UNIVERSITY OF CALIFORNIA MEDICAL CENTER LIBRARY SAN FRANCISCO Dr. Howard I. Hawdsley \. A MANUAL OF PHYSICAL MEASUREMENTS BY JOHN O. [R_EED, PH.D. PROFESSOR OF PHYSICS IN THE UNIVERSITY OF MICHIGAN AND KARL B. GUTHE, PH.D. PROFESSOR OF PHYSICS IN THE UNIVERSITY OF MICHIGAN FOURTH EDITION, REVISED AND ENLARGED QC37 R3L l\\3 GEORGE WAHR, PUBLISHER ANN ARBOR, MICHIGAN 1913 COPYRIGHT, 1902 BY JOHN O. REED AND KARL E. GUTHE COPYRIGHT, 1906 BY JOHN O. REED AND KARL E. GUTHE COPYRIGHT, 1912 BY JOHN O. REED AND KARL E. GUTHE THE ANN ARBOR PRESS PREFACE. This manual has been prepared to meet the needs of students beginning work in the Physical Laboratory of the University of Michigan. Such a book must inevitably possess a certain local coloring peculiar to the conditions it has been designed to meet. A manual equally suited to all laboratories, has not been and probably will not be written. Each laboratory reflects in greater or less degree the individual trend of the man who stands at its head; and its exercises and methods are the result of an ex- tended process of adaptation and assimilation. Hence it happens that one laboratory is largely devoted to the study of the phe- nomena of light, another to those of electricity, and a third to those of elasticity, heat, or electrochemistry, as the case may be. The moral of all this is, that the practice and traditions of each laboratory are best conserved by a text representative of its own methods, and if no better reason should be found, perhaps this may serve to explain the appearance of this, another laboratory manual. The exercises herein described embody the work required of students in Physics and in Engineering in their first course in Physical Laboratory Practice. Such a course is expected to oc- cupy three laboratory periods of two hours each for one semester, and embraces some thirty-six to forty of the exercises in this manual. Owing to the diversity of the work prescribed in the various courses in Engineering, no one student is expected to complete all the exercises in this book in a single semester. In accordance with the practice in the University of Michigan, it is expected that the laboratory work shall be supplemented by lectures upon the theory of the exercises, and recitations upon the work actually done and the results obtained. In this way it is believed that the student is brought to a clearer understanding of the significance of the exercise and of the accuracy attainable vi PHYSICAL MEASUREMENTS under given conditions. To this end the exercises are numbered consecutively throughout the text, and those under any specific subject are preceded by sufficient theory to render the formulae and methods clear to persons familiar with the fundamental principles of Physics as set forth in any standard textbook. Being designed for beginners in the Physical Laboratory, this manual makes no claim to completeness, either in subject mat- ter or in exposition. The aim has been to furnish a coherent and logical series of graded exercises in Physical Measurement, such as will best furnish an introduction to Practical Physics, and at the same time afford opportunity for developing ability in record- ing and interpreting observations, and skill in the manipulation of delicate and sensitive apparatus. For convenience of reference a series of tables of the more important physical constants, of squares, cubes, square roots and multiples of TT, of the logarithms of numbers, and the trigonomet- ric functions have been added. A thorough drill in the use of logarithmic tables in the computation of results, should form a feature of any successful course in Laboratory Practice. To this end an orderly method of procedure in such computation has at all times been insisted upon. The authors have drawn freely from many standard works on Practical Physics, notably .from those of Kohlrausch, and Stew- art and Gee in General Physics, and from Carhart and Patter- son's Electrical Measurements. In conclusion we wish to thank our colleagues, Professors Carhart and Patterson, for helpful suggestions and criticisms during the preparation of the work. University of Michigan, March, 1902. FROM THE PREFACE TO THE SECOND EDITION. The necessity for a second edition of this book has presented an opportunity for a careful revision of the text, both in the elimination of errors and in the addition of certain features which experience has shown to be desirable and necessary to make the manual truly representative of modern laboratory practice. In making these additions the needs of the average student have been kept constantly in mind both as regards his previous prep- aration and the requirements laid upon him by his subsequent University work. In the first part several articles are devoted to the measure- ment of angles, and a chapter has been added upon Surface Tension and Viscosity. In order to meet more fully the demands made upon students of Mechanical and Electrical Engineering the chapters upon Heat and Electricity have been practically rewritten. Several of the articles in these chapters contain new and important matter, notable among which are the discussion of the ballistic d'Arsonval galvanometer, and the exercises involving the use of the potentiometer and of the thermoelement. While the additions have in general been such as to render the work more advanced in character, with the possible exception of some exercises in the measurement of angles, still it is hoped that the book will not be found less useful for elementary work than before. The forms for recording results and the outlines for computation have abundantly justified the wisdom of their insertion in the immense saving of time and energy to the busy instructor. While it has been urged by some that students readily and intuitively devise explicit, symmetrical and logical arrange- ments for their data and computations, such students have as yet entirely escaped our observation. September, 1906. PREFACE TO THE THIRD EDITION. The third edition of this manual has been prepared mainly "because it was felt that the arrangement of the subject matter should correspond more closely to that found in the authors' COLLEGE; PHYSICS, recently published by the Macmillan Company. The book has also been thoroughly revised, the treatment been changed in a number of places and a few exercises, notably some elementary exercises in electricity, .been added. The authors are greatly indebted to their colleagues, Professor H. M. Randall and Mr. W. W. Sleator, for assistance in reading the proof sheets of the present edition. JOHN O. RSED. KARL E. GUTHE. August, 1912. TABLE OF CONTENTS. INTRODUCTION. ARTICLE. PAG- 1 Benefits of laboratory work I 2 Instruments I 3 Record of observations .' . 2- 4 Graphical methods 2 5 Errors of observation 5 6 Probable error 6 7 Influence of errors upon the result 7 8 Interpolation 9- 9 Hints on computation 9- CHAPTER I. FUNDAMENTAL MEASUREMENTS. 10 Fundamental magnitudes II LENGTH. 1 1 Contact measurements 1 1 12 Methods of subdivision 12 13 The micrometer screw 12 14 Exercise i. The micrometer gauge 12 15 Exercise 2. The 'spherometer 13 16 The vernier 15 17 Exercise 3. The vernier 15 18 Exercise 4. The vernier caliper 16- 19 Line measurements 17 20 The cathetoimeter 17 21 Optical micrometers 17 22 The dividing engine 21 23 'Exercise 5. The dividing engine 22 : X PHYSICAL MEASUREMENTS ANGLE. ARTICLE. PAGE. 24 Measurement of angle 23 25 Definitions 23 26 Exercise 6. The protractor 24 27 Exercise 7. Angles from trigonometric functions 25 28 Correction for eccentricity 26 29 Telescope and scale 27 30 Exercise 8. The optical lever 28 31 The lever tester 30 32 Exercise 9. Constants of the level 31 33 Small angles 'by the filar micrometer 33 34 Exercise 10. Angular measurement by the filar micrometer 34 35 The sextant 34 36 Exercise n. Measurement of angles by the sextant. 37 MASS. 37 The balance 37 38 Determination of the resting point 38 39 Sensibility of the balance 39 40 Exercise 12. To make a single weighing 39 41 Reduction to weight in vacuum 41 42 Exercise 13. Double weighing 41 TIME. 43 Period of vibration 42 44 Method of coincidences 43 45 Exercise 14. Period of torsional pendulum 47 46 Exercise 15. The barometer 47 CHAPTER II. ELASTICITY. pAGE 47 Definitions 48 Hooke's law .....!!.!..!..!..!!.!...! 40 49 Coefficients of elasticity 50 50 Coefficient of volume elasticity so CONTENTS XI ARTICLE. 51 52 53 54 55 56 57 59 PAGS. Young's modulus 51 Simple rigidity 51 Exercise 16. To verify Boyle's law 54 Exercise 17. The Jolly balance 57 Exercise 18. Young's modulus by stretching 59 Exercise 19. Verification of the laws of bending 60 Exercise 20. Young's modulus by flexure 66 Exercise 21. (Simple rigidity 67 Exercise 22. Simple rigidity of a brass wire from torsional vibrations 69 CHAPTER III. PENDULUM EXPERIMENTS AND MiOMENT OF INERTIA. 60 The simple pendulum 71 61 Exercise 23. Law of the simple pendulum 71 62 Exercise 24. Computation of g 73 63 Exercise 25. Moment of inertia of a connecting rod 73 64 Exercise 26. Moment of inertia from torsional vibrations 75 65 Exercise 27. The ballistic pendulum 78 CHAPTER IV. DENSITY. 66. Definition 81 67 Exercise 28. Density from mass and volume 81 68 Exercise 29. The pyknometer 81 69 Exercise 30. Mohr's balance 82 CHAPTER V. SURFACE TENSION AND VISCOSITY. 70 Characteristics of a liquid 84 71 Exercise 31. Measurement of surface tension 84 72 Exercise 32. Surface tension from capillary action 86 73 Coefficient of viscosity 87 74 Exercise 33. Coefficient of viscosity by flow through a capillary tube . ..88 x ii PHYSICAL MEASUREMENTS CHAPTER VI. MEASUREMENTS IN SOUND. 75 Exercise 34. Velocity of sound in metals (Kundt's method) 91 76 Exercise 35. Computation of Young's modulus 93 77 Exercise 36. Rating a turning fork. Graphical method 93 CHAPTER VII. MEASUREMENTS IN HEAT. ARTICLE. 78 Effects of heat 96 THERMOMETRY. 79 Thermometry 96 80 Exercise 37. Determination of the fixed points of a thermometer 97 81 -Stem correction 100 EXPANSION. 82 Coefficient of linear expansion 100 83 Exercise 38. Coefficient of linear expansion of a solid 101 84 Expansion of liquids 104 85 Exercise 39. Coefficient of expansion of a liquid by the dilato- meter 106 86 Air free water 109 87 Exercise 40. Constant volume air thermometer 109 CAI,ORIMETRY. 88 Definitions 112 89 Specific heat by method of mixtures 113 90 Exercise 41. Water equivalent of a calorimeter 114 91 Exercise 42. Specific heat of copper 116 92 Correction for radiation 117 93 Exercise 43. Heat of fusion of water 120 94 Exercise 44. Heat of vaporization of water at boiling point 121 95 Exercise 45. Melting point and heat of fusion of tin 124 VAPOR PRESSURE. 96 Measurement of vapor tension 126 97 Exercise 46. Vapor tension of ether 127 98 Exercise 47. Vapor tension of water at various temperatures. . 127 CONTENTS X1H CHAPTER VIII. ELECTRICAL MEASUREMENTS. UNITS AND STANDARDS. ARTICLE. PAGE. 99 Resistance 130 100 Current 134 101 Electromotive force 134 102 Quantity o"f electricity 136 103 Capacity 136 104 Selfinductance 137 INSTRUMENTS. 105 Keys 138 106 Galvanometers 139 107 The astatic galvanometer 140 108 The d'Arsonval galvanometer 140 109 Methods of observation 142 1 10 Shunts 142 in Exercise 48. Calibration of a galvanometer by Ohm's law 143 112 Exercise 49. Figure of merit of a galvanometer 144 1 13 Ballistic galvanometers 147 1 14 Constant of ballistic galvanometer 148 115 Exercise 50. Determination of the constant of a ballistic gal- vanometer 148 1 16 Voltmeters and ammeters 149 ELEMENTARY EXERCISES. 117 Exercise 51. Cells in series and in parallel 150 118 Exercise 52. Kirchhoff's laws 151 119 Exercise 53. Resistances in series and in parallel 153 MEASUREMENT O? RESISTANCE. 120 Exercise 54. Resistance by substitution 1,54 121 Exercise 55. Resistance by voltmeter and ammeter 15,5 122 Exercise 56. Very high resistances by direct deflection 156 123 The Wheatstone bridge 157 124 Exercise 57. Resistance by Wheatstone bridge box 158 125 Exercise 58. Resistance by slide-wire bridge 160 126 Exercise 59. Resistance of a galvanometer Thomson's method 162 127 Exercise 60. Resistance of a galvanometer Second method.... 163 128 Exercise 61. Resistance of an electrolyte 164 XIV PHYSICAL MEASUREMENTS ELECTROMOTIVE FORCE AND POTENTIAL DIFFERENCE. ARTICLE. 129 The voltmeter 167 130 Exercise 62. Electromotive force of a cell 167 131 Exercise 63. Electromotive force by potentiometer method.... 168 132 The potentiometer 170 133 Exercise 64. Calibration of a voltmeter 171 134 Exercise 65. Thermo-electromotive force of a thermoelement.. 172 ELECTROMOTIVE FORCE AND RESISTANCE OF BATTERIES. 135 Electromotive force and difference of potential 173 136 Exercise 66. Terminal potential difference as a function of external resistance 175 137 Exercise 67. Electromotive force and internal resistance, volt- meter and ammeter 176 138 Exercise 68. Electromotive force and internal resistance by condenser method 177 139 Exercise 69. Internal resistance by method of Nernst and Haagn 179 MEASUREMENT OF CURRENT. 140 Measurable effects of a current 181 141 Law of electrolysis 181 142 The copper coulometer . 182 143 Exercise 70. Calibration of an instrument by use of copper coulometer 182 144 Exercise 71. Calibration of ammeter by standard cell 183 COMPARISON OF CAPACITIES. 145 Exercise 72. Comparison by direct deflection 184 146 Exercise 73. Method of mixtures 186 MEASUREMENT OF INDUCTANCE. 147 Exercise 74. iSelfinductance of a coil compared with a standard 187 148 Exercise 75. Mutual inductance of two coils . . 189 CHAPTER IX. MAGNETIC MEASUREMENTS. MAGNETIC FIELDS. 149 Magnetic fields 192 150 Exercise 76. Determination of H (First method) 193 151 Exercise 77. Determination of H (Second method) 198 CONTENTS XV MAGNETIC PROPERTIES OF IRON AND STEEL. 152 Magnetic permeability 200 153 Exercise 78. Commutation curve for iron and steel 201 154 Exercise 79. Hysteresis curve for iron and steel 205 CHAPTER X. OPTICAL MEASUREMENTS. CURVATURE. PAGE. 155 Curvature of optical surfaces 208 156 Exercise 80. Radius of curvature of a lens by the spherometer. 208 157 Exercise 81. Radius of curvature by reflection 210 158 Exercise 82. Focal length of lenses 213 159 Exercise 83. ! Lens curves 215 MAGNIFYING POWER. 160 Exercise 84. Magnifying power of the telescope 216 161 Exercise 85. Magnifying power of microscope 218 INDEX OF REFRACTION. 162 Exercise 86. Index of refraction of lenses from radii of curva- ture and focal lengths 219 163 Exercise 87. Index of refraction by means of a microscope.... 220 THE SPECTROMETER. 164 Description 222 165 Adjustments of the spectrometer 224 166 Reflecting surfaces 226 167 Exercise 88. To measure the angle of a prism 228 168 Angles by method of repetition 230 169 Exercise 89. Index of refraction of a glass prism 232 DIFFRACTION. i/o Exercise 90. Wave lengths of sodium light by diffraction grat- ing 234 171 Exercise 91. Constant of a diffraction grating 235 172 Dispersion, normal and prismatic 236 173 Exercise 92. Dispersion curve for a prism 237 X vi PHYSICAL MEASUREMENTS TABLES. I. Atomic weights of some elements 239 II. Density of water at different temperatures 239 III. Density of mercury at different temperatures 240 IV. Density of various bodies 240 V. Reduction of barometer readings to oC 241 VI. Coefficients of elasticity 241 VII. Viscosity and surface tension of liquids at 2OC 242 VIII. -Moments of inertia 242 IX. Boiling point of water under different barometric pressures.. 243 X. Heat constants 243 XI. Vapor tension of liquids 244 XII. Index of refraction for sodium light 244 XIII. Wave lengths of lines in solar spectrum 244 XIV. Electrical resistance of metals 245 XV. Electrical conductivity of solutions at i8C 245 XVI. Numbers frequently required 246 XVII. Numerical tables 247, 248 XVIII. Trigonometric functions 249, 250 XIX. Logarithms ; 251, 252 ^T O > ,r. Let the corresponding values of y be F and y. Then if these values lie near enough together so that we can assume that F--V is proportional to X ,r, we find 9. Hints on Computation. The following suggestions re- garding the computation of results will be found useful. (a) Taking the Mean. In taking the mean of a set of read- ings, it is necessary to average only those parts of the various- readings which differ among themselves. Thus if ten readings have the first three figures 264, and differ only in the tenths it is clearly unnecessary to average more than the tenths. (b) Significant Figures. The student should avoid carrying, a result out to a large number of decimal places, far beyond the point where the figures have any significance whatever. Thus if six readings of the barometer be 743-3, 743- 2 , 743-3. 743- 1 * 743- 2 > 743.3 mm., the mean is 743.233 mm., of which but four figures are- significant and the tenths are uncertain since they cannot be read every time alike. If this reading be corrected for temperature the result should likewise be given to tenths of a millimeter but no* farther, since nothing is known beyond that. In general the re- sult should be carried to so many figures that the last, owing to errors, makes no pretension to accuracy, while the next to the last may be regarded as reasonably accurate. (c) Approximations. In many cases where it is necessary to- compute results from values, some of which are very small in- IO PHYSICAL MEASUREMENTS comparison to others, the labor may be greatly reduced by approx- imation formulae. Some of the more useful are given below, where a, b, c and d are to be regarded as very small in compari- son to unity. ic m =i tnd (II) = i n= V* d (ia) (i &) (ic) (d) Supplementary Tables. At the end of the book will be found a series of tables of mathematical and physical constants. The student will find it of advantage to consult these freely in the course of his work. While the values of the physical constants contained in these tables have been selected from reliable sources, the student is warned against the error of assuming that a value obtained in the laboratory is necessarily wrong, because it differs slightly from that given in the table. CHAPTER I. FUNDAMENTAL MEASUREMENTS. 10. Fundamental Magnitudes. Most physical quantities may be expressed either directly or indirectly in terms of the fundamental units of mass, length, and time. The first prob- lem of physical measurement therefore has to do with the methods for evaluating certain quantities in terms of these fundamental magnitudes. Closely related to such fundamental measurements are the methods employed for the measurement of angles and of the pressure exerted by the atmosphere. On account of the great importance of these quantities the methods for their deter- mination are included here. The instruments and processes de- scribed in this chapter being essentially those employed in the exercises which follow, a few words may be devoted to these fun- damental measurements. LENGTH. 11. Contact Meaurements. Many instruments for measure- ing length with a greater or less degree of accuracy involve the necessity of bringing the instrument into contact with the object to be measured. This may be effected by bringing the object between two jaws, one of which is movable, or between a movable part of the instrument and a fixed plane of reference. In any case accuracy of measurement requires that the contact between the object and the instrument shall be exact, and that the distance between the points of contact shall represent the true length of the object to be measured. Hence it is of importance to see that the contact points of such instruments are scrupulously clean and true, and that the ends of the body are accurately faced and free from adhering matter of any kind. Such measurements are termed contact measure- ments. 12 PHYSICAL MEASUREMENTS 12. Methods of Subdivision. Instruments for refined meas- urements of length usually involve the principle of the microm- eter screw or of the vernier, or a combination of the two. Prom- inent among such instruments may be mentioned the micrometer gauge, the spherometer, the vernier caliper, the cathetometer, the micrometer cathetometer, the comparator, and the dividing engine. Of these the micrometer gauge and the spherometer employ the principle of the micrometer screw, the vernier caliper and the ordinary cathetometer employ the principle of the vernier, while the micrometer cathetometer, the comparator and the divid- ing engine employ a combination of the two. It is not the inten- tion to describe the working of each of these instruments in de- tail, but so to give fundamental principles upon which each instrument is based, as to enable the student to make the applica- tion for himself. *' 13. The Micrometer Screw. In the micrometer screw we have a screw of fine thread working in a nut and furnished with a graduated head divided into some convenient number of aliquot parts. A complete rotation of the head advances or withdraws the screw by an amount equal to the distance between its adjacent threads, that is by some fraction of a centimeter, or of an inch. This small length is further subdivided by means of the divisions on the graduated head, so that the least .count of the instrument, that is, the least length directly measurable by it, is the distance between the threads, divided by the number of divisions on the head. Still finer readings may be made by estimating tenths of these divisions. 14. Exercise i. The Micrometer Gauge. In the microm- eter gauge the end of the screw works against a fixed jaw. .The number of complete turns of the screw is read from the stem of the instrument and the frac- tion of a turn from the gradu- ated head. In use the least count of the instrument is first deter- mined and recorded. The end of - * the screw is next brought into FUNDAMENTAL MEASUREMENTS contact with the fixed jaw by slight pressure and the zero reading taken. The object to be measured is then brought between the jaws and the screw turned down until contact is made as before and the reading is again taken. The difference between this read- ing and the zero reading gives the thickness of the object, ex- pressed in the units marked upon the stem. In determining the zero and final readings the mean of five readings is to be taken in each case. In more accurate instruments undue pressure upon the jaws is avoided by means of a ratchet head which slips as soon as contact is made. In using such instruments always turn slowly by means of this head and stop as soon as the ratchet slips. FORM OF RECORD. Exercise i. The Micrometer Gauge. . Date Object measured Pitch of screw Micrometer gauge No Least count Zero readings Final readings Mean Mean Thickness 15. Exercise 2. The Spherometer. In the spherometer the screw works in a nut supported by a frame having three legs of equal length, so placed that when the four points rest upon a plane the three feet form an equilateral triangle about the point of the screw at the center. The instrument is usually placed on a square of good plate glass. Notice of contact between the point of the screw and the plane is given by the instrument's hobbling or rocking on the plane. The screw is then care- fully turned back until this hobbling just ceases. The zero reading is then taken. The object to be measured is then placed beneath the mid- dle point and the screw turned down until contact is made, and the Fig. 2. PHYSICAL MEASUREMENTS reading taken as before. The difference between the zero and final readings gives the thickness of the object. In the more delicate form of the instrument, made by the Ge- neva Society, (Fig. 3), the screw point is connected by a system of light levers, to a delicate pointer which rises when contact is made. Readings are taken when the pointer rises to a fixed mark. In use avoid touching the graduated head with the fingers. Turn by means of the milled head provided for that purpose. An extremely d >e 1 i- cate means of determin- ing the position of con- tact in the use of the spherometer, is by means of the interfer- ence fringes of sodium light. The spherometer is placed upon a piece of good plate glass and a J small piece of glass with B a good plane surface is placed under the central leg. When the surface of the glass is lighted by a sodium flame, the in- terference fringes appear at once. The slightest increase in pres- sure causes the lines to shift their position, thus indicating the position of contact. FORM OF RECORD. Exercise 2. The Spherometer. To measure the thickness of a piece of glass. Pitch of screw Spherometer No Object measured Zero readings ' Fi'naY readings Date Least Count. Mean Thickness of Mean 01 23456789 10 1 I I I I I I I I I I I I FUNDAMENTAL, MEASUREMENTS 15 1 6. The Vernier. The vernier is a device for subdividing the least division of a scale. _! It consists of a short subsidi- ary scale placed in front of, Flg * 4 * and in contact with the scale of the instrument, and is usually so divided that n divisions of the vernier correspond to n i divisions of the scale. The least count, i. e., the least subdivision which may be read by the use of the vernier, is i/n of a scale division. Thus, if S be the least division of the scale and V the least division of the \ernier then Fig. 5. V or Least count: S V = (12) In some cases n divisions on the vernier are made equal to one less than some multiple of n divisions of the scale; the formula then becomes nV(.an i) S whence Least count: aS V 5* (13) as before. To read the vernier, first read the units of the scale up to the zero of the vernier; to this reading add as many nths of a scale division as are indicated by the vernier division which coincides with a scale division. Thus in Fig. 4 the reading is 2.6 scale divisions. 17. Exercise 3. The Vernier. Determine the least count of the verniers on five or more instruments. i6 PHYSICAL, MEASUREMENTS FORM OF RECORD Exercise 3. To determine the least count of the verniers on five different instruments assigned by the instructor. Date. Name of Instrument Value of S 1 8. Exercise 4. The Vernier Caliper. The vernier caliper (Fig. 6), is an instrument in which the principle of the vernier is applied to the measurement of length. It consists of a pair of steel jaws, one of which is attached to the scale, the other to the vernier which slides upon the scale. In some instruments the movable jaw is pro- ind slow motion screw for fine- adjustment. Fig. 6. '^Q-^ Most ins truments are adapted to inside measurements also, by means of a pair of rounded lugs attached to the ends of the jaws. When a separate scale is not provided for inside measurements, the thickness of these lugs must be added to the indicated reading. In use the value of a scale division S, and the least count, are first determined and recorded. The jaws are next brought to- gether and the zero reading taken. The object to be measured is then placed between the jaws and the mean of several readings taken. The difference between the zero and final readings gives the length of the object. FORM OF RECORD Exercise 4. ' The Vernier Caliper. To measure the diameter and length of a brass cylinder. Object measured.... Vernier caliper No. Zero readings Date Least count Diameter readings Length readings Mean 'Mean Diameter Mean Length FUNDAMENTAL MEASUREMENTS 17 19. Line Measurements. In the previous exercises on meas- urements of length it will be noted that in each case the meas- urement has been effected by contact measurements, that is, by bringing the measuring instruments into contact with the object to be measured. Frequently this is neither practicable nor desirable. In such cases line measurements are employed ; that is, the distance between two lines drawn upon a body is determined, by focusing a microscope or telescope upon the lines in succes- sion, and noting the readings of the vernier attached to the instru- ment. Line measurements are made by means of the cathetom- eter, the comparator and the dividing engine, and are used in all cases where great accuracy is desired. 20. The Cathetometer. The cathetometer is an instru- ment for measuring the difference in height between two points. It consists of a vertical standard, (Fig. 7), upon which slides a ring carrying a telescope and furnished with a vernier and a clamp for holding the tele- scope at any height desired. The instrument is provided with a level for bringing the axis of the telescope into a horizontal line, by means of adjusting screws. In most instru- ments the level may be rotated through a right angle about the standard as an axis, in order to secure the vertical position of the latter. The telescope is focused first upon one point and the reading taken, and then lowered by means of the clamp to the level ^ of the second point, and set upon it, and Fig. 7. the corresponding reading determined. For ease and accuracy in making the settings, the ring carrying the telescope is usually furnished with a slow motion screw. The difference between the two settings gives the vertical distance between the two points. 21. Optical Micrometers. A micrometer is a device for measuring minute lengths or changes in length with great accu- racy. Such instruments are constantly employed in physical measurements and are made in a variety of forms. Since most jg PHYSICAL MEASUREMENTS such instruments embody a combination of a micrometer screw and an optical system of some sort, it has seemed fitting to class them as optical micrometers. (a) Eye-piece micrometer. Perhaps the simplest form of such micrometer consists of a finely divided scale, ruled upon glass and placed upon a diaphragm between the two lenses in the eye-piece of a mi- croscope. The diaphragm must be so adjusted that the scale divisions are seen sharply defined on looking through the eye-piece turned toward a bright window. On introducing the eye-piece again into the microscope and focusing it upon some minute object, the image will be seen sharply outlined upon the ruled surface and the size of the image may be read off directly in terms of the units of the scale. By placing upon the stage of the microscope a test plate, or object micrometer, containing divisions of a known length, the magnifying power of the instrument and the corresponding value of a single division of the eye-piece scale may be directly determined. Once the value of a division is known the eye-piece may be used to measure lengths by simply counting the number of scale divisions covered by the magnified image. (b) Screzv micrometer. In the screw micrometer (Fig. 9), the tube of the microscope is moved in a line at right angles to its axis, by means of a micrometer screw, and the different parts of the image may be brought successively upon the cross hair of the microscope. The advantage of such an instrument is that its range of measurement may be extended to several centimeters. When fitted to a microscope of low power and mounted upon a vertical standard the instrument forms a micrometer catheto- meter (Fig. 10), and may be used to determine the distance between adjacent points with great accuracy. It may also be used to measure small changes in length as in the determination of Young's modulus by stretching, (Exercise 18). (c) Filar micrometer. In the filar micrometer (Fig. n), FUNDAMENTAL MEASUREMENTS IQ the micrometer screw is used to displace across the field of view a glass plate carrying a pair of fine lines, or a light frame upon which are stretched a pair of spider threads, either parallel and close together or crossing at an acute angle. In most instruments there are also two fixed threads or hairs, at right angles to one Fig. 9. Fig. 10. another. All the threads or cross hairs must lie in the focal plane of the eye-lens and be seen sharply defined when the micro- meter is turned toward the bright sky. When the micrometer is placed in the microscope or telescope and the instrument focused upon some object, the measurement of any featare of the image is effected by bringing the movable hairs successively into coincidence with the points under con- 20 PHYSICAL MEASUREMENTS sidcration and noting the corresponding readings upon the micrometer head. In some instruments the count for the integral number of turns of the screw is shown in the field of view by means of a toothed index, where the moving hairs pass from one tooth to the next for one complete revolution of the head. Every fifth notch is made deeper than the others to assist in the reading. The zero may be taken at any definite point in the field as best suits the convenience of the observer. Usually the divided head is held in place by means of a friction washer or a lock nut which Fig. ii. allows it to be adjusted for any desired zero. If there be a fiducial mark in the field of view to be used as a zero, then when the parallel hairs include this mark symmetrically between them the divided head should read zero, or be adjusted until it does. The micrometer here described is used in the micrometer micro- scopes on dividing engines and comparators, in the reading micro- scopes of spectrometer and other finely divided circles, in tele- scopes on cathetometers and spectrometers, and on astronomical telescopes. The filar micrometer is also used for measurement of small angles, as shown in Article 33. It is to be remembered that lengths measured by a micrometer placed in the eye-piece of a microscope are made upon the magni- fied image and are therefore related directly to the magnifying power of the instrument. Consequently in any series of measure- ments this magnifying power must be kept constant, i. e. t neither FUNDAMENTAL MEASUREMENTS 21 the microscope objective nor the distance between the objective and the micrometer must be changed. It is also to be observed that in making any setting the hairs must be brought up to the final position from the same direction in each case, in order to avoid the lost motion of the screw. If this position has been passed, the screw should be run back through at least two turns and brought up with greater care. Usually the hairs are carried from left to right across the field by direct pressure of the screw, and displaced in the opposite direc- tion by means of a spring as released by the screw. Hence the best direction for approach to a setting is that in which the spring is compressed. 22. The Dividing Engine. In the dividing engine the micrometer screw is mounted in a rigid metallic bed so as to rotate freely, and carries upon it the movable nut which advances or recedes as the screw rotates. Attached to this nut is the traveling carriage upon which is mounted the device for ruling short lines transverse to the length of the screw. This ruling device may be operated either by hand or automatically by the machine itself, and is so arranged as to make every fifth and tenth line dis- tinctively longer than the others, thereby facilitating the reading of the graduated divisions. In modern machines the pitch of the screw is usually one milli- meter and the divided head allows the screw to be advanced through any desired fraction of this length. Usually the smallest division of the head may be still further^ subdivided by a small vernier. In most machines the carriage is provided with one or more micrometer microscopes, whereby the operator is enabled to extend the graduation to lengths beyond the length of the screw. In the machine shown in Fig. 12, the graduating device and the reading microscopes are fixed and the table carrying the surface to be graduated or the scale to be examined, moves with the motion of the screw. Otherwise the relations are the same as those de- scribed. The dividing engine may also be used to measure lengths smaller than that of the screw with great accuracy, as soon as the pitch of the screw is known. To determine this, a standard scale 22 PHYSICAL MEASUREMENTS is supported horizontally beneath the reading microscope, and accurately parallel to the length of the screw. This adjustment is secured when the scale is seen in sharp focus throughout its en- tire length as the microscope is moved over it, and when the inter- section of the cross hairs in the microscope cuts the divisions of Fig. 12. the scale at corresponding points throughout. The microscope is then set upon some definite division of the standard scale and the number of turns of the screw needed to carry the microscope to the next similar division is carefully determined. From the value of the scale division measured and the observed rotation, the pitch of the screw is readily computed. The dividing engine is also useful for measuring accurately small lengths, for calibrating thermometer or other capillary tubes, for study of the errors of standard scales, for de- termining the constants and errors in the ruling of diffraction gratings, etc. 23. Exercise 5. The Dividing Engine. In order to meas- ure with the dividing engine the distance between any two fixed points, the line connecting the points in question is placed accu- rately parallel to the screw of the instrument and readings taken first upon one point and then upon the other. The difference between the means of the two sets of readings gives the distance sought. In making any setting of the microscope care must of course be taken to avoid lost motion of the screw. FUNDAMENTAL MEASUREMENTS 23 A smoothly planed, rectangular iron bar has ruled upon one of its sides two fine lines at right angles to its length. A third fine line ruled lengthwise of the bar, intersects the transverse lines normally, and enables the operator to set the bar accurately parallel to the length of the screw. The temperature of the bar is noted and at least five readings made upon each of the transverse marks. The observed distance is to be corrected for temperature by means of Table X. FORM OF RECORD Exercise 5. The Dividing Engine. To determine the distance between two marks on an iron bar. Date Temperature Reading A. Reading B. Difference Distance corrected to 2OC. ANGLE. 24. Measurement of Angle. One of the most common meas- urements to be made in the .physical laboratory is the determination of an angle. This may be the angle included between two lines on a plane surface, or that subtended by two points in space, or the angle swept out by a body as it rotates about an axis. For the determination of angular magnitudes many devices are em- ployed of which only the more important will be described here. 25. Definitions. In circular measure an angle is defined as the ratio between its subtending arc s, and the radius r, or e= --. (14) The unit angle in circular measure is the radian, that is, an angle whose subtending arc is equal to the radius. Angles are also measured in degrees, minutes, and seconds ( ' "). The de- gree is the angle subtended by 1/360 part of a circumference. PHYSICAL MEASUREMENTS From this it follows that the entire angle that may be described about a point is 2 TT radians, or 360, and consequently i radian = ~~ = 57-295& 7T i degree radians. In theoretical formulae angles are usually expressed in radians, while measuring instruments give the values in degrees and frac- tions of a degree, hence frequent use must be made of the above equivalents in reducing from one system to the other. Most measuring instruments are graduated in degrees and minutes and furnished with verniers or reading microscopes for subdividing the least division of the scale. The degree of accuracy required in the specific problem in hand must determine the re- finements in measurement to be employed. 