned in the burner Q. The products of combustion after passing through the tubes traversing the water jacket escape through the vent Y. The temperature of the gas as it enters the burner and the tem- perature of the products of combustion as they leave the calorimeter are given by the thermometers T" and T'". A stream of water flows from the supply pipe D into a small reservoir kept at constant level by means of the overflow pipe 0. From this regulator the water passes down the tube U through the control valve V, thence through the water jacket B, thence through Cf- and the discharge nozzle IT into the measuring vessel U. The temperatures of the water as it enters and as it leaves the calorimeter are given by the thermometers T' and T. Water vapor formed by the combustion of the gas CALORIMETRY 245 condenses on the inside of the combustion chamber and escapes through the outlet J into the measuring vessel TF. The flow of water and of gas is so adjusted that the tempera- ture of the products of combustion escaping at Y is approxi- mately the same as the temperature of the gas entering the burner at P. Let V represent the volume (reduced to standard conditions) of the gas burned during a certain time. Let the mass of water which passes through the calorimeter during this time be denoted by «z„, and let its temperatures on entering and on leaving be represented by t' and t respectively. Let the mass of steam condensed during the combustion be represented by m„ and let the temperature at which it condenses and the tem- perature of the condensed steam as it leaves the calorimeter be denoted by t^ and t^ respectively. Then the heat value of the gas S is given by the equation V where h is the heat equivalent of vaporization of water. If m„ and Trig are measured in grams, v in liters, and temperatures in degrees centigrade, then S is given in gram calories per liter or kilogram calories per cubic meter. Manipulation and Computation. — After assembling the apparatus, connect D to the water supply so that any leak in the calorimeter will make itself evident. The flow of water into the apparatus must always be sufficiently great to overflow through the pipe 0. With gas valve at the burner P closed, connect the gas regulator to the gas supply and notice whether the index of the meter moves. If it does, seek out the leak and remedy it. With the water still flowing through the apparatus, take the burner out of the calorimeter, light the gas, and replace the burner. If the gas is lighted while the burner is inside the com- bustion chamber, there is danger of an explosion. Have the top of the burner from 12 to 15 cm. above the lower opening to the combustion chamber. The damper Z should be from one half 246 PRACTICAL PHYSICS to completely open, depending upon the draught required for the flame. Arrange the flow of water by means of the valve V and the flow of gas by means of the valve P so that the thermometers T" and T'" indicate practically the same temperature. For ordinary illuminating gas the proper rate of flow of water is from 1.0 to 1.5 liters per minute. After all of the thermometers indicate nearly stationary tem- peratures, note simultaneously the gas meter reading and the temperatures indicated by the thermometers T&nd T' . Then immediately place suitable vessels U and W so as to catch the warmed water escaping from 5" and the condensed steam escap- ing from J. Note the temperatures of the ingoing and the out- going water every 15 seconds until two or more liters of water have flowed into the vessel U. Then remove the vessels Cand TTand at the same time take the gas meter reading. Note the temperature % of the condensed steam in W. Determine »i„ and m^ by weighing. From the difference between the two gas meter readings to- gether with the temperature and pressure of the gas passing to the burner, the value of v is found by means of the fundamental law of gases. The temperature is given by the thermometer T" . The pressure is the sum of the barometric reading and the height of mercury corresponding to the difference in the levels of water in the manometer V. All of the data are now at hand for substitution in (238.) By substituting a properly designed lamp for the gas burner, Junker's calorimeter can be used for finding the heat value of a liquid. CHAPTER XV THERMODYNAMICS It is found that whenever TF units of mechanical energy are entirely used in producing heat, the amount of heat produced is always the same, being independent of the particular way in which the energy is used to produce the heat ; that whenever H units of heat are entirely used in producing mechanical energy the amount of mechanical energy produced is always the same, being independent of the particular way in which heat is used to produce the energy ; and that if W units of mechan- ical energy produce H units of heat, H units of heat produce W units of mechanical energy. These three facts may all be indicated by the one equation W= JH, (239) in which J represents the number of units of mechanical energy that are required to produce one unit of heat. This J" is given the name mechanical equivalent of heat. Its value depends only upon the units in terms of which the mechanical energy and the heat are measured. Exp. 56. Determination of the Mechanical Equivalent of Heat by Rowland's Method Object and Theory of Experiment. — One method of de- termining the mechanical equivalent of heat is to measure the amount of heat developed when a given amount of mechanical energy is used to stir water vigorously. In the apparatus used by Joule and improved by Rowland this stirring is done in the 247 248 PRACTICAL PHYSICS ■J^^^> inner vessel (Fig. 