26. Exercise 6. The Protractor. In determining the angle included between two lines drawn upon a plane surface, as a sheet of paper of a black- board, one of the simplest methods is to use a protractor (Fig. 13). This is a gradu- ated circle with an arm rotat- ing about its center. The cen- ter of the instrument is placed over the point of intersection of the two lines in question and the edge of the movable arm is brought into coinci- dence with the lines in turn and the corresponding read- ings of scale and vernier noted in each case. The difference between these readings is the angle sought. Measure the Fig. 13. the angles of a triangle and check work by taking the sum. Meas- ure each angle three times. FUNDAMENTAL MEASUREMENTS FORM OF RECORD Exercise 6. The Protractor. To measure the angles of a triangle. Date. . Angle Mean B Sum =: 27. Exercise 7. Angles from Trigonometric Functions. A very simple method for the approximate determination of an angle is by means of measurements which enable us to compute either its sine or tangent, preferably the latter. Thus if it be desired to know the angle subtended at the eye or- at the ob- jective of a telescope by a length /, placed symmetrically normal to the line of sight, at a distance L, a simple consideration shows that the angle is given by tan = 2 2.L, d5) from which may be readily evaluated. Determine, both in radians and in degrees, the angle subtended by some object in the laboratory at three different distances, and compare results thus obtained. To what degree of approximation may angles of i, 2, 3, 5, and 10 be set equal to their tan- gents. FORM OF RECORD Exercise J. To determine the angle subtended by an object at three different distances. Object Da tan te I L tan 0/2 Difference i 2 3 in degrees in radians i 2 3 5 10 tan Difference PHYSICAL MEASUREMENTS 28. Correction for Eccentricity. Instruments intended for accurate work are provided with two or more verniers for the purpose of eliminating the error in reading the graduated circle iue to eccentricity. This ;rror is due to the fact that the renter of the graduated circle and the center about which it or the vernier arms rotate, are never exactly coincident. This error is usually small, yet for instruments of preci- sion it must be eliminated. This is readily accomplished by reading "the circle at two points, 180 apart and combin- ing the readings for every observation. Thus in Fig. 14, let C be the true center of the graduated circle, and let C' be the center of rotation either for the circle or for the arm carrying the vernier. Then when the vernier stands at A, the axis of the telescope points in the direction of CA, while the reading at A corresponds to the direction C'A, and the error e, due to the displaced center is the angle CAC or A'C'A. The true reading is the arc ZA' instead of ZA. On the other hand, the reading on vernier B, corresponds to a direction C'B, while the telescope looks in the direction CB, and the error e, is BC'B' as before. In the first case the true reading R, is given by (16) and in the second by whence R = R + 180 B e A+ ( 180 (17) the error due to eccentricity may be completely eliminated by taking the mean of two readings 180 apart. In practice it 777 , 777 FUNDAMENTAL MEASUREMENTS 2J is well to designate the right hand vernier as A and record the de- grees as read from this vernier only, averaging the minutes and seconds as read from the two. It may also be shown that the error due to eccentricity may be eliminated by taking the mean reading of any number of equidistant verniers. 29. Telescope and Scale. In measuring small anguilar displace- ments as indicated by the tilting or rotating of a mirror about an axis the most common and efficient method is that of the telescope and scale. A telescope of low magnifying power (Fig. 15) is furnished with a scale, usually in millimeters, on which the figures are both inverted and perverted. This scale is so arranged that it may be held near the telescope and at right angles to its optical axis, in either a horizontal or vertical position. A small plane mirror m, whose angular displacement is to be determined, forms a virtual image of the scale which is viewed by the telescope. The scale is made as nearly parallel to the surface of the mirror as may be and the telescope adjusted until the scale division nearest the axis of the instrument is seen sharply defined on the cross hairs. If now the mirror and its normal be rotated through an angle 0, then by the law for .the reflection of light, the reading r, on the scale as seen in the telescope will form an angle of 2 9 with the axis of the instrument. If D be the distance from the scale to the mirror then Fig. 15. tan 2 D (18) from which the value of 6 is readily determined. For accurate measurements the deflections are read first upon one side and 28 PHYSICAL MEASUREMENTS then upon the other, and the mean value for r used in computa- tion. The equation (18) also shows that the sensitiveness of the method increases as the distance D is increased. In many cases the ratio between the deflections is all that is needed. In such cases if two scale readings r t and r 2 have been observed we have tan tan (19) or, since for small angles the tangents may be set equal to the angles themselves, we see that the angular deflections are directly proportional to the scale readings. The method of telescope and scale is of universal application in work with galvanometers, magnetometers, radiometers and in all cases where slight angular displacements are to be determined. An exceedingly delicate instrument embodying the same principle is found in the optical lever. 30. Exercise 8. The Optical Lever. The optical lever is a device for measuring small lengths by means of the displacements of a beam of light reflected from a plane mirror. It consists of a stout bar (Fig. 16), supported up- on four short, blunt pointed legs, of which one, at one end, is a mi- Fig. 16. crometer screw, for pur- poses of adjustment and calibration. The bar carries at its middle point a plane mirror capable of rotation about a horizontal axis, and which may be clamped in any desired position. The instrument is placed upon a piece of good plate glass, in front of and some four or five meters distant from a reading tele- scope furnished with vertical scale. The telescope is focused upon the reflected image of the scale as seen in the mirror, and the FUNDAMENTAL MEASUREMENTS 29 micrometer screw adjusted till all hobbling ceases. The mirror is then rotated until the central division of the scale falls upon the horizontal cross hair in the telescope. In making these adjust^ ments care must be taken to free the plate and the points of the feet from all dust particles or lint, as otherwise the zero point will vary after the instrument has been tilted or moved on the plane. After these adjustments have been made the object, as a small piece of microscope slide, is placed under the middle of the lever, the instrument is tilted so as to stand upon three legs, and the reading on the scale noted. The lever is then tilted in the oppo- site direction, placing a small weight if necessary upon the end previously elevated, and the scale reading again noted. The two positions of the lever are shown in Fig. 17. Then if the scale m Fig. 17. readings in the two tilted positions of the mirror be r and r s , if L be the distance from scale to mirror, / the half length of the lever and a the angle of tip in each case, we have for d, the thickness of the glass to be measured d = /sin a = la 20 since a is small. PHYSICAL MEASUREMENTS Also since the total angular displacement of the mirror is 2 a, the reflected beam is turned through 4 a, and consequently = 2 tan 2 a = 4 tan a and d = (21) (22) (23) Make at least three determinations of r and r 2 and take the mean. Determine / by means of the dividing engine. FORM OF RECORD H.vtfcise 8. The Optical Lever. To measure the thickness of a piece of glass. Date Object measured I Mean Y\ Tz ^^ d 31. The Level Tester. The spirit level is an instrument for testing a line or a surface as to its deviation from the true horizontal. It consists of a tube (Fig. 18), nearly rilled with Fig. 18. ether or alcohol and closed at both ends. The inner surface of the upper side of the tube is accurately ground into a circular arc of large radius. When the tube is brought into the true horizontal FUNDAMENTAL MEASUREMENTS 3! the enclosed air bubble, seeking the highest point of the arc, stands at the middle of the tube. Any change from the horizontal is indicated by the bubble moving toward the higher end. A scale of equal divisions is graduated upon the tube for reading the position of the bubble. In accurate instruments the tube is held in its frame by adjust- ing screws so that its ends may be raised or lowered to bring the bubble into the middle when the level is placed upon a horizontal plane. By the horizontal screws the tube may be shifted laterally so as to bring its axis parallel to the line whose inclination is to be tested. In sensitive levels the vertical adjustment is seldom perfect, but it is more convenient to eliminate this error in use than to correct it by adjustment. This is done by reading the level in the direct and reversed positions each time and taking half the difference between the readings as the true position of the bubble. 32. Exercise 9. Constants of the Level. The value of one division of the level and the radius of curvature of the tube are termed the constants of the level. These may be de- termined by means of the level tester shown in Fig. 19. This ^SS^S^^S^^^^^gg^^S:^ Fig. 19. consists of a stout cast iron frame with a heavy horizontal metal bar hinged at one end so as to move in a vertical plane, and supported by a micrometer screw at the other end. By one turn of the screw the bar may be tilted through an angle < whose value in radians is given by (24) J-t where p is the pitch of the screw and L the length of the bar. 32 PHYSICAL MEASUREMENTS The level is placed upon the bar and the readings on the mi- crometer and on the ends of the bubble recorded. The readings on the east and west ends of the bubble in the direct and reversed positions may be designated by e, w, e' and w f respectively. The screw is turned until the bubble moves over three or more divi- sions and the readings repeated. Proceed in this way by successive steps of from three to five divisions throughout the length of the graduation. Reverse the level and repeat the readings for the other half of the tube. From the data thus obtained compute the value of in, the distance the bar has been raised to move the bubble over one scale division. From the known values of L and m we may at once compute 8, the angular value of one scale divi- sion of the level, from the relation --f M where 8 is expressed in radians per division. Reduce this value to seconds per division. If there be n scale divisions graduated on the tube, then n 8 is the angle subtended by this part of the tube at the center of the circle of radius R, of which it forms a short arc. If the length of the scale be /, then from the definition of an angle in circular measure we have = -L. (26} I LI * = ^ = 7T^ (27) FORM OP RECORD. Exercise p. The Level Tester. To determine the value of a scale diiision, and the radius of curvature of a spirit level. Date L= /= Level direct Level reversed Bubble Screw Bubble Screw e w e w' Mean value of m = FUNDAMENTAL MEASUREMENTS 33 33. Small Angles by the Filar Micrometer. Small angles may be measured with great accuracy by means of a telescope of large focal length in which the ordinary eye-piece has been replaced by a filar micrometer. For laboratory experiments the telescope should have a focal length of at least 50 centimeters and should also have a slow motion in azimuth. A large reading telescope will do very well. The angle to be measured may be that subtended by two divisions of a brilliantly lighted scale or better by two small pin holes in a piece of tin foil placed before a small lamp. After the tele- scope has been sharply focused the micrometer is revolved in the tube until the transverse hair connects the two points in ques- tion, in order that the distance measured may be the actual distance between them. By the slow motion screw of the telescope the fixed hair is brought upon the image of one of the points. The micrometer head is then turned till the movable hair coincides with the fixed hair and the reading noted. The movable hair is then brought upon the other image and the read- ing upon the head again recorded, care having been taken to avoid lost motion of the screw in making the settings in each case. The difference between the two readings gives the angle in terms of the divisions of the micrometer head and the focal length of the telescope objective. Let m be the measured dis- tance between the .images and F the focal length of the objective, both expressed in millimeters ; then <, the angle sought, is ex- pressed in radians by 0=^. (28) If r is the least reading of the micrometer, then <$>' , the least angle that can be measured is (29) from which it is clear that the delicacy of measurement increases as the focal length of the objective increases. 34 PHYSICAL MEASUREMENTS 34. Exercise 10. Angular Measurement by the Filar Mi- crometer. Take a piece of stiff, dark blue card board, 35 cms square, and cut from its center a hole 5 cms in diameter. Over this paste a piece of rather heavy tin foil in which have been pierced two small holes not more than 5 mms apart. Place the card board in front of a fishtail gas burner at a distance L, of at least 20 meters from the telescope. The bright yellow images are seen sharply defined upon the deep blue field. Repeat the meas- urements at least five times, setting the fixed hair first upon one image and then upon the other. Return the final value of the angle in seconds. Measure L and compare the measured value with the computed value of the angle subtended by the holes at the center of the objective. Compute also the least angle meas- urable with the apparatus. FORM OF RECORD Exercise 10. The Filar Micrometer. To measure the angle subtended by two points distant meters from the telescope. Date, 1st Image Readings 2nd Image Difference M Mean value of m $ Observed Computed I Difference 35. The Sextant. The sextant is an instrument for meas- uring the angle subtended by two distant points or the angular elevation of a point above the horizon. It consists of a light frame (Fig. 20), carrying a telescope BF, a graduated arc AC, to the plane of which the axis of the telescope is parallel, and two FUNDAMENTAL MEASUREMENTS 35 20 - mirrors / and H, whose planes are normal to the plane of the arc. The mirror /, or index glass, is attached to the index arm IB, which rotates about / as a center and carries a vernier along the arc AC. The hor- izon glass //, is fixed in po- sition and is silvered on its lower half and transparent on its upper half, so that on looking through the tele- scope, objects are seen through the upper half in the direction HP, while in the lower half are seen by reflection objects in the di- rection of HP'. Let it be required to meas- ure the angle PHP' subtended at H by the points P and P'. The telescope is pointed toward P and sharply focused. If the planes of the two mirrors are parallel the two lines of sight are parallel, and the two images of the object at P are coincident. If now the instrument be rotated about the line HP as an axis until the plane of the graduated arc passes through P' , and the index arm be then moved so as to bring the image of'P' also into the field of the telescope, the two images may be made to coincide. In this case the light from P' has been reflected from the two mirrors 7 and H, and consequently it has been deviated through an angle equal to twice the angle between the mirrors. 1 The half degree marks on the arc are numbered as whole degrees, and hence the reading of the index gives directly the angle PHP f . Figure 21 shows the usual form of the instrument, in which the various parts will be recognized -without difficulty. The ver- nier arm is furnished with clamp and slow motion screw for making the final settings. It is important for accurate work that the light from the two objects should be as nearly as possible of the same intensity. To this end shades of colored glass are 1 College Physics, Article 442. 36 PHYSICAL MEASUREMENTS attached to the instrument which may be rotated in front of either mirror at will. Adjustments of the Sextant. In order that its readings may be reliable the sextant must be accurately adjusted and carefully handled. The instrument should fulfill the following conditions: (a) The plane of the index glass should be normal to the plane of the graduated arc. This adjustment is secured by plac- ing the eye near the index glass and adjusting the glass until the graduated arc and its reflected image appear to be in the same plane. (&) The horizon glass should also stand normal to the plane of the graduated arc. This adjustment is attained when the two Fig. 21. images of a distant object can be made to coincide exactly. (c) The axis of the telescope should be parallel to the plane of the graduated arc. This adjustment is correct when the images of two stars, about 120 apart may be made to coincide exactly on either of two cross hairs, equidistant from the field of view. The foregoing adjustments, when once carefully made should not require further attention. The instrument will be adjusted by the instructor in charge and beginners are cautioned against attempting such adjustments for themselves. (d) The index arm should read zero when the two mirrors are parallel. This adjustment is rarely perfect, and when made FUNDAMENTAL MEASUREMENTS 37 is subject to frequent changes. It is therefore better to correct for the error, thus introduced, and allow for it in all readings. It is also necessary to determine this correction every time the sextant is used as it varies from day to day. (c) Index correction. The correction for the index error is determined by focusing the telescope upon some distant point and bringing the two images into exact coincidence. In this position the mirrors are parallel and the vernier should read zero. If it does not, suppose the reading to be A ; then for an angle which gives the scale reading B, the true value is B A. In some cases the value of A is negative, or the arm must be moved back be- yond the zero in order to bring the images into coincidence. For this reason the graduation is continued slightly beyond the zero, as shown in Fig. 21. 36. Exercise n. Measurement of Angles by the Sextant. (a) Measure the angle subtended by two distant points. (b) Measure the elevation of a point using an artificial hori- zon. FORM OF RECORD Exercise n. To measure the angle betiveen two points by the sextant. Index reading Mean Index error. Angle Date. True angle MASS 37. The Balance. Strictly speaking all determinations of mass are indirect. The balance is an instrument for the com- parison of masses. In its simplest form it consists of a light beam turning readily about its middle point and carrying at its ends two scale-pans of equal weight. When disturbed the system oscillates about its position of equilibrium to which it finally re- turns. When masses are placed in the pans, it is evident that the original position of equilibrium will be resumed, only when the moments of the forces due to the action of gravity upon these PHYSICAL MEASUREMENTS masses are equal. If now, we assume the arms of the bal- ance to be equal, we may set the masses equal to each other when their moments have been shown to be equal ; i. c., when the balance resumes its origi- nal position of equilibrium. In balances of precision (Fig. 22), the beam and scale pans are hung from accurately ground knife-edges resting up- on agate plates. When not in use, the knife-edges are re- Fig. 22. lieved from the weight of the pans and beam by means of an arresting device. This must always be used when weights are to be changed, or articles to be weighed are to be placed upon the scale pans or removed from them. The oscillations of the balance are observed by means of a long slender pointer moving in front of a graduated scale. Care should be taken to raise the system from the knife-edges, only when the pointer is at the middle of the scale. A system of light levers is usually placed under the pans, to maintain the bal- ance in equilibrium for small differences of weight in the pans, and to prevent undue movements of the beam during rough weighing. The beam is generally divided from the middle out- ward, into n equal divisions, the last one coinciding with the knife- edge over the pan. By sliding upon this beam a small wire rider weighing n milligrams, weighings may be made directly to mil- ligrams. For subdivision of the milligram the method of oscil- lations is used. 38. Determination of the Resting Point. Owing to the loss of time incident upon waiting for the balance to come to rest, it is usual to determine the final position of the pointer from a series of its successive turning points. Since the vibrations of the system are more or less damped, it is necessary to take the initial and final readings of the pointer on the same side. If we call FUNDAMENTAL MEASUREMENTS 39 the central division of the scale zero, and readings to the right and left respectively plus and minus, then the resting point is found by averaging the means found for each side separately? For example, if the readings are Left Right + 104 9.2 + 10.0 -8.9 + 9-7 Mean 9.05 Mean +10.03 Resting point = - IO -3 ,_ _j_ Q 4g This means that the pointer would finally come to rest at a point about 0.5 of a division to the right of the zero. The resting point should be determined both before and after making a weigh- ing, and should remain constant if the balance be properly ad- justed. In some balances the scale divisions are numbered continuously from left to right. In the use of such instruments the readings are taken directly and the positive and negative signs are avoided, a method which seems more convenient than the former. If the scale divisions are not numbered, call the middle mark ten. 39. Sensibility of the Balance. If the rider be now placed upon the first division of the beam and the resting point deter- mined as before, then the difference between these two resting points is the number of scale divisions through which the beam has turned for a difference in weight of one milligram. This is by definition the sensibility of the balance, and is usually termed the sensibilty for zero load in the pans. Since the sensibility varies with the load, it is always necessary to determine it for the specific load upon the pans. 40. Exercise 12. To Make a Single Weighing. A single weighing will not afford an accurate determination of mass, since the equality of the lengths of the arms is tacitly assumed, and this method, though commonly used, should not be employed in meas- urements of precision. Moreover it does not require a smaller number of observations than the much more precise method of double weighing described below. However, where only relative .40 PHYSICAL MEASUREMENTS weights are required, as in determinations of specific gravity, or in chemical analysis, single weighing is permissible, and is there- fore described here. First determine the resting point for zero load. Next place the object to be weighed upon the left hand scale pan and an esti- mated equivalent weight upon the right hand pan. Release the arrest very slightly and note the indication of the pointer. If the weight be too small it should be sufficiently increased to turn the pointer to the opposite side on the next trial. In this way the true weight may be rapidly approximated to the nearest gram, then to the nearest centigram. After this the balance case should be closed, the rider applied and the -pan arrests turned down. Having found the weight to the nearest milligram, the balance is set swinging and the resting point is determined. The rider is then shifted one whole division on the beam, so as to bring the resting point on the other side of the zero position, and the rest- ing point again determined. If now we call the three resting points p 0f p, and P, where P corresponds to the weight greater than the true weight, then d W, the fraction of a milligram to be .added to smaller weight W , is (30) where s is the sensibility for the given load. The true weight is therefore Thus suppose / = + 0-49 p = + 0.77 P 0.15 then /> />o 0.28 J=P = W =-3 milligram. Care should be taken to avoid error in counting up the weights upon the scale pan. An excellent method is to write down the weights from the vacant spaces in the box and then check each weight as it is returned to the box. FUNDAMENTAL MEASUREMENTS 4! FORM OP RECORD. Exercise 12. To make a single weighing. Balance No Date. Object to be weighed Approximate weight P Po= ; S = P = Weight of W-\- 41. Reduction to Weight in Vacuum. The weight of a. body in vacuum is 1 W = w(i+~') (32) where w is the observed weight of the body, d its density, and a and D the densities of the air and of the weights respectively. In weighing a quantity of water with brass weights, this correction amounts to about 1.06 milligram, for every gram. It is there- fore imperative, in the case of calibration of glassware by means of water or in similar problems to make the reduction to vacuum. 42. Exercise 13. Double Weighing. 2 In order to eliminate any inequality between the two arms of the balance the object may be weighed first in" the left hand pan and then in the right. The true weight, W Q , is given by the equation Wo = V Wi Wz, or, since for every good balance W \ and W ' 2 are very nearly equal, W.= y 2 (W l +W^. (33) In practice the operations may be carried out as follows : Let 'P , p, and p' be the resting points for zero load, load on the left, and load on the right respectively; and let W be the approxi- mate weight, i. e., the weight sufficient to balance the load to within one milligram; then since 1 College Physics, Article 76. 2 For detailed treatment of the balance and its use, see Stewart and Gee, Practical Physics, Vol. I, pp. 63-94. 42 PHYSICAL MEASUREMENTS and W, = W- (34). From this it is seen that in double weighing the determination of p is not required. The sensibility may be determined with the load in either pan. It will be instructive to determine IV and W 2 separately by single weighing, since the square root of their ratio will give us the ratio of the lengths of the arms of the balance. It should be noted that the method of double weighing, as described, requires no larger number of observations than the single weighing and is far more accurate. FORM OP RECORD. Exercise ij. To weigh a piece of brass, by the method of double weighing. Balance No ........... Date .......... Object to be weighed .............. Approximate weight, W = Load left Load right i mg. added p= p'= log 0.5 = log (/ /') = W cologj log A W TIME. 43. Period of Vibration. In the case of a system vibrating freely about its position of equilibrium, the time elapsing between two successive passages of the system through the same point in its path in the same direction, is defined as its period of vibration. Most measurements of time in the physical laboratory consist in determining the period of vibration of some system, as a pendu- lum, magnetic needle, galvanometer needle, etc. The problem FUNDAMENTAL MEASUREMENTS 43 presents itself most frequently as the determination of the period of a pendulum, and although what follows is applied directly to this end, the method is equally applicable to the case of any freely vibrating system. 44. Method of Coincidences. The process usually adopted is a modification of Borda's method of coincidences. The essence of the method consists in determining the time- necessary for the pendulum to complete some large number of vibrations ; from this, if the number of vibrations be known, the period of a single vibration is at once deduced. Perhaps the simplest way is to note the time occupied in counting 100 or 1000 vibrations. This meth- od however, is both tedious and inaccurate, since owing to its monotony the observer is liable to make an error of one or even of ten vibrations in counting a large number. To this source of error is added the uncertainty of beginning and ending the count exactly upon the second. In order to avoid the first source of error the pendulum is made to keep count of its own vibrations when compared with a clock beating seconds. . The second error is minimized by attach- ing a pointer to the pendulum and observing its passage over a scale, or in work of greater accuracy, by viewing the pendulum through a telescope and noting the time of its passage over the cross-wires in the focal plane of the eye-piece. It is best to ob- serve this passage when the pendulum is in the middle of its swing, and moving with its maximum velocity. The clock is connected electrically with a sounder and the beats are thus made audible throughout the room. The pendulum to be timed is set swinging through a small arc not to exceed 3. The observer at the telescope notes the transits of the pendulum image from left to right over the cross-wires and awaits the coincidence of such a transit with the click of the clock. He then notes carefully the number of seconds elapsing before the next passage of the image over the cross-wires in the same direction and estimates, as well as possible, the fraction of a second in tenths, thus gaining a roughly approximate period. Suppose the period is found to be somewhere between 2.3 and 2.5 seconds. 44 PHYSICAL, MEASUREMENTS A coincidence is again observed and the seconds counted con- tinuously until a number of fairly good coincidences have been observed. From this observation a closer approximation to the period can be obtained. Thus if the period is near 2.5 seconds, good coincidences will occur in 5, 10, and 15 seconds, i. c., after 2, 4 and 6 complete vibrations of the pendulum. If on the other hand the period be nearly 2.3 seconds, then fairly good coinci- dences will be noted at 7 and 16 seconds, and a good one at 23 seconds, the intervals corresponding to 3, 7, and 10 vibrations. The imperfect coincidences at 7 and 16 seconds are due of course to the fact that the interval of 7 seconds is o.i second greater than the time needed for 3 swings, and that of 16 seconds is less by o.i second than that needed for 7 swings of the pendulum. In this way it is possible to determine the provisional period accurately to tenths of a second. The observer may distinguish between the grade of coincidences, by underscoring in his record, a mod- erately good coincidence with one line and an excellent one with two. Several trials should be made and, if necessary, 50 or even 75 seconds counted, in order to determine this period with accuracy. Suppose it has been found to be 2.3 seconds. The observer again awaits a good coincidence, noting the seconds by calling each one up to and including the second of coincidence, zero. He then walks to the clock, counting the seconds as he goes, and on the tenth second reads the time, noting the seconds first, then the minutes and then the hour. This recorded coincidence is ob- viously ten seconds later than the observed one, but by counting ten seconds each time before reading the clock, the interval be- tween the coincidences is preserved and no error is introduced. A second coincidence is observed as soon as possible and re- corded in the same way. Now the difference between the first, and second readings of the clock gives the number of seconds corresponding to some integral number of swings of the pendu- lum, and a glance at the approximate period is usually sufficient to show what this number is. In case of doubt divide the time by the provisional period and take the nearest integer as the number of vibrations. (Why?) The number of seconds divided bv the FUNDAMENTAL MEASUREMENTS 45 number of vibrations, gives now the period to a closer degree of approximation than before. A third coincidence is observed and recorded as before. The interval between this coincidence and the first corresponds to a still larger number of integral swings of the pendulum. This larger number is found by dividing the seconds by the period last deduced, and the new value of the period is computed as before. Thus by successive observations we find intervals corresponding to a larger and larger number of vibrations, using in each case the period last found. In this way the period of a pendulum may be readily and rap- idly determined to thousandths of a second. After a little practice the student is able to judge a coincidence accurately to o.i of a second. Thus in the given example after 20 minutes of observa- tion the pendulum would have made something over 500 vibra- tions, and the time needed for this number of vibrations would be determined to 0.2 of a second, or the period would be accu- rate to thousandths of a second. Example. The following example will make the method and computation clear : SIMPLE PENDULUM. February 8, 1899. Length, 130 cms. Good coincidences,^, 14, 16, 23. Approximate period 2.3 seconds. Coincidences Interval No. of Period 3 h. 19 min. 52 sec. in seconds Vibrations in seconds 20 15 23 10 2.3 20 38 46 2O 2.30O 21 26 94 41 2.293 (21 50) 118 52 2.269 ??. 22 30 158 69 2.2S9 8 23 57 245 107 2.28o 7 24 29 277 121 2.289" 30 19 627 274 2.288 s 30 51 659 288 2.288 1 33 22 810 354 2.288 1 It is to be observed that the period T gradually approaches a limiting value which becomes constant to thousandths of a second as soon as the number of observed vibrations, n, reaches a definite value. If the maximum error, e, made in taking any coincidence be o.i of a second, then the maximum error possible in any number of observed seconds is 0.2 of a second. Hence to have the period T constant to thousandths of a second, we must make 46 PHYSICAL MEASUREMENTS o.2/n less than o.ooi, or n must be greater than 200. Obviously n must be larger, the larger e becomes. How large must n be taken if e = zt 0.2 seconds ? In case any observation gives a result sharply at variance with the others, the difficulty lies either in the arithmetical work, or in a false reading of the clock. The latter error renders the observation useless; it should be bracketed as indicated above, and the next taken with greater care. The advantage of the method is that no single error in reading the clock can perma- nently vitiate the result. Instead of determining the number of vibrations by dividing the seconds by the last value of T deduced, it is much simpler to use the preceding intervals and vibrations as measures of the new. Thus the second interval 46, is manifestly double the- first; hence the number of vibrations must be twice as many, or 20. In the third, the interval, 94 seconds, is twice the second + 2 sec- onds; the excess, 2 seconds, corresponds to an additional vibra- tion; hence n = 2 X 20 + I =41. In the seventh determination the interval, 277 seconds, may be evaluated for n in a number of ways ; for example, from the third we may have, after multiply- ing by three : sees. vibs. 282 = 123 277 277 = 121 or from the third and fifth thus : 94= 4i 158= 69 252= no 277 +25=+! i 277 = 121 For the measurement of small intervals of time the tuning fork furnishes an accurate and convenient method. For the practical application of this method see subsequent articles. FUNDAMENTAL MEASUREMENTS 45. Exercise 14. Period of Torsional Pendulum. 47 FORM OF RECORD. Exercise 14. To determine the period of a torsional pendulum, to o.ooi of a second. Record as indicated above, p. 45. 46. Exercise 15. The Barometer. The barometer, (Fig. 23), consists of a closed tube of glass of uniform bore about 80 cms long, filled with mercury and in- verted in a dish containing mercury. The free sur- face of the mercury in the vessel is in communica- tion with the outer air. In the cistern barometer, the reservoir (Fig. 24), has a bottom of leather which is adjustable by means of a thumb- screw. A small ivory point extending downward from the upper surface of the reservoir, forms the zero-point from which the measurements indicated on the scale are reckoned. Before reading the barometer the mercury in the cistern must be so adjusted by means of the screw that the surface just touches the ivory point. The upper part of the tube is then gently tapped to free the mercury surface from the sides of the tube, and the vernier ad- Fig. 24. justed by means of the thumbscrew at the side, until, on looking through the slit in the barom- eter case, the upper part of the meniscus is seen to be just tangent to the line joining the sharp edges at the front and back of the vernier. In the instrument made by Haak, of Jena, we have an auto- matic adjustment of the mercury in the cistern. The zero-point is the tip of a vertical tube connecting with a lower, auxiliary reservoir. Air is forced into the lower reservoir by means of a bulb, thus driving mercury into the reservoir proper and covering the zero point. On releasing the bulb the mercury flows out until the tip of the zero point tube is again exposed ; the barometer is Fig. 23. 48 PHYSICAL MEASUREMENTS then in adjustment for reading. The scale is etched directly upon the tube of the barometer, and fractions of a millimeter may be read with ease by means of the adjustable ring, at the top, which carries a fine line for subdividing the millimeter divisions. Readings on the barometer must be corrected for temperature effects, and are reduced to oC by the use of the following formula : x H, = Ht [i (a -&)*], (35) where H is the barometric height at oC; H t is the barometric height at tC; a is the coefficient of cubical expansion for mer- cury, (01 = 0.000181 per degree), b is the coefficient of linear ex- pansion for the material of the scale ; ( for glass, b = 0.0000085 per degree, for brass, b 0.000019 per degree). FORM OF RECORD. Exercise 15. Adjust and read each barometer three times; cor- rect for temperature and compare readings. Barometer No. i Barometer No. 2 Date. Least count Least count Readings Readings Mean Mean Reduced to o C Reduced to o C Express the barometric pressure in dynes per square centimeter. 1 For reducing the barometric reading to standard conditions, viz. o C, sea level, in latitude 45, we have the complete formula where g is the acceleration due to gravity at the place of observation and #45 980.63 cm per sec. per sec. CHAPTER II. ELASTICITY. 47. Definitions. Elasticity is that property of matter by virtue of which a body resists the action of a force tending to change its shape or bulk, and resumes its original shape or bulk after the force is removed. If a body possess elasticity of shape it is called a solid ; if it possess no elasticity of shape it is called a fluid. Any change either of size or shape, produced by the action of a force upon an elastic body, is called a strain, and is measured by the relative change produced. The reaction against the de- formation is called a stress and results in the appearance of a force resisting further deformation. After equilibrium ensues the applied and the resisting forces are equal. Hence stress may be evaluated in terms of the applied force per unit area of the cross-section upon which the force is exerted. In the metric system the unit of stress is the dyne per square centimeter. Fluids possess perfect elasticity of bulk, i. e., they return exactly to their former bulk on removal of the compressing force. Solids do not all recover their initial shape with equal promptness. In some cases the return is much retarded, especially after repeat- ed or long continued distortion. This retardation is commonly termed elastic after effect, and is quite noticeable in metals. For every solid there is a limiting distortion beyond which the body, when freed from the distorting force, no longer completely re- gains its former shape. In engineering practice the elastic limit is usually measured in terms of the stress producing this limiting distortion. 48. Hooke's Law. When an elastic body is distorted within its limit of elasticity, the opposing force called out by the distor- tion, tending to restore the body to its original condition, is pro- portional to the distortion. This is known as Hooke's Law, 50 PHYSICAL MEASUREMENTS and as originally stated, "ut tensio sic vis," expresses the pro- portionality between the distortion and the restoring force. The applications of this law are very numerous including every form of elastic reaction against strains produced by external agencies. 49. Coefficients of Elasticity. In general twenty-one co- efficients would be needed to express completely the elastic nature of any solid. If however, the solid be isotropic, these twenty-one coefficients reduce to two : the coefficient of volume elasticity, e, and the coefficient of rigidity, n. The general expression for these coefficients is the quotient arising from dividing the stress by the strain. 50. Coefficient of Volume Elasticity. In the case of the co- efficient of volume elasticity e, we have the stress measured by the applied pressure p, divided by the compression produced, where compression denotes the change in volume v, divided by the original volume V ', or (37) In the case of a gas, the volume is at all times a function of the pressure to which it is subjected. Hence for gases it should be noted, that the coefficient of elasticity is to be defined in terms of the change in pressure and the corresponding change in volume. Since these changes are conceived as being very small, if we assume a volume of gas V , to be subjected to a change in pressure dp, producing a corresponding change in volume dV, then for a gas, dV It should be observed that the expression for the strain, dV /V , denotes a dilatation if positive and a compression if negative. ELASTICITY 51 The coefficient e f however, has reference simply to the absolute value of the ratio Vdp/dV, and is therefore independent of the sign. The coefficient of elasticity of volume is the only one pos sessed by fluids, and is of special interest in all cases involving the propagation of disturbances through fluid media. 51. Young's Modulus. In solids in the form of wires or rods, subjected to longitudinal forces tending to produce either elonga- tions or compressions, we are interested in the relative elongation or compression /, produced in length L, and cross section a, by a force of P dynes, when the body is free to contract or expand laterally. In general, longitudinal expansion is accompanied by lateral contraction and longitudinal compression by lateral dis- tention. The measure or modulus of the elastic behavior of a solid under such conditions is known as Young's Modulus, and may be defined as the ratio between longitudinal stress F/a f and longitudinal strain l/L ; that is, u FL M =~aT (39) or Young's Modulus is numerically equal to that force which, when applied to a wire of unit cross section, would be suffi- cient to stretch it to double its length, provided of course, that the elongation remained proportional to the force at all times. 52. Simple Rigidity. Besides the elasticity of volume, solids have, as we have seen, elasticity of shape as well. If a solid be so distorted that its shape alone is changed, it is said to have undergone a shear. Thus if we conceive all the particles in one plane in a body to be fixed, and all the remaining particles to move in planes parallel to this plane, and by amounts propor- tional to their distances from this plane, such a distortion consti- tutes a shear. The stress caused by such a shear is called a shearing stress, and the coefficient of rigidity or the simple rigidity is the quotient obtained by dividing the shearing stress by the shearing strain. In order to learn how these quantities may be experimentally PHYSICAL MEASUREMENTS determined, let us consider a circular cylinder (Fig. 25), of radius r, held vertically by a rigid clamp at the upper end and subjected to a torsional twist at the lower end. The effect of such a twist is to produce a shearing strain throughout the cylinder. Imagine the cylinder to be made up of a large number of tubes, one inside the other, and cut at right angles to the axis into a large number of circular sections. Each circular section would be composed of a large number of concentric rings. Now the shearing strain increases regularly from above downward from section to section, and when the low- er end of the cylinder has been twisted through an angle 0, the dis- tortion or shear for the outer ring of the lower section will be the arc r0, and the shearing strain is meas- ured by the ratio of the shear to the distance of the sheared surface from the fixed end, or by rO/L ; or in gen- eral, the shearing strain at any point in a cylinder is the circular displacement at that point divided by the distance from that point to the fixed end of the cylinder. Again since the cylinder under stress is in equilibrium, the mo- ment of the couple producing the shear must be balanced by the moment of the couple called out by the shear; or by Hooke's law, it must be proportional to the shear itself, hence also pro- portional to rO/L. Now by definition, the simple rigidity n, is the proportionality factor between shearing stress and shearing strain; hence we have the shearing stress = nrO/L. Let us now consider the entire lower circular section of the cylinder, and in that section, a ring of radius x and of width dx. The shear will be xQ/L, and the shearing stress will be nxB/L. The area of the elementary ring is 2 ?r x dx, hence the force, due to the shearing stress and equal to stress times area, is 2 TT n & x 3 dx a r = ; This force acting with a lever arm of x, gives an elementary mo- ELASTICITY 53 ment of shearing- force for the elementary ring of width dx, equal to 2,irnQx z dx (40) For the entire section, the moment of the shearing force will be the sum of the elementary moments for all elementary rings, whose radii vary from zero, at the center, to r at the surface of the cylinder, and so the moment of the torsional couple called out by the shear, is found by integrating the expression in equa- tion (40), or = r Jo 2 TT n 9 r* 4L or JT = ^IH (42) whence iL^- "= -- (43) It must be observed that is here expressed in radians. If is measured in degrees, what correcting factor must be intro- duced ? In case of equilibrium ^~is equal to the moment of the tor- sional couple producing an angular twist 6, in a cylinder of radius r, and length L. By making the angular twist equal to unity we have the moment per unit angle and finally, by reducing the length L and the angular twist 0, each to unity we have *= (45) 54 PHYSICAL MEASUREMENTS The quantity t is called the modulus of torsion ; it is numer- ically equal to the moment of the couple required to produce unit angular twist in a wire of unit length. Summary: We have now defined and derived expressions for the following quantities: Coefficient of volume elasticity e = V -777 (38) Young's modulus.. M = - r (39) a I Torsional moment ............ J^~ - (42) Simple rigidity ............... n = (43) Torsional moment per unit twist <-/ - - (44) Modulus of torsion t = (45) In the experiments which follow several of the above quantities will be measured in one or more different ways. 53. Exercise 16. To Verify Boyle's Law. The apparatus consists of a cylindrical reservoir, (Fig. 26), formed of a glass tube some 25 cm long and 3 cm in diameter, into which is sealed a uniform tube B, some 30 cm long and 12 mm in diam- eter, closed at the top by a square-cut plug carefully cemented in, and at its lower end extending to the bottom of the larger tube. At the lower end of the reservoir is sealed on a second tube A, 5 mm in diameter and about 200 cm long. This tube is bent back upon itself about 10 cm below the reservoir, so as to be vertical and parallel to the tube B. It is terminated at its upper end by a small thistle bulb, for convenience in filling the reservoir with mercury. The instrument is mounted upon a suitable sup- port carrying a scale graduated to millimeters at the side of the tube A, throughout its entire length. At the bottom the scale ELASTICITY 55 fin A-- stands between the two tubes so that readings upon the height of the mercury in the tubes A and B, may be made from opposite edges of the same scale. A small side tube is sealed to the reservoir near the top by means of which air may be forced into the reservoir from a small force pump. Be- fore beginning the experiment the instru- ment is so adjusted that the shorter tube B, is about half filled with mercury when the air in the reservoir is at atmospheric pressure. Air is next driven in through the side at C, by means of the pump, until the mercury almost fills the long tube A. The air in B is now under a pressure measured by the barometer col- umn plus the difference in height of the mercury in the tubes A and B, and is cor- respondingly compressed. Varying pressures and corresponding volumes are successively secured by al- lowing small quantities of air to escape through the tube C. Readings are made upon the height of the mercury in A and B, and upon the lower end of the plug in B. Readings should be continued until the mercury in the tube A falls to the level of the mercury in the reservoir. The air must be allowed to come to the tem- perature of the room after each setting before the reading is taken. (Why?) If it be desired to take readings at pres- sures below that of the atmosphere, the pump is reversed and the air partially exhausted. According to Boyle's law, the product of the pressure and the volume of gas is a constant, for a constant temperature 1 , or P.V = C. (46) 1 College Physics, Article 77. PHYSICAL, MEASUREMENTS By definition, the coefficient of volume elasticity for a gas is dP dV Also from (46) by differentiation, we get or dP dV (47) (48) Thus we see that for a perfect gas undergoing isothermal changes, the coefficient of volume elasticity e, is at all times equal to the pressure P. For the purpose of our experiment let a and b denote the observed heights of the mercury in the tubes A and B. Let P represent the atmospheric pressure at the time of the ex- periment, p = a b, the applied pressure, and let V represent a quantity proportional to the result- ing volume of the air enclosed in the tube B. Then equation (46) becomes 80_ 60. 4Q. 3SL ^ 20 40 GO/ f . . 08 or = C/V. (49) (50) If now we plot the observed values of p on the Y axis and the reciprocals of the corresponding volumes on the X axis, (Fig. 27), our curve is represented by the 'equation y = Cx P, (51) Fig. 27. the equation of a straight line cut- ting the Y axis at a point P below the origin. This point may be considered the true origin, measured from which the values of y denote the successive values of e f corresponding to the related values of V. The tangent of the angle included between the curve and the X axis is equal to C. If all quantities be expressed in the ELASTICITY 57' proper units, then C becomes the gas constant. Under what con- ditions does e approach zero? FORM OF RECORD. Exercise 16. To verify Boyle's law. Reading on plug Barometer Date. Tube A Tube B colog V Plot values of p and \/V. . Determine the value of P from the curve and compare the result with the barometer reading at the time of the experiment. 54. Exercise 17. The Jolly Balance. In the Jolly balance the elastic body is a spiral spring mounted upon a suitable support and carrying at its lower end a pan to receive the substance under experiment. To this pan is attached a second pan to carry the substance when immersed in the water. The support con- sists of two telescoping nickel-plated tubes mounted upon an adjustable tripod base. The inner tube to which the spring is at- tached, is actuated by means of a rack and pinion A (Fig. 28), and is graduated in. mil- limeters at its upper end through about 50 cms of its length. A vernier $, on the upper end of the outer tube,reading to tenths of a millimeter gives the elongation of the spring under any load when the indicator B attached to the lower end reads zero. This indicator consists of a small rod of alumi- num furnished with two short cross arms, one at either end of a short vertical glass tube enclosing the rod and supported by an adjustable clamp. The glass tube is whitened at the back and at its middle has on the inside a fine black line passing en- tirely round it. The aluminum rod carries p^ 2 S. 58 PHYSICAL MEASUREMENTS at its center a small cylinder bearing three equidistant black lines, the middle one of which is made to coincide with the line on the glass tube when the spring is brought to the zero position. An adjustable support carries a small vessel containing distilled water. After the support is once adjusted it should remain in the same position throughout the experiment, since in this way the pan will always be immersed to the same depth in the water. In use the support is adjusted and the reading r , taken with both pans empty. The substance under experiment is then placed in the upper pan, the support adjusted and a second reading r lf is made. The substance is then transferred to the lower pan, the ad- justment made and the reading r 2 , taken. Then for the density of the substance we have where d is the density of the water used. 1 Derive and explain this formula. For substances lighter than water a small piece of metal heavy enough to sink the substance, is placed in the lower pan and kept there during all three readings. The density is then computed as above. For liquids the lower pan is removed and a suitable sinker, usually of glass is attached to the hook by a fine platinum wire. Readings are made with the sinker in the air, r , with the sinker immersed to a definite depth in the water r , and with the sinker immersed to the same depth in the liquid, r 2 . Derive and explain the formula in this case. It is necessary to exercise care in the use of the apparatus in order to secure trustworthy results. Air bubbles must be re- moved from the substance and from the lower pan before readings are taken. Determine the density of three solids : brass, zinc and paraffin, and of a solution of copper sulphate. Compare results with those obtained in preceding exercises. 1 College Physics, Article 69. ELASTICITY 59 FORM OF RECORD. Exercise ij. Density by means of Jolly balance. Density of Date, ........ ........ ........ Density ...... Mean ........ ........ ........ 55. Exercise 18. Young's Modulus by Stretching. An iron bracket firmly attached to the wall near the ceiling supports a long brass wire which carries on its lower end, by means of a clamp, a cage for the reception of weights. Near the lower end of the wire is attached a needle upon the point of which is focused a micrometer cathetometer reading to o.ooi mm. In order to elimi- nate the yielding of the supporting bracket, a small rod hung from it is loosely attached to the wire, and bears on its lower end a small metal square or flag, near the needle point, so that both flag and point appear in the field at the same time and readings may be made upon them successively. An adjustable table is placed under the cage before weights are added and then gradually lowered, to avoid subjecting the wire to sudden jerks. A weight of two kilograms is left permanently upon the cage in order to free the wire from kinks and to insure uniformity of stretching. Readings are taken upon the flag and the point, F , P Q , with only the zero load, two kilograms, on the cage, the mean of three readings being used in each case. Let The table is then raised beneath the cage and two kilograms added. The table is then lowered and readings made as before. Fn / i JT i k. In this way readings are successively taken for weights of two, four and six kilograms in addition to the zero load. After this the weights are removed, two kilos at a time, readings being made as before until the zero load remains. From the values thus found 6o PHYSICAL MEASUREMENTS the stretch for two kilograms is computed for each reading. from the reading with 6 kilograms added, we have Thus The mean of the values of / thus obtained, is taken as the stretch produced by two kilos. Having determined L and a, we have FL 2000 . 980 M = r~ = - - dynes per cm. Determine the effect upon the final result, (a) of an error ot I mm, made in the measurement of L ; (&) of an error of o.ooi mm, made in the determination of each, / and r. From a preliminary determination of these three quantities, compute the greatest error permissible in each one of them, if the result is to be accurate to within one per cent. L r Weights FORM OF RECORD. 18. Youngs modulus by stretching. Date. Flag Point Computation : log 2000 = log 980 = log L = CO log 7rr'= co log / = log M = Mean. . 56. Exercise 19. Verification of the Laws of Bending. The bending of a beam supporting a weight is a function of the weight w, supported, of the three dimensions of the beam /, b, d, and proportional to a constant C, which depends upon the material of the beam and the manner in which it is supported. It is pro- posed in this exercise to determine experimentally these relations. ELASTICITY 6 1 Let us assume for the purposes of the investigation, the general expression for the bending B, B =Cw a I 1 d y combining the observations as in (a). Take as the final value for ft the mean of the ten values found as above. Record as in (a), substituting / for w. (c) Vary b and determine B", for a constant load of 150 grams. Use bars i, 2, 3, 4, with constant depth. Measure bi, b<2, b 3J b, by means of the micrometer gauge. Determine for bi, b. 2 , b 3 , b, the corresponding values B'\, B" 2 , B" z , B'\. Apply formula (56), computing as in (a) and (b). Record as before. (d) Vary d and determine B for a constant load, w=i$o grams. Use bars with constant breadth. For depths d l} d 2 , d s , d, determine the bending /", BJ" B z "', B^ f . Compute from formula (57). Record as before. O) Insert final values of a, fi, y, e, in the general formula. Formulate the laws of bending in words. Curves: The observations may be graphically represented by curves. In cases of direct proportionality between the bending and the quantity which was varied in the experiment, plot this 66 PHYSICAL, MEASUREMENTS quantity as abscissa and the corresponding bending as ordinate. The curve should be a straight line. If the exponent of the vari- able quantity is not unity a straight line may be obtained by plotting the logarithms. Plot the curves for each of the four series of observations. 57. Exercise 20. Young's Modulus by Flexure. As we have learned in the last exercise, the bending of a bar of length I, breadth b, and depth d, under a load w, is expressed by the equation *-*: (58) It has already been observed that the constant C, depends upon the mode of support and the material of which the beam is com- posed. It is shown in the theory of elasticity that when the beam is supported by its ends the value of the constant relating to its mode of support, is 1/4, and when supported from one end its value is 4. There remains therefore, the undetermined part of our constant C, depending upon the material of the beam. It may be shown from mathematical analysis that this part of the con- stant is \/M, where M is Young's modulus for the material in question. If the bar be supported at the ends as usual, then we ma write 1 B== or If now we insert in this formula the dimensions of bar No. i, we shall find for any bending B and the related load w, a value for M very nearly that obtained in Exercise 18 as Young's modulus for the material of the bars. 1 It may 'be shown directly that the expression for M as given above, is true for all bars supported at the ends and loaded at the middle. Such proof would, however, exceed the limits set for this text. See Stewart and Gee, Vol. I, pp. 162-195. ELASTICITY 67 Moreover it appears that Young's modulus is concerned here if we consider attentively what takes place in the bending of a bar. We see that upon the under side the bar is stretched, while upon the upper side a compression must ensue. Now the resistance to this stretching stress on the one side and to the compressing stress on the other, form. the two parts of a couple tending to right the bar under its load, and the stress divided by the strain gives again Young's modulus for the material in question. FORM OF RECORD. Exercise 20. . . To determine Young's modulus of brass and steel by the method of flexure. I. For brass. Date ............ From Exercise 19 (a) w = ...... log w = ...... / = ...... log P = ...... b = ...... colog 4 = ...... d = ...... colog b = ...... B = ...... colog d s = ...... colog B = ...... M= ...... - logM = ...... II. For steel. (a) First part as on page 64. (b) Computation as above. 58. Exercise 21. Simple Rigidity. As shown in equation (43) the expression for the simple rigidity of a cylinder under torsional stress is where ^l s the moment of the torsional couple needed to produce an angular twist of 6 radians in a circular cylinder of length L and radius r. In order to determine n we must measure the four quantities ^\ B, L, and r. This is most readily done by means of the following apparatus. A metal rod (Fig. 30) some 150 to 200 cm in length, is sup- ported horizontally and held firmly at one end by a clamp. The 68 PHYSICAL MEASUREMENTS other end is held clamped in the axis of a light aluminum wheel, some 20 cm in diameter, which is mounted on ball bearings, and graduated on its face through 180. The wheel is furnished with a double vernier for reading its position at any time, and carries on its rim a flat steel tape to which is attached the holder for the weights producing the torsion. The manipulation is as follows : A zero load of some 50 grams is first placed in the pan and the zero readings taken. A load of loo grams is then added and the reading again taken. The differ- Fig. 30. ence between the two readings is the angular twist for 100 grams. The load in the pan is again increased by 100 grams and the read- ing made. Half the difference between this reading and the first is likewise the twist for 100 grams. The load in the pan is in- creased in this way by successive steps of 100 grams until 500 grams have been added and then diminished by similar steps until the zero load remains. The twist for 100 grams is computed from each reading by combining it with the zero reading as shown above. The mean of the values thus found is the angular twist 6, expressed in degrees, for a load of 100 grams. The length of the rod L, and the radius of the wheel /, are to be determined by measurement. Determine the radius of the rod from five measure- ments of the diameter by means of the micrometer gauge. ELASTICITY FORM 01? RECORD. Exercise 21. To determine the simple rigidity of two metals. Date Computation : Load Readings Diff. for 360 . L . m . g . I CQ erns 100 gms. log 1,60 . . i TO " r , r log L . . 2^0 " M- 100 log tn -2CO " a 080 log a - Mean T= log / . colog 7' "olog it* colog r* = log n n . . 59. Exercise 22. Simple Rigidity of a Brass Wire from Torsional Vibrations. We have learned from Hooke's law that, within the limits of elasticity, the restoring force called out by any distortion is simply proportional to that distortion. An im- portant consequence of this law is, that if a heavy body be sus- pended by a wire and the wire be twisted through a moderate angle and then released, the restoring force is continually propor- tional to the distortion. The motion induced is simple harmonic and consequently the vibrations of the body are isochronous. Now if / be the moment of inertia of the body, and-^the moment of the torsional couple divided by the angular twist, then T, the time of a complete vibration of the system, is given by the equation from which (60) (61) We have also seen from (44) that the coefficient of simple rigidity is defined by the equation TJ* ft Y c ^ r=: "HT whence, equating values for ^"ancl solving for n, we have =*#. ( fe > an expression involving only quantities amenable to measurement. 70 PHYSICAL, MEASUREMENTS The apparatus consists of a heavy metal cylinder suspended by a brass wire and furnished with a light pointer moving over a horizontal circular scale. By means of an adjustable clamp the length of the torsional pendulum may be varied within wide limits. The times of vibration 7\, T 2 , T s , for lengths L lf L 2 , L 3 , are determined to thousandths of a second by the method given in Article 44. The moment of inertia I, of the cylinder is com- puted from its mass M, and radius R. The radius of the wire is determined by means of the micrometer gauge. The insertion of any pair of related values of T and L in formula (62) gives a value for n. Compute the three values and return the mean as the value found for n. FORM OF RECORD. Exercise 22. Simple rigidity of brass wire from torsional vibra- tions. Date .................. First part as given under Exercise 14 Computation : Second part thus: 8*1 L log M log / R= log L I =. colog r* r= colog T 2 log n M CHAPTER III. PENDULUM EXPERIMENTS AND MOMENT OF INERTIA. 60. The Simple Pendulum. For practical purposes the pendulum may consist of a lead ball about 3 cm in diameter sus- pended by a fine steel wire some three meters in length. The wire is supported from the upper end of a substantial wooden bar two and a half meters in length about 10 cm wide and 5 cm thick, fastened firmly to the wall in a vertical position. The bar has let into its front face a brass strip ruled to millimeters and read- ing continuously throughout its length of 250 centimeters. Over this bar slides a wooden clamp which may be set at any position, and carries on its front a short, horizontal brass rod 10 cm long and i cm in diameter, furnished at its outer end with a diametral slit set vertically, and adjustable by means of a set screw. In this way the fine steel wire of the pendulum may be clamped in the slit of the rod at any point and the length of the pendulum varied between 50 and 300 centimeters. Such a pendulum approximates very well an ideal simple pen- dulum. If h be the distance from the lower side of the clamp slit, to the center of the ball and r the radius of the ball then the length I, of the equivalent ideal simple pendulum is given by (63) Compute the error produced by omitting the last term of this formula in the above pendulum, when h is 100 cm. 61. Exercise 23. Law of the Simple Pendulum. The ob- ject of this exercise is to investigate the relation between the period of a simple pendulum and its length. Since the most casual ob- servation shows that the period of a pendulum is some function PHYSICAL MEASUREMENTS of its length, we may assume as a general expression for the exist- ing relation (64) "where C and m are constants to be determined by experiment. Passing to logarithms and solving for m, as in Exercise 19, we find for in the value log T 3 log Ti log l a log/! (65) From a series of observations on five pendulums of different lengths we get, by combining as in Exercise 19, ten values of m. The mean value of m so determined, when inserted in the equa- tions connecting related times and lengths, gives five independent equations for C, of the form log C = log T m log /. (66) The mean value of C thus determined, and the mean value of m when inserted in equation (64), give the relation sought. The exercise is to be completed as follows : (cc) Determine to thousandths of a second the period of vibra- tion of a pendulum for five lengths 120, 140, 160, 180 and 200 cms. (&) Compute from these observations the values of m and C as described above, observing the arrangement adopted on page (65). (c) Plot curve, using values of log T as ordinates and those of log / as abscissae. FORM OF RECORD. Exercise 23. To determine the () Use form of record given under Art. 44. (&) / log / T log T m of the simple pendulum. Date C PENDULUM EXPERIMENTS 62. Exercise 24. Computation of g. From the well known, formula for the time of vibration of a simple pendulum 2 \^ we see that or C = 2 7T 47T (67) (68) (69) From the mean value of C found in Exercise 23, compute the value of g. The value of g found at a place, H meters above sea level, may be reduced to sea level by means of the equation g g -j- 0.0003086 H (70) Compare this reduced value g w ith the normal value y at sea level and latitude , given by Helmert's formula 70 = 978.03 (i -j- 0.005302 sin 0.000007 sin 2 ) FORM OF RECORD. Exercise 24. To compute value of g. C log C = Date. log 4 = log 7T 2 colog C' = log flr 9 63. Exercise 25. Moment of 'Inertia of a Connecting Rod.. The moment of inertia of a body about its center of gravity may easily be determined, provided the body can be vibrated as a physical pendulum from a point whose distance from the center of gravity is accurately known. 74 PHYSICAL MEASUREMENTS A connecting rod (Fig. 31) is well adapted for this experi- ment. The mass M of the rod is measured once for all and given to the student by the instructor. The center of gravity is located upon a line C which is found by balancing the rod upon a knife edge. This line should be carefully marked on the bar. Fig. 31. If the rod is now hung from a knife edge, placed inside the circular hole at either end, it forms a physical pendulum whose time of vibration is given by the equation T=2 ^ww (72) where / is the moment of inertia about the point .of suspension and h the distance from the point of suspension to the center of gravity of the rod. 1 Let the rod be swung first about the point A whose distance h from the line C must be accurately measured. Determine the period 7\ by the method of article 44. Then r ' where / is the moment of inertia about the point A. From this equation I is calculated. Next, suspend the rod from point B whose distance from C is h. 2 . If the time of vibration in this case be T 2 we have V where I 2 is the moment of inertia about point B and is calculated from the last equation. 1 College Physics, Article 55. MOMENT OF INERTIA 75 The moment of inertia about the center of gravity is 2 h = h Mh\ (73)- Similarly Calculate 7 from both equations and take the mean as the final result. FORM OF RECORD. Exercise 25. To find the moment of inertia of a connecting rod about its center of gra-vity. M= ...... hi = ...... 9 = ...... hi = ...... Determination of Ti Determination of T* (Record as in article 44) /i ...... /a = ...... Io= ...... Io= ...... Mean h = ...... 64. Exercise 26. Moment of Inertia from Torsional Vi- brations. It is frequently of importance to determine the moment of inertia of a body whose form is irregular or such as to render its determination by computation difficult or impossible. In such case the moment of inertia may be determined by the method of torsional vibrations. Suspend the body by a stout wire so as to swing freely about a vertical axis, with its surface horizontal. If twisted through an angle it will tend to return to its position of equilibrium, and in doing so will execute simple harmonic vibrations of period T. If I be the moment of inertia of the body and J^ the moment of the restoring couple per unit angular twist, then from equation (60) we have If now there be added to the body a ring whose moment of inertia 2 College Physics, Article 51 ; Eq. 134. 76 PHYSICAL MEASUREMENTS I T , may be readily calculated from its dimensions, (Table VIII), then the period of the system becomes (74) Eliminating ^"from these two equations we have ( 75 ) Again if to the original system there be added a pair of cylin- ders, each of mass M, and radius r, symmetrically placed, each at distance a, from the axis of rotation, the period of the combined system becomes ( 76) where / c , the moment of inertia of a single cylinder about the axis of rotation, is given by the equation 1 Mr 3 /e = - +Ma 2 (77) Combining equations (60) and (76) we find / = 2/o. r ^ r ( 7 8) Finally, if to the original system we add both ring and cylinders and determine the period of vibration T 3f of the system, we secure a third value for / in the same way as above, from the equation . nr (79) The body to be investigated may have the form of a rectangular 1 College Physics, Article 51. MOMENT OF INERTIA 77 bar as the magnet of a magnetometer, or the form of a circular cylinder or wheel, or it may be of any irregular outline, so long as it be furnished with a stout hook or other device for suspending it from an axis that shall pass vertically through its center of grav- ity. Its surface when suspended should be horizontal, and should have marked upon it lines to insure the exact placing of the ring and cylinders. The position of the centers of the cylinders should be clearly indicated in order to facilitate the determination of a. A line connecting these centers must pass through the axis of rotation of the body. The ring should be of rectangular cross section, with sides and edges well polished, and a diameter clearly marked upon each. The cylinders should be accurately turned, as nearly alike as pos- sible, and have sufficient mass to insure a distinct difference in the period of vibration when they are added. The ring should have a mass of at least one kilogram. The exercise comprises the following measurements : (a) Measure the external and internal diameters of the large ring by means of the vernier caliper and compute the external and internal radii, 1\ and r 2 . (b) Measure on the dividing engine the distance between the centers of the two cylinders and compute the length a, from the axis of rotation of the system to the center of either cylinder when placed upon the body. (c) Measure the diameter of the cylinders and compute the mean radius r. (d) Determine the mass of large ring M T , and of either of the two cylinders Af c . (e) Determine to thousandths of a second the time of vibra- tion of the body alone, T ; of the body and ring, 7\ ; of the body and cylinders, T 2 ; of the body, ring and cylinders, T 3 . 78 PHYSICAL MEASUREMENTS FORM OF RECORD. Exercise 26. To determine the moment of inertia of a body by torsional vibrations. (a) Enter results as in Art. 44. (k) Value Mr ?r Me a Ic Value r T\ T, 2Y+T Computation : colog ( colog ( 2 logMrL COlog 2 i= TiT log /r = logT' = 1 = 65. Exercise 27. The Ballistic Pendulum. When two bod- ies collide in such a way that their velocities at the instant of col- lision lie along the common normal at the point of contact we know from Newton's Third Law of Motion that the sum of the momenta of the two bodies is unchanged by the collision. If m and m 2 be the masses of the two bodies, z\ and v 2 , v\ and v\ their respective velocities before and after collision, then mi Vi + W 2 Va = (80) Newton also showed that under the foregoing conditions the relative velocities, before and after impact, bore a constant ratio to each other, or if we assume v to be the velocity before impact of the more rapidly moving body, we have 1/1 v\ e (vi %) (81 ) where e is a quantity depending upon the material of which the bodies are composed. Investigations by Hodgkinson, 1 and by Vincent 2 show that e is influenced in some degree by the initial velocity. The quantity e is termed the coefficient of restitution, and is always less than unity. In the laboratory the foregoing principles are easily verified by the study of tne collisions between suspended spheres, where the direction of motion may be controlled and the respective velocities 1 Hodgkinson, Report of British Association, 1834. 2 Vincent, Proc. Cambridge Phil. Society, Vol. X, p. 332. MOMENT OF INERTIA 79 - may be readily measured. For convenience of adjustment the spheres are taken of the same size, and the masses varied by mak- ing them of different material. The apparatus (Fig. 32), consists of two spheres of wood, ivory, iron or lead, about 9 cms in diameter, hung on bifilar suspensions, about 250 cms in length, and so adjusted as to swing accurately parallel to the graduated arc ss f . The spheres carry at the lower ends of their vertical diameters small pointers by means of which their posi- tions may be read off on the graduated arc. When at rest the spheres must touch lightly, and their centers lie on an arc parallel to the scale. Read the position of the pointers, and displace one of the spheres, say the lighter one, through an arc of not more than 20. Read off the displacement BC = c accurately from the graduated scale, and allow the sphere to swing back against the one at rest. Determine as accurately as possible the extreme positions of the two spheres after impact, taking the mean of at least five simultaneous readings for each. Two observers are required. Repeat the experiment by allowing the heavier sphere, raised through a smaller arc to swing against the lighter one and taking readings as before. In all cases take care to avoid giving a spin- ning motion to the released sphere. The masses of the spheres are to be determined by the student unless furnished by the in- structor in charge. From Fig. 32 it is seen that 1 Fig. 32. and also whence ? = 2g A B 2 OB (82) (83) (84) 1 College Physics, Article 25. 8o PHYSICAL, MEASUREMENTS and the velocity v lt of the impinging sphere is directly 'propor- tional to the chord C B = c, of the arc through which it swings. For angles not exceeding 20 the arc may be set equal to the chord to within one-third of one per cent. We may therefore insert readings from the graduated arc in equation (80) instead of the actual velocities. Apply equation (80), in which v 2 is to be taken equal to zero, and v lf the velocity of the impinging sphere, is to be regarded as positive in each case, and the reverse direction as negative. FORM OF RECORD. Exercise 27. The ballistic pendulum. To prove the law of conservation of momentum. Date. . Sphere 1 .Material 2 Material No. z/ v\ CHAPTER IV. DENSITY. 66. Definition. Density is denned as mass per unit volume, and is therefore, in the c. G. s. system, measured by the number of grams per cubic centimeter. The International Bureau of Weights and Measures however, for practical reasons, chooses the milliliter as the unit of volume, where the liter is denned as the volume occupied by one kilogram of distilled water at 4C (= 1000.05 cm 3 ). The density of water at this temperature be- comes then strictly equal to unity, or one gram per milliliter. According to this definition of unit volume, the numerical value for the density of any substance becomes identical with that denot- ing its specific gravity. In what follows no distinction will be made between the two definitions, since the degree of accuracy attainable by the methods described below, does not warrant such a distinction. 67. Exercise 28. Density from Mass and Volume. FORM OF RECORD. Exercise 28. To determine the density of a brass cylinder from its volume and mass. Density of brass cylinder No Date.. Length Diameter Mass, Mean Mean Volume Density. 