110) of a calorimeter. From the inner walls of this vessel pro- ject vanes W between which the paddles JPP have just room to turn. These paddles are fas- tened to a piece of brass tubing that carries at its upper end a disk which is driven by the belt from the small motor seen at the right. The vessel O is supported below on a point with very little friction and on top carries a disk D. Around this disk is lapped a cord which passes over a pulley P and carries at its end a mass M. If there were nothing to prevent it, the weight of M would cause G to turn until a projection on D came against one of two stops between which it plays. But the motion of the paddles PP throws water against the vanes F'F' so rapidly that when the adjustments have been properly made O remains nearly at rest, the projection on D playing between the two stops. Let M denote the mass of M and d the diameter of D. Then Mff X ^dis the torque that iHf exerts in keeping from turning with the paddles. When the paddles have turned n times they have turned through an angle of 2 irn radians. From the propo- sition in elementary dynamics which states that the work done by a rotating body is measured by the product of its rota- tion in radians and the torque which opposes that rotation, we have then Fig. 110. THERMODYNAMICS 249 W= irnMgd, (240) where W denotes the mechanical energy used in stirring the water. Let TO denote the mass of water in the calorimeter ; e, the water equivalent of the vessel C, the paddles, and the immersed part of the thermometer ; tj, the initial temperature of the water in the calorimeter; t^, its temperature after the paddles have made n turns, and R the net amount of heat lost from C by radiation. Then from (206), n=.(m + eXh - «i) + ^> (241) where H denotes the amount of heat developed by the churning of the water. From (239), (240), and (241) we have then J = nrnMgd ^242) \m + e)(t^—t^)+-R' Manipulation and Computation. — In order to deter- mine the radiation correction R it is necessary to know how the readings of the two thermometers compare. Adjust the Beckmann thermometer as directed on p. 160, and suspend both thermometers in a bath of water near the temperature of the room. Stir the water occasionally, and after a time record the reading of each thermometer. Meantime take the diameter of D with a caliper and meter stick, see that the pulley P runs easily, and be sure that the vessel and the paddles are dry and weigh them together. Then fill to within a few milli- meters of the top with water at a temperature some 3° or 4° below the temperature of the room, and weigh again. Assemble the apparatus, set the motor running, and by means of the screw A move the motor until the tension of the belt is such as to keep the projection on I) playing about halfway between its stops. At some chosen instant read the thermometer T and imme- diately thereafter the speed counter 8. For some ten or fifteen minutes after that instant read Tand T-^^ every minute — always 250 PRACTICAL PHYSICS reading one of them half a minute after the other. At the end of this time open the switch that supplies power to the motor, note the reading of the speed counter, and continue reading the thermometers for five or ten minutes. During this time the water in the calorimeter ought to be kept stirred. This can be done by turning the paddles steadily and very slowly by hand. The paddles must not be turned faster than about one revolu- tion in two minutes. On the same sheet plot two curves coordinating temperature and time — one for the therinomter 2'and the other for T^ — and by Regnault's metjiod determine B. In finding n note that the speed counter reads 1 for every four turns of the paddles. Determine Mhy weighing and e by (210). Without throwing out the water or repeating the weighings make three determinations and find the mean. y Exp. 57. Determination of the Mechanical Equivalent of Heat with Barnes's Constant Flow Current Calorimeter Object and Theory op Experiment. — In text-books on General Physics it is shown that when a steady electric current of /amperes flows from one to the other of two points between which there is a potential dif- ference of F" volts, in t seconds there is trans- formed between those two points from elec- tric energy into heat the amount of energy FiQ. m. W= IVt joules = in ■ 107 ergs. (243) THERMODYNAMICS 251 If the. current, the potential difference, the time, and the heat produced can be accurately measured, (243) suggests a method of determining the mechanical equivalent of heat. In this experiment the electric current flows through a wire coiled inside of a glass tube B^B^ (Fig. 111). Through this same tube is a steady flow of water. The heat developed by the electric current warms the water during its passage through the tube so that the thermometer 7j indicates a higher tempera- ture than T^- If m grams of water escape in t seconds, and during the passage through the tube are raised from tempera- ture T^ to temperature T^, the amount of heat developed in the wire during the same t seconds is, by (206), H= m{T^ - T^} calories. (244) On substituting in (239) the values of W and E from (243) and (244) we obtain J = — — ergs per calorie. (245) »w(ij — ij) Since the temperatures F^ and T^ are practically steady dur- ing the time observations are being taken there are no correc- tions for the thermal capacity of wire, glass tube, thermometers, nor anything else. With a good flow of water and the mean of the temperatures of the inflowing and outflowing water within 5° C. of the room temperature the heat losses by conduc- tion and radiation are negligible. Manipulation and Computation. — The rate at which water flows is kept constant by a small reservoir inside a larger jacket shown at the top of Fig. 111. The water sup- ply is so arranged that water is always overflowing gently from the small reservoir, and the head of water is therefore kept constant. FiQ. 112. 