68. Exercise 29. The Pyknometer. The pyknometer in its simplest form consists of a glass vessel (Fig. 33), provided with 82 PHYSICAL MEASUREMENTS an accurately ground stopper, perforated through- out its length. Before use it is to be thoroughly cleaned and dried with alcohol or ether. It is then carefully weighed. Call this weight W^. The pyknometer is then rilled with distilled water, placed in a bath of water at 30 C. and allowed to remain there five minutes. (Why?) It is then wiped dry, and its weight, W lf determined. After Fig. 33. cleaning and drying as before, the pyknometer is filled with the liquid under examination, brought to the temperature of 30 C, wiped dry and weighed. Call this weight W 2 . Derive and apply formula for the density of a liquid. Correct for temperature, by multiplying d Q , the value found, by d lf the density of the standard 1 (for water at 3OC, d 1 = 0.9958 g per cc). Unless the values are to be carried to more than three decimal places, the buoyant force of the air may be neglected. Instead of making all the weighings at 30, any other temperature may be taken provided it be higher than the temperature of the room where weighings are made (Why?) FORM OF RECORD. Exercise 29. Determine the density of two liquids. Pyknometer No Date Density of Corrected density, d, of = 69. Exercise 30. Mohr's Balance. In Mohr's balance (Fig. 34), the beam is di- vided into ten equal parts of which the last coincides with the end. Upon this end is hung by means of a fine platinum wire a small sinker containing a thermometer. The weights consist of four pairs of riders weighing respectively m, o.i m, o.oi m, o.ooi m grams. The instrument is usually so adjusted that the sinker is exactly coun- terpoised in air. When immersed in water 1 College Physics, Article 69. Also Stewart and Gee, Practical Physics, Vol. I, p. 119. DENSITY 83 at 4 C the buoyant force on the sinker is equal to the weight of the largest rider. When the sinker is put into any other liquid the weights needed to equal the buoyant force upon the sinker in each case are in direct proportion to the densities of these liquids. If in be called unity then the density of the liquid can be read directly from the beam. The balance must show for water at t C the density corresponding to this temperature as given in Table II. If, instead of this density d, it show a density d lf then all results must be multiplied by d/d . It is obvious that the instrument may be used as an ordinary balance as well, and the densities of solids and liquids referred to water at any tem- perature may be determined by means of it with equal ease. FORM OF RECORD. Exercise jo. Mohr's balance. Redetermine the densities of the substances used in Exercises 28 and 29. i. For solids: Date ...... , Density of .................. Weight in air ............... Weight in water ............ Temperature of water. Corrected density For liquids : Date. Density of Temperature Buoyancy of sinker in water Buoyancy of sinker in liquid Density. . CHAPTER V. SURFACE TENSION AND VISCOSITY. 70. Characteristics of a Liquid. A liquid is a body which has no elasticity of shape, or which yields continuously under the action of a shearing stress. It is characterized by consider- able mobility of its molecules, by a distinct, free upper surface, usually of the meniscus shape when confined in a tube, and by the existence in that free surface of a specific stress or tension not found elsewhere in the body. As a result of this tension the liquid behaves as if it were enclosed in an elastic bag which tends to contract indefinitely and compress the liquid into as small a volume as possible. If the liquid exist in the form of a film then the two sides of the film exhibit this tension in like degree and the film tends to contract indefinitely unless prevented by the application of ex- ternal force. If the external force required to keep a film, / units in width, in equilibrium be F dynes, then 7 1 , the surface tension of the film, measured in dynes per centimeter, is given by the equation 1 2Tl = F (85) or T = 7T (86) As the surface tension of any liquid is greatly modified by the presence of any substance dissolved in it, it is of the highest importance in experiments on surface tension that all surfaces coming in contact with the liquid be absolutely clean, and that the liquid itself be pure. 71. Exercise 31. Measurement of Surface Tension. The apparatus consists of a Jolly balance and several small frames of 1 College Physics, Article 89. SURFACE TENSION different width made from short pieces of fine glass tubing or fine platinum wire, with the ends bent so as to form three sides of a rectangle, all lying in the same plane. Suspend one of these frames from the spring of a Jolly balance by a fine thread or wire after removing the pan, and adjust so that the sides of the frame hang vertically and the ends just dip into the liquid. Take the reading on the Joly balance, making at least three adjustments and taking the mean. Next form the film by raising the vessel containing the liquid until the horizontal side of the frame is immersed and lowering the vessel again to its former position. The tension in the film tends to pull the frame back into the liquid and elongates the spring. Raise the frame by the rack and pinion movement until the indicator has reached its former zero position. Note the reading. Make four determinations with different lengths of the film for each frame, using frames of three different widths. Determine the constant of the spring of the balance by placing gram masses upon the pan and noting the elongation per gram. Calculate the force m g, in dynes exerted by the spring of the bal- ance to hold the film in equilibrium. The surface tension of the film in dynes per centimeter width, is given by T= ^ (87) FORM OF RECORD. Exercise ji. To determine the surface tension of water, and of a soap solution at room temperature. Constant o Mass f Jolly Balance Reading Width of Frame i 2 3 4 Zero Re Water a dings Soap Solution Force in dynes to stretch spring one scale division, = . . . . Surface Tension Frame No. Water Mean Soap Solution 86 PHYSICAL MEASUREMENTS 72. Exercise 32. Surface Tension from Capillary Action. If a freshly drawn glass tube of small diameter be held vertically and lowered into clean water, the surface action of the water drags the liquid up the tube to a height h, at which the weight of the liquid column just balances the vertical component of the force due to surface tension, and equilibrium ensues. If r be the radius of the tube and the angle of contact between the liquid and the tube and d the density of the liquid, then for equilibrium we have 2TrrTcos9 = Trrhdg + vdg (88) where v is the volume of the liquid forming the meniscus. If the liquid wet the tube 6 may be set equal to zero, and if we assume the radius to be so small that the cup formed by the men- iscus is hemispherical, then (89) and the expression for T becomes or where h' is the distance from the surface of the liquid to a point r/3 above the lowest point of the meniscus. Place a freshly drawn glass tube, of less than one millimeter diameter, in a glass cup with straight sides and pour in two or three centimeters depth of distilled water. Move the tube up and down through a range of a centimeter or two so as to make sure that the meniscus is formed against a wet surface. Measure with a finely divided scale or better with a micrometer cathetometer, the height h'. Mark the point on the tube reached by the water column and with a fine file cut the tube at this point. Measure the diameter of the tube with the micrometer microscope, taking the mean of at least three readings. Note the temperature and insert in the formula the appropriate value for d. Compute the value of the surface tension for pure water at the observed tern- VISCOSITY 87 perature. Repeat the experiment with a dilute solution of fuch- sine in alcohol. FORM OF RECORD. Exercise 32. To determine the surface tension of distilled zuater and of an alcoholic solution of fuchsine at room tempera- ture, Temperature .................... Date ................ Readings Diameter of tube Surface of liquid Meniscus I T for water = ...... T for fuchsine solution = ...... 73. Coefficient of Viscosity. As has been shown in a pre- vious article, a liquid yields continuously under the action of a shearing force. An ideal liquid would offer no resistance to such a force but would suffer its molecules to glide past each other without any loss of energy. Most liquids however offer some resistance to a shearing force and this property by virtue of which a liquid resists the relative motion of its parts is termed viscosity. The coefficient of viscosity is defined as the constant ratio of the shearing stress in a fluid to its time rate of change of shear- ing strain. The latter is given by l/Lt, equal to v/L. If F be the force acting on one of two parallel surfaces A of a fluid, separated by a distance L cm and moving with a velocity v rela- tive to each other, the coefficient of viscosity of the fluid is L The coefficient of viscosity is therefore numerically equal to "the tangential force on unit area of either of two horizontal planes at unit distance apart, one of which is fixed, while the oth- er moves with the unit of velocity, the space between being filled with the viscous substance." 1 This quantity may be measured either by determining the flow of a viscous fluid through a tube of small bore, or by determin- ing the resistance offered by the fluid to the passage of a solid body through it. 1 Maxwell, Theory of Heat. 88 PHYSICAL MEASUREMENTS 74. Exercise 33. Coefficient of Viscosity by Flow through a Capillary Tube. It is shown in works on physics 1 that the vol- ume of a liquid discharged in time t, through a capillary tube of radius r, and length /, under a pressure of p dynes per square cen- timeter, is given by the equation 8r)l (93) where V is the volume, d the density, 77 the coefficient of viscosity of the liqui'd, and h the height from which it flows. From this it follows that ?r/>r 4 ifhdg r 4 * h d 3 g r* v i -t, (94) when m is the mass of the liquid discharged through the tube. To determine this coeffi- cient in absolute meas- ure, each of the above quantities must be ex- pressed in absolute units. Select a capillary tube of uniform bore and clean it thoroughly by running through it re- peatedly a solution of sodium bi-chromate and strong sulphuric acid, and afterward washing it well with distilled water. Fasten the tube, by means of a rubber stopper, to the lower end of a large separating flask, as shown in Fig. 35. Fig. 35- Fig. 36. Pour in some of the liquid to be studied and allow it to flow through the capillary for a short time. Now close the stopcock, 1 Poynting and Thompson, Properties of Matter, Chapter XVIII. VISCOSITY 8c> fill the flask with the liquid under examination, and arrange for constant head during the flow, by inserting in the upper end of the flask a stopper through which passes a piece of glass tube of wider bore, reaching well down into the liquid as shown in the figure. Place under the lower end of the capillary a wide beaker, or flat dish, and pour in sufficient liquid to cover the end of the capil- lary tube. Weigh this vessel with its contents. Replace the beak- er, open the stopcock and allow the liquid to flow for a definite time t. Weigh the beaker and its contents a second time and determine m, the mass of liquid discharged. Determine the dens- ity of the liquid if necessary by means of Mohr's balance, the lengths / and h, by means of a meter stick and the radius of the tube by filling a measured length of it with mercury, weighing it, and computing the average radius from the length, mass, and density of the mercury. Use three capillary tubes of different radii. For a determination of the specific viscosity, i. e., the ratio between the viscosity of the liquid and that' of water at oC, or at room temperature, the dimensions of the capillary tube need not be known. By performing the experiment, with the apparatus unchanged, first with water and afterward with the liquid, we* have = j*** . (95) n 2 #2 iMifi or ^=4^ (96) if t, the time of flow be made the same in each case. The exercise may be varied by allowing equal volumes of the two liquids to flow through the capillary, and comparing the times required, from which we have = TT. (97) 77 2 d~t~ In this case a tube as shown in Fig. 36 is very convenient. Draw the liquid into the tube until it rises above the upper mark.. PHYSICAL MEASUREMENTS Then let it run out and note the time it takes from the instant it passes this mark until the liquid enters the capillary. The time may be determined conveniently by means of a stop watch. FORM OF RECORD. Exercise jj. To 'determine the coefficient of viscosity of water at room temperature. Temperature of water No. of tube Length Mass of mercury Volume of tube Average radius Date 2 3 No. of tube Time of flow Mass of beaker after Mass of beaker before Mass of water i 2 3 Mean = CHAPTER VI. MEASUREMENTS IN SOUND. 75. Exercise 34. Velocity of Sound in Metals. (Kundt's Method.) A brass rod held firmly clamped at its middle point when stroked with a resined cloth vibrates longitudinally like the air in an open organ pipe when sounding its fundamental tone. The middle of the rod being rigidly fixed is obviously a node and the length of the rod is therefore the half wave-length in brass of the sound produced. If the end of the rod be brought into con- tact with an enclosed column of air whose length may be varied at will, it is possible so to adjust the length of the air column as to render it capable of vibrating in unison with the rod. In this case the enclosed air column having been thrown into stationary vibration behaves as a resonator closed at both ends; it must therefore contain at least one, and usually contains a number of half wave-lengths, of the sound in air, produced by the rod. From the fundamental equation connecting velocity, frequency and wave-length, we have 1 where N is the frequency and V , v, A, and A' denotes the velocities and wave-lengths of the same sound in brass and in air respect- ively. From this we have at once r=sv. 7 (99) where v, the velocity of sound in air must be corrected for tem- perature according to the formula 2 r t = v Q V ( i -\- at) ( 100) where a for air at ordinary humidity, may be taken as 0.004 P er degree C. 1 College Physics, Article 104. 2 College Physics. Article 114. 92 PHYSICAL MEASUREMENTS In practice a brass rod about one centimeter in diameter and one meter long is clamped in a vise at its middle point and bears at one end a small disk of paper. A glass tube about 5 cms in diameter and 150 cms long (Fig. 37), has one end closed air- tight by a sheet of rubber membrane tied smoothly over the end, while the other end is furnished with an adjustable piston sliding freely in the tube. The walls of the tube are lightly dusted throughout with fine cork filings or amorphous silica. The tube is placed horizontally upon two V-shaped wooden supports so that the rubber membrane presses lightly against the disk of stiff paper on the end of the rod. The rod is set in vibration by chafing it gently with a piece of cloth or chamois skin covered with pow- dered resin. The cloth or chamois skin should be held between the thumb and fore finger of each hand and pressed lightly against Fig- 37- each side of the rod. When the rod is properly clamped a very slight pressure is sufficient to produce a loud, clear tone. Adjust the piston in the outer end of the tube until the enclosed air vibrates freely on stroking the rod. This is indicated by the powder being tossed about in the tube and falling in the charac- teristic figures shown above. The nodes are indicated by small rings of powder and the antinodes by transverse layers or striae. The value of A'/2 is found by measuring over a number of circles from center to center, and dividing the distance by the number of spaces or loops measured, remembering that the distance from node to node is equal to A'/2. It will be noticed that a node is found both at the rubber diaphragm and at the piston. Why? Measure over as large a number of loops as possible and com- pute A'/2. Tap the tube lightly, rolling it over and over until the powder is evenly distributed, and repeat the determination. Take at least five separate sets of measurements. Avoid heating the rod by undue pressure or by continued rubbing. EXPERIMENTS IN SOUND 93 FORM OF RECORD. Exercise 34. To determine the velocity of sound in brass. Temperature Length of rod Date Number of loops Distance A' X between nodes 2 2 \ \' v Velocity of sound in brass Mean. 76. Exercise 35. Computation of Young's Modulus. From the equation for the velocity of sound in any medium v= . -4- where e is the coefficient of elasticity, and d the density of the medium in question, we may compute e at once from the data obtained in previous experiments. In the case of longitudinal waves transmitted through solids the coefficient of elasticity in- volved is M, Young's modulus, whence M = V* d . FORM OF RECORD. (102) Exercise 55. To compute Young's modulus for brass from velocity of sound and density. Date V as found in Exercise 34 = d as found in Exercise 28 = Compare result with that obtained in Exercises 18 and 20. 77. Exercise 36. Rating a Tuning Fork. Graphical Meth- od. A tuning fork, one prong of which is armed with a fine sharp tip of flexible sheet copper, is mounted at right angles to a 94 PHYSICAL MEASUREMENTS metallic cylinder. The cylinder is carried upon an axis furnished with a thread so that when rotated it is advanced longitudinally at the same time, so that the tracing point generates a spiral upon the surface of the cylinder. If the surface of the cylinder be slightly smoked, the tuning fork set in vibration and the cylinder rotated uniformly the tracing point describes a sinusoidal curve upon its surface. The number of vibrations executed in a sec- ond being thus automatically recorded by the fork, it is only necessary to indicate the beginning of the successive seconds upon the curve in order to read the vibration frequency of the fork directly from the surface of the cylinder. In practice the cylinder is covered with a sheet of firm smooth paper pasted smoothly on by gumming one end of the paper and pressing the gummed surface upon the other. The paper must not be stuck upon the cylinder. The paper is then smoked uni- Fig. 38. formly and lightly by means of a gas flame passed back and forth near the paper while the cylinder is continuously rotated, allowing only about an inch of the tip of the flame to touch the paper. Care should be taken not to smoke the paper too black. The fork is then adjusted in its holder so that the point just touches the paper at the highest part of the cylinder and at the left end of the cylinder so that when the latter is rotated the fork seems to move from left to right along the cylinder. The cylinder must rotate from the tracing point. The time intervals are recorded upon the paper by connecting the cylinder and the fork to the secondary terminals of an induction coil, the primary circuit of which contains a suitable battery and is closed by a pendulum beating seconds. Consequently when the tracing point rests upon the paper and the coil is put in action a spark passes from the point to the cylinder each time the circuit is closed, i. e., every second. The passage of this spark leaves a small spot on the smoked surface thus marking the time very accurately. The ap- pearance of the sparks should be like that shown in Fig. 38. EXPERIMENTS IN SOUND 95 vSometimes the coil will give two or three sparks instead of one. (Why?) In this case read from the last one. The fork should be so adjusted that when the point touches the paper and the fork" is properly bowed it will continue to vibrate for at least twelve seconds before coming to rest. When all the adjustments are made the coil is put in action, the fork bowed and the cylinder rotated uniformly while the fork continues to vibrate. The fork is then bowed again and the cylinder rotated as before. When the paper is filled the fork is removed, the paper cut along the lap f parallel to the axis of the cylinder almost but not quite apart ; the cylinder is then turned over and the paper broken loose. In this way only can the paper be removed from the cylinder without spoiling the record. The paper is then passed, face upwards, through a fixing solu- tion of shellac in alcohol and then dried. In a few minutes the curve is ready to be examined. Count the waves for ten seconds and record. In counting the waves always take an even number of seconds. FORM OF RECORD. Hxercise 36. To determine the frequency of a tuning fork by the graphical method. Date Seconds Number of Vibrations : : N CHAPTER VII. MEASUREMENTS IN HEAT. 78. Effects of Heat. In general, all physical properties of a body, except its mass and weight, are changed by adding to it heat energy. In this chapter only the more distinctive of such changes will be considered, among which the following may be noted : (a) The temperature of the body rises. (b) The body undergoes a change in volume; in general an increase in volume of the body attends an increase in tempera- ture, if the pressure remain constant. (c) The body exhibits a change of pressure, if its volume be kept constant. (d) The body may change its state or condition, as for example, ice changes to water and water to steam upon the addi- tion of definite quantities of heat. Other changes, such as changes in the molecular, optical, electrical or magnetic properties of a body, incident upon the addition of heat energy to it, are usually investigated only in so far as such changes are dependent upon changes of temperature. The percentage variation of any physical property of a body per unit change of temperature is termed temperature coeffi- cient, and is more or less characteristic of the substance of which the body is composed. The determination of such charac- teristic coefficients falls more naturally under the chapters deal- ing with those properties. THERMOMETRY. 79. Thermometry. If two bodies possessing different tem- peratures be brought into thermal union, heat flows from the one THERMOMETRY 97 of higher temperature to the one of lower temperature, and in general the flow of heat is such as to produce and maintain_an_ equilibrium of temperature in the body, or system of bodies, if thermally insulated from all other bodies. Temperature is defined quantitatively in terms of the increase in pressure of hydrogen gas of constant volume, under the as- sumption that equal increments of temperature produce equal increments of pressure in the gas. As points of reference in the measurement of temperature the two fixed points for water, the freezing and the boiling points under standard conditions, have been chosen. In the centigrade scale the temperature of melting ice is called o. and the temperature of steam forming freely under a pressure of 760 mm of mercury, is taken as 100. Temperatures are usually measured by means of mercury-in- glass thermometers. Such thermometers possess numerous dis- advantages as compared with the gas or air thermometer, promi- nent among which are the following: (a) Inequality of the bore of the glass tube. (&) Inequality of the scale. (c) Neither glass nor mercury expands equally and uniform- ly throughout any large range of temperature. (d) The instability and uncertainty of the fixed points of the thermometer, arising either from slow changes going on in the thermometer itself or from sudden and large variations in tem- perature incident upon the. use of the thermometer. From these causes it is evidently a matter of first importance to verify the readings of a mercury-in-glass thermometer, with which any accurate work is to be attempted. 80. Exercise 37. Determination of the Fixed Points of a Thermometer. The fixed points of an ordinary thermometer are usually in error by some fraction of a degree and these errors when determined, form the basis of correction to be applied to all subsequent readings made with the instrument. The fixed points 9 8 PHYSICAL MEASUREMENTS must be frequently determined. These determinations fall under two heads : (a) The Boiling Point. The appara- tus (Fig. 39), consists of a brass vessel partly filled with water and having in its upper part double walls so arranged that the steam passes up through the inner cylinder, down through the outer space and escapes from a short tube near the bottom. The thermometer is passed snug- ly through a close-fitting cork, into the inner cylinder. The bulb is not allowed to come in contact with the water and should be surrounded by wire gauze to prevent overheating. If possible, almost the entire filament of mercury should be enclosed by the steam issuing freely un- der atmospheric pressure. The tempera- Fig. 39. ture of the steam is found from the baro- metric pressure under which the water boils, by means of the following approximate formula t = 100 + 0.0375 (B 760) , (103) where B denotes the barometric pressure in mm, corrected to oC. For accurate values of boiling point at different pressures see Table IX. If the thermometer reads t + b, instead of the calculated t de- grees, the correction to be applied to this reading is b degrees. No readings should be taken until the steam issues freely from the tube at the bottom. This tube should be kept entirely open and the water should not be boiled too violently. (Why?) (b) The Zero Point. The thermometer is now removed from the boiling point apparatus and as soon as the temperature has dropped to 50 is plunged into a clean vessel containing a mixture of distilled water and pure ice so that the mercury thread is en- tirely surrounded by the mixture. When the reading is taken THERMOMETRY 99 the thermometer should be raised just high enough to show dis- tinctly the upper end of the mercury filament. In reading care should be taken to avoid parallax. As soon as the mercury has fallen to i, note the reading every minute until it has remained stationary for five minutes. Take the final readings as the zero reading of the thermometer. What would be the correction for the zero point, if the reading is found to be -}- a degrees ? The zero point thus found is termed the depressed zero, since it is usually lower than the value found for the zero point, if it be determined just before the boiling point is taken. This difference is due to the fact that the glass contracts slowly for a consider- able time after being heated and allowed to cool, and is thus unable to follow immediately sudden changes in temperature. The depressed zero point is, however, the one to be used for calibration. Under the supposition that the bore of the tube and the scale are uniform between the two fixed points, the value of a scale part may be found in terms of the mercury in glass scale by dividing t degrees by (t + b a). Calculate by this method the value of the tenth, twentieth, thirtieth, etc., divisions of the scale. Compare the values so found with a calibration obtained by direct comparison of the thermometer with a standard thermometer. 1 For this comparison both thermometers should be placed in a large bulk of water in such a way as to keep their bulbs close to- gether, and the water constantly stirred during the comparison. Change the temperature of the water by steps of 5 at a time from o up to 50, and read both thermometers each time after the water has been well stirred. Compare the corrected table obtained by the comparison with the standard, with the calibra- tion obtained by calculation. For accurate determinations of temperature it is necessary to determine the depressed zero point immediately after taking the temperature in question. The temperature must be reck- *For the reduction of the mercury in glass scale to the normal (hydrogen) scale see Chappuis, Rapp. Congr. internat. Paris, 1900, Vol. I, p. 142. 100 PHYSICAL MEASUREMENTS oned from this depressed zero, even though it is not a constant for different temperatures. Determine the fixed points of an ordinary thermometer and calibrate it from o to 50 C. FORM OP RECORD. Exercise 37. To determine the fixed points of a thermometer. Thermometer No. Boil Reading ing point Correction Zero Reading point Correction Calibi Compared -ation Computed Value of one scale part = 81. Stem Correction. In order that the reading of the ther- mometer may give the exact temperature of the body in question, it is necessary that the entire mass -of mercury should be at the temperature of the body. Frequently a portion of the filament extends above the region whose temperature is sought, and the reading will be too high or too low according as the temperature of the filament is above or below that of the body to be measured. The relative coefficient of expansion for mercury in Jena nor- mal glass is 0.000157 per degree C. If t of the filament are out- side, then the difference in height of the filament for a difference of i C will be 0.000157 t. If f be the mean temperature of the filament exposed, and ^ the temperature indicated by the ther- mometer, then the correction to be added to the indicated reading is 0.000157 (f x * ) t degrees; t is determined by an auxiliary thermometer. EXPANSION. 82. Coefficient of Linear Expansion. The coefficient of linear expansion of a substance is defined as the increase per unit length, per degree increase of temperature. Thus if the substance be in the form of a rod or wire, and / be its length at temperature f , and / 2 its length corresponding to t 2 , then j3, the coefficient of linear expansion, is by definition If one of the temperatures be taken as o C, the corresponding EXPANSION 101 length / , and the length at any other temperature /, be / t , then the above formula reduces to (105) or (io6) 83. Exercise 38. Coefficient of Linear Expansion of a Solid. The solid whose coefficient of expansion is to be deter- mined is in the form of a long thin rod R, Fig. 40, which is held Fig. 40. by means of thin rubber stoppers S $, at the center of a brass tube T, about five centimeters in diameter, and of approximately the same length as the rod. The tube is furnished with inlet and out- let tubes for the passage of water or steam. Thermometers are placed at the ends of the tube in two openings a a. The tube rests on two V-shaped blocks, which together with the block b, are fastened to the stout board B. The block b carries an adjustable screw whose axis is in line with that of the rod, and which has on its inner end a small glass cap, to prevent as far as possible any loss of heat from the rod by conduction. The screw is turned till the glass cap presses firmly against the end of the rod, and thus acts as the fixed point from which the length of the rod is measured. The variations in length are measured by means of a series of IO2 PHYSICAL MEASUREMENTS levers pressing against the other end of the rod. The lever L, mounted in a slot in the block G, rotates about O, and actuates the tilting mirror M, through the right angled arm P. Four shallow grooves, o' ' , o" , o'" , o lv , are cut in the surface of the block at right angles to the level L, in which the points of the tilting mirror M, may be placed so as to stand astride of the slot. The rotation of the mirror is read off by means of a telescope and a vertical scale. Loss of heat by conduction from the rod is prevented by insert- ing a small piece of glass between it and the point of the lever L, while the tube is covered with a layer of asbestos, and screens of asbestos are placed at either end to prevent loss by radiation. By means of the screw Sc, the block with its system of levers is pushed forward and the apparatus adjusted so that, with ice water flowing through the tube, the mirror stands vertically, and the zero reading is taken. If the distance between and the point of contact of the lever with the glass plate be called l lt and 1 2 be the distance between O and F, then when the rod expands by an amount a, for a differ- ence in temperature t, the lever will rotate through an angle <, and its ends will describe short arcs a and b, such that a = 0/1 (107) and b = $k (108) Also, since the point F moves upward through the short arc b, the mirror is tilted through an angle 0, such that b = o p (109) where p is the distance o' F. By eliminating b and from the above equations we have 9= ^ (no) />/! Since the quantity l 2 /p / x is a constant of the instrument we may write = ka . (in) The angle is evaluated from the reading a, observed in the EXPANSION IO3 telescope, and the distance D, between the telescope and the mir- ror by means of equation (18). Since k, a, and D are known, the value of a may be readily determined, and from the relation (112) we have Determination of k. To determine the constant k, put glass slides of known thickness p f measured by means of the sphero- meter, between the end of the rod and the lever and observe the resulting deflection, while the temperature and the position of the block G, remain unchanged. Solve for k from the relation (114) Use two different slides and two positions of the mirror. Since for accurate determinations of k, it is desirable that the glass slides should not be too thin, the telescope and scale should be brought nearer to the mirror, in order to keep the readings on the scale. In determining the deflection during the experiment, the sensi- tiveness may be increased at will by increasing the distance be- tween the mirror and scale. The temperature of the rod is changed by passing water of the desired temperature from a larger vessel through the tube. As soon as the temperature becomes constant, as indicated by the con- stancy of the scale reading, take this reading and that of the ther- mometers. Thermometer readings must be corrected for the ex- posed filament. Change the temperature of the rod by steps of about 20 from o to 100 C. In order to obtain the last point steam is passed through the tube and the barometer reading noted. Since the change in length is a very small fraction of the IO4 PHYSICAL MEASUREMENTS original length of the rod, this length may be determined with sufficient accuracy by means of a millimeter scale. Since in the instrument as described above a small portion of the rod at each end is not at the exact temperature of the water or steam, a slight error will be introduced, especially if the differ- -ence between the temperature of the rod and that of the room be large. The determination may be made much more accurate by holding the rods, which are cut slightly shorter than the tube, in the center of the tube by wire frames, and inserting in the rubber stoppers short pieces of nickel-iron whose coefficient of expansion is almost negligible. In a similar type of apparatus the block with the system of levers is replaced by a spherometer screw, passing through a nut on a support solidly fastened to the base of the instrument. The end of the screw is brought into contact with the movable end of the rod and the position of the screw is read from the spherometer scale. Readings are repeated with the rod at differ- ent temperatures. From these readings the expansion of the rod can easily be calculated. The form of record, given below, must of course be altered correspondingly, if this type of apparatus be used. FORM OF RECORD. Exercise 38. To determine the coefficient of linear expansion of a metal and of glass. Date I. Determination of k. Thickness of glass slide a D tan 2 & k II. Determination of /?. Length of rod, Temperature a D tan 2 84. Expansion of Liquids. In determining the coefficient of expansion of a liquid, we are met by the difficulty that the liquid must be contained in some sort of a receptacle, the material of which expands at the same time as the liquid, and hence the apparent expansion of the liquid is always a differential effect, EXPANSION IO5, being the difference between the increase in volume of the recep- tacle and that of the expanding liquid. By measuring the height of two communicating columns of the same liquid at different temperatures the coefficient of cubical expansion of the liquid may be determined without reference to the expansion of the contain- ing vessel. For the attainment of accurate results, however, the apparatus becomes too complicated for use in an elementary course 1 . It is therefore customary to measure the relative expansion of the liquid in a vessel, usually of glass, and calculate a the absolute coefficient of expansion of the liquid from this value and from the known coefficient of cubical expansion g of the vessel. This latter coefficient is found either by determining {$, the linear coefficient for the glass of which the vessel is made, and putting a 3 /?, or by using a liquid whose absolute coefficient is known. Thus let a be the relative coefficient of cubical expansion of the liquid in glass, a its absolute coefficient, g the coefficient of cubical expansion for glass, Fthe apparent volume of the liquid at t C, V\ the real volume at t 'C, Fo its volume at o C. Then we have the following relations : and from which by eliminating F and t, we have a = a + g^,l ("8) KO .- , or, since F/F is very nearly unity 1 Preston, Theory of Heat, p. 170. 106 PHYSICAL MEASUREMENTS 85. Exercise 39. Coefficient of Expansion of a Liquid by the Dilatometer. A dilatometer consists of a bulb with an ac- curately graduated stem of uniform capillary bore. In work of extreme precision it is necessary to calibrate the stem throughout in order to correct for irregularities of cross section. The volume of the bulb at o C, and that of one scale division on the stem must be accurately determined, as well as the coefficient of cubical expansion of the bulb. These form the constants of the instrument. The exercise is divided into three parts : (a;) Determination of the volume of one division of the capillary stem. Bring into the graduated stem an amount of mercury sufficient to fill it nearly full, and measure the length of the filament in terms of the scale divisions, using a mirror scale or reading microscopes. Let the observed length of the filament be / scale divisions. On account of the meniscus the length measured from end to end will be slightly too large. For capillary tubes it is sufficient to subtract from the observed reading 0.4 h, where h is the height of the meniscus in terms of a scale division. Let the mass of the mercury filament be m grams, at the temperature t. One gram of mercury weighed in air, occupies at t C, the volume v' 0.07355 (i -j- 0.000181 cm 3 (120) so that the volume corresponding to one scale division is (b) Determination of the volume of the bulb. The volume of the bulb may be determined by taking the mass of the dilatom- eter, first when empty and dry, and secondly when filled with air-free water up to scale part /' in the capillary tube, at a EXPANSION temperature not below 15 C. Let the mass in the first case be iii , and in the second w t , then m, the mass of the water, is m = m r 11 t< One gram o'f water, weighed with brass weights in air, occupies very nearly (2.00106 d) cm 3 , where d is the density of the water at temperature t , as given in Table II. The volume of the bulb and the capillary tube up to the zero mark of the stem, at this temperature is therefore, J/i = [w(2.ooio6 d)~ V v] cm 3 (121) The volume F , at o is readily found from this by applying the equation where g may be taken as 0.000025. If it be desired to determine the coefficient of cubical expansion of the glass vessel more accurately, it is best to fill the dilatometer with pure mercury and determine the readings l\ and l\ at two different temperatures, say t and o C. If M be the mass of the mercury, and the values of i/ (equation t2o) at the two tempera- tures v\ and z"' , then V, = Mv\ l\v and V* = Mi/* l'*v from which g can easily be calculated. (c) Coefficient of expansion of a liquid. Fill the dilatometer with the liquid under investigation and immerse it in a large glasi vessel containing water. Vary the temperature of this bath and observe the resulting height of the liquid in the stem of the instru- ment. Be sure to leave the instrument in the bath long enough to insure thermal equilibrium between its contents and the bath. Stir constantly to keep the temperature of the bath uniform in all parts. Add small quantities of warm or cold water, if necessary, to keep the temperature of the bath constant. loS PHYSICAL MEASUREMENTS Take four or five different temperatures, t lf t zt t s , , and observe the corresponding readings on the stem, l it I 2f Denote the related volumes by V^, V Z) ,then etc. (122) Plot volumes and temperatures, and calculate the mean co- efficient of expansion from (123) In case the expansion of water is to be studied in the neighbor- hood of 4 C, the bulb must be quite large or the capillary tube of smaller bore than in the average instrument, as the expansion of water at these temperatures is very slight. Plot the apparent and the real volumes as ordinates and the temperatures as abscis- sae. Compute a table for the specific volumes, V/M, for the dif- ferent temperatures. FORM OF RECORD. Exercise 39. To determine the coefficient of expansion of a liquid by means of the dilatometer. Date. (a) Value of one scale division of stem. Mass of mercury ...., number of divisions = . . . ., v (fc) Volume of bulb at o C. Mass of dilatometer empty , Temperature. . 'Mass of dilatometer filled , m , I' = 'Coefficient of expansion of the Mass of dilatometer filled Mass oi dilatometer lass. Reading V (c) Coefficient of expansion of Reading t V EXPANSION 109 86. Air-Free Water. Air-free water is used in many cases for finding the volume of glass instruments. Such water may be prepared by boiling distilled water for half an hour. Another method for free- ing water from the absorbed air is shown in Fig. 42. The water is contained in a flask which is closed by a rubber stopper through which passes a capillary tube of fine bore, extending almost to the bottom of the vessel. A water aspirator con- nected to the side tube produces a partial vacuum in the flask and air is forced in through the capillary tube and rises through the water. This air, rising through the water under diminished pres- sure, produces a state of unstable equi- librium in the air previously absorbed in the water and this absorbed air can now be seen rising in the form of small bubbles. The process can be hastened by giving the flask from time to time a smart blow with the hand. 87. Exercise 40. Constant Volume Air Thermometer. If the volume of a given mass of gas be kept constant and its temperature be varied, the pressure of the gas upon the walls of its containing vessel will vary according to the formula p t = p (i -{- a t) . (124) where p and p t are the pressures at o and t C, respectively, and a is the pressure coefficient of the gas at constant volume. We know that for a perfect gas a is the same as the coefficient of voluminal expansion for the gas at constant pressure. 1 For all permanent gases the value of the pressure coefficient is nearly the same. In order to determine a, the gas, usually air, is kept at constant volume, its temperature is varied and the resulting pres- Fig. 42. 1 College Physics, Article 163. no PHYSICAL, MEASUREMENTS sures observed. When properly calibrated the instrument furn- ishes the means of determining temperatures as a function of the observed pressures, and hence is termed a constant volume air thermometer. The air is contained in the bulb A, Fig. 43, which is attached to a fine capillary tube a, bent twice at right angles, and joined at its outer end to a larger tube B about one centimeter in diameter. Fig. 43- The tube B has at the point of junction with the capillary tube a fine pointer p of colored glass fused into its side, which serves as a point of reference for the height of the mercury column in B, and to which the mercury must be brought each time before a reading is taken. A mark on the capillary may serve the same purpose. In the simpler forms of the apparatus the -tube B is connected by a thick walled rubber tube to a second glass tube C, which is capable of considerable movement up and down a graduated scale placed between the two tubes. By means of this scale the level of EXPANSION 1 1 1 the mercury in each tube may be read off and the corresponding pressure at any temperature upon the gas determined. In this way the volume of the enclosed air is kept constant except for the change in volume of the bulb due to expansion, which must be allowed for in the final computation. In order to determine the pressure coefficient of dry air at constant volume, the bulb is first carefully cleaned, dried, filled with dry air, and surrounded with melting ice. The tube C is then moved up or down until the surface of the mercury in B just touches the tip of the colored pointer. In this position the tube C is clamped and the reading on the surface of the mercury in each tube is noted. If the difference in these readings be h , and we call the barometric reading b, then the pressure upon the gas will correspond to a barometric height (115) The bulb is next surrounded by steam and the tube C adjusted so as to bring the mercury again to the tip of the pointer, and a second reading taken. Care must be taken to wait long enough before reading to allow the air in the bulb to reach the temperature of the steam, a condition clearly indicated by the level of the sur- faces of the mercury columns remaining constant. If we denote the difference in level between the two columns by h, the air is under a pressure corresponding to a barometric height H = b + h. (126) In measuring b and h it is sufficient to reduce both to the same temperature throughout the experiment, but for finding t, the temperature of the steam, it is necessary to reduce the barometric reading to o C, in order to make use of Table IX. In all ma- nipulations of the instrument great care must be exercised to pre- vent any mercury from entering the capillary tube, and being drawn into the bulb. 112 PHYSICAL MEASUREMENTS From Gay Lussac's law 1 deduce the following formula for the pressure coefficient of a gas at constant volume where g denotes, as usual, the coefficient of voluminal expansion for the glass bulb: 2 H. H What is the formula for any temperature t, as given by the air thermometer ? Insert the thermometer in water of different temperatures and compare the calculated temperatures with those measured by a mercury-in-glass thermometer. Plot temperatures and pressures. In the derivation of the above formula it is assumed that the volume of the capillary connection from the bulb to the tip of the pointer is negligible as compared to the volume of the bulb. This assumption is the more nearly justified the larger the bulb and the finer the bore of the capillary tube. FORM OF RECORD. Exercise 40. To determine the pressure coefficient of dry air by means of the constant volume air thermometer. Barometric pressure. Temperature Readinj B Pressure in cm of Hg CALORIMETRY. 88. Definitions. Temperature is to be sharply distinguished from quantity of heat. The former has reference only to the condition of a body affecting the sensation of warmth and cold and has no reference to the amount of matter involved. In quantity of heat account must be taken both of the temperature 1 College Physics, Article 161. 2 For the derivation of complete formula, see Kohlrausch, Physical Measurements, 3d English from 7th German ed., pp. 93-97. CALORIMETRY 113 of the body and of its mass. The unit of temperature is the de- gree centigrade. The unit of quantity of heat is the calorie. A calorie is the quantity of heat required to raise the temperature of one gram of water iC. The thermal capacity of a body is numerically equal to the number of calories required to raise the temperature of the body one degree centigrade. The thermal capacity of the substance, of which the body is composed, is its thermal capacity per unit mass, or it is numerically equal to the heat in calories required to raise the temperature of one gram of the substance one degree centigrade. If we call the thermal capacity of a substance c, the mass of the bod) M, then the heat needed to raise the temperature of the body from t. 2 to t degrees is given by H = cM(h fc) calories ( 128) From the definition of the calorie it follows that the thermal capacity of water is taken as unity. Though c for water varies slightly with temperature 1 , it will be considered as constant in the following exercises. The specific heat, s, of a substance is the ratio of the thermal capacity of the substance to that of water, or (129) Since c w is unity the specific heat of a substance is numerically equal to its thermal capacity. The specific heat of any substance is different for different temperatures and hence as usually given, it denotes the mean value for the specific heat between certain temperature limits. 89. Specific Heat by Method of Mixtures. If two sub- stances of masses M^ and M 2 , at temperatures and t 2 and having thermal capacities c and c. 2f be brought into contact, they will come to some intermediate temperature t, such that the number of calories given out by the first is exactly equal to the number ^College Physics, Article i/i. 114 PHYSICAL MEASUREMENTS gained by the second, provided of course, that no heat has been added from the outside, or been lost externally through conduction or radiation. Then the equation for the heat ex- change is t t z ). (130) If the second substance be water, then M 2 the expression becomes = c w and (131) In actual practice it is impossible to avoid loss of heat both by radiation and by conduction. If the water be contained in a vessel or calorimeter, then the latter receives heat along with the water and finally comes to the common temperature /. If M be the mass of the calorimeter and c its thermal capacity the amount of heat needed to raise its temperature from f, to t degrees is H = cM(t f 8 ) This quantity must be added to the right hand member of equa- tion (130). The calorimeter usually consists of a beaker, stirrer and thermometer for each of which cM must be calculated and their sum be taken in the above equation. If we set = Z M = 2 sM (132) then m is the mass of water which has the same thermal effect as the calorimeter. This mass is called the water-equivalent of the calorimeter. Therefore the total heat gained by the water and calorimeter will be c w (m + M w ) (f * 2 ) and we have M,) (f fQ d33) 90. Exercise 41. Water-Equivalent of a Calorimeter. The calorimeter is a cup of thin metal, preferably of aluminium. CAI.ORIMETRY 115 which is placed inside a large vessel upon a flat piece of cork or other poor conductor. In the calorimeter are a stirrer and a ther- mometer. Let m be the water-equivalent in grams of the calorim- eter including stirrer and thermometer; also let the calorimeter contain M w grams of water at a temperature t 2 ; suppose the re- sulting temperature, due to adding M grams of water at t lf finally comes to be t. Then the exchange of heat is represented by the equation <-'w(^w + "0 (fa =r w Mi (/ fi) (134) or, solving for m M, (135) In practice weigh the calorimeter empty and dry, then fill about one-third full with water at a temperature about fifteen degrees above the temperature of the room and weigh again. The differ- ence is M w . Next add water at a temperature, about ten degrees below room temperature, until the resulting temperature after vig- orous stirring is about room temperature. The temperature of the cold and warm water should be carefully determined, just before mixing, by means of a thermometer reading to tenths of a degree centigrade. The resulting temperature is to be taken only after the thermometer reading has become constant. Repeat the experiment twice, taking the mean of the three results as the water-equivalent. For the correction due to radiation see Art. 92. FORM OF RECORD. Exercise 41. To determine the water-equivalent of a calori- meter. Weight of Vessel with vessel l water 1st M w t* fi t Vessel with water 2d MX m Compare this result with that obtained by multiplying the mass of the calorimeter by the specific heat of the metal, as given in Table X. In the case of a thermometer the exact masses of mer- IIO PHYSICAL MEASUREMENTS cury and glass are unknown. But the thermal capacities for equal volumes of mercury and glass are nearly the same, namely 0.47 calories per cc. To find the water-equivalent of a thermom- eter it is therefore sufficient to multiply the immersed volume of the thermometer by this number. 91. Exercise 42. Specific Heat of Copper. The piece of copper whose specific heat is to be determined is heated in a brass tube (Fig. 44), which is surrounded by a steam jacket. The copper is hung by a thread in the middle of the tube and the top is closed by means of a cork carrying a thermometer. The heater sits upon a wooden board having a hole of the diameter of the inner tube and directly beneath it. This board slides upon Fig. 44. a support provided with a similar hole and so arranged that the two holes coincide when the board is pushed in as far as possible. Immediately under the hole in the support is placed the calori- meter so that the heated body may be passed directly from the tube through the support into the calorimeter. The sliding board is to shield the calorimeter from heat during the heating of the copper. In use the dry copper is weighed and hung in place, the hole in CALORIMETRY 117 the support closed by the sliding board and steam passed through the jacket until the temperature of the interior becomes constant, t^. This heating usually requires about twenty minutes. Mean^ while the calorimeter with the contained water is carefully weighed. The temperature of the water in the calorimeter is then read, t 2 . The calorimeter is put in position, the sliding board pushed in, and the heated copper lowered gently into the vessel beneath. The calorimeter with its contents is then re- moved, the water thoroughly stirred and the highest temperature t, carefully noted. In order to avoid the necessity of correcting for radiation, it is well to have the temperature of the water in the calorimeter some 4 or 5 below the temperature of the room at the beginning the experiment. Apply formula (133). FORM OF RECORD. Exercise 42. To determine the specific heat of copper. Date. Mass of Calorimeter Mi M* m t* t Calorimeter with water Specific heat = 92. Correction for Radiation. In the preceding experiments the temperatures were so chosen as to render correction for radiation and absorption unnecessary. In experiments requiring a greater degree of accuracy this loss or gain of heat by the cal- orimeter must be taken into account. This is best done by noting times and temperatures for an interval of at least five minutes be- fore the instant at which the experiment proper begins, that is, the instant at which the ice, steam or metal enters the calorimeter. Readings should be taken every twenty seconds. Plot the times as abscissae and the temperatures as prdinates, (Fig. 45). After the beginning of the experiment the temperature rapidly rises or falls, and having reached a maximum or a minimum it will practically become a linear function of the time for a few minutes, showing the rate of radiation or absorption. iiS PHYSICAL MEASUREMENTS According to Newton's law of cooling, the rate of loss or gain of heat energy by a body due to radiation, varies directly as the difference in temperature between the body and surrounding ob- jects. From the straight parts of the curve determine this rate r, for a difference in temperature of one degree. Next determine rlie average temperature of the calorimeter for the time interval (ab t Fig. 45), during which the mixing occurs. This is done by dividing ab into a number of small intervals, taking the sum of the Minutes Fig. 45- temperatures belonging to each, and dividing by the number of in- tervals. The difference between this average and room tempera- ture multiplied by the rate r, gives the correction to be applied with its proper sign, to the observed reading at the time b. A numerical example will make the method clearer. The readings of the thermometer were begun ten minutes before CALORIMETRY 119 the heated body was introduced into the calorimeter, and the read- ing continued for twenty minutes after mixing took place. The room temperature was 19. 45 C. Read. 2i.97 .94 .92 .89 .87 -84 .83 79 76 74 71 Min. Read. >Min. Read Aver. Min. o i8.04 10 i8.i9 (Calc.) i .06 ii 19 .8 19 .0 20 21 2 .07 12 21 .O 20 .4 22 3 -09 13 21 .7 21 1 23 4 .10 14 21 .9 21 .8 24 5 -12 15 22 .0 6 .13 16 22 .04 21 .95 22 .02 25 26 7 -15 17 22 .04 .16 l8 22 .02 22 .04 22 .03 27 28 9 .17 19 22 .00 10 20 21 .97 22 .OI 21 .98 29 30 Average temperature, Before mixing: i8.i2 C. 2Oth to 3oth minute: 21. 83 C Rate of change of temperature per minute : "Rpfnrp mivino-. Io.I7 18.04 o.oi 20th to 30th minute: ?L 2171 _ o >026C , p , erminute> 10 Rate per minute per degree, Before mixing: - OI 44 2oth to 3oth minute : 19.45 18.12 0.026 = o.on C per minute per degree. = o.on C per minute per degree. 21.83 1945. Average temperature of the calorimeter from the loth to 2Oth minute : 2i.46C. Decrease in temperature due to radiation in 10 minutes, 10(21.46 19.45)0.011 = 0.22 C Highest temperature corrected for radiation : 21.97 + 0.22 = 22. 19 C. Total temperature change, corrected : 22.19 18.19 r=4.ooC. J2O PHYSICAL MEASUREMENTS 93. Exercise 43. Heat of Fusion of Water. Water absorbs definite amounts of heat energy on passing from the solid to the liquid, and from the liquid to the gaseous state. The quantities of heat per gram thus absorbed are termed the heat of fusion and the heat of vaporization. The number of calories necessary. to change one gram of ice at oC. to water at oC. is therefore numerically equal to the heat of fusion of water. This may be measured in various ways. One of the simplest is by the method of mixtures ; i. e., a known mass of ice at o, is added to a definite mass of water at a known temperature, and the temperature of the water at the end of the melting enables us to compute the amount of heat consumed in melting the ice. Thus let M 2 grams of ice at zero, be added to M grams of water at ^ and let the temperature at the end of the melting be t; also let the water-equivalent of the calorimeter, stirrer and thermometer be m, and let / denote the heat of fusion of ice as de- fined above. Then since the water formed by the melting of the ice must be warmed to t degrees, we have the heat lost by the cal- orimeter and its contents equal to the heat absorbed by the ice and the ice-water. Hence M 2 / + r w M 2 * = f w (M, + m) (*i (136) from which since c w equals unity, we have the numerical equality i= w.+g ".-o_,. (I37) . The apparatus consists of a calorimeter, a thermometer, and a circular stirrer covered with wire gauze to keep the pieces of ice under water while melting. Weigh the calorimeter, fill nearly full of water at a temperature t lt about fifteen degrees above room temperature and weigh again. The difference is the mass of water M . Break clean ice into small pieces and add to the water sufficient dry ice to bring the temperature of the calorimeter and its contents to about fifteen degrees below room temperature, CALORIMETRY 12 T when all the ice is melted. Stir vigorously throughout the opera- tion, read the temperature t, as soon as the ice is all melted, and weigh the calorimeter and its contents once more. The differ- ence between the last two weighings gives the mass of the ice M z , that was added. FORM OF RECORD. Exercise 43. To determine the heat of fusion of water. w Alone eighings of With M t calorimeter : With M! + M a M l Dal M a e. . m ti t Heat of fusion of water = 94. Exercise 44. Heat of Vaporization of Water at Boil- ing Point. The heat of vaporization at the boiling point is the number of calories per gram required to change water at that temperature into steam at the same temperature. Conversely if one gram of steam at this temperature be condensed into water the same number of calories will be liberated. Thus if M 2 grams of steam at a temperature t 2 , having been conducted into a cal- orimeter of water equivalent m, containing M grams of water at temperature t^ produce by condensation and cooling .a result- ant temperature t, we may write our equation of heat thus : M 2 L or, since c w equals unity -j- m) (f (138) (139) where L is the heat of vaporization of water. The water equiva- lent m, of the calorimeter, may be found experimentally as before, or by multiplying the mass of the calorimeter by its specific heat. The determination may be made by either of the following methods : First Method. Steam generated in a suitable flask is passed 122 PHYSICAL MEASUREMENTS Asbestos Wood- Fig. 46. through a wide tube 15 cms long and 3 cms wide, (Fig. 46), in which the water from condensation is caught and retained. From this it passes directly into the water in the calorimeter. The mass of steam condensed, M 2 , is determined from the increase in weight of the calorimeter. The calorimeter having been care- fully dried and weighed is filled nearly full of water at a tempera- ture some fifteen or twenty de- grees below that of the room and again weighed. The difference in weight is M . After the steam passes freely from the vertical tube leading from the water trap, the calorimeter and its contents are brought into place and the steam passed directly into the water until its temperature is some fifteen or twenty degrees above the temperature of the room. The water should be vigorously stirred during the condensation. The calorimeter is then removed and the stirring continued until the temperature reaches a maximum, when the reading t, is taken and recorded. The temperature t 2 , of the steam entering the cal- orimeter 'is to be determined by taking the barometric reading at the time of the experiment, and referring to Table IX. A third weighing determines the mass of steam condensed, M 2 . Since M 2 is usually a small mass any loss of water due to drops adhering to the exit tube from the water trap- leads to a relatively large error in the mass of steam condensed, and should be taken into account for accurate work. If steam be allowed to enter the calorimeter too rapidly, the tube leading into it is covered on its inner surface with minute drops which are difficult to recover. Owing to the large influence of the above sources of error the fol- lowing method is to be preferred. Second Method. The apparatus (Fig. 47), consists of a closed copper vessel or calorimeter, provided with a stirrer and an open- ing for a thermometer ; inside the closed vessel is a second smaller vessel into which steam is passed and there condensed. The CALORIMETRY 123 calorimeter proper having been carefully dried and weighed, is filled nearly full of water at a temperature some 15 to 20 be- low room temperature and again weighed. The difference is M. Steam is then generated in a small glass retort connected with the inner vessel into which the steam passes and condensing gives up its heat to the calorime- ter and its contents. The tem- perature of the cold water t lt is to be taken just before the steam enters the calorimeter. Steam is allowed to pass in until the resulting temperature rises as much above room temperature as the initial temperature of the water was below it. The flame is then removed and the tempera- ture t, carefully determined. The amount of steam condensed is found by weighing -the small retort before and after the experi- ment. The difference is the mass of steam condensed, M 2 . The temperature t.,, of the steam entering the calorimeter is to be de- termined from the barometric reading as before. Care must be taken to prevent radiation from the retort and the flame under it from reaching the calorimeter. Stirring should be continued after removal of the flame until the temperature ceases to rise. Fi g- 47- FORM OF RECORD. Exercise 44. To determine the heat of vaporization of water. Date ................ Weight of calorimeter = ........ Specific heat of calorimeter = ........ Water equivalent in ........ Weight of calorimeter with water =. ........ Weight of retort before experiment = ........ Weight of retort after experiment = ........ -I/, tt 1 L 124 PHYSICAL MEASUREMENTS 95- Tin. -/ ~- Exercise 45. Melting Point and Heat of Fusion of The tin is contained in an iron vessel I, Fig. 48, which is closed by a cover, provided with a slit 5 to admit the stirrer S, and carrying a narrow tube T, to receive a thermom- eter reading to 360 C, or a thermo- element. The tube should extend well into the tin, and contain a small quantity of mercury to insure good thermal contact. In case a thermom- eter is used the stem correction (Ar- ticle 81) must not be neglected. It is best to place the iron vessel inside a larger one, or surround it with an asbestos screen in order to avoid irregularities in cooling due to draughts of air. Heat the vessel slowly until all the tin is fused and the tempera- ture has risen to about 280 C. Turn out the flame and take temperature readings every minute, stirring the molten metal Fig. 48. T, t t C D t; Fig. 49. constantly until it solidifies. Continue the readings until the temperature has fallen to about 180 C. Plot times and corrected temperatures. The curve obtained will be similar to that shown in Fig. 49. A study of the curve thus found will show the fol- lowing facts : i. Solidification occurs where the curve becomes practically CALORIMETRY 125 horizontal. The mean temperature reading for this part of the curve is the melting point of tin. 2. To find the heat of fusion of tin, determine the rate of cool- ing before the melting point is reached and after the metal is all solidified. These rates are given by the expressions * "~ and t z ti T \ ~ T f " where T denotes temperature, and t denotes time. t z t i If now we call the specific heat of molten tin ^ = 0.064, and that of solid tin s 2 = 0.060, and if M represents the mass of tin, and m the water-equivalent of the iron vessel, stirrer, etc., then the quantity of heat Q lt lost by radiation during the interval t 2 f lt is, since c w equals unity, given by the numerical equality Q 1 =(Ms l + m) ( Ti Tz) ( 140) and that lost during the interval t' 2 t\, is 0,= (M * + )( TV -TV) - (141) and for R, the rate at which heat is lost we have the two values , N 1 8 -f m) *-; : ti /i (142) Take the mean of these two values for R. Continue the straight parts of the curve where only cooling occurs, until they intersect the line of constant temperature in the points c and d. We may then assume that during the time t = C D, corresponding to the era of constant temperature c d, the process of solidification developed sufficient heat to supply the 126 PHYSICAL, MEASUREMENTS loss due to radiation and thus maintained the temperature con- stant. This quantity of heat is ML, if L be the heat of fusion of tin. Again the quantity of heat radiated, during this interval is Rt, hence ML = Rt (143) and Rt d44) FORM OF RECORD. Exercise 45. To determine the melting point and heat of fusion of tin. Date Weight of vessel =. Weight of tin = Water equivalent ==. Time Temperature TV 7Y=... Fusing point of tin = . . . . L = .. VAPOR PRESSURE. 96. Measurement of Vapor Tension. The vapor tension of a liquid is the pressure, measured in millimeters of mercury at o C, exerted by its saturated vapor produced by evaporation in a vacuum. An increase 'in temperature produces an increase in vapor tension, but vapor tension increases at a more rapid rate than the temperature. When the pressure of the saturated vapor above a liquid becomes equal to the atmospheric pressure exerted upon its surface, the liquid begins to boil. We are thus in position to measure vapor tension in two ways : (a) By measuring the depression of a barometric column of mercury, due to saturated vapor formed in a vacuum. (b) By determining the boiling point under different pres- sures. The curve showing the vapor tension as a function of the temperature, is called the vapor tension curve. VAPOR PRESSURE 127 o 97, Exercise 46. Vapor Tension of Ether. The apparatus, Fig. 50, consists of a U-shaped tube with unequal arms, of which the shorter, about 50 cms in length, is closed and the other open. The short arm and part of the longer arm of the tube is first filled with mercury, the mercury being boiled in the tube to expel the air and then a small quantity of air free ether is introduced into the shorter arm. Insert the tube into a tall beaker filled with water and furnished with a ther- mometer and a stirrer. Vary the temper- ature of the water between 10 and 40 C, and measure the vapor pressure for each temperature with a cathetometer or with a meter stick. Care must be taken not to raise the temperature of the water at any time much above the normal boiling point of ether. Enough ether should be in the tube so that at any temperature obtained some of it remains in liquid form. Note the barometric reading. Plot the total pressure in cms of mercury as a function of temperature and determine from the curve the boiling point for a pressure of 760 mms of mercury. FORM Of RECORD. Exercise 46. To determine the vapor tension of ether. Fig. 50. Temperature Reading on short arm On long Date, Difference P in cms of Hg Normal boiling point 98. Exercise 47. Vapor Tension of Water at Various Tem- peratures. As pointed out in the previous exercise, a liquid be- gins to boil as soon as its vapor tension equals the pressure upon the liquid. The easiest way therefore to determine the vapor tension of a liquid at temperatures near its boiling point, is to 128 PHYSICAL, MEASUREMENTS vary the pressure upon the liquid and determine the temperature at .which it will boil under the new pressure. The apparatus Fig. 51, consists of a metallic flask A, contain- ing the boiling liquid, and furnished with a tight fitting rubber stopper, through which pass a thermometer and the glass tube leading to the condenser C. A second flask B is connected to the upper end of the condenser and to a stopcock tube to which Fig. 51. the small mercury manometer M is fused. This tube can be closed against the external air by the stopcock V. The reading of the barometer at the time of the experiment plus or minus the difference in the height of the mercury in the two arms of the manometer gives the pressure under which the liquid boils, and the thermometer gives the corresponding temperature. A simple method of increasing the pressure above the liquid is to start with the stopcock V closed when the flame is placed under the flask A. Soon the pressure will rise. If necessary the pressure may be reduced by small steps by quickly turning the stopcock around 180 degrees. VAPOR PRESSURE 129 In order to prevent radiation from the thermometer bulb towards the cooler outside it is advisable to surround it by a cylinder of asbestos. Vary the pressure over ranges assigned by the instructor, and note the temperatures at which boiling takes place in each case. Put glass beads in the flask A, to avoid bumping. Plot pressures and temperatures. FORM OF RECORD. Exercise 47. To determine the vapor tension of water at temperatures in the neighborhood of the boiling point. Date. Barometer reading... Thermometer No. . . . Temperature Manometer Tube a I Tube b CHAPTER VIII. ELECTRICAL MEASUREMENTS. UNITS AND STANDARDS. 99. Resistance. The practical unit of resistance is the ohm. It is represented by the resistance offered to an unvarying cur- Fig. 52. rent by a column of mercury at the temperature of melting ice, 14.4521 grams in mass, of a constant cross-sectional area and of length 106.3 centimeters. For practical purposes standard ohms and multiples of the ohm are made of coils of wire, usually of manganin, an alloy whose resistance varies but little with the temperature, and which has a small thermo-electromotive force against copper, mounted in suitable protective cases. The makers usually furnish certificates for these coils issued by the testing laboratories of the different countries. Fig. 52 represents some types of the Ger- man or Reichsanstalt standards. Their re- sistance varies slightly with humidity. This has led to the construction of standards in hermetically sealed Fig. 53. ELECTRICAL UNITS 13! cases filled with oil which has been carefully freed from moisture and air. These standards were proposed by the Bureau of Stand- ards in Washington and are called the N.B.S. standards of resist- ance, (Fig. 53). For ordinary measurements, resistance boxes containing a number of coils of insulated wire, wound on bobbins non-induct- ively, 1 are in general use. The top of the box containing the coils is usually of ebonite and car- ries on its upper surface a number of heavy brass blocks so arranged that connection between adjacent blocks may be made by means of plugs inserted between them. The ends of the separate coils (Figs. 54 and 55), are fastened to the ends of adjacent blocks, so that when. any plug is removed the cur- Fig. 54- rent passes from one block to the next by passing through the connecting coil. In this way any resistance may be added by removing the proper plugs. When the plug is inserted the current passes from block to block through the plug itself, the resistance of which must of course be negligible. On this account the plugs must fit accurately and be kept bright and clean, the ebonite must at all times be kept free from dust or moisture and never be allowed to stand in the sun, as the ebonite disintegrates slowly under the action of sun- light, a conducting layer of sulphur forms on the surface, and the efficiency of the coils is impaired. The plugs should at all times be handled by their hard rubber tops, they should be inserted with a gentle twisting motion, and should all be loosened before the box is returned after use. *Two methods of non-inductive winding are used; in the one case the required lengjh of wire is measured off, doubled upon itself and then wound so doubled, upon the spool. In the second method the wire is wound in one direction only in each layer, but in opposite direction in consecutive layers. By this method the capacity of the coil, which by the first method may be quite appreciable, especially in coils of high resistance, is much reduced. (Chaperon.) 132 PHYSICAL MEASUREMENTS In the common form of resistance boxes there are only four coils for each decade, namely of I, 2, 3 and 4 units, or I, i, 2 and 5 units (Fig. 86). Fig. 55- In the "decade" resistance boxes the inconvenience of adding up the resistances of the separate coils has been avoided, see Fig. Fig- 56. Fig. 57. 56. In each row of blocks representing a decade a plug is placed between two blocks and inserts in the circuit as many units as are indicated by the number on one of the blocks. UNITS 133 An ingenious method in which only four coils are used foi each decade has been devised by Leeds and Northrup. The ar- rangement is shown by Fig. 57. In each decade there is one re- sistance coil of i unit, one of 2, and two coils of 3 units ; for example if a plug be inserted between the two blocks, marked "4" the current must pass through resistances I and '3', while 2 and 3 areshortcircuited. These boxes combine the advantage of the decade plan with cheap- ness of construction. In the "dial" resistance boxes, built on the decade plan, the blocks are arranged in a circle and contact with a central ring is made by a slid- ing contact rotating around the center of the circle, (see Fig. 58). F^ 58. Previous to the adoption of the ohm various other units of re- sistance were employed and resistance boxes representing these earlier standards are sometimes met in practice. The most com- mon are the British Association Unit, proposed by the British Association in 1864, an d the legal ohm adopted at the Paris Con- gress in 1884. The relation between the ohm and these units is as follows: i ohm = 1.01358 B. A. units = 1.0028 legal ohms, i B. A. unit = 0.9866 ohm. i legal ohm = 0.9972 ohm. In reporting work in measurements of resistance the student should state explicitly which units have been used. The term rheostat is used for an unknown resistance of either fixed or variable value, and employed in work with currents ex- ceeding o.i ampere. The ends of the lead wires connecting the different instruments should be bright and clean and clamped firmly by the binding posts, since loose connections offer high and variable resistance. It is a bad practice to connect two wires by twisting them together. 134 PHYSICAL MEASUREMENTS 100. Current. The practical unit of current is the ampere. The ampere is represented sufficiently well for practical purposes by "the unvarying current which will deposit silver from silver nitrate at the rate of 0.001118 grains per second." 101. Electromotive force. The practical unit of electromo- tive force is the volt. The volt is the electromotive force which steadily applied to a conductor whose resistance is one ohm will produce a current of one ampere. The Cadmium cell is chiefly used as the practical standard of electromotive force. This cell has for its positive electrode mercury covered by a paste of mercurous sulphate and a saturated solution of cadmium sulphate, and for the negative electrode cadmium amalgam, placed in the other leg of the H-shaped vessel, (Fig. 59), and covered by a saturated CdSO 4 Crystals Hg 2 S0 4 CdSO 4 Crystals Cd Amalgam Fig- 59- solution of cadmium sulphate. Saturation of the cadmium sulphate solution is secured by an excess of cadmium sulphate crystals. The glass tubes are hermetically sealed. The E. M. F. of this cell at the temperature tC is given by the formula E t = 1.0183 0.00004 (t 20) volts. (i45) The legal form of this cell is not portable and suffers the addi- tional disadvantage of a slight lag in the E. M. F., since the dens- ity of the solution adjusts itself to a new temperature but slowly, some time being required for the cadmium sulphate crystals to dissolve or to crystallize out. This cell should never be placed on a short circuit. (Why?) Some older forms of standard cells are still in frequent use. ELECTRICAL UNITS 135 The Clark cell, the first efficient form of standard cell, differs from the cadmium standard cell by the use of zinc and zinc sul- phate in place of cadmium and cadmium sulphate. In other respects its construction is identical with that of the cadmium cell. The E. M. F. of the Clark cell is given by the formula H t = 1433 0.00119 (t 15) volts. (146) The Carhart-Clark cell is an unsaturated form of Clark cell and its E. M. F. is about E t = 1.44 0.00056 (t 15) volts. (147) The Western Company manufactures portable unsaturated cadmium cells, which are extremely convenient on account of their negligible temperature coefficient. Their E. M. F. is very con- stant and about 1.019 volts; the exact value being readily deter- mined by comparison with a primary standard. Standard cells are used in all cases where it is necessary to determine the absolute value of an E. M. F. or of a 'potential difference. In many cases however, we wish simply to compare deflections of a galvanometer. In others the conditions may be so chosen that no current passes through the galvanometer. Such a method is termed a zero method. In experiments employing the zero method the Leclanche battery may be used. Where steady deflections are required however, a cell of constant electromo- tive force is necessary. In such cases we may use either a Daniell cell or a storage battery. To set up a Daniell cell (Fig. 60), first fill the porous cup con- taining the amalgamated zinc, or negative electrode, two-thirds full of zinc sulphate solution and wait until the solution begins to moisten the outside of the cup, before placing the cup in the glass jar containing the copper, or positive electrode, and the copper sulphate solution. In order to avoid changes in the internal resistance during the experiment it is well to short circuit the cell for ten minutes before using. Fl - 6o - After use the cell must be taken apart, the zinc sulphate poured 136 PHYSICAL MEASUREMENTS back into its bottle, the porous cup thoroughly washed and the zinc rubbed clean. Copper oxide is usually deposited on the zinc as a black film. This may be readily rubbed off while it is moist, but. if it be allowed to dry it adheres firmly and renders it difficult to amalgamate the zinc again. The E. M. F. of a Daniell cell is about i.i volts and this value may be used in all cases where great accuracy is not required. When a constant E. M. F. of more than one volt is required, a storage battery (E = 2.2 volts) may be employed. On account of the very low internal resistance of a storage battery great care must be exercised to avoid short circuiting the cell. The student using a storage battery should always leave one of the electrodes disconnected until the instructor has seen and approved of the arrangement of the apparatus. 