252 PRACTICAL PHYSICS After the apparatus is set up as shown in Fig. Ill and the water is started, the electric connections are to be made as in- dicated in Fig. 112. B^ and B^ are the binding posts shown in Fig. Ill, and Wis the wire inside the glass tube. Fis a volt- meter, A an ammeter, R a rheostat, and UH' the terminals of an electric circuit. In connecting the ammeter and voltmeter care must be taken that the positive wire is connected to the side marked -f- . If it is not known which terminal is positive, one wire may be connected and then the other flicked quickly across the other terminal. After making the electric connections and adjusting the flow of water and the electric current to suitable values, open the switch in the electric circuit. After a few minutes, when the readings of the thermometers have become steady, record their readings every minute for four or five minutes. Make all thermometer readings to hundredths of a degree. Close the switch, and when the thermometers have again become steady, put under the out- let a weighed vessel and at the same instant start a stop watch. After fifteen seconds read the voltmeter, af cer fifteen more the ammeter, after fifteen more on© thermometer, and after fifteen more the other thermometer. Continue taking readings in the same order every fifteen seconds for five or ten minutes. At the end of this time remove the vessel from under the outlet and at the same instant stop the watch. Find the mass of the water that flowed through. To get (^x — T^), subtract the difference between the averages of the temperatures indicated by the two thermometers before the electric current was turned on from the difference between their average readings while the current was flowing. Take five sets of observations for different rates of flow of water and different values of electric current. TABLES 254 PRACTICAL PHYSICS TABLE 1. — Conversion Factors Length 1 centimeter = 0.39371 inch 1 inch = 2.53995 cm. 1 meter = 3.2809 feet Ifoot = 0.30479 m. 1 kilometer = 0.62138 mile 1 mile = 1.60931 Km. 1 micron = 0.001 mm. = 0.0000394 inch Irail = 0.001 inch = 0.00254 cm. Area 1 sq. cm. = 0.15501 sq. in. 1 sq. in. = 6.4514 sq. cm 1 sq. m. = 10.764 sq. ft. 1 sq. ft. = 0.092900 sq. m. Volume ' 1 cu. cm. = 0.061027 cu. in. 1 cu. in. = 16.386 cu. cm. 1 cu. m. = 35.317 cu. ft. 1 cu. ft. = 0.028315 cu. m. 1 liter = 1.76077 pints 1 quart = 1.13586 liters 1 gram = 15.43285 grains 1 kilogram = 2.20462 lb. Mass 1 grain = 0.064799 gram 1 lb. (7000 grs.) = 0.45359 Kg. Angle 1 radian = 57.296 degrees | 1 degree = 0.017453 radian Density 1 g. per c. c. = 62.425 lb. per cu. ft. 1 lb. per cu. ft. = 0.016019 g. per c.c. Force 1 dyne = 0.000072331 poundal 1 g. wt. = 0.0022046 lb. wt. 1 poundal = 13825 dynes 1 lb. wt. = 453.59 g. wt. 1 cm. g. unit = 2.3731 X 10-8ft. lb. units Moment op Inertia 1 ft. lb. unit = 421390 cm. g. units TABLES 255 Stress 1 dyne per sq. cm. = 0.067197 poundal per sq. ft. 1 g. wt. per sq. cm. = 2.0482 lb. wt. per sq. ft. 1 cm. of mercury at 0° C. = 13.596 g. wt. per sq. cm. = 0.19338 lb. wt. per sq. in. 1 poundal per sq. ft. = 14.8816 dynes per sq. cm. 1 lb. wt. per sq. ft. = 0.48824 g. wt. per sq. cm. 1 in. of mercury at 0° C. = 34.533 g. wt. per sq. cm. = 0.49117 lb. wt. per sq. in. 1 erg = 2.3731 x lO"" ft. poundals 1 joule = 10' ergs = 23.731 ft. poundals Ig.ctn. = 7.233 x 10-5 ft. lb. Work or Energy 1 ft. poundal = 421390 ergs 1 ft. lb. = 13825.5 g. cm. = 1.35485 joules 1 H. P. hour = 2685600 joules Power 1 watt = 10' ergs per sec. = 23.731 ft. poundals per sec. = 44.23 ft. pounds per min. 1 force de oheval = 75 Kg. m. per sec. = 0.9863 horse power 1 ft. poundal per sec. = 421390 ergs per sec. 1 ft. lb. per min. = 0.13825 Kg. m. per min. 1 horse power = 745.96 watts = 1.0139 force de cheval Thermometric Scales C = f(F-32) I J? = f C+32 Unit Quantity of Heat 1 g. calorie = 0.0039683 B. T. U. | 1 B. T. U. = 252.00 g. calories Mechanical Equivalent or Heat * 1 g. calorie = 4.19 joules = 426.9 Kg. m. = 1400.6 ft. lb. 1 B. T. U. = 1055 joules = 778.1 ft. lb. Logarithms logio ^ = 0.43429 loge N \ log^ N= 2.3026 log,„ iV * Computed with the value of g at Greenwich. 256 PRACTICAL PHYSICS TABLE 2. — Densities of Solids and Liquids Since density varies with the temperature and with the specimen, these numbers are to be regarded as approximations only. Substance Aluminium NH^Cl . . Antimony . Asbestos Asphalt . Beeswax . Benzene . Bismuth . Brass . . Brick Bronze . CaClj . . CS2 at 20° Chalk . . Coal Copper .... CuSO, Cork Diamond ... Ether at 0° C. German Silver Glass Glycerin . . Gold, pure Granite . . Graphite . Ice at 0" C. cast Iron pure . . steel . . wrought Ivory . . . , Lead (cast) Grams PER c.c. 2.7 1.52 6.71 <2.0 J 2.8 51.0 J 1.8 0.96 0.70 9.80 (7.7 I 8.7 51.6 I 2.1 8.6 2.2 1.264 (1.8 12.8 (1.2 1 1.8 8.92 2.27 0.24 3.52 0.736 8.62 ^2.5 J 3.9 1.26 19.32 (2.5 I 3.0 2.8 0^9167 (7.0 I7.7 7.86 57.6 /7.8 (7.79 17.85 (1.83 )1.92 11.34 Lbs. per OU. FT. 170 95 419 125 175 62 110 , 60 44 612 480 540 100 130 540 140 78.9 110 175 75 110 557 142 15 220 45.9 538 150 250 79 1206 150 190 140 57.22 440 480 491 470 490 486 490 114 120 708 Substance Lime Limestone . . . Marble Mica Mercury at 0° C. Nickel Oil, Linseed . . . Oil, Olive . : . . ParaiBn Phosphorus Platinum . Porcelain . KjC^rO, . . . Quartz . . . Resin .... Sandstone . Shellac Silver pure , mint Slate Soapstone j . . . Solder (soft)- . . NaCl Sulphur, rhombic Tin Turpentine . . . Vulcanite .... Water at 4° C. . ash . . . cherry . oak . . . Woods sea- soned Zinc . ZnSO, pme. . poplar .walnut Grams PER c.c. 52.3 ?3.2 J 2.5 J 3.0 j2.6 12.8 (2.6 J 2.9 13.596 8.90 0.94 0.91 (0.87 jO.93 1.83 21.5 2.4 2.72 2.70 2.65 1.07 (2.2 (2.5 (1.1 U.2 10.53 10.3S 2.7 2.7 8.9 2.15 2.07 7.29 0.87 1.22 1.000013 0.75 0.67 (0.7 jl.O 0.5 0.4 0.7 7.15 2.0 Lbs. FEB CU. FT, 140 200 150 190 160 175 160 180 848.7 556 59 57 54 58 114 1340 1.50 170 169 165 67 140 150 70 75 657 648 170 170 555 134 129 455 54 76 62.4252 47 42 45 62 31 25 45 446 125 TABLES 257 TABLE 3.