1 02. Quantity of electricity. The practical unit of quantity is the coulomb. The coulomb is the quantity of electricity trans- ferred by a current of one ampere in one second. 103. Capacity. The practical unit of capacity is the farad. The farad is the capacity of a condenser which is charged to a potential of one volt by a quantity of one coulomb. The micro- farad = icr 6 farad, is commonly used as the measure of capacity. Secondary standards of capacity are made in the form of con- densers with solid dielectrics. A large number of sheets of tin- foil interleaved with mica or paraffined paper are placed in a bath of melted paraffin in a vacuum chamber, to remove air bubbles, and allowed to cool. Alternate sheets of tin-foil are then con- nected and joined to one binding post, the remainder to another, thus forming the two ends or terminals of the condenser. A sub- divided condenser is made by connecting all the leaves of one set t0 ne bar> Earth, ( Fi 6l )' - as a terminal, and dividing the leaves of the other set into some number of divisions each of which is connected to a sep- Fig. 61. arate bar which in turn may be ELECTRICAL, UNITS 137 connected to a second binding post by means of a plug. When any division is to be used the plug is inserted for that division. In another form, made by Leeds and Northrup, (Fig. 62), the subdivisions of the condenser are placed between two ad- joining crossbars, thus allow- ing their use in series as well as in parallel. Where must the plugs be inserted to obtain I mf., 0.4 mf. or 0.25 mf ? Fig. 62. 104. Selfinductance. The practical unit of selfinductance is the henry. A henry is the selfinductance in the circuit when the E. M. F. induced in the circuit is one volt, while the inducing current varies at the rate of one ampere per second. The usual form of selfin- ductance (Fig. 63) ), consists of two coils in series, one fixed and the other movable about a diameter of the fixed coil as an axis. The movable coil may be rotated through 180 and in this way the selfinductance may be varied considerably. The instrument is calibrated empir- ically. The coils are wound on wood, and metal is, so far as possible, entirely avoided in the pj g 6 3 construction. Other standards of inductance (Fig. 64), usually of single value only, are wound upon spools of non- magnetic and nonconducting material, such as mahogany, marble or soapstone. The last is, however, generally magnetic in slight degree, and standards wound upon soap- stone should be mistrusted. Fig. 64. 138 PHYSICAL MEASUREMENTS INSTRUMENTS. 105. Keys. In most forms of apparatus for making electrical measurements, it is necessary to allow the current to pass through the measuring instrument for but a short space of time. Keys are provided for the purpose of closing and opening the circuit as may be desired. The ordinary key is so arranged that when pressed down, contact is made between two platinum points and the circuit is closed. On releasing the key the circuit is opened. For closed circuit work, plug keys or knife switches of various forms are used. In work with the Wheatstone's bridge it is necessary to close the circuit through the galvanometer, after the current in the arms of the bridge has reached a constant value. For this pur- pose a double key (Fig. 65) is provided, in which the contacts for the circuit through the bridge arms, and that through the gal- vanometer, are made successively in the order mentioned. Such a key is termed a successive contact key. Fig. 65. Fig. 66. It is frequently necessary to reverse the direction of the current through an instrument without loss of time. This is most con- veniently effected by means of the Pohl's commutator (Fig. 66). This consists of four cups containing mercury connected by cross wires as indicated in the figure, and a light frame of wires by which two other cups at the end of the block may be put in con- nection with the pair of cups on either side. If the source of current be connected to the pair of binding posts at the ends of the block, and the galvanometer to the two binding posts on either GALVANOMETERS 139 side, the current is passed through the instrument in one direction or the other by tilting the frame from side to side. All mercury contacts should be kept clean and should have the ends of the metal dipping into the mercury well amalgamated. 1 06. Galvanometers. The space about a magnet or about a wire carrying a current of electricity is called a magnetic field. Such a field is conceived to be filled with lines of magnetic induc- tion. The direction of these lines is assumed to be the direction along which a free north seeking pole would tend to move. The lines of magnetic induction are said to run out from the north pole of a magnet, curve round through the air and re-enter the magnet at the south pole. In the case of a wire carrying an elec- trical current the lines of induction are concentric circles sur- rounding the wire. Whenever two magnetic fields are brought near to each other there arises a stress between them, tending to turn the fields into such a position that they will mutually include the greatest num- ber of lines of induction. Obviously the moment of this stress will be greatest when the two systems of lines of induction stand at right angles to each other. This is the position adopted in gal- vanometers and electro-dynamometers. In instruments de- signed to measure currents, at least one of the magnetic fields must arise from the current flowing through a coil of wire and hence must vary as the strength of the current. 1 The other magnetic field may be produced by a .permanent magnet as in the galvanometer, or by a second coil carrying a current, as in the electro-dynamometer. One of the magnetic fields must be capa- ble of rotation. Galvanometers may be divided into two classes : 2 (a) Galvanometers with stationary coil and movable system of magnets; needle type; (Fig. 67). 1 College Physics, Article 257. - College Physics, Article 260 and 261. 140 PHYSICAL MEASUREMENTS (b) Galvanometers with stationary magnets and movable coil; d'Arsonval type. (Fig. 68.) In all measuring instruments the deflecting mo- ment due to the current to be measured, must be balanced against a restoring or directing moment which tends to restore the system to its original position. When the system thus subjected to the action of two moments comes to rest we know that the moments of the deflecting and restoring forces are equal and opposite in direction. The angle of deflection is determined in one of several ways and the current determined as a function of this angle. The directing moment may be due to the action of an independent magnetic field upon the movable magnetic needle, or to the torsional Fig. 67. moment of the suspending wire. The sensitiveness of a galvanometer may be increased by de- creasing the directing or restoring moment. 107. The Astatic Galvanometer. In the astatic galvanom- eter the moving system is composed of two magnets or systems of magnets of nearly equal strength placed one above the other with poles opposed. One of the needles is placed within a coil, the other needle either outside, or, better still, enclosed in a second coil through which the current flows in the opposite direction. In this way the magnetic moment of the system with respect to the earth's field is greatly reduced. On the other hand a magnet placed under or over a movable magnetic needle may be made to exert any desired directive mo- ment. Such a magnet is termed a controlling magnet. Some very sensitive galvanometers employ both an astatic system and a controlling magnet, the latter being placed in such a direction as to oppose the magnetic field of the earth. 1 08, The d'Arsonval Galvanometer. The d'Arsonval gal- vanometer, (Fig. 68), is more convenient for ordinary measure- ments. The coil (Fig. 69) is suspended by means of a fine metal wire or ribbon, and is attached to the base of the instru- GALVANOMETERS 141 ment by another wire or metallic spiral spring. The current en- ters and leaves the coil by these upper and lower suspensions. The magnetic field of the permanent magnet is directed from N to S across the gap in which the coil is suspended. The current flowing through the coil sets up a magnetic field whose direction may be found by the right hand rule 1 , and is indicated in the fig- ure by the crosses X, where the lines of induction enter the plane of the paper and by the dots , where the lines come out. The Fig. 68. Fig. 69. coil tends to place itself so that its lines of induction inside the coil are parallel to those of. the field of the stationary magnet. Thus with the current flowing as indicated, the side ab tends to- rise out of the paper. The coil comes to rest when the deflecting moment, due to the action of the fields, equals the torque iir the suspension. The finer the suspension, the more sensitive is the instrument. Deflections are usually observed by means of mirror and scale. The d'Arsonval galvanometer has the great advan- tage that on closed circuit considerable damping effect is pro- duced by electro-magnetic induction in the moving coil. Why is this? The degree of damping depends largely upon the resist- 1 College Physics, Article 256. 142 PHYSICAL MEASUREMENTS ance of the circuit, being larger, the smaller the resistance. For each galvanometer a resistance of the circuit can be found, for which the swing of the coil changes from an oscillatory to an aperiodic motion. This limiting resistance is called the critical resistance of the circuit, and it can be shown that with this crit- ical resistance the action of the galvanometer is quickest. The instrument is also practically independent of the surround- ing magnetic field and is consequently free from disturbances arising from variations in the intensity of this field, which often prove very troublesome in galvanometers of the needle type. The d'Arsonval galvanometer has the additional advantage that it may be placed in any position, while instruments of the needle type must be set so as to have the plane of the coils in the magnetic meridian. The needle type however, has usually greater sensitive- ness, although the d'Arsonval type is sufficiently sensitive for most purposes. 109. Methods of Observation. In some instruments the mov- ing system is furnished with a light pointer playing over a scale, but in the more sensitive galvanometers the deflections are ob- served by means of a mirror attached to the moving system. This mirror may be either concave or plane. In the first case an illum- inated slit is focused by the mirror upon a semi-transparent scale and the deflections are read directly from the scale. In the sec- ond case a telescope is employed to view the image of a scale reflected in a plane mirror (see Article 29). Both methods are in common use. no. Shunts. It frequently happens that the current to be measured will produce a deflection too great to be observed, or in some cases it may even endanger the instrument itself. In such cases we may reduce the current flowing through the galvanom- eter by means of a resistance connected in parallel with it. This resistance is called a shunt. Let g be the resistance of the galvanometer, s that of the shunt, then the resistance of the two GALVANOMETERS 143 circuits in parallel is gs , and by Ohm's law, 1 if / denote the 9 ~T~ S total current and 7 g the current through the galvanometer, we have or If we observe the current I g in the galvanometer, then the total current is I e . 9 "*" s ; where the factor g + s is called the multiply- s s ing power of the shunt. Let the galvanometer resistance be n times that of the shunt, (g==ns), then the multiplying power is . If n be 9, 99 or 999, the corresponding values of the multiplying power are 10, 100, and 1000. The makers frequently furnish with the galvanometer shunt-boxes containing resistances equal to 1/9, 1/99, and 1/999 of that of the galvanometer, in which case the observed current 7 g , is equal to o.i, o.oi, or o.ooi /. in. Exercise 48. Calibration of a Galvanometer by Ohm's Law. The object of this experiment is to determine whether the deflections of a galvanometer are proportional to the current flowing through it, or if that is not the case, to ascertain how the deflections vary with the current. To obtain currents which stand in a definite relation to each other we apply Ohm's law, by connecting the terminals of the galvanometer to different points of a circuit through which a constant current is flowing. Then the potential difference, P. D., between the extremities of a re- sistance r, through which a current i is flowing, is equal to ir, and this potential difference causes the current through the gal- vanometer producing the observed deflection. The simplest arrangement is to use a battery of constant E. M. F. 2 and to send the current through a straight wire, as for ex- *It should be kept in mind that Ohm's law applies to constant cur- rents only. 2 The student may observe as a general rule, that a battery of constant E. M. F. is always to be used where constant deflections are to be obtained while ordinary cells may be employed for zero methods or ballistic methods. 144 PHYSICAL MEASUREMENTS Q ample, the wire of a slide wire bridge, (Fig. 71), and connect the galvanometer to two points, P and A, on this wire, where A denotes the position of the contact maker. Since the current through the wire must remain constant, no matter where the gal- vanometer is attached, it is evident that the galvanometer should have a very high resistance as compared with that of the wire. Explain this. The resistance of the wire of a slide wire bridge is usually about 0.2 of an ohm. If the galvanometer resistance is less than 2000 ohms, a resistance box R 2 , with sufficient resist- ance to bring the re- sistance of the galva- nometer up to 2000 ohms, should be put in series with it. The re- sistance box R! is inserted in the battery circuit in order to reduce the potential difference between P and A to a value such that with the maximum length of wire used in the experiment the deflections of the galvanometer will still be on the scale. Move the point A, an ordinary contact maker, along the wire by steps of 5 cms , from 5 to 95 cms on the scale and observe the successive deflections of the galvanometer. Next reverse the bat- tery current and repeat the observations. The mean of the de- flections and the corresponding lengths between P and A are plotted. The resulting curve will be a straight line, if the deflec- tions of the galvanometer are proportional to the current flowing through it. What principle besides Ohm's law has been applied in this experiment? FORM OF RECORD. Exercise 48. To calibrate a galvanometer by means of Ohm's law. Galvanometer No Date Resistance boxes Ri R Length Deflection a Deflection b Mean Deflection 112. Exercise 49. Figure of Merit of a Galvanometer. The figure of merit /, of a galvanometer is the ratio of the current GALVANOMETERS 145 to the deflection which it produces and is measured by that cur- rent which will produce a deflection of one scale division. In case the scale is movable the distance of the scale from the mirror must be speci- fied, or, still better, the result should be calculated for a distance of 100 cms. This exercise furnishes a sim- ple application of Ohm's law. The exercise may be performed in either of the following ways : (a) The arrangement is as shown Fig. 72. in Fig. 72. Let B denote a battery of constant electromotive force E; R a very high resistance; g and b the resistances of the galvanometer and battery .respectively ; then 1 = if no shunt is needed, and R+9+b ' (150) (151) if a shunt is used. If the galvanometer show a deflection d, on the passage of a current I gJ through it, we may assume, in the great majority of cases, that this deflection is proportional to the current, and we have the relation fd = I ' (152) The factor / is the figure of merit of the galvanometer. Hence in the first case d and in the second case f=^- = s (153) d54) Usually b as well as g, and still more L- , are negligible in comparison with R, and the formulae become much simpler. 146 PHYSICAL MEASUREMENTS (&) For sensitive galvanometers which have no permanent shunt the following method is convenient: Let B, (Fig. 73), be a cell of constant electromotive force. E, and close the circuit through P and Q. Then take the potential difference over Q to produce the deflection of the galvanometer. Q must be very small in comparison with R. Now the ap- plied potential difference is ^ E and our formula for / becomes Fig. 73- d55) P + Q In practice vary the resistance R between 150000 and 250000 ohms and observe the deflections; in case (b) the ratio may also be varied. The term sensitiveness of a galvanometer is frequently used in a different sense. It may be defined as the resistance which the circuit must have in order that one volt may produce unit deflec- tion. This is termed the ohm sensitiveness of the galvanometer, (156) The potential difference per unit deflection is called the volt sensitiveness of the galvanometer, a factor of the high- est importance in most electrical measurements, as for example in work with the Wheatstone bridge, the potentiometer or the thermal couple. If in Fig. 73 the electromotive force of the battery be known, then the volt sensitiveness of the galvanometer can be shown directly to be (157) In order to give a definite meaning to this term, the resistance GALVANOMETERS 147 in the galvanometer circuit (d'Arsonval galvanometer) should be the critical resistance. Many makers give the sensitiveness at the terminals of the galvanometer, but this is too indefinite since frequently, owing to excessive damping, the galvanometer can- not be used upon short circuit. FORM OF RECORD. 'Exercise 40. To determine the figure of merit of a galvanom- eter. Galvanometer No g = Distance of mirror from scale Date. Q E R Q P P + Q d f f(R + 9) 113. Ballistic Galvanometers. While in ordinary galvanom- eters it is required to observe deflections due to a steady current flowing through the instrument, or to prove the absence of a cur- rent from the absence of a deflection, it is often necessary to measure the quantity of electricity passing through the galvanom- eter, as in measuring the quantity of electricity stored in a con- denser. When a condenser is discharged through a galvanometer the current rises rapidly to a maximum, and then decreases to zero. In such a case a galvanometer having a coil with a large moment of inertia must be employed. Such an instrument is termed a bal- listic galvanometer and its advantage consists in this, that the coil remains practically at rest until the entire quantity of electricity has passed through it. In this way the full force of the magnetic thrust is effective in starting the coil which moves off as if started by a blow. For small angular deflections the quantity of electric- ity may be set proportional to the deflection. 1 Here we observe the maximum deflection attained by the system on the first throw of the needle, and not a constant deflection. The quantity of 1 Carhart and Patterson, Electrical Measurements, pp. 207-213. I 4 8 PHYSICAL MEASUREMENTS electricity per unit deflection is called the constant of the ballistic galvanometer, or if the quantity Q gives a deflection d, then c= & (158) 114. Constant of Ballistic Galvanometer. The constant of a ballistic galvanometer, especially in instruments of the d'Ar- sonval type, depends largely upon the resistance of the galvan- ometer circuit. In the following exercise it is determined with the galvanometer on open circuit, in which case it matters not how much resistance we add to the galvanometer. However the value thus found cannot be applied to the instrument when used upon a closed circuit, without taking into account the damping effect. This must be remembered in Exercises 74, 78 and 79. 115. Exercise 50. Determination of the Constant of a Bal- listic Galvanometer. To determine the quantity Q giving a cer- tain deflection d, use a battery of known electromotive force E, as a standard cell, and charge a condenser of known capacity by depressing the key to the point b (Fig. 74). Then by releasing the key, the condenser is disconnected from the battery and discharged through the galvanometer. Special keys, used for such experiments, known as charge and dis- charge keys, are shown in Fig. 75. Fig. 74. Letting c represent the constant of the ballistic galvanometer, Q the quantity discharged, and d the de- flection, then s-\ r* /- (159) Fig. 75- GALVANOMETERS 149 where C is the capacity of the condenser and H the E. M. F. of the standard cell Change C by small steps and plot Q and d. FORM OP RECORD. Exercise 50. To determine the constant of a ballistic galvan- ometer. Date E C Q d c Galvanometer. . . Condenser Temperature Standard cell. . Mean =:.... 116. Voltmeters and Ammeters. Voltmeters and ammeters are usually portable galvanometers of the d'Arsonval type. Wall instruments designed to measure high voltages or large currents, as in electric lighting or power stations, are usually attached to the switch board directly and give continuous indication as to the pressure and strength of the current furnished. Voltmeters and ammeters are direct-reading instruments, that is, they are so Fig. 76. Fig. 77- calibrated as to show directly upon an arbitrary scale the differ- ence of potential existing, or the current flowing between any two points to which they may be connected. The best known instruments of this class are those designed by Weston. In these the directing couple is furnished by two spiral springs of phosphor-bronze. Rapid damping is secured by the use of aluminum frames upon which the coils are wound. The I5O PHYSICAL, MEASUREMENTS voltmeter (Fig. 76) is commonly provided with two scales, one for high and one for low voltages. In the figure the low voltage terminal on the negative side is marked 15. Back of the needle is a mirror, and placing the eye in such a position that the image of the needle is hidden by the needle itself the error, due to par- allax, in reading the scale is avoided. The Weston instruments are durable and remarkably accurate and the student should thor- oughly familiarize himself with them before attempting to use them independently. ELEMENTARY EXERCISES. 117. Exercise 51. Cells in Series and in Parallel. This simple exercise, designed to give the student some practice in the use of voltmeters and ammeters, consists in the study of the ef- fect which a different grouping of cells has upon the current in an electric circuit. Use new drv cells for this exercise. Fig. 78. (a) Connect a voltmeter V to the terminals of a cell B, being careful to join the binding post of the voltmeter, marked -)-, to the positive pole (Fig. 78). Then replace the cell at B, first, by two cells in series and, second, by two cells in parallel. Take the readings in the three different cases and note the effect. Should the reading with the two cells in series be exactly twice that ob- tained with one cell? EXERCISES (b) Arrange a circuit, consisting of a cell B, a rheostat R of about TO ohms and a milammeter A, (Fig. 79). Again re- Fig. 79- place the cell at B, first, by two cells in series and, second, by two cells in parallel. Note the effect of the different arrangements upon the reading and discuss the result. FORM OF RECORD. Exercise 51. Study the effect of different arrangements of cells. Voltmeter No, Ammeter No. Voltmeter reading Date Ammeter reading One cell Two cells in series Two cells in parallel Discussion of results. 118. Exercise 52. Kirchhoff's Laws. Kirchhoff's laws 1 may be studied in the following manner: (a) Arrange a battery B in series with three rheostats R lt R 2 and R s , (Fig. 80). Place an ammeter or milammeter successively in the positions a, b, c and d. Take the readings of the ammeter in each case. Does the current change in passing through a re- sistance? (b) Arrange the rheostats in parallel (Fig. 81), and take the 1 College Physics, Article 271. 152 PHYSICAL MEASUREMENTS readings of the ammeter, successively in positions a, b, c and d. Discuss the results as examples of KirchhofFs first law : (c) With the rheostats in series (Fig. 80) place a voltmeter t> b ! i Fig. So. Fig. 81. successively over ab, be, cd and ad and read the differences of potential in each case. (d) With the rheostats in parallel (Fig. 81) take the readings of the voltmeter over ab, ac, and ad. Discuss the results as ex- amples of KirchhofFs second law : It should be noted that an ammeter is always placed in series with the circuit and that a voltmeter is placed as a shunt between the two points whose potential difference is to be measured. FORM OF RECORD. Exercise 52. KirchhofFs laws. Date. Position of ammeter Rheostats in series Position of ammeter Rheostats in parallel a a 7 d d Position of voltmeter Position of voltmeter ab ab be ac cd ad ad Discussion of results. ELEMENTARY EXERCISES 153 ng. Exercise 53. Resistances in Series and in Parallel. In order to prove the laws of resistance 1 , a circuit is arranged consisting of a battery of two cells, a milammeter and three resistance boxes in series, with resistances R^ = 5, R 2 = 8 and R & 20 ohms respectively. Note the reading of the ammeter /. Take two boxes out of the circuit and adjust the resistance R in the remaining box until the ammeter reading / is the same or nearly the same as before. Be careful to have always some re- sistance in the circuit so as to avoid shortcircuiting the cells. The resistance R should agree with the equation R = Ri -j- Ra -j- Ra ( l6o) Next, arrange the three resistance boxes in parallel, making the resistance of each 15 ohms. Note the reading of the ammeter 7 V Remove one of the boxes and make the resistances of the remaining two boxes 15 and 10 ohms respectively. Note the read- ing of the ammeter, I 2 . Finally leave only one box in the circuit and adjust its resistance R' until the current I\ is exactly or nearly equal to / ; adjust again to R" until the current is r z = h. The results should agree with the equation -4= 2 i- (161) Give a sketch of the arrangement of apparatus for each of the different cases. FORM OF RECORD. Exercise 55. To study the laws of resistance. Date a. Resistances in series : R = I b. Resistances in parallel : R'= R s = r,= 1 College Physics, Article 276 and 277. 154 PHYSICAL MEASUREMENTS MEASUREMENT OF RESISTANCE. 120. Exercise 54. Resistance by Substitution. The ap- paratus is arranged as shown in Fig. 82. B is a battery of con- stant E. M. F., x the unknown re- sistance, R a resistance box, G a galvanometer, K a key which may connect the galvanometer either to x or to R. The battery circuit is first closed through the resistance x, and the galvanometer. If the deflection be too great, the controlling magnet or the torsion head may be so turned as to bring the deflection back upon the scale, or a shunt may be applied to the galvanometer. Do not use a Fig. 82. high resistance in the circuit instead of a shunt (Why?). Assuming that the deflections are porportional to the current we have i = c 1 1 = c . (162) where r is the resistance of the whole circuit except x, and c is the proportionality factor. Next send the current through R instead of .r; then E R + r (163) If now R be adjusted until the deflections in the two cases are equal, then x = R. (164) If R cannot be adjusted so as to produce exactly the same de- flection as x, interpolate between the two nearest values above and below. Measure three resistances separately and in series. RESISTANCE 155 FORM OF RECORD. Exercise 54. To measure three resistances by substitution. Date I Deflection with R 2 4 , ohms. Galvanometer Resistance of Deflection with x Temperature Resistance box. Deflection with + Xi -f- X* = 121. Exercise 55. Resistance by Voltmeter and Ammeter. This exercise consists in measuring at the same time, the cur- rent flowing through a wire and the difference of potential at its terminal points. Then if x be the resistance of the wire, we have by Ohm's law V x-=. (165) The potential difference at the terminals is measured by means of a voltmeter whose resis- tance must be very high in comparison with the resist- ance to be measured, as other- wise the current through the resistance x, will not be the * total current I, measured by <> b x a Fig. 83. the ammeter. This is easily seen by applying the shunt rule, (166) where 7? v is the resistance of the voltmeter. If the resistance x is too large, it is better to include the ameter, whose resistance is very small, together with the resistance x between the terminals of the voltmeter, as shown by the dotted lines in Fig. 83, and then subtract the resistance of the ammeter from the resulting value of x. Measure the resistance of an incandescent lamp. Put a rheostat in series with the lamp and the ammeter. Cut out resistance from the rheostat step by step until the lamp has reached its full candle power. Observe at each step the readings of the volt- meter and ammeter. Calculate the resistance for each current. 156 PHYSICAL MEASUREMENTS and the watts absorbed by the lamp; also the watts per candle when the lamp is burning at its normal voltage. FORM OF RECORD. Exercise 55. To measure the resistance of an incandescent lamp by voltmeter and ammeter. Date Voltmeter No Milammeter Name of Lamp ' Candle power V 1 R Watts Watts per candle = 122. Exercise 56. Very High Resistance by Direct De- flection. This method is a modification of the last. Instead of an ammeter, a galvanometer (Fig. 84) is used whose figure of merit is determined as in Exercise 49. The formula for the resistance then becomes x=j- d g, (167) since / / d. Usually g, the resistance of the galvanometer, is negligible in comparison with the high resistance X. High resistances of this kind are insulation resistances, as for example, the resistance offered to the passage of a current by the insulation of a cable or of a con- denser. It is advisable to insert a high resistance H.R. in series with the galvanometer, in order to pro- tect the instrument from excessive currents in case of a break in the insulation. If the deflections vary notably with the time, it is advis- able to keep the key K closed and g 4 take a series of readings at definite time intervals. // X represents the resistance of a condenser, short circuit the galvanometer by means of a shunt key, before closing the key K, and afterwards open the short circuiting key to observe the steady deflection. RESISTANCE 157 Date. .... Time V d X FORM OF RECORD. Exercise $6. To measure the resistance offered by the insula- tion of a commercial condenser. ' Condenser Galvanometer Figure of merit, (Exercise 49) Resistance of the galvanometer 123. The Wheatstone Bridge. The most accurate methods for measuring resistance are based upon the Wheatstone bridge principle. In its simplest form the Wheatstone bridge consists of a network of six conductors joining four points, A, B, C, D, (Fig. 85), so arranged that each point is joined to each of the other three points by separate conductors. Let one of the conductors contain a source of E. M. F. ; four of the others will form a divided circuit while the remaining one, containing a galvanometer, will form a bridge between the two' parallel conductors. Let R^, R 2 , R s , R be the resistances forming the four branches of the divided circuit, and suppose them to be so adjusted that no cur- rent flows through the galvan- ometer Zero method. Then it may be shown that the resist- ances satisfy the relation R^ E*. R* (168) For let the potentials of the four points be represented by F & , F b , F c , F d , then since there is the same fall of potential be- tween A and B, whether we pass by one route or the other, and since with a constant current the fall of potential is at all times proportional to the resistance passed over, we have Fa Fc and Fa Fj Fa V b R, (169) 158 PHYSICAL MEASUREMENTS But since no current passes through the galvanometer P. Fc=Fa Fd therefore R whence ^ = #r (I7I) From the above relation it is evident that if three of the resist- ances be known the fourth may at once be determined. In fact it is necessary to know but one resistance and the ratio between the other two. Since the current through the parallel branches should become steady before the potential at C and D are tested, it is necessary to close the galvanometer key last in every case. A successive contact key is best adapted to this work. What would be the effect of selfinductance in one of the branches? It will be found advantageous to follow Maxwell's rule : x "Of the two resistances that of the battery and that of the galva- nometer connect the greater resistance so as to join the two greatest to the two least of the four other resistances." This insures the greatest sensitiveness of the apparatus. vSince both the potentials at C and at D depend directly upon the difference of potential between A and B, the absolute value of the latter or a slow change in the E. M. F. of the battery will not affect the balance of the galvanometer. A Leclanche cell may therefore be used as the source of current. The temperature of the room must be carefully noted. 124. Exercise 57. Resistance by Wheatstone Bridge Box. A very convenient form of apparatus for the measurement of resistance by the Wheatstone bridge method is a box (Fig. 86), containing three known resistances connected in series, one of which may be given any value between one ohm and the maximum, Maxwell, Electricity and Magnetism, jd edition, Vol. I, p. 478. RESISTANCE 159 00000000 .50 20 10 10 5 2 1 1 usually 10,000 ohms. The other two resist- ances form the pro- portional arms of the Wheatstone bridge and contain but a few coils, usually I, 10, 100 and 1000 ohms. In this way the ratio between the known and the unknown re- sistances may be Fig. 86. varied from 1000 to o.ooi. The galvanometer is generally con- nected to the outer ends of the proportional coils and the battery to the remaining two binding posts. Frequently the two keys and a galvanometer are included in the case, thus making it a very compact and convenient instrument for the measurement of re- sistance. (Fig. 87.) This form of Wheatstone bridge is also known as the Postoffice Box. In practice, especially where the unknown resist- ance is not even approxi- mately known, it is well to begin with equal resistances in the proportional arms R 2 and R. If the unknown re- sistance RI is too large the deflection on closing the galvanometer will be in one direction, if too small it will be in the opposite di- rection. First find two values for R 3 which change the direction of the de- flection of the galvanom- eter, and keep in mind the meaning of the direction of the deflection. After having thus found the approximate value of the unknown resistance, change the ratio of the propor- tional arms so as to obtain the smallest value of the ratio R 2 /R 4 Fig. 87. i6o PHYSICAL MEASUREMENTS that R 3 will allow and still produce a balance, and then make the final determination. Sometimes it may be necessary to interpolate between the values of R s . Measure the resistance of three pieces of wire, and calculate the resistivity of each metal from the form- ula P = R j~ , (172) where R is the resistance, / the length and a the cross-sectional area of the wire. FORM OF RECORD. Exercise 57. To determine the rcsisthnty of three metals. Postoffice box No Galvanometer R 3 Date Temp. R=RA I of w diam. ire Temp of room Specimen of wire R 3 R, a P ::::::;::::: 125. Exercise 58. Resistance by Slide- Wire Bridge. From equation (171), it is seen that only the ratio R 3 /R, and one re- sistance R 2 , need be known, to effect the measurement of an un- known resistance. The ratio may be furnished by the two parts of a wire of uniform cross-section. In the slide-wire bridge the sum of the lengths of the two wires representing R s and R is kept con- stant, usually i ocx) mms, and the ratio of their lengths is changed by moving one of the galvanometer terminals along the wire. For this purpose a contact maker K f (Fig. 88). is substituted for the galvanometer key K 2 in the Wheatstone bridge. If on closing K and making contact at K r , no current flows through the gal- vanometer, we have, neglecting the very low resistance of the copper straps, - <'"> Fig. 88. RESISTANCE l6l or letting the reading at D on the wire equal o^ mms, then (I74) To obtain accurate results we must interchange x and R. in order to correct for possible errors due to variations in the cross- section of the wire, or to unsymmetrical placing of the scale, or to a constant error in the reading of the position of the contact maker. After exchanging x and R, a new reading, a 2 , is obtained ; then or combining (174) and (175) TOCO -+-(at >) , -. (176) R 1000 ( _o_: /^ 25 a/ Jlesisi l an( ?e 4 / O / Daniell Cell 7 02.T oso Current 1/2 Amps. ig 175 200 External Resistance Fig. 101. 