— Specific Gravity of Water at Different Temperatures Referred to Water at 4° C. °c. Sp. Gk. "C. Si". Gk. "C. Sp. Gr. °u. Sp. Gr. °c. Sp. Gr. -4 0.99945 17 0.99882 38 0.99303 59 0.98382 80 0.97191 -3 58 18 8(34 39 268 60 331 81 129 -2 70 19 845 40 233 61 280 82 066 -1 79 20 825 41 195 62 228 83 004 87 21 804 42 157 63 175 84 0.96941 1 93 22 782 43 117 64 121 85 876 2 97 23 759 44 077 65 067 86 812 3 99 24 785 45 035 66 012 87 747 4 1.00000 25 710 46 0.98993 67 0.97957 88 682 5 0.99999 26 684 47 949 68 902 89 616 6 97 27 657 48 905 69 846 90 550 7 93 28 629 49 860 70 790 91 483 8 88 29 600 50 813 71 733 92 416 9 82 30 571 51 767 72 674 93 348 10 74 31 540 52 721 73 615 94 280 11 64 32 509 53 674 74 555 95 212 12 54 33 477 54 627 75 495 96 143 13 42 34 444 55 579 76 435 97 074 14 29 35 410 56 530 77 375 98 005 15 14 36 372 57 481 78 314 99 0.95934 16 0.99899 37 337 58 432 79 253 100 863 TABLE 4. — Specific Gravities of Aqueous Solutions of Alcohol % Specific Gravity at I % Specific Gravity at Alcohol Alcohol BY "Weight BY Weight 10° . 20° 30° 10° 20° so° 0.99975 0.99831 0.99579 55 0.91074 0.90275 0.89456 5 .99113 .98945 .98680 60 .89944 .89129 .88304 10 .98409 .98195 .97892 65 .88790 .87961 .87125 15 .97816 .97527 .97142 70 .87613 .86781 .85925 20 .97263 .96877 .96413 75 .86427 .85580 .84719 25 .96672 .96185 .95628 80 .85215 .84366 .83483 30 .95998 .95403 .94751 85 .83967 .83115 .82232 35 .95174 .94514 .93813 90 .82665 .81801 .80918 40 .94255 .93511 .92787 95 .81291 .80433 .79553 45 .93254 .92493 .91710 100 .79788 .78945 .78096 50 .95182 .91400 .90577 258 PRACTICAL PHYSICS TABLE 5. — Specific Gravities of Aqueous Solutions at 15° Referred to Water at 4° C. c. % HCl HNO3 HiiSO, NaOH NaCl CUSO4 ZnSOi AT n°.5 % 0.9991 0.999 0.9991 0.999 0.999 0.999 0.999 0.9987 5 1.0242 1.029 1.0334 1.056 1.035 1.050 1.0.52 1.0184 5 10 1.0490 1.058 1.0687 1.111 1.072 1.103 1.108 1.0388 10 15 1.0744 1.089 1.1048 1.166 1.110 1.161 1.168 1.0600 15 20 1.1001 1.121 1.1430 1.222 1.150 1.225 1.236 1.0819 20 25 1.1262 1.154 1.1816 1.277 1.191 1.307 1.1047 25 30 1.1524 1.187 1.223 1.133 1.382 1.1282 30 35 1.1775 1.220 1.264 1.387 1.1526 35 40 1.2007 1.253 1.307 1.442 1.1780 40 45 1.287 1.352 1.496 1.2041 45 50 1.320 1.399 1.548 1.2313 50 55 1.350 1.449 1.2593 55 60 1.377 1.503 1.2883 60 65 1.402 1..559 1.3183 65 70 1.424 1.616 •1.3494 70 75 1.443 1.675 1.3813 75 80 1.461 1.733 80 85 1.479 1.785 85 90 1.497 1.819 90 95 1.514 1.839 95 100 1.530 1.838 100 TABLE 6. — Reduction of Arbitrary Hydrometer Scales Light Liquids Scale Reading Heavy Liquids Haume Beck Cartier Baum6 * Baum^t Beck Twaddell rip. Ur. Sp. Gr. Sp. Gr. Sp. Gr. Sp. Gr. Sp. Gr. Sp. Gr. 1.000 1.000 1.000 1.000 1.000 0.971 5 1.035 1.036 1.030 1.025 i.noo 0.944 10 1.073 1.074 1.062 1.050 0.967 0.919 0.970 15 1.114 1.116 1.097 1.075 0.936 0.895 0.936 20 1.158 1.161 1.133 1.100 0.907 0.872 0.905 25 1.205 1.210 1.172 1.125 0.880 0.850 0.876 30 1.257 1.262 1.214 1.150 0.854 0.829 0.849 35 1.313 1.320 1.259 1.175 0830 O.SIO 0.824 40 1.375 1.384 1.308 1.200 0.807 0.791 45 1.442 1.4.53 1.360 1.225 0.785 0.773 50 1.517 1..530 1.417 1.250 0.764 0.756 55 1..599 1.616 1.478 1.275 0.745 0.739 60 1.691 1.712 1.545 1.300 0.723 65 1.795 1.820 1.619 1.325 0.708 70 1.912 1.920 1.700 1.3.50 * Original scale for liquids denser than water. t Newer or so-called " rational " scale. TABLES 259 TABLE 7. — Specific Gravities of Gases and Vapors Referred to Water at 4° C; also to Air and Hydrogen at 0° C. and 760 mm. of mercury pressure. All results are given for a pressure of 760 mm. of mercury. Specific Gravity ueferred to Substance FOEMTTLA 'T'lJ'M1JT!''P ATTTTJir T! J. \ri^ia \j " X tSla.lrmKA.l.UlX,Ci Vj Water Air Hydrogen Air 0.0012931 1.0000 14.445 Ammonia .... NHg 0.0007616 0.5890 8.508 Carbon dioxide . co; 0.001965 1.520 21.955 Chlorine ci, 0.0031674 2.4500 35.382 Coal gas . . . . j 0.000421 0.3256 4.715 0.000667 0.5158 7.452 Hydrogen .... H2 0.0000895 0.0692 1.000 Nitrogen N, 0.0012546 0.9701 14.013 Oxygen • • • . . 0, 0.0014292 1.1052 15.964 Acetic Acid . . . CH3COOH 125 0.00414 3.2 46.2 250 0.00269 2.08 30.0 Amyl bromide . . C,H„Br. . 152 0.00703 5.43 ■ 78.5 196 0.00604 4.67 67.5 295 0.00412 3.18 46.0 360 0.00340 2.63 38.0 Ammonium NH.Gl . . 300 0.00128 0.986 14.23 chloride* . . 360 0.00122 0.944 13.63 448 0.00120 0.932 13.45 Iodine h 448 0.01130 8.74 126.9 680 0.01064 8.23 118.8 855 0.01043 8.07 116.5 1043 0.00906 7.01 101.2 1275 0.00753 5.82 84.0 1468 0.00657 5.06 73.0 Nitrogen Np, . . . 4.2 0.00335 2.588 37.36 peroxide. . . 49.6 0.00294 2.27 32.77 60.2 0.00269 2.08 30.03 70.0 0.00248 1.92 27.72 90.0 0.00222 1.72 24.83 100.1 0.00217 1.68 24.25 154.0 0.00204 1.58 22.81 ♦Ammonium cliloride vapor gives abnormal vapor densities only when in presence of moisture. 260 PRACTICAL PHYSICS TABLE 8. — Coefficients of Friction Substance Static Coefficient /x Kinetic Coefficient h Metals on metals (dry) Metals on metals (wet) Metals on metals (oiled) Wood on wood (dry^ * Wood on wood (dry) f Leather belt on wood pulley .... Leather belt on iron pulley from 0.2 to 0.4 from 0.15 to 0.3 from 0.15 to 0.2 from 0.5 to 0.7 from 0.4 to 0.6 from 0.45 to 0.6 from 0.25 to 0.35 from 0.18 to 0.35 from 0.14 to 0.28 from 0.14 to 0.18 from 0.2 to 0.3 from 0.18 to 0.3 from 0.3 to 0.5 from 0.2 to 0.3 * Motion in direction of fiber. + Motion normal to fiber of sliding- block. TABLE 9. — Elastic Constants of Solids N. B. — Flexural Eesilience per unit volume equals one ninth the Tensile Eesilience per unit volume. Young's Elastic Bkeakino Simple Tensile Substance Modulus Limit Stress Rigidity Resilience dynes lbs. dynes lbs. dynes lbs. dynes sq.cui. lbs. sq.in. erps cu.cm. ft. lbs. sq.cm. sq.ln. sq.cui. sq.in. sq.cm. sq.lu. cu.ft. Multiply by 10" 10« 108 108 108 103 10" 106 10* 1 Brass : cast . . . 6.5 9 4.5 6 20 30 2.4 3.5 16 330 wire . . . 10 14 11 16 60 80 3.7 5.4 60 1300 COPPKR : annealed . 10 14 3 4 31 43 5 100 east. . . . 12 17 4.5 6.3 18 25 4.0 6.0 8 170 wire . . 12 17 7 10 40 55 4.5 6.5 20 420 Glass .... 6.5 9 2.3 3.2 2-9 3-12 2.4 3.5 4 80 Iron: annealed . 21 30 5 7 50 70 6 130 cast . 12 17 7 10 15 20 5.3 7.6 20 420 wire . . . 19 26 20 30 60 85 8.0 12.0 100 2000 wrought . 20 28 20 30 40 55 7.7 11.0 100 2000 Steel : Bessemer . 22 31 33 46 70 100 250 5200 cast . . . 20 28 40 60 8.0 12.0 t5600 tl20000 hearth . . 21 30 70 100 wire . . . 19 26 *40 *60 110 150 *420 *8800 Woods : oak .... 1.0 1.4 2.3 3.2 t5 t7 27 560 pme . . . 1.1 1.6 2.4 3.3 t4 t5 26 540 poplar . . 0.5 0.7 1.5 2.2 t3 t4 23 480 * TJnannealed. t Parallel to grain. % Hardened. TABLES 261 TABLE 10. — Viscosities of Liquids 17 denotes the coefficient of viscosity in c.g.s. units, ssq, S20i ^tc, the specific viscosity, or viscosity relative to water at 0° C, 20° C, etc, (a) Water at Different Temperatures Temp. n «0 Temp. 1 ^0 0° 0.01809 1.000 30° 0.00812 0.449 5 0.01530 0.846 40 0.00664 0.367 10 0.01826 0.733 50 0.00570 0.315 15 0.01150 0.636 60 0.00487 0.269 20 0.01016 0..562 70 0.00424 0.235 25 0.00903 0.499 (b) Aqueous Solutions of Sugar of Various Concentrations at 30° C. % Sugar =20 % SUGAE 2» %Si(:ar ^20 2 1.0521 12 1.4110 22 2.0552 4 1.1104 14 1.5092 24 2.2454 6 1.1840 16 1.6196 26 2.4540 8 1.2576 18 1.7484 28 2.7055 10 1.3312 20 1.8895 30 3.0674 (c) Various Commercial Lubricating Oils In the following table specific gravities are taken at 20° C, and viscosities are given in c.g.s. units. Tkade Na.me Summer Lubricating . Extra Lard Oil .... Std. Gas Engine . . . Solar Red Atlantic Red Union Thread Cutting Arctic Engine Rarius Cylinder .... Vax Cylinder Renove Polar Ice Machine . . 600 End Cylinder . . . Alaska Cylinder . . . Diamond Paraffin . . . EC Test Cylinder . . . Capitol Cylinder . . . Vacuoline Engine. . . No. 1 Dynamo .... Golden Engine .... Sp. Gb. 25° C. 50° C. 75° C. 100° C. 125° C. 150° C. 0.913 .910 .907 .905 .905 .905 .904 .900 .895 .893 .887 .885 .885 .882 .880 .877 .876 .870 .866 0.77 .146 .27 .26 .16 .116 .10 .11 .186 .076 .142 .10 .048 0.148 .068 .092 .090 .069 .06 .053 .157 .29 .054 .078 .44 .289 .37 .066 .053 .029 0.056 .045 .052 .052 .048 .04 .034 .084 .115 .032 .046 .14 .183 .038 .012 .143 .038 .032 .022 0.045 .032 .035 .034 .082 .08 .026 .056 .065 .027 .031 .08 .10 .035 .07 .076 .030 .026 .019 0.024 .025 .027 .025 .026 .026 .023 .04 .043 .025 .023 .05 .067 .042 .046 .024 .024 .018 0.023 .023 .025 .024 .024 .021 .021 .036 .038 .022 .021 .042 .047 .037 .042 .022 .022 .017 262 PRACTICAL PHYSICS TABLE 11. — Corrections for the Influence of Gravity on the Height of the Barometer (a) Reduction to Latitude 45° From 0° to 45° the corrections are subtractive ; from 45° to 90° the correc- tions are additive. Bakometrio Height in mm. Reduced to 0° C. Lat. 670 680 690 700 710 720 730 740 750 760 770 780 Lat. 0° 5° 10° 15° 20° 25° 30° 35° 40° 45° mm. 1.74 .71 .63 .50 .33 .12 0.87 .59 .30 .00 intn. 1.76 .73 .65 .53 .35 .13 0.88 .60 .31 .00 mm. 1.79 .76 .68 .55 .37 .15 0.89 .61 .31 .00 mm. 1.81 .79 .70 .57 .39 .17 0.91 .62 .31 .00 inm. 1.84 .81 .73 .59 .41 .18 0.92 .63 .32 .00 imn. 1.86 .84 .75 .61 .43 .20 0.93 .64 .32 .00 mm. 1.89 .86 .78 .64 .45 .22 0.95 .65 .33 .00 mm. 1.92 .89 .80 .66 .47 .23 0.96 .66 .33 .00 mm. 1.94 .91 .83 .68 .49 .25 0.97 .66 .34 .00 mm. 1.97 .94 .85 .70 .51 .27 0.98 .67 .34 .00 mm. 1.99 .96 .87 .73 .53 .28 .00 0.68 .35 .00 mm. 2.02 1.99 .90 .75 .55 .30 .01 0.69 .35 .00 90° 85° 80° 75° 70° 65° 60° 55° 50° 45° (b) Reduction to Sea Level Corrections are subtractive. Barometric Height in mm. Deduced to 0° c. Elevation 660 680 700 720 740 760 770 m. mm. mm. mm. mm. mm. mm. mm. 100 0.01 0.01 0.01 0.01 0.02 200 0.03 .03 .03 .03 .03 0.03 300 .04 .04 .04 .04 .04 400 0.05 .05 .05 .06 .06 .06 500 .06 .07 .07 .07 .07 .07 600 .08 .08 .08 .08 .09 700 .09 .09 .10 .10 .10 800 .10 .11 .11 .11 .12 900 .12 .12 .12 .13 1000 .13 .13 .14 .14 TABLES 263 TABLE 12, — Boiling Point of Water under Different Barometric Pressures (a) Temperatures in Degrees Centigrade and Pressures in Millimeters of Mercury °c. .0 .1 -.2 .3 .4 .5 .6 .7 .8 .9 90 525.4 527.4 529.4 531.4 533.4 535.5 537.5 539.6 541.6 543.7 91 545.7 547.8 549.9 551.9 554.0 556.1 558.2 560.3 562.4 564.6 92 566.7 568.8 571.0 573.1 575.3 577.4 579.6 581.8 584.0 586.1 93 588.3 590.5 592.7 595.0 597.2 599.4 601.6 603.9 606.1 608.4 94 610.7 612.9 615.2 617.5 619.8 622.1 624.4 626.7 629.0 631.4 95 633.7 636.0 638.4 640.7 643.1 645.5 647.9 650.2 652.6 655.0 96 657.4 659.9 662.3 664.7 667.1 669.6 672.0 674.5 677.0 679.4 97 681.9 684.4 686.9 689.4 691.9 694.5 697.0 699.5 702.1 704.6 98 707.2 709.7 712.3 714.9 717.5 720.1 722.7 725.3 727.9 730.5 99 733.2 735.8 738.5 741.2 743.8 746.5 749.2 751.9 754.6 757.3 100 760.0 762.7 765.5 768.2 770.9 773.7 776.0 779.2 782.0 784.8 (b) remperatures in Degrees Fahrenheit and Pressures in Inches of Mercury °F. .