137. Exercise 67. Electromotive Force and Internal Re- sistance by Voltmeter and Ammeter. This method is a modifi- cation of the preceding. A milammeter is joined in series with the resistance box R in Fig. 100. The reading for H is made with the key open and then readings for ' and / are taken simultane- ously, with K closed. Since we have (191) (192) Vary the resistance of the ammeter circuit and take three dif- ferent currents for each cell, but all of such a value that E' re- mains nearly equal to /2. If R and the resistance of the amme- ter be known, equation (190) may be used as a check formula, giving check values r', which should agree with those of r. In the case of dry cells the current is frequently so small that it can- ELECTRIC CEUvS 1/7 not be read accurately. In this case compute r from the values of E, ', and R. Since the polarization is quite rapid in some cells, it is better to read E immediately after the key K is opened, than at the beginning of each observation. The key is closed just long enough to take the readings. Name of cell X |/ R + R r / X. . ... FORM OF 1 RECORD. Exercise 6j. To determine electromotive force and internal re- sistance of five different cells by voltmeter and ammeter. Voltmeter No ____ Ammeter No Resistance box 138. Exercise 68. Electromotive Force and Internal Re- sistance by Condenser Method. The foregoing method has two disadvantages : First, the voltmeter reading is only approxi- mately equal to the E. M. F. of cells of high internal resistance, and second, the key K must remain closed until the readings can be taken, thus permitting of considerable polarization. In the condenser method the first objection is entirely overcome and the time needed for the observation is much shorter than in the pre- vious method. The principle of this method is to obtain a deflec- tion of a ballistic galvanometer proportional to the E. M. F. or to the difference of potential of the cell. Two methods may be employed, (a) As shown in Fig. 102, a condenser of capacity C, and a ballistic galvanometer G are substituted for the voltmeter in Fig. 100. When the key K^ is pressed down the condenser is charged with a quantity of electricity proportional to the difference of a potential at the ter- minals of the battery, and when the key swings back to the upper contact a deflec- tion is obtained proportional to the charge, i. e., proportional to the difference of po- tential of the battery terminals. In order to charge the condenser with the E. M. F. Fig. 102. of the battery, keep key K z open and operate only key K To PHYSICAL MEASUREMENTS obtain deflections proportional to the difference of potential ' on closed circuit, close K 2 , charge and discharge the condenser and immediately afterwards open K 2 . Since the discharge is completed in a small fraction of a second it is not necessary to wait for the completion of the swing of the galvanometer coil before opening K 2 . A "pendulum apparatus" which automatically operates four keys, closing the circuit, charging and discharging the condenser and finally opening the circuit, is described in Carhart and Pat- terson's Electrical Measurements, pp. 107-109. This instrument is very useful for this exercise, not alone because it rapidly per- forms the four operations needed, but also because in successive observations always the same time interval elapses between the closing of the circuit and the charging of the condenser. (b) The apparatus is arranged as in Fig. 103. The key K is a sliding contact which when passing from a to c makes for a short time connection between a and b and charges the condenser C through the galvanometer, the resulting deflection being proportional to the difference of potential at the terminals of the battery. To obtain a de- flection proportional to key KI should be open; to obtain a deflection with ', K^ is closed, and K is turned from a through b to c. This closes the circuit as soon as the slider connects a and b and at the same time the condenser is charged. As the slider advances the gal- vanometer circuit is broken and shortly after also the current through R. Since very short time is needed to establish the cur- rent through R and charge the condenser fully, the momentary contact between a and b suffices. It is evident that there can be only slight polarization during this brief interval. The condenser is charged at the instant the circuit is closed. KI should be opened before the slider is pushed back to its orig- inal position. After each charge the condenser must be discharged IQ 3- ELECTRIC CELLS 179 completely by the closing of key K 2 , which should be kept closed for a short time to remove all absorbed charge if such be present. In order to express the E. M. F. in volts, the condenser is also charged from a standard cell and discharged through the gal- vanometer. Let the deflection in this case be d 2 , and let d be the deflection given by the cell under investigation when on open circuit, and d^ when the cell is closed through the resistance R. The deflections are proportional to the quantities of electricity passing through the galvanometer, and these are themselves equal to the product of the difference of potential at the terminals of the condenser, times the constant capacity, C. Therefore if 8 be the E. M. F. of the standard cell, then and d, (193) (i94) It should be observed that the internal resistance of batteries, and especially of dry batteries, appears not to be a constant, but to decrease with a decrease of R, that is with increasing current furnished by the cell. This is probably due to polarization, whose effect is neglected in the formulae given above. 1 Use three different resistances for each cell, such that d is about d/2. FORM OF RECORD. Exercise 68. To determine electromotive force and internal re- sistance of five cells by the condenser method. Date. Condenser Name of cell d cf R E r Resistance Deflection, standard cell Temp of standard cell E. M. F. of standard cell.. 139. Exercise 69. Internal Resistance by Method of Nernst and Haagn. In this method the internal resistance of the cell is determined by means of an alternating current, while the cell itself furnishes no current whatever. In this way, if the 1 Guthe, Physical Review, VII, p. 193, 1898. i8o PHYSICAL MEASUREMENTS period of the alternating current is sufficiently small, the disturb- ing effects of polarization are entirely avoided. The alternating current is furnished by the induction coil /. C. (Fig. 104), which charges the condenser C alternately to positive and negative poten- tials. These alternations are transmitted to a Wheatstone bridge, of which the condensers C L and C<> form two arms, and the noninductive resistances R and r form the other two, where r is the resistance of the battery B. The resistance R is so ad- justed as to produce minimum sound in the telephone receiver which re- places the galvanometer in the ordi- nary Wheatstone bridge. The applied difference of potential produces a current i, through the re- Fig - I04 - sistances R and r. Condenser C\ will be charged with a quantity Cjr, and condenser C 2 with a quantity C. 2 iR. For a minimum sound in the telephone these charges are equal or (195) or (196) In practice it is best to connect a slide-wire bridge in series with the battery and connect the telephone with the contact maker. We have then instead of r in one arm of the Wheatstone bridge, r -(- R', where R' is the resistance of the part of the wire from the battery B to the contact maker. The resistance of the other portion of the slide wire forms of course part of R. The formula becomes in this case r = ^R-R' (197) FORM OF RECORD. Exercise 6p. To determine the internal resistance of five cells by the method of Nernst and Haagn. Name of cell d C 2 R R' CURRENT l8l MEASUREMENTS OF CURRENT. 140. Measurable Effects of a Current. Since it is obviously impossible to preserve standards of current, it is also impossible to determine currents by direct comparison with a standard as in the case of resistance or electromotive force. It is necessary therefore to employ certain known effects of the current for pur- poses of measurement. The following are those chiefly used in laboratory practice, (a) Electromagnetic effect as applied in galvanometers, voltmeters and ammeters. These instruments have been in frequent use for the measurement of currents in the previous experiments. If any doubt as to their accuracy arises they should be calibrated by one of the methods which follow. (&) Chemical effect, as applied in the copper or silver coulom- eter. (c) The production of a potential difference at the ter- minals of a known resistance, as exemplified in Exercise 71. 141. Law of Electrolysis. According to Faraday's law 1 the quantity of a substance deposited by an electric current is pro- portional to the quantity of electricity passing through the elec- trolytic cell. A steady current of one ampere will deposit 4.025 grams of silver or 1.1838 grams of copper in one hour. The elec- trochemical equivalent s, of a substance is the ratio of the mass of the substance deposited to the quantity of electricity flowing through the coulometer. Since the quantity of electricity is the product of current and time it is easy to deter- mine the average current flowing through an electrolytic cell, by dividing the number of grams of the substance deposited in time t by the weight which would have been deposited by unit current in the same time. Thus if iv be the number of grams de- posited, then (198) Whenever it is desired to calibrate an instrument for measur- ing current by means of the coulometer, it is necessary to deter- 1 College Physics, Article 283. 182 PHYSICAL MEASUREMENTS mine the average reading of the instrument during the time the current flows, and by comparing this reading with the average current, to determine the constant of the instrument or the cor- rection to be applied in the case of direct reading instruments. 142. The Copper Coulometer. Of the various forms of coulometers the simplest is the copper coulometer (Fig. 105). Its manipulation demands considerable care. The coulometer con- sists of two copper electrodes in the form of plates or spiral wires, immersed in a solution of CuSO 4 . This solution must be kept in a separate bottle and be used for this experiment only. 1 That part of the kathode on which the copper is to be deposited must be kept perfectly clean and must not be touched with the fingers. If not clean, the plate must be dipped into a strong solution of potassium cyanide then washed well with water and then dipped into strong alcohol and dried. All these operations should be performed as rapidly as possible since moist copper oxydizes easily in the air. In case the kathode is perfectly clean it may be used without further preparation. 143. Exercise 70. Calibration of an Instrument by use of Copper Coulom- eter. First weigh the kathode to o.i milli- gram. Then set up the coulometers, two in series, in order to check the result. In the circuit place a battery of high, and if possible, of constant E. M. F., a large vari- able resistance, the instrument to be cali- brated, and a key for opening and closing the circuit. The resistance should be adjusted beforehand so as to give the current that is to be sent through the coulometers. The current density may be as high as 1.5 amperes per square deci- meter of the electrodes. Close the circuit for a definite time, say thirty minutes, and note the reading of the instrument every min- ute. After the circuit is opened wash the kathode in plenty of water, rinse in alcohol, dry and weigh to determine the mass of Fig. 105. 1 T i he following solution is recommended: CtiSO*, 15 grams; 5 grams; alcohol, 5 grams; water, 100 grams. CURRENT 183 the deposit. Apply equation (198) for the calculation of the cur- rent. FORM OF RECORD. Exercise jo. To calibrate a by copper Goniometer. Date Name and number of instrument.... Formula of instrument.. Wt. of kathode before Wt. of kathode after Gain Average gain Average current Coulometer I Coulometer II 1 Reading " Average reading of Inst Constant (or correction) of Inst.. 144. Exercise 71. Calibration of Ammeter by Standard Cell. A current may be measured by comparing the difference of potential at the terminals of a known resistance r, through which it flows, with a known E. M. F. Let A (Fig. 106), be an ammeter to be calibrated ; B a battery of constant E. M. F., and j an adjust- able resistance for varying the current. R i and R 2 are two resist- ances, each very large in comparison with r. The standard cell is set in such a way as to oppose the difference of potential at the terminals of R if The resistances R-L and R 2 are so adjusted that on closing K' after K, no current will flow through the galvanometer; then (199) where I is the current flowing through r, and i that through J? and R*. I is sensibly the same current as that through the ammeter. Moreover 8 = iRi therefore St.C. (201) It is evident that under the conditions given the current must 1 84 PHYSICAL MEASUREMENTS be equal to or greater than E s / r , in order to make a balance possi- ble. 1 Calibrate an ammeter using at least five different currents. FORM OF RECORD. Exercise 71. To calibrate ammeter No Date Resistance r. Temperature. ... E. M. F. . . . Zero point of ammeter. . . . Standard cell No.. .A Observed .mmeter reading Corrected for zero pt. ff, R, I computed Correction COMPARISON OF CAPACITIES. 145. Exercise 72. Comparison by Direct Deflection. The simplest method of determining the capacity of a condenser con- sists in comparing the deflections of a ballistic galvanometer (Fig. 74), caused by the discharge of the quantities of electricity stored in a standard condenser and in that under investigation, when each has been charged to the same difference of potential. From Exercise 50, = QJc = EC, (202) (203) (204) One of the condensers C lt is a standard condenser of known capacity. x For a method for measuring small currents see Carhart and Patter- son, Electrical Measurements, p. 172. CAPACITY If the capacity of a condenser is so large as to give too large deflections, when discharged through the galvanometer, the experiment may be arranged as shown in Fig. 107. A battery of constant E. M. F. sends a current through two resistances J^ t and R 2) the sum of which must be kept constant. The condenser is connected over one of these resistances R^, and charged with the difference of poten- tial at its terminals equal to i R lt If Fi &- I0 7- the deflection on discharging is too large the value of R is to be reduced, R. 2 being increased by the same amount. Then if C and C 2 represent, respectively,, the standard and the unknown capacity -i ^4^ Rz \ R, - i R \Ci . , i and a 2 = therefore (205) (206) Instead of two resistance boxes the wire of a slide wire bridge may be used. Not infrequently condensers are found whose capacity depends very largely upon the time of charge. Such condensers are termed absorbing condensers. The quantity of electricity stored up in such condensers may be considered as consisting of two parts (a) the free charge, (&) the absorbed charge. In discharg- ing through a galvanometer the free charge will pass first and then the absorbed charge will follow. Part of the latter will affect the throw of the galvanometer and the "effective" capacity will there- fore depend to a certain degree upon the period of the galvanom- eter, being the larger the larger the period. To obtain results independent of the period the discharging circuit must be opened a few hundredths of a second after contact with the galvanometer is made. 1 Thus we measure only the free charge which in absorb- Zeleny, Physical Review, Vol. 22, p. 65, 1906. i86 PHYSICAL MEASUREMENTS ing condensers is frequently less than one half of the quantity of electricity observed by applying the above method after long con- tinued charging. To study the effect of absorption the student should take different intervals for charging and note carefully the resulting effect upon the observed capacity. FORM OF RECORD. Exercise 72. To measure the capacities of four condensers. Date. . Resistance boxes . . Cell.. I 146. Exercise 73. Method of Mixtures. The apparatus is arranged as shown in Fig. 108. In the middle of the figure is Fig. 1 08. shown a double throw switch, by means of which the points a and of may be connected either to b and b f or to c and c' . By. means of the first connections the condensers and C are charged with quantities of electricity equal to R i iC l and'R 2 iC 2 re- INDUCTANCE i8 7 spectively. On reversing the switch these quantities mix and there will, in general, be a deflection of the galvanometer result- ing from the difference of these two charges on pressing the key k. Adjust the resistances R^ and R 2 until there is no deflection of the galvanometer ; then and Ri i d R 2 i C 2 z = C\ . -r> (207) (208) Name of condenser Time R\ R\ c sees ienser. . The battery should have a relatively high E. M. F. ; six to ten Leclanche elements may be used. The disturbing influence of absorption may be studied as in the preceding experiment. FORM OF RECORD. Exercise 75. To compare two condensers by the method of mixtures. Galvanometer Resistance 'boxes MEASUREMENT OF INDUCTANCE:. 147. Exercise 74. Selfinductance of a Coil Compared with a Standard. Arrange the apparatus as in Fig. 109, in which the two selfinductances form each one arm of a Wheatstone bridge. Let their re- sistance and selfinductance be R : and L lf R 2 , and L 2 respectively. The re- maining two arms are formed by the noninductive resistances R ?i and R t . First adjust R 3 and R 4 for constant current until the galvanometer shows no deflection on closing k, that is, Fig. 109. R* (209) Then, keeping k closed, open K. There will in general be a de- l88 PHYSICAL MEASUREMENTS flection of the galvanometer due to the difference of the E. M. F. produced by the selfinductances in the branches Z^ and L. 2 . Vary the selfinductance of the standard, until upon opening K no de- flection is obtained. Then the points connected to the galvanom- eter will be at the same potential as well during the steady as during the variable state of the current. Let the difference of potential over R^ or R 2 for a constant cur- rent be B ; then the current through R^ is i i = 'E/R 1 , and through R 2J i, = n/R 2 . Now the value of the E. M. F. due to selfinduc- tance is 1 '=-* (2IO) In the case under consideration, the value of e on opening K at any moment, is and <=-& sr=-j$r (2I2) But after the adjustment of the standard, e equals e 2 and dH/dt is the same for both e^ and e. 2 , and hence the condition for no deflection during -the variable state of the current is given by the equation ~R~ ~ ~R~ or ^ x = "k* ~R~ = ^ ~R~ ' In practice care must be taken to place the two selfinductances in such a position that they will not influence each other. To ob- tain accurate results the battery must possess a high voltage. The best results are obtained by using an alternating current and sub- stituting for the galvanometer a telephone receiver as in Exercise 6 1, or by rectifying the current before sending it through the gal- vanometer. In an instrument designed by Ayrton and Perry-, called the secohmmeter, 2 this rectification of the current is ef- fected by means of a double commutator. J College Physics, Article 333. 2 Carhart and Patterson, Electrical Measurements, p. no. INDUCTANCE 189 FORM OF RECORD. Exercise 74. To measure the selfinductance of Apparatus Voltage of battery 148. Exercise 75. Mutual Inductance of two Coils. Accord- ing to definition the coefficient of mutual induction or, the mutual inductance of two coils is given by the equation 1 (214) Date R 3 ft Reading of standard L where e is the counter E. M. F. in the secondary due to the mutual inductance. M, and i, the current in the primary. Let the induced E. M. F. produce a current through a ballistic galvanometer whose resistance is g ; the current i' ' , so produced, will at any moment be , where r + g is the total resistance of the secondary circuit. The total quantity of electricity passing through the galvanometer s where the integral is to be taken between o and /, the final value of the steady current in the primary circuit. Therefore the total quantity in the secondary circuit when the primary circuit is closed, becomes, neglecting its direction If the current through the primary be suddenly reversed the value of Q is doubled. From the formula for the ballistic galvanometer we have Q = cd, where c is the constant of the galvanometer. Therefore 1 College Physics, Article 332. igO PHYSICAL MEASUREMENTS This formula is based upon the assumption that no counter electromotive force is generated in the galvanometer. But this assumption is not correct, especially in the case of instruments of the d'Arsonval type. In this case a counter electromotive force is set up, due to the cutting of lines of magnetic induction by the coil swinging in the field of the permanent magnet. Since the total number of lines cut, or the change of flux through the coil is nearly proportional to the actual deflection d, the resulting quantity of electricity 1 is nearly equal to - , and tending to flow in the opposite direction, would, if acting alone on the cir- cuit, produce a deflection d 2 , in the opposite direction from d, while if there were no inductive effect in the coil, the deflection would be simplv We obtain, therefore, for the actual deflection ^- (2I9) In order to obtain accurate results we must therefore add the quantity k/c to the actual resistance r + g ; in other words, we must here use the apparent resistance of the galvanometer, k ' g -| -- . (220) instead of its ohmic resistance g. The expression for M thus becomes (22I) The apparent resistance g' , of the galvanometer may be deter- mined in the following manner. As described above, produce a deflection of the galvanometer, and note the deflection. Let the total resistance of the secondary circuit be r + g, where r denotes 1 College Physics, Article 330. INDUCTANCE 191 the resistance outside the galvanometer. Next introduce in series a resistance r' ' , such that with the same primary current the de- flection is now reduced to one half its former value. Then c(r-\-g'}d _ I and (222) The deflection is proportional to the current in the primary and inversely proportional to the resistance of the secondary cir- :uit. To show these relations dearly make the following experiments: The arrange- ment is shown in Fig. no. Let P represent the primary and the secondary coil. The primary is in series with a battery B, a variable resistance R and an ammeter A. To the secondary coil are joined a resistance r and a ballistic galvanometer. (1) Vary / keeping r constant. (2) Vary r keeping / constant, curves should be straight lines. Fig. no. Plot values of d and /. Plot d and i/(r + g') ; both FORM OF RECORD. Exercise 75. To determine the mutual inductance of, Date, Ammeter (I] / Bal ) r co d listic gi nstant M ilvanon (2 r ieter Determination Capacity . . of constant ) / CO d istant M Electromotive Deflection force Constant . Apparent resistance : r . di r'.. d a' . . CHAPTER IX. MAGNETIC MEASUREMENTS. MAGNETIC 149. Magnetic Fields. In most of the foregoing experiments the action between a magnetic field produced by a current and another magnetic field was used to determine electrical quantities. In the following experiments magnetic fields and their effect upon the magnetic properties of iron and steel will be studied. A magnetic field is perfectly determined if at every point the in- tensity H of the field and the magnetic induction B = ^ H of the field be known. The intensity may be compared to a stress in an elastic body, the induction to a strain 1 . In the case of air, since /A equals unity, these two quantities are numerically equal, but it should be kept in mind that they are different physical quantities. The intensity of a magnetic field at a given point may be meas- ured by the force per unit pole strength acting on a pole placed at this point. The unit of intensity of magnetic field is the dyne per unit pole and is called the gauss. If the magnetic field be produced by an electric current it is best to calculate the intensity at any point from the current. In the case of a circular current of radius r it is 2 H = y / gauss (223) In the case of a very long solenoid it is s H = 4^ 1 auss ( 22 4) where n' is the number of turns of wire per unit length of the solenoid. In both equations 7 is expressed in amperes. 1 College Physics, Article 242. 2 College Physics, Article 257. 8 College Physics, Article 319. MAGNETIC FIELDS 193 The magnetic induction B at any point is measured by the number of lines of induction per unit crosssection and its unit is therefore the line per square centimeter. The magnetic flux $ through a given area is simply the number of lines of induction through this area A, or 1 (225) 150. Exercise 76. Determination of H. (First Method). The lines of magnetic induction due to the earth's field run from South to North, although deviating in some places by an ap- preciable angle from the geographical North and South line. This angle is called the angle of declination. The lines of induction are also inclined towards the horizontal plane, making in Ann Arbor an angle of 72 with the horizon. This is the angle of magnetic inclination or the magnetic dip. It should be noted that neither the magnetic declination nor the dip is constant. They not only vary from place to place over the earth's surface, but they also vary slightly from year to year and from day to day in the same place. The magnetic field of the earth may be conceived as the resultant of two components, one horizontal H, and one vertical V. Then , the angle of dip, is given by the equation tan = 77-- (226) ri It is of great interest to determine the value of the horizontal component of the earth's magnetism in terms of the fundamental units of length, mass and time, since all the practical magnetic and electrical units are based on these 2 . A relation between the magnetic moment of a magnet and the 1 College Physics, Article 324. 2 College Physics, Articles 258 and 404. IQ4 PHYSICAL MEASUREMENTS strength of the magnetic field in which it is situated may be de- rived in either one of two ways : ( i ) Method of deflections. Let a magnet NS, (Fig. in) be placed with its axis on the magnetic East- West line 1 and on this line in the same horizontal plane, at a meas- ]JV ured distance from it place a very small magnetic needle ns, suspended by a fine silk Fig. in. thread or quartz fiber. Let m and / be the pole strength and half-length of magnet NS. Let * m' and /* represent the same for magnet n s. Also let d 'be the distance between the centers of the magnets, H the horizontal component of the earth's magnetic field, M the magnetic moment of the magnet NS = 2ml, M' the magnetic moment of the magnet n s = 2m'/'. The magnet ns will be deflected from its normal position by the angle such that the turning moments due to the earth's field and that due to the influence of the magnet NS are equal 2 . The force due to the earth's field on one of the poles of ns is m'H, and the lever arm on which it acts is /' sin ; so the turning moment on the whole magnet is 2m'l'H sin = M'H sin < . (227) The force on the north-seeking pole of ns due to the south- seeking pole of NS is, according to Coulomb's law of the inverse squares -- ( J nm ,. 2 , I' being considered negligible in comparison 1 To find the magnetic East-West line, suspend in the center of a plane coil of wire a magnetic needle. Turn the coil until on sending a current through it the needle is not deflected. The plane of the coil is then in the magnetic East-West line. 2 The angle is determined by mirror and scale. Suppose the distance of the mirror from the scale to be D and the deflection d, then tan 2$ MAGNETIC FIELDS 195 with d. The force of the north-seeking pole of NS is in the opposite direction and equal to 77 ^ j so that the whole force on the north-seeking pole of the needle is : D m m x i 1 = ~ ' L(d-/) a *~ (d + 1 J ( } and if / be small in comparison with d, and since p. is unity, (229) The turning moment due to this force is FJ' cos , and since the turning moments on the two poles of ns are equal and in the same direction the total turning moment exerted on ns by the magnet NS is 2 FJ' cos or 8mm'//' MM' 7P COS0^=2 . -p COS . (23O) But the turning moments produced by the forces exerted by the earth and the magnet NS must be equal since the needle is in equilibrium. So the expressions for these turning moments may be set equal to each other, or MM' M Hsm = 2 cos0 (231) whence M d 3 = . tan = A. (232) (2) Method of oscillations. The law of vibration of a mag- netic needle in a magnetic field is the same as that of the physical pendulum. If we suspend the deflecting magnet NS, so as to swing freely, its period is IK where T is the period of a complete vibration and K is the mo- ment of inertia of the magnet. MH is the torque when the PHYSICAL MEASUREMENTS magnet is at right angles to the force. In order to find K the same method is applied as in Exercise 26, by putting a brass ring of known moment of inertia K' on the magnet so that the axis of rotation and the axis of the ring coincide. Let the period of vibration of the new system be T', then and from the equations (233) and (234) K' (235) Equations (232) and (235) furnish two expressions involving M and H. %=* (236) and MH B, from which or on substituting the values of A and B, It is obvious that by this method the value of H is deter- mined in the fundamental units of mass, length and time. The instrument used for this experiment is called a magnet- ometer. It consists of a closed case furnished with windows to enable the observer to measure the deflection of the needle by means of the mirror which it carries. Place the deflecting magnet at a certain distance from the small needle as described above and observe the deflection. Turn the magnet end for end and again observe the deflection, which will now be in the opposite direction. Repeat these two operations with the magnet at the MAGNETIC 197 same distance on the opposite side of the needle and take the mean of the four observations as the deflection <. In case the length of the deflecting magnet is not negligible in comparison with the distance d, we may take it into account. Kohlrausch has shown that for a bar magnet the distance between the poles is very nearly 5/6 of the length of the magnet and this value should be used for 2/. The formula for H becomes 2TT 2K'd (239) In the vibration method the time of vibration may be determined as in Exercise 13, or by the use of an ordinary stop watch if less accuracy is required. The stop watch should however be com- pared with a standard clock. How may the magnetic moment of the deflecting magnet be determined from the foregoing formulae? FORM OF RECORD. Exercise 76. Determination of H. \ L J Determination of -77 H Distance d Position of magnet. d Deflection tan 2 tan L/ensf'th of magnet Reversed b Reversed Computation of ;r (2) Determination of M H: (a) Determination of T. (&) Determination of T'. (c) Data for moment of inertia of ring. 'Mass of ring Outer diameter of ring. Moment of inertia of ring Computation of MH. 3) Computation of H. Inner. 198 PHYSICAL MEASUREMENTS 151. Exercise 77. Determination of H. (Second Method). In formula (233), T 2. the torsional moment of the suspending wire was neglected. The complete formula for a given magnet is or (240) where <3\ is the moment of the torsional couple of the suspending wire, and c and c' are constants. These relations are illustrated in the following experiment. Let a short magnetic needle be sus- pended at the center of a solenoid whose length is at least twenty times its diameter, and whose axis is parallel to the magnetic meridian. Observe the period of vibration of the needle, first when swinging in the earth's field H, and then when a magnetic field H lf due to a known current / through the soleniod, is super- posed upon the field of the earth. Let the period of vibration of the system in the first case be T and in the second T . Then (241) (242) Vary the current and observe T it T z , T s , T, etc. ; also the periods of vibration T\, T' 2 , T' s , etc., when the current through the solenoid is reversed. Then plot i/T 2 as ordinates attd the mag- netic field strength produced by the currents as abscissae, the latter positive or negative according to its direction 1 . The curve consists 1 lf the current be increased in the negative sense beyond a certain value tfhe magnet turns through 180. Why? MAGNETIC 199 of two straight lines meeting at the point 0' ', Fig. 112. The time corresponding to the ordinate B is then the period of vibration corresponding to the rigidity of the suspension alone. If 0' be considered as the origin and the strength of field be plotted on O'x it is evident that O'B is the value of the horizontal component of the earth's field, and that the strength H' of any field may be readily determined by allowing the magnet under experiment to vibrate in this field and determining its period of vibration. Then H' (243) where c and c' are to be substituted from the foregoing observa- tions. A X _.3V -ff=Q19 --LI-I- Strength of Magnetic Field Fig. 112. In practice a storage cell of constant E. M. F. is used to furnish the current through the solenoid. A resistance is joined in series to allow the current to be varied within wide limits. The periods of vibration may be determined by means of a stop watch. From equation (224) the strength of field at the center of the solenoid s * * 1 " T 10 L where N is the number of turns of wire and L the length of the solenoid. The current / is computed from the electromotive force of the cell and the total resistance. 2OO PHYSICAL MEASUREMENTS FORM OF RECORD. Exercise 77. Determination of H. (Second Method). Dimensions of solenoid : R I T r n Determined from curve c ZJr MAGNETIC PROPERTIES OF IRON AND STEEL. 152. Magnetizing Field and Permeability. When a bar of unmagnetized iron or steel is introduced into a magnetic field the distribution of lines of magnetic induction is greatly altered. The number of lines per square centimeter at any point in the baj* is considerably larger than it was at the same point before the introduction of the iron. In other words B which is called the induction is much increased. At the same time the intensity of the field inside the iron is, in general, much smaller than it was before at that place. This intensity which is called the magnetiz- ing intensity may be considered as being the result of the super- position of the original field and the demagnetizing effect of the ends of the iron bar. This effect is in the opposite direction to the original field and thus the intensity of the field is reduced. The demagnetizing effect of the ends may be neglected only in the case of a long bar. An excellent method of avoiding this dis- turbing influence of a magnetized piece of iron consists in using a ring instead of a bar and in producing the field by means of a solenoid wound around this ring. Since there are no ends in this case the magnetizing field is exactly equal to the field calculated from the equation (224) H 10 1 = 10 L I where N is the total number of turns of wire and L the length of the ring shaped solenoid which is equal to 2-n- times the average radius of the ring. The magnetic permeability ^ of iron is the ratio between the *For a more complete treatment of the properties of ferromagnetic substances and a description of the various methods of measurement, see J. A. Ewing, Magnetic Induction in Iron and other Metals. MAGNETIZATION OF IRON 2O I induction produced and the magnetizing intensity of the field, or B *= JT (244) As shown in exercise 79 this ratio depends greatly upon the pre- vious history of the piece under experiment. In order that the permeability of a substance shall give definite information con- cerning its magnetic quality, it has been agreed to express the permeability of iron and similar substances as the above ratio when the substance is originally unmagnetized and then subjected to an increasing magnetizing intensity, (Exercise 78). Even un- der these conditions the permeability of a given sample of iron is not a constant but varies with the magnetizing intensity. Under the influence of variable or alternating magnetic fields iron and steel show what is termed magnetic hysteresis, and the hysteresis curve (Exercise 79) should be determined for all material which is intended to be used in alternating current apparatus, for example in alternators or transformers. All substances which have a large permeability approaching in magnitude that of iron are called ferromagnetic substances. 153. Exercise 78. Commutation Curve for Iron and Steel, [t is of interest to deter- mine the dependence of B upon the intensity of the magnetizing field produced by the current alone, when the intensity is slowly increased from o to 30 or 40 gauss. It will be seen from Fig. 113, that B increases slowly at first, then very rapidly, then slowly again, until it finally in- creases at the same rate as HI that is, the pres- ence of the iron does not introduce any additional lines of induc- tion. The iron is then said to be saturated. 