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 194 20.68 20.73 20.77 20.82 20.80 20.90 20.95 29.99 21.04 21.08 195 21.13 21.17 21.22 21.26 21.30 21.35 21.39 21.44 21.48 21. .53 196 21.58 21.62 21.67 21.71 21.76 21.80 21.85 21.89 21.94 21.99 197 22.03 22.08 22.12 22.17 22.22 22.26 22.31 22.36 22.40 22.45 198 22.50 22.54 22.59 22.64 22.69 22.73 22.78 22.83 22.88 22.92 199 22.97 23.02 23.07 23.11 23.16 23.21 23.26 23.31 23.36 23.40 200 23.45 23.50 23.55 23.60 23.65 23.70 23.75 23.80 23.85 23.89 201 23.94 23.99 24.04 24.09 24.14 24.19 24.24 24.29 24.34 24.39 202 24.44 24.49 24.54 24.59 24.64 24.69 24.74 24.80 24.85 24.90 203 24.95 25.00 25.05 25.10 25.15 25.21 25.26 25.31 25.36 25.41 204 25.46 25.52 25.57 25.62 25.67 25.73 25.78 25.83 25.88 25.94 205 25.99 26.04 26.10 26.15 26.20 26.25 26.31 26.36 26.42 26.47 206 26.52 26.58 26.63 26.68 26.74 26.79 26.85 26.90 26.96 27.01 207 27.07 27.12 27.18 27.23 27.29 27.34 27.40 27.45 27.51 27.56 208 27.62 27.67 27.73 27.79 27.84 27.90 27.95 28.01 28.07 28.12 209 28.18 28.24 28.29 28.35 28.41 28.46 28.52 28.58 28.64 28.69 210 28.75 28.81 28.87 28.92 28.98 29.04 29.10 29.16 29.21 29.27 211 29.33 29.39 29.45 29.51 29.57 29.62 29.68 29.74 29.80 29.86 212 29.92 29.98 30.04 30.10 30.16 30.22 30.28 30.34 30.40 30.46 264 PRACTICAL PHYSICS TABLE 13. — Pressure of Saturated Aqueous Vapor In millimeters of mercury °c. Pressure °o. Pressuke °o. Pressure °C. Pressure °C. Pressure 1 4.91 31 33.37 61 155.95 91 545.77 121 1539.25 2 5.27 32 35.-32 62 163.29 92 566.71 122 1588.47 3 5.66 33 37.-37 63 170.92 93 588.33 123 1638.96 4 6.07 34 39.52 64 178.86 94 610.64 124 1690.76 5 6.51 35 41.78 65 187.10 95 6-33.66 125 1713.88 6 6.97 36 44.16 66 195.67 96 657.40 126 1798.35 7 7.47 37 46.65 67 204.56 97 681.88 127 1854.20 8 7.99 38 49.26 68 213.79 98 707.13 128 1911.47 9 8.55 39 52.00 69 223.37 99 733.16 129 1970.15 10 9.14 40 54.87 70 233.31 100 760.00 130 2030.28 11 9.77 41 57.87 71 243.62 101 787.59 131 2091.94 12 10.43 42 61.02 72 2.54.30 102 816.01 132 2155.03 13 11.14 43 64.31 73 265.38 103 845.28 133 2219.69 14 11.88 44 67.76 74 276.87 104 875.41 134 2285.92 15 12.67 45 71.36 75 288.76 105 906.41 135 2353.73 16 13.51 46 75.13 76 301.09 106 938.31 136 2423.16 17 14.40 47 79.07 77 313.85 107 971.14 137 2494.23 18 15.33 48 83.19 78 327.05 108 1004.91 138 2567.00 19 16.32 49 87.49 79 340.73 109 1039.65 139 2641.44 20 17.36 50 91.98 80 354.87 110 1075.37 140 2717.63 21 18.47 51 96.66 81 369.51 111 1112.09 141 2795.57 22 19.63 52 101.55 82 384.64 112 1149.83 142 2875.30 23 20.86 53 106.65 83 400.29 113 1188.61 143 2956.86 24 22.15 54 111.97 84 416.47 114 1228.47 144 3040.26 25 23.52 55 117.52 85 433.19 115 1269.41 145 3125..55 26 24.96 56 123.29 86 450.47 116 1311.47 146 3212.74 27 26.47 57 129.31 87 468.32 117 1354.60 147 3301.87 28 28.07 58 135.58 88 486.76 118 1399.02 148 3392.98 29 29.74 59 142.10 89 505.81 119 1444.55 149 3486.09 30 31.51 60 148.88 90 525.47 120 1491.28 150 3581.23 TABLE 14. — Pressure of Saturated Mercury Vapor In millimeters of mercmy °c. Pressure °o. Pressure °c. Pressure °c. Pressure °C. Peesbuee 0.00047 10 0.00080 20 0.00133 70 0.050 120 0.779 2 .00052 12 .00089 30 .0029 80 .093 130 1.24 4 .00058 14 .00009 40 .0063 90 .165 140 1.93 6 .00064 16 .00109 50 .013 100 .285 150 2.93 8 .00072 18 .00121 60 .026 110 .478 160 4.38 TABLES 265 TABLE 15. — The Wet and Dry Bulb Hygrometer From Smithsonian Tables Let the temperature of the atmosphere given by a dry bulb thermometer be denoted by f C, and let the reading of a wet bulb thermometer be de- noted by (t-M). In the following table, corresponding to the various values of M given in the top line, we have given the pressure (in mm. of mercury) of the aqueous vapor in the atmosphere at the temperature t° C, i.e. the pressure that would be exerted by the aqueous vapor in the atmos- phere if the temperature were reduced to the dew point. Difference between tiii Dry and Wet Bulb E EADINGS i°C. 1 2 8 4 5 6 7 8 9 10 4.6 8.7 2.9 2.1 1.3 1 4.9 4.1 3.2 2.4 1.6 2 5.3 4.4 8.6 2.7 1.9 1.1 0.3 3 5.7 4.8 3.9 3.1 2.2 1.4 0.6 4 6.1 5.2 4.3 3.4 2.6 1.8 0.9 5 6.5 5.6 4.7 3.8 2.9 2.1 1.2 6 7.0 6.0 5.1 4.2 3.3 2.4 1.6 7 7.5 6.5 5.5 4.6 3.7 2.8 1.9 1.1 0.2 8 8.0 7.0 6.0 5.0 4.1 3.2 2.3 1.4 0.6 9 8.6 7.5 6.5 5.5 4.5 3.6 2.7 1.8 0.9 10 9.2 8.1 7.0 ^ 6.0 5.0 4.0 3.1 2.2 1.3 11 9.8 8.7 7.6 6.5 5.5 4.5 3.5 2^6 1.7 12 10.5 9.3 8.2 7.1 6.0 5.0 4.0 3.0 2.1 1.2 0.3 13 11.2 10.0 8.9 7.6 6.5 5.5 4.5 3.5 2.5 1.6 0.7 14 11.9 10.7 9.4 8.3 7.1 6.1 5.0 4.0 3.0 2.0 1.1 15 12.7 11.4 10.1 9.0 7.8 6.6 5.5 4.5 3.4 2.5 1.5 16 13.5 12.2 10.9 9.7 8.4 7.3 6.0 5.0 4.0 3.0 1.9 17 14.4 13.0 11.7 10.4 9.1 8.0 6.7 5.6 4.5 3.5 2.4 18 1.5.4 13.9 12.5 11.2 9.9 8.8 7.4 6.3 5.1 4.0 3.0 19 16.3 14.9 13.4 12.0 10.7 9.4 8.1 6.9 5.7 4.6 3.5 20 17.4 15.9 14.3 12.9 11.5 10.2 8.8 7.6 6.4 5.2 4.1 21 18.5 16.9 15.3 13.8 12.4 11.0 9.6 8.4 7.1 5.9 4.7 22 19.7 18.0 16.4 14.8 13.3 11.9 10.5 9.1 7.8 6.6 5.4 23 20.9 19.2 17.5 15.9 14.3 12.8 11.3 10.0 8.6 7.3 6.1 24 22.2 20.4 18.6 17.0 15.3 13.8 12.3 10.9 9.4 8.1 6.8 25 23.5 21.7 19.9 18.1 16.4 14.8 13.3 11.8 10.3 9.0 7.6 26 25.0 23.1 21.1 19.4 17.6 15.9 14.3 12.8 11.3 9.8 8.4 27 26.5 24.5 22.5 20.7 18.8 17.1 15.4 13.8 12.3 10.8 9.3 28 28.1 26.0 24.0 22.0 20.1 18.3 16.6 14.9 13.3 11.8 10.2 29 29.8 27.6 25.5 23.5 2L5 19.6 17.8 16.1 14.4 12.8 11.2 30 31.5 29.3 27.1 25.0 22,9 21.0 19.1 17.3 15.5 13.9 12.3 266 PRACTICAL PHYSICS TABLE 16. — CoeflScients of Linear Expansion of Solids Substance Temp. ° C. a Substance Temp. " 0. a Aluminium . . 40 0.000023 Iron (soft^ . Iron (cast) . 40 0.000012 Brass to 100 0.000019 40 0.000011 Copper 40 0.000017 Lead .... 40 0.000029 German silver to 100 0.000018 Nickel . . . 40 0.000013 Glass (crown). to 100 0.000008 Silver .... 40 0.000019 Glass (flint) . to 100 0.000009 Zinc 40 0.000029 TABLE 17. — Coefficients of Cubical Expansion of Liquids Substance Tempekatukb Pi ^2 Ps Alcohol * . -39 to 27° C. 0.001033 0.00000145 27 to 46 0.001012 0.00000220 Analin . . . 7 to 154 0.000817 0.00000092 0.000000000628 Glycerin. . 0.000485 0.00000049 Mercury . . 24 to 299 0.0001818 0.00000000018 0.00000000035 Water . . . to 25 -0.00006106 0.000007718 -0.00000003734 25 to 50 -0.00006542 0.000007759 -0.00000003541 50 to 75 -0.00005916 0.000003185 0.00000000728 75 to 100 -0.00008645 0.000003189 0.00000000245 * 93.8% (by volume) pure. TABLE 18. — Heat Values of Various Fuels h indicates the number of gram calories of heat developed by the complete oxidation of one gram of substance, h' indicates the number of gram calo- ries developed by the burning of one liter of gas measured at 0° C. and 760 mm. pressure. If water is one of the products of combustion, the heat value is given when the water is in the liquid form. Substance h SriJSTANCE 7i' Cane Sugar Carbon (charcoal) . . 3866 8080 4140 5500-9000 9692 10200-11500 4000-4500 4000-5000 Acetylene Benzene vapor . . . Coal gas Dawson gus Hydrogen Natural gas (Ind.) . Water gas . . '. . . (carbureted) 14460 33496 1000-1400 5500-7000 3090 9500 2000-3500 3500-7000 Coal Naphthalin . Petroleum . . Peat Wood* . . . ♦Containing 10-12 % of moisture. TABLES 267 TABLE 19. — Specific Heats of Solids and Liquids Unless otherwise stated, the following values express the mean specific heats from 0° to 100° C. SUUBTANCE Sp. Heat Substance Sp. Heat Aluminium Alcohol, ethyl Antimony 0.219 0.685 0.050 0.030 0.093 0.250 0.093 0.316 0.095 0.576 0.161 0.117 0.193 0.108 0.117 Lead Mercury Marble Nickel 0.032 0.033 0.216 0.113 Brass Paraffin (solid 0°-40°) (liquid 50°-100°) Platinum Rock salt Sandstone Silver Sugar Turpentine Tiu Vulcanite Zinc 0.560 Calcium sulphate . . . Copper Copper sulphate .... German silver ... Glycerin (15°-50° C.) . Glass (crown) Glass (flint) Granite Iron (wrought) .... Iron (steel) 0.710 0.033 0.219 0.224 0.056 0.304 0.467 0.056 0.331 0.094 TABLE 20. — Melting Points and Heat Equivalents of Fusion SUBBTANGE Melt- ing Point Heat Equiv. OF Fusion Substance Meit- ING Point Heat Equiv. OF Fusion Beeswax Benzol Bismuth Bromine Cadmium Glycerin Ice Iodine Iron, cast (gray) .... Iron, cast (white) . . . Lead C. 61.8 5.4 266.8 -7.3 320.7 13 0.0 115 1200 1100 326 Cal. . perg. 42.3 30 12.6 16.2 13.7 42.5 80 11.7 23 33 5.4 Mercury Naphthalin Nickel Palladium Paraffin Phenol Platinum Silver Sulphur Tin Zinc 0. -29 79.9 1450 1500 52.4 25.4 1779 999 115 233 415 Cal. perg. 2.8 35.7 4.6 36.3 35.1 24.9 27.2 21.1 9.4 14.3 28.1 268 PRACTICAL PHYSICS TABLE 21. — Boiling Points and Heat Equivalents of Vaporization Substance Boiling Point Hkat Equit. OF VaP. Substance Boiling Point Heat Equiv. OF Vap. Acetic acid C. 188 56.6 182.5 78 64.5 Cal. perg. 84.9 125.3 93.3 205 267.5 Benzol .....-- C. 79.6 61 35 350 100 Oal. per g. 93.5 Acetone .... Analin Alcohol, ethyl . Alcohol, methyl . . . Chloroform . Ether, ethyl . Mercury . . Water . . : 58.5 90.5 62 536 TABLE 22. — Thermal Emissivities of Different Surfaces The following results were obtained by Bottomley for a cooling copper globe surrounded by air at atmospheric pressure in an inclosure kept at a constant temperature of 14°.5 C. The emissivities are expressed in gram calories of heat lost per second per square centimeter of surface per degree centigrade excess of temperature of the body above the temperature of the surroundings. Temperature op Globe in °C. Surface Polished Bright Surface Polished Bright and Thinly Lacquered Surface Thinly COATED WITH Lampblack 21 165 X 10-6 246 X 10-6 278 X 10-6 22 170 250 281 23 174 254 284 24 178 257 287 25 181 260 290 26 184 263 293 27 187 265 296 28 190 268 299 29 192 271 301.5 30 . 194 273 304.5 31 196 276 307 32 198 278 310 33 199.5 280 313 34 201 282.5 316 35 202 285 320 36 203.5 287 323 37 205 290 326 38 206.5 292 329 39 207 294 332 40 208 297 335 41 299 338 42 301.5 341 TABLES 269 TABLE 23. — The Greek Alphabet Letteu Name LETrER Name Letter Name A, a Alpha I, t Iota P.p Rho B,/3 Beta . K, K Kappa 5,^ Sigma ki Gamma A, X Lambda T, T Tau Delta M,ti Mu Y,v Upsilon E,£ Epsilon N, V Nu ^,4> Phi z,t Zeta S,f Xi X, X Chi Eta 0,0 Omicron *,^ Psi Theta n, TT Pi n, (0 Omega INDEX Absorbing power, 215. Acceleration, 63, 66, 69. Accuracy required, 6. Air thermometer, 188. Alcoholimeter, 103. Angle, 42. Approximations, table of, 7. Area, 54. August's psychroraeter, 206. Balance, analytic, 25. Jolly-Lin ebarger, 97. Mohr-Westphal, 99. Balancing columns, metboii of, 183. Ballistic pendulum, 64. Barnes's current calorimeter, 250. Barometer, 53, 176, 262. Baume'a hydrometer scale, 258. Bearings and journal, friction between, 80, 82. Beck's hydrometer scale, 258. Beckmann thermometer, 159. Belt and pulley, friction between, 77. Boiling point of a solution, 173. Boiling point tables, 263, 268. Borda's method of coincidences, 67. Breaking stress, 119, 260. Brittleness, 120. Bulk modulus, 119. Bullet, speed of, 64. Bunsen's effusiometer, 107. Calibration of an hydrometer, 102 a thermometer, 160, 166. standard masses, 86. Caliper, 17, 21. Calorimetry, 207. Cartier's hydrometer scale, 258. Cathetometer, 22. Center of percussion, 65. Chronograph, 33. Chronometer, 33. Clock, 33, 34. Coefficient of elasticity, 118. of expansion , 176, 266. Coefficient of expansion — Continued. of air, 188. of a rod, 179. of glass, 186. of mercury, 183. of friction, 75, 260. of restitution , 64. of viscosity, 143. Coincidences, method of, 39, 68, 72. Cold test of an oil, 170. Combustion bomb, 237. Compound pendulum, 69. Concentration and boiling point, relation between, 173. Constant errors, 3. Constants, number to keep in empirical formula, 11. Conversion factors, 254. Cooling, law of, 209. Correction, 4. Correction factor of a planimeter, 54. Coulomb's method for viscosity, 148. Cubical expansion, 176, 256. Current calorimeter, 250. Damping constant, 148. Daniell's hygrometer, 203. Datum circle of planimeter, 59. Densimeter, 103. Density, 88, 256. by immersion, 95. by measurement and weighing, 89. of an unsaturated vapor, 198. with Jolly balance, 97. with Mohr-Westphal balance, 99. with pyknometer, 90, 92. Depressed zero, 157. Determinate errors, 3. Dew point, 203. Differences of various orders, 12. Dilatometer, 186. Direct measurements, 1. Distance measurements, 16, 42. Divided circle, eccentricity in mounting, 61. 271 272 INDEX DWiding engine, 18. Dynamometer, oil testing, 80, 82. Eccentricity in mounting of circle, 61. Eifusiometer, 107. Elasticity, 118, 260. Elongations, method of middle, 38. Emissivity, 215, 268. Empirical equations, 10, 124. Errors, 2. Expansion, 176, 266. See also Coefficient of expansion. Exposed stem correction, 158. Eye and ear method, 35. Eyepiece micrometer, 20. Filar micrometer microscope, 19. Fire test of an oil, 170. Flash and stop watch method, 36. Flashpoint of an oil, 170. Fly wheel, change of speed, 63. Friction, 75, 260. Gases, fundamental law of, 176. Golden's oil-testing dynamometer, 80. Gravity, acceleration due to, 66, 69. specific. See Specific gravity. Greek alphabet, 209. Heat equivalent, 208. of fusion, 231, 267. of vaporization, 235, 268. Heat value of coal, 237, 266. of gas, 243, 266. Hempel's combustion bomb, 238. Humidity, 202. Hydrometer, 102, 258. Hygrometry, 202. Hypsometer, 164. Immersion, specific gravity by, 95. Indeterminate errors, 3. Indirect measurements, 1. Inertia, moment of, 110. Jolly's air thermometer, 188. spring balance, 97. Joly's steam calorimeter, 229. Joule's method for mechanical equivalent of heat, 247. Journal and bearing, friction between, 80, 82. Junker's calorimeter, 243. Latent heat, 208. Law of cooling, 209. Least count of a vernier, 21. Length, 16, 42. Level trier, 49. Lever, optical, 42,47. Limit of elasticity, 118. Linear expansion, 176, 179, 266. Linebarger's spring balance, 97. Logarithms, accuracy, 6. Lubricated journal, friction at, 80, 82. Mass, 25, 86. Maximum elongation, 38. Mechanical equivalent of heat, 247. Melting point table, 267. Mercury vapor pressure, 264. Meter bridge, 168. Meter stick, 16. Meyer's method for vapor density, 198. Micrometer, 16-20. Middle elongations, method of, 38. Mixture, method of, 221. Modulus of elasticity, 119, 260. of resilience, 140, 260. Mohr-Westplial balance, 99. Moment of inertia, 110. Miiller's optical lever, 179. Newton's law of cooling, 209. New York Board of Health tester, 171. Notation, 15. Oil, test of, 148, 170. Oil testing machine, 80, 82. Omitted transits, method of, 35. Optical lever, 42, 47. Oscillation, 34. Parallax, 2, 156. Passages, method of, 36. Pendulum, ballistic, 64. compound, 69. seconds, 33. simple, 66. Percussion, center of, 65. Period of oscillation and vibration, 34. Permanent set, 118. Planimeter, 54. Plotting, 9, 13. Poiseuille's method for viscosity, 143. Pressure, temperature coefficient of, 188. Projectile, speed of, 64. Psychrometry, 202. INDEX 273 Pulley, correction for friction of, 76. Pulley and belt, friction between, 77. Pyknometer, 90. Pyrometers, 156. Qualitative experiments, 1. Quantitative experiments, 1. Radiating power, 215. Badiation, constant, 210. correction for, 209. Radius of spherical surface, 45. of spirit level, 49. Regnault's method of correcting for radi- ation, 209. Regnault's method for vapor pressure, 197. Resilience, 140, 260. Resistance thermometer, 166. Restitution, coefficient of, 64. Results, methods of expressing, 8. Rider, of a balance, 26. Rigidity, 119, 134, 137, 260. Rowland's method of correcting for radi- ation, 212. Rowland's method for mechanical equiva- lent of heat, 247. Rumford's method of correcting for radi- ation, 214. Salinimeter, 103. Seconds pendulum, 33. Sensibility of a balance, 28. Sensitiveness of a spirit level, 49. Series, summing, 15. Set, 118. Shear, 119. Simple pendulum, 66. Simple rigidity, 119, 134, 137, 260. Slide modulus, 119. Slide rule, accuracy, 6. Slide wire bridge, 168. Solution, boiling point of, 173. Specific gravity, 88, 256-259. by immersion, 95. with effusiometer, 107. with Jolly balance, 97. with Mohr-Westphal balance, 99. with pyknometer, 90, 92. Specific heat, 207, 267. by cooling, 219. by mixture, 221. with stationary temperature, 226. with steam calorimeter, 229. Speed, 63, 04. Spherometer, 17, 45. Spirit level, sensitiveness, 49. Standard masses, calibration of, 86. Stem exposure correction, 158. Stop watch, 33. Strain, 118. Stress, 118. Summation notation, 15. Telescope, adjustment, 44. Temperature, 155. Tenacity, 119. Tensile coefficient of elasticity, 119, 120, 128, 2C0. Thermodynamics, 247. Thermometer, air, 188. calibration, 100, 166. errors, 156. resistance, 166. Thickness of a thin plate, 42. Thurston's oil testing machine, 83. Time, 32. Transits, method of omitted, 35. Trustworthy figures, 4. Twaddell's hydrometer scale, 258. Ultimate resilience, 140. Vapors, 194, 264. Variable errors, 3. Velocity, 63. Verification of a barometer scale, 53. Vernier, 20. Vibration, 34. Vibrations, method of, 26. Viscosity, 143, 148, 261. Watch, 33, 34. Water equivalent, 208, 224. Water vapor pressure, 194, 196, 264. Weighing, 25. errors, 30. precautions, 32. Weight dilatometer, 186. Wet and dry bulb hygrometer, 205, 265. Wheatstone bridge, 168. Young's modulus, 119, 120, 128, 260. Zero circle of planimeter, 59. point of balance, 26. reading of caliper, 17.