12000 &ooo 4000 2O2 PHYSICAL MEASUREMENTS The method here described is known as the ballistic or ring method and has the advantage, that the magnetic circuit con- tains no air gaps which offer large magnetic resistance and therefore ex- ert strong demagnetizing effects. (Ar- ticle 152). The metal is given the form of a ring of uniform cross sec- tion, Fig. 114. It is best to have the cross section of the ring approach the form of a rectangle and the diameter large as compared with the thickness. Fig. 114. The ou ter and inner diameters of the ring must be measured with care, and the volume of the ring obtained from its loss of weight in water. The cross section A is readily found from the volume by dividing by the average circumference. The ring is then covered with insulating tape upon which is wound a layer of wire covering the entire ring and forming the primary coil. The number of turns N if in the primary coil must be carefully determined. Over this is wound the secondary coil, five to twenty turns, N 2 . The apparatus should be arranged as shown in Fig. 115. In the primary circuit are joined in series a stor- age battery, an am- meter, a rheostat R, the primary coil of the ring and a com- mutator for reversing the current through the coil. The second- ary coil is connected to a ballistic gal- vanometer whose constant and appar- ent resistance are Fi S- :I 5- known (see exercise 75). It is most convenient to make the total galvanometer resistance the critical resistance (see page 142). MAGNETIZATION OF IRON 203 The constant may also be determined by means of a known mutual inductance M ; in this case c = - , where d is the de- ars flection produced in the secondary by closing the current / in the primary. In this case, however, the total resistance of the second- ary must remain the same, either by keeping the secondary of the induction coil in the circuit or by substituting an equal resist- ance for it. What formula must then be used instead of (249) ? What is the numerical factor, if M be expressed in millihenry s (10 c. g. s. units) and / in amperes (lO" 1 c. g. s. units) ? If we vary the number of lines of magnetic induction in the solenoid, the electromotive force induced in each turn of the sec- ondary is at any moment' equal to the rate of change of the flux threading through the circuit, or If there are A\ turns of wire in the secondary coil then e = - N *'7T' (246) The current i, passing through the galvanometer is then ,- = - * (247) r 2 d t where r z is the total resistance of the secondary circuit. The total quantity of electricity Q, passing through the ballistic gal- vanometer is Cidt, corresponding to a change of $ lines of induction, or where c is the constant of the galvanometer and d the throw. Express c in c. g. s. units. Since the constant is usually given in micro-coulombs per scale unit, one micro-coulomb being io~ 7 2O4 PHYSICAL MEASUREMENTS c. g. s. unit, and the resistance in ohms, one ohm being equal to 10 c. g. s. units, the formula becomes * = IO - jvT'- (249) Since B denotes the number of lines of induction per unit area the increase of B corresponding to a deflection d of the galvanome- ter when the current is varied in the primary, is given by (250) To obtain the commutation curve it is important to start with an unmagnetized ring. It is best to demagnetize the ring by sending first a rather large current, say three amperes, through the primary, increasing the resistance R, and then reversing the direction of the current. This must be done several times, tak- ing care to decrease the current each time before the reversal, until the current has become very small. On breaking the cir- cuit the ring will contain no residual magnetism. A more con- venient way is to connect the primary to the terminals of a small alternator driven by a belt. Let the alternator attain its full speed and then throw off the belt. As the speed decreases the current decreases, and the iron is as before subjected to cycles of con- stantly decreasing magnetic intensity. Arrange the apparatus, as shown in Fig. 115. Adjust the re- sistance so as to give a small current through the primary coil. Reverse the current and observe the first ballistic throw of the needle d\ reverse again and observe d'\ ; the mean, rf ly corre- sponds to a reversal of the current /, read from the ammeter. This is repeated, increasing the current step by step, until the deflections increase very slowly with increasing current. Compute H from equation (224), and B from (250), remembering that but half the average deflection must be taken, since the current is changed each time by 2.1 instead of by /. Plot B and H ', the curve will resemble that shown in Fig. 113. From the curve it appears at once that /JL is not a constant, but varies greatly with the degree of magnetization. MAGNETIZATION OF IRON 205 FORM OF RECORD. Exercise 78. Commutation curve for iron. Galvanometer Constant of galvanometer Ring Outer diam Volume of ring / d' Crosssection Turns in primary Turns in secondary Date Res Inne d" Ammeter I 1 istance r 2 To Ave H rage d d/2 B A* 154. Exercise 79. Hysteresis Curve for Iron or Steel. If instead of reversing the current in the previous exercise we should begin with a large current, decrease / by small steps, ob- serve the corresponding throws of the galvanometer and plot B with respect to H, we obtain an entirely new curve owing to the effects of residual magnetism in the iron or steel. The experiment thus allows us to study the lagging of the in- duced magnetism be- hind the magnetizing field intensity. Thus if, in the beginning, the iron were well magnetized, the start- ing point on the curve would be the highest point to the right, (Fig. 116), and on re- ducing the current to zero, there would still remain in the iron the lines of induction re- presented by the posi- tive ordinate. The Fig- 116. iron is still magnetized, but on reversing the current the value of B decreases very rapidly, becomes zero for a small negative cur- 2O6 PHYSICAL, MEASUREMENTS rent, and finally reaches a value symmetrical to that at the start- ing point, when the current in the negative direction reaches the same value it. had at the beginning in the positive direction. On decreasing the current again to zero and increasing it in the positive direction to the original value, the curve will have a form resembling closely the first branch. The closed curve is called a hysteresis curve. The value of B for .zero current is termed the remanence and the value of H for zero B is called the coercive force of the iron. The arrangement of the apparatus is similar to that used in the preceding exercise. The commutator in the primary is con- 'nected as shown in Fig. 117. Start the experiment with the com- mutator in the position indicated by the dotted lines and with switch K closed. Adjust rheostat R, until the largest current to be used in the experiment is reached. The iron has then at- tained its maximum magnetization, indicated by the highest point on the hysteresis curve. If now K be opened, the current is suddenly de- creased, owing to the addi- tion of resistance in rheostat R lt Adjust R to differ- ent values, beginning with small resistance and in- creasing them for every fol- lowing observation. We ob- tain every time on opening K a deflection corre- sponding to the total change in the induction, due to the decrease of current to a value I, determined by the resistance in R . Finally the current is broken by making R infinite. In this way we obtain points on the hysteresis curve corresponding to a decrease of the magnetiz- ing field from a maximum value to zero. MAGNETIZATION OF IRON 2O/ To obtain the points corresponding to a reversal of H, the commutator is connected so as to join a to c, and a' to c' '. R remains adjusted for the largest current and the commutator is then suddenly reversed. In this way a magnetic field is produced in the iron in the opposite sense, the strength of the field depend- ing- upon the resistance in R^. This resistance should now be reduced from very high values to smaller and smaller ones and finally to short circuit. The deflections d will now be in l.he opposite direction to that before, unless the galvanometer or the battery connection be reversed. It is evident that by this method we obtain points for but one-half of the hysteresis curve. Com- plete the curve by drawing the other half. The origin of the axis of induction B, is found by taking as the ordinate for maxi- mum induction the A B corresponding to one-half of the maxi- mum deflection. Use equation (250). FORM OF RECORD. Exercise /p. Hysteresis curve for iron or steel. Date. . Record apparatus as in preceding exercise. H CHAPTER X. OPTICAL MEASUREMENTS. CURVATURE. 155. Curvature of Optical Surfaces. It is shown in treatises on optics that the effect of a mirror or of a lens of any form, con- sists in impressing upon the wave-front of the luminous disturb- ance a curvature directly related to the curvature of the mir- ror or lens in question. By definition the curvature at any point in a curve is the reciprocal of the radius of the osculating circle at that point. Since the effects produced by mirrors or lenses are to be predicted from a knowledge of their constants, it becomes a matter of importance to measure the curvature of an optical surface, in other words to determine its radius of curva- ture. The surface most commonly employed in optical construction is that of the sphere since only the largest mirrors or lenses pos- sess surfaces differing noticeably therefrom. .Concave or convex mirrors may therefore be regarded as parts of spherical shells, with the inner or outer surface polished as the case may be. The radius of curvature of such a mirror is obviously the radius of the sphere, of which the mirror forms a part. In lenses both surfaces are to be regarded as parts of spheres of definite radii. In the case of a plane surface the radius is of course infinite. 156. Exercise 80. Radius of Curvature of a Lens by the Spherometer. The experiment consists in determining the ra- dius of curvature from a careful measurement of the amount by which the lens surface departs from a plane, i. e., by measur- ing the sagitta 1 . If we place a spherometer upon a lens with the three feet resting upon the surface of the lens, we may imagine a plane passed through the lens, cutting from it a seg- 1 Preston, Theory of Light, p. 80. CURVATURE 209 A} nient whose base is a circle passing through the three feet of the instrument. At right angles to the base of this seg- ment stands the micrometer screw of the spherometer, and by taking readings, first upon a plane surface and then upon the lens, the sagitta of the curve, that is the distance the central foot of the instrument is above or below the plane containing the other three feet, may be accurately determined. Thus in Figure 118 is shown in perspective the lens, the spherometer in place, and the imaginary segment, ABC 2 C^. Above is the equilateral triangle formed by the three feet C lf C 2 , Co, while B at the center, marks the point where the radius BO, pierces the base of the segment. Then BB=s, is the sagitta, EC 2 s=d, the distance from the point of the micrometer screw to the center of any leg, and C^O=BO=R, the radius of the sphere of which the lens is a part. By geometry we have or whence BE (2# BE) = EC,' rf'4-r 2S (251) (252) (253) The distance d is usually measured once for all on the dividing engine or comparator, and is called the constant of the instrument. It may also be determined in terms of I, the length of one side of 2io PHYSICAL MEASUREMENTS the equilateral triangle .formed by the feet of the spherometer as follows : Press the instrument firmly upon a piece of stiff paper until the positions of the three are left sharply defined. The length of the sides of the triangle may then be accurately meas- ured by the vernier caliper. Then from Fig. 118, = (254) 44 3 whence by substitution R=J I^ + ~r (255) In practice the spherometer is first placed on a piece of plate glass and the zero reading accurately determined. It is then transferred to the lens and the readings upon the lens are made, care being taken to prevent the feet from slipping off the lens. The difference between the zero and the final readings gives the value of s. From the known value of d, the value of R is at once computed, or the value of / may be determined as shown above and the value of R computed from the equation (255). FORM OF RECORD. Exercise 80. To determine the radii of curvature of -five lenses, by the spherometer. d I Date Zero Lens i Lens 2 Lens 3 Lens 4 Lens 5 Mean Mean Mean Mean Mean Mean s R s R .? R s R .? R Measure both sides of each lens. 157. Exercise 81. Radius of Curvature by Reflection. The radius of curvature of a polished spherical surface may be determined by means of purely optical considerations if we em- ploy the phenomena and formulae relating to spherical mirrors. Assume that the convex spherical surface mm' (Fig. 119), is CURVATURE) 211 placed before the telescope T } at a distance A, and that it receives light from two brilliant objects L and Z/ sym- metrically placed with respect to T. There will be formed in the mirror mm' ' , two virtual, erect and diminished images of the objects L and L' '. Owing to the inversion Fig. lip. of these images by the telescope they are seen inverted in T. A small scale ss', placed in contact with the lens enables the observer to read off directly the apparent distance ss' between the two images. Now since the rays from L and L', after reflection at s and s' enter the telescope and seem to come from the images / and /', the normals Cs and Cs' will, if produced, bisect approx- imately the angles LsT and L's'T, and to the same degree of approximation, PQ = I/2LI/ where P and Q are respectively the intersections on LU of Cs and Cs' produced. Let ss'=s; OT, the distance from the lens to the objective of telescope, -.A; OC = R and LL'=L. Then from the triangles PQC and ss'C we have PQ CT ss' '" CO L/2 s or R 2 As L, 2S (256) (257) (258) In practice the lens is held in a clamp supported upon a tripod base, one foot of which bears an adjusting screw for tilting the lens about a horizontal axis. This foot should stand in a line parallel to the axis of the telescope, and normal to the lens sur- face. Two small lamps are placed at L and U with their flames turned edge-wise to the lens. The telescope and lens are set up on two tables at least three meters apart, the lens facing the most 212 PHYSICAL MEASUREMENTS brightly lighted window in the room. The telescope is focused upon the lens surface until the scale ss' is sharply defined. One observer then takes one of the lamps and moves it slowly back and forth and up and down along the line TL, until the other catches sight of the moving image in the telescope. It is to be noted that the image in a convex mirror is erect and is seen inverted owing to the inversion in the telescope ; this inversion applies to the motions of the lamp as well, so that if the light moves to the right, the image seen in the telescope moves to the left and vice versa. Should the image fail to appear when the above directions are followed, the lens holder should be ro- tated slightly about its vertical axis until the image appears in the field. The image is then brought to the level of the scale by means of the adjusting screw in the foot of the lens holder. A black cloth placed close behind the lens renders the image much more bright and distinct. Care must be taken to avoid confusion of the true images from the front surface of the lens, with the pair of erect images seen in the telescope which are due to reflection from the back of the lens ; these images are originally inverted owing to the concave surface, and are erected by the telescope. Which pair of images must be chosen in case of a concave lens ? What change is needed in the formula? It will usually be found necessary to change the focus of the telescope very slightly in order to fix sharply the position of the image on the scale. This difference in focus becomes the more marked the more nearly the lens surface approaches a plane. The method is therefore best adapted to lenses of large curvature. The small scale may be dispensed with by pasting upon the lens two strips of paper with straight edges, parallel and facing each other. The perpendicular distance between the edges of the strips is then carefully measured with the vernier caliper and recorded. The lamps are then so adjusted that their respective images just disappear behind the edges of the paper. The measured dis- tance is then equal to s. The distance A and L should be meas- ured with a steel tape or a long stick and a metric rule. Measure by this method the radii of curvature of three lenses. CURVATURE 2I 3 FORM OF RECORD. Exercise 81. To determine the radii of curvature of three lenses by the method of reflection. Date Lens No. A L s R 158. Exercise 82. Focal Length of Lenses. x From the well known formula for the focal length of a lens i i , i T = T + T (259) we may deduce an important relation under the condition that the object and image remain at a fixed distance, greater than $f, from each other. Let / be the distance between the object and the screen upon which the image is received. Then there will- be two positions of the lens for which a sharp image is projected upon the screen, one near the object giving an enlarged image, with the lens at a distance p from the object and q from the image ; and another nearer the screen giving a small but bright image. In this position the distances p and q are interchanged, so that now the lens is at a distance q from the object. Let a be the distance between these two positions of the lens. Then whence P + Q = I, and q p = a /4-a / q = and p = substituting in (258) we have 2 F-tr (260) (261) The apparatus consists of an optical bench about two meters long, provided with a scale reading to millimeters and two sup- 1 College Physics, Article 458. - Owing to the fact that the distances p and q are not measured from the same point, but from the two principal points of the lens, this formula is not strictly accurate ; the error is, however, not large. For the correction due to this approximation, see Glazebrook and Shaw, Practical Physics, P. 350. 214 PHYSICAL MEASUREMENTS ports to carry the screen and the lens. The object is placed at the zero end of the scale and at a suitable height above it so that the object, the center of the lens and the middle of the screen are all in the same straight line. In a thin board at the zero end of the scale is cut a hole 3 cm in diameter. This is closed by a piece of ground glass, which is strongly lighted by an incandescent bulb. A watch hand of elaborate design placed against the glass on the side toward the lens, forms a well defined, dark object upon a bright field. A rough approximation to the value of / may be obtained by placing a piece of white paper in front of the object and bringing the lens toward it, until there is formed upon the paper an image of the window bars opposite, or of the trees and buildings out- side. The reading of the lens carrier gives at once the approx- imate focal length. Why is not this the true value of /? The screen should then be placed at a distance from the object not less than five nor more than seven times this rough value of /. (Why?) The lens is now shifted until a sharp image is projected upon the screen. The mean of five settings is taken as the position of the lens. The second position of the lens for a sharp image is then determined in the same way. The difference between these mean values is a, and this value with its related value of the set- ting of the screen I, will, when substituted in the formula, give a value for /. At least three different settings of the screen should be used and the mean of the three values of / returned as the focal length of the lens. In the case of a concave lens, there can, of course, be no real image. Therefore, in order to use this method, it is necessary to combine the concave lens with a convex lens of suitable cur- vature, and determine first the focal length of the combination and then the focal length of the convex lens separately. If F be the focal length of the combination, and /' that of the auxiliary lens then the focal length of the concave lens /, is given by the relation T _ i i T~~T" F or CURVATURE 215 FORM OF RECORD. Exercise 82. To determine the focal lengths of five different lenses. Date Lens Screen (0 Lens ist position Lens 2nd position a /' a 2 4l f 159. Exercise 83. Lens Curves. We have seen from Ex- ercise 82 that for every setting of the screen there are in general two positions of the lens for which a sharp image is ob- tained. If now we plot the settings of the screen as ordinates and the corresponding settings of the lens as abscissae, we obtain what is known as the lens curve, Fig. 120 From our nomencla- ture the equation of the curve is Focal length of lens = Fig. 120. (263) Is this the equation of an hyperbola ? If so, what are its asymp- totes? What is the physical interpretation of each? Use for this experiment a lens of about 15 cm focal length. Determine at least fifteen separate positions of the screen with their related settings of the lens. Plot the curve and draw the asymptotes. -2l6 PHYSICAL MEASUREMENTS Take pains to obtain as many as four or five points near the bend of the curve, i. e., where the two images approach each other. What is the value of y for the lowest point of the curve ? Deter- mine the focal length of the lens from the curve. FORM OF RECORD. Exercise 83. Lens curves. Lens Screen (y} Lens Date Lens Focal length = Plot curve. MAGNIFYING POWER. 1 60. Exercise 84. Magnifying Power of the Telescope. The telescope in its simplest form consists of two lenses, the ob- ject-glass or objective L, a convex lens of long focus, and the Fig. 121, eye-piece U , a short focus lens either convex or concave. The distance from the object to the instrument is always great as compared with the focal length of the objective and the image is consequently smaller than the object in all cases. In case the eye-piece is a convex lens, (Fig. 121), this small image is viewed MAGNIFYING POWER 2I/' directly by the eye-piece as an object placed nearer the lens than its focal distance. The result is a magnified virtual image of the image. The effect of a telescope is to increase the visual angle sub- tended by a very distant object, that is, to bring an image of the object near the eye, so that when this image is viewed by the eye- directly, the visual angle subtended by it is larger than that sub- tended by the object, in the ratio F/25, where F is the focal length of the objective and 25 cms represents the distance of dis- tinct vision for the normal eye. 1 This relation is readily seen from Fig. 122, where the objec- tive L forms an image of a distant object upon a screen. An eye at the center of the objective would see both , Fig. 122. image and object as of the same size, since the subtended angles are equal. If however, the eye approach the screen, the angle subtended by the image will increase until at a distance of twenty- five cms from the screen the image will appear larger than the object in the ratio F/2$, as given above. If the eye be brought nearer to the image in order to increase the magnification, its power must be increased by the use of a lens as a simple magnifier. Such a lens is termed an eye- piece. The magnification produced by the eye-piece is 2$/f where the focal length of the eye-piece is /. The total magnification of the two lenses forming the telescope is therefore the product of the two, or F/f. In case the object is not at a great distance, F is no longer the focal length of the objective, but is the distance from the objective to the image formed by it. Consequently F/f changes with the distance of the object from the telescope. The ratio F/f is called the magnifying power of the telescope, and is most readily measured as follows: A long scale is set up at one end of the room and so lighted that the divisions shall be seen sharp and clear. The telescope 1 College Physics, Article 476. 218 PHYSICAL MEASUREMENTS is focused upon the scale so as to give a sharp image. The ob- server next looks through the telescope with the right eye and views the scale directly with the left eye. A little adjustment of the direction of the telescope and a little patience will enable the observer to see the two images formed by the two eyes, overlapping, so that he sees at the same time, (Fig. 123), the complete scale, and projected upon it, the magnified divisions of the scale image. By careful adjustment of the telescope the lengths of these magnified divisions may be read di- rectly in terms of the divisions of the scale. Thus, sup- J7 pose the half division from 4 to 4^ is seen projected upon the scale, its upper edge appearing to be at 4.10 and its lower edge at 8.15. It is clear that one half g< 123> division seems to cover 4.05 divisions, hence the mag- nifying power is 8.10. Care should be taken to avoid touching the telescope or its support during the measurements, as well as to avoid moving the head while comparing the upper and lower edges of the image for coincidence with the scale divisions. Measure the magnifying power of the telescope at distances of 4, 7, 10, and 15 meters from the scale. Next remove the field combination, by unscrewing the telescope at the first joint from the eye-piece, and taking out the lens found there. Repeat the measurement as above. What is the purpose of the field combination ? How does the magnification vary with the distance? FORM OF RECORD. Exercise 84. To determine the magnifying power of a tele- scope. Date Distance ffom scale Magnifying power with combination without combination Effect of distance upon magnifying power. 161. Exercise 85. Magnifying Power of Microscope. In a precisely similar way the magnification of a compound micro- scope may be measured by placing upon the stage of the instru- MAGNIFYING POWER merit an object micrometer, usually one containing a millimeter subdivided into tenths and hundredths, and focusing it sharply with good transmitted illumination. A millimeter scale held at the distance of distinct vision from the eye-piece is next placed in position and adjusted until, on looking through the microscope with both eyes open, the magnified image is seen projected upon the scale, and certain prominent lines of the two scales are made to coincide. The computation is identical with that in Exercise 8 4 . An alternative method is to place on the stage of the microscope the object micrometer, and by means of a camera lucida, or an Abbe illuminating prism, project the image of the scale directly into the eye-piece and view the two images with a single eye. In the case of instruments provided with a micrometer eye-piece the magnification is determined by measuring with the microm- eter the size of the image of known divisions on the object scale. The magnification is then the ratio between the size of image and the size of object. Most microscopes are provided with a millimeter scale on the side of the draw tube. Measure the magnification of the micro- scope for at least three positions of the tube. How does the mag- nification vary with the position of the eye-piece? FORM 01? RECORD. Exercise 85. To determine the magnifying pozcer of a micro- scope. Date Microscope No Object micrometer No Reading on draw tube Magnifying power 1 I 2 2 3 3. Effect of extending tube INDEX OF REFRACTION. 162. Exercise 86. Index of Refraction of Lenses from Radii of Curvature and Focal Lengths. It is shown in geomet- rical optics that the focal length of a lens for any wave length is a function of the index of refraction p, of the glass for light of 22O PHYSICAL MEASUREMENTS that wave length, and of the radii of curvature of the lens. This relation is given by the equation 1 -j = (n-i) 6^-;-~-). (264) If we have the values of /, r and r 2 for any lens we may com- pute the index of refraction at once from the above formula. The focal length having been obtained by means of white light, the resulting value of /*, will of course refer to no definite color, but will in general correspond to the brightest part of the spectrum, i. e., to the part between the lines D and . FORM OF RECORD. Exercise 86. From the values of f } r^ and r 2 for the lenses measured in exercises 80 and 82, compute the mean index of re- fraction for each lens. Date Lens i\ r 2 f ^ How may equation (264) be simplified, when one of the radii is infinite? When the two radii are equal? 163. Exercise 87. Index of Refraction by Means of a Mi- croscope. It is shown in works on physics 2 that if a point A, (Fig. 124), be viewed vertically through a transpar- ent plate of thickness OA and refractive index /*, the point will appear to be raised to some position / in the verti- cal, such that OA = //, O7, or t / 1 ' 1 i / A Fig. 124. where AI = a. In this way the index of refraction of a transparent plate, or of a. layer of fluid may be determined by means of a microscope furnished with a scale and vernier on its tube. 1 For the derivation of this formula and its interpretation see College Physics, Articles 454-457. 2 College Physics, Article 448. INDEX OF REFRACTION 221 In practice the microscope, fitted with a low power objective is focused upon a mark on a piece of stiff paper, or better upon a scratch in a piece of flat metal, held upon the microscope stage by means of clips or bits of wax. The instrument having been sharply focused upon some prominent feature of the scratch, the position is taken by reading the scale and vernier on the tube. The transparent plate, usually a plate of glass some 5 mm thick, is next placed upon the stage above and immediately in con- tact with the scratch in the plate. The microscope is again focused upon the same feature of the scratch through the plate, and the reading taken as before. The microscope is then focused upon the upper surface of the plate and the reading made. From these three readings, each being the mean of at least five separate settings, the values of OA and OI are readily determined and the value of /x computed from the formula. For liquids a small flat-bottomed dish is fastened to the micro- scope stage by two bits of wax, and the instrument focused upon a scratch on the upper surface of the bottom. The liquid is added by means of a medicine dropper to a depth of from 3 to 5 mm, and the reading taken upon the same scratch through the liquid. A few grains of lycopodium powder are then sifted upon the surface of the liquid, the microscope focused upon a grain of the floating powder and the reading taken as before. For liquids the instrument must of course stand vertical. In case the readings differ by as much as 0.06 mm, the mean of a larger number of readings must be taken. The depth of the liquid may be increased after each determination, and readings through the liquid and on top of the liquid give data for a new value of //,. Determine by this method the refractive indices of two pieces of glass and of distilled water. FORM OF RECORD. Exercise 87. To determine the indices of refraction of glass and of distilled water by means of a microscope. Date. Reading on scratch Through subst. On top t ta A* ::::::: 222 PHYSICAL, MEASUREMENTS THE SPECTROMETER- 164. Description. A spectrometer consists essentially of an achromatic telescope, a graduated circle and a collimator, or tel- escope with the eye-piece replaced by an adjustable slit, for pro- ducing a beam of parallel rays. The telescope is mounted upon a substantial tripod base, so as to move freely about the axis of rotation of the instrument. The collimator is usually fixed in position, but has slight freedom of movement for purposes of ad- justment. The graduated circle may or may not rotate about the axis of the instrument, but it must in any case be capable of being firmly clamped at will, and be provided with verniers or reading microscopes to determine the position of the telescope at any time. In the Geneva Society instrument shown in Fig. 125, the prism table is provided with a " ' radial arm carrying a ver- nier along the graduated circle, by means of which the rotation of the table may be accurately determined. By means of a set screw the table may be clamped to this arm or released from it, and when free it may be raised or lowered, or rotat- Fi - I2 5- ed at will about its own axis for purposes of adjustment. It is also provided with three levelling screws, symmetrically placed about its axis for bringing the surface of prism or grating parallel with the axis of rotation. Both the radial arm and the arm carrying the telescope are provided with clamp and slow motion screws for accurate setting upon any desired feature in the field of view. Through adjust- ing screws attached to their supports, both telescope and colli- mator are capable of slight motion in a vertical plane in order to bring their optical axes accurately perpendicular to the axis of rotation of the instrument. The telescope is also provided with a Gauss eye-piece, for THE SPECTROMETER 223 illumination of the cross-hairs in making certain adjustments and measurements. This eye-piece (Fig. 126), has an opening in one side and a piece of transparent, plane parallel glass set across the tube at an angle of 45 to the axis, by which light entering the opening is reflected parallel to the optical axis of the telescope. When the cross-hairs are brought into the focal plane of the objective, light passing them leaves the objective as parallel rays and after striking a plane surface placed normal to the axis of the telescope, is reflected directly back into the telescope forming in its focal plane a dark image of the cross-hairs themselves. A far more delicate and useful device is the Zeiss-Abbe eye-piece, (Fig. 127), in which a small, totally reflecting prism is inserted in the focal plane of the telescope at one side of the field of view, and provided with an adjust- able slit on the side toward the objective. The Fig. 126. slit and cross-hairs lie in the same plane, and when the telescope is focused for infinity, light entering through a small window in the side of the eye-piece is reflected by the prism through the slit and emerges from the objective as par- allel rays. A plane surface placed normal to the axis of the telescope returns the light as from an infinite distance and shows a sharp image of the slit in the focal plane of the telescope. By this arrangement the functions of collimator and observ- ing telescope are combined, the re- flected image is sharp and unmis- I2 7- takable, and the system offers many important optical advantages, some of which will be mentioned later. 165. Adjustments of the Spectrometer. Before the spectrom- eter is readv for use a number of adjustments must be made, 224 PHYSICAL MEASUREMENTS some of which must be repeated at frequent intervals, while others should be needed but rarely. It is not expected that the beginner should attempt such adjustments for himself, but rather that he may obtain an intelligent idea of the working of the instrument from a careful study of them. The adjustments are given in the order that they should be made by an observer on beginning work upon a new instrument. (a) The cross-hairs. The eye-piece has at its focus a pair of fine hairs, termed cross-hairs, which must be sharply seen by the eye on looking into the telescope. If the cross-hairs are not sharp, the eye-piece must either be drawn out or pushed in with a gentle twisting motion until they are seen sharply defined on a white field when the telescope is turned toward the window. This is the first adjustment to be made in every case, and as the adjust- ment is slightly different for different persons, it must be made each time before any work is attempted with the telescope. (b) The telescope. The telescope must be focused for parallel rays. In instruments provided with a Zeiss-Abbe eye-piece this adjustment is readily effected by placing' in front of the objective a piece of glass with a good plain surface ; a piece of French plate mirror glass will do very well. After the slit is illuminated the telescope is focused upon the reflected image until it is seen clearly defined. The telescope is then in adjustment. Although the same end may be accomplished by use of the Gauss eye-piece, the result is not so satisfactory since the images are always faint and lack- ing in distinctive features by which to judge the accuracy of focus. A small dust particle adhering to the hairs may sometimes furnish the desired criterion. (c) The collimator. The telescope is next turned so as to look directly into the collimator. A small lamp is placed behind the slit, which is slightly opened, and the outer end of the colli- mator either pulled out or pushed in until the slit is seen sharply focused in the telescope. The images of slit and cross-hairs must show no parallax, that is, there must be no apparent motion of slit and cross-hairs with reference to each other as the head is slightly moved from side to side while looking into the telescope. The collimator is now in adjustment, since the telescope focused THE; SPECTROMETER 225 for parallel rays shows the slit sharply defined, and the rays emerging from the collimator objective must therefore be parallel, It is well to mark this position of the collimator tube in order that it may be replaced with little trouble in case it should be accidentally displaced. In some instruments the collimator tube may be clamped in position, once it is adjusted for focus, but this arrangement is neither common nor necessary. For laboratory work under ordinary conditions it is usually more expedient to focus and adust the collimator beforehand and require the student to confine his manipulations to the telescope and prism. (d) The optic axes of telescope and collimator. i. The axes of telescope and collimator must be perpendicular to the axis of rotation of the instrument. This adjustment is most readily secured by the use of one of the collimating eye-pieces mentioned above, together with a small plate of plane parallel glass silvered on both sides. If the Gauss eye-piece be employed, it is inserted in the tele- scope and focused upon the cross-hairs, care being taken to leave the glass reflector as nearly vertical as possible. A small lamp on an adjustable support is then brought up to within a few inches of the opening and so adjusted both laterally and vertically that on placing a square of good mirror glass over the objective the image of the diaphragm carrying the cross-hairs is seen filled with light, and on focusing the telescope slightly the cross-hairs are seen sharply defined on a bright field. A slight tilting of the mir- ror glass will show by the disappearance of the circle of light that the illuminated circle is due to reflection from it and not from the posterior surfaces of either of the two intervening lenses. The small plate of plane parallel glass is then mounted upon the table of the spectrometer, either upon an adjustable table of its own, or upon two pieces of soft wax, so as to stand as nearly vertical as may be, and with its face parallel to a line connecting two of the three levelling screws of the prism table. It can then be rotated slightly about this line by means of the third screw. The plate is then turned so as to throw into the telescope the light issuing from the objective, and adjusted until the image of the cross-hairs is again in the field and brought into coincidence with 226 PHYSICAL MEASUREMENTS the hairs themselves. If on rotating the plate through 180 the images are again in coincidence the plate stands parallel to the axis of rotation, and the axis of the telescope is normal to it. If this be not the case then the reflected image from the second side must be brought into coincidence with the cross-hairs, by altering by equal amounts the positions of the plate and telescope. On reversing the plate a similar procedure will soon give the desired adjustment. Finally the telescope is turned to face the collimator, the slit of which has been illuminated and turned horizontal. The level of the collimator is then adjusted until the image of the slit coincides with the horizontal cross-hair of the telescope. If the parallel plate be now turned to reflect the image into the telescope, the slit image should remain on the horizontal hair as the tele- scope follows it around the circle. 2. The optic axes must intersect the axis of rotation of the instrument. This adjustment in many instruments is cared for by the maker, and the telescope and collimator are incapable of motion about a vertical axis. In others the two tubes are adjust- able about a vertical axis, and are very liable to be displaced. The adjustment is most readily secured by placing over the objective of the telescope a cap containing a lens whose focal length is slightly less than the distance from objective to the center of the table. A bullet suspended by a cocoon fiber is then brought directly over the center of the table by means of a suitable sup- port, and the telescope turned about its vertical axis until the image of the thread falls upon the intersection of the cross-hairs. Its vertical axis is then firmly clamped, the lens removed, and the telescope swung round facing the collimator, which is then ad- justed till the slit image likewise falls in the center of the field, when the collimator is also firmly clamped. The adjustment should not be disturbed except for special reason, and the student when rotating the telescope about the circle, should take hold close up to the circle, and not by the outer end. 1 66. Reflecting Surfaces. A reflecting surface intended for optical work should be accurately ground and polished. Many surfaces polished on cloth show innumerable fine ripples when THE SPECTROMETER 227 examined in good light, at a distance of about 15 inches from the eye. Such surfaces are worthless for the formation of optical images. Surfaces of prisms should be true planes, standing normal to the base of the prism and should extend up to the edges of the prism. Many otherwise good prismatic surfaces are rounded off at the edges by careless work in polishing. A perfect prism surface should show no curvature at the edges when tested by interference fringes upon a true plane. It is frequently desirable to examine a reflecting surface in order to see whether or not it is clean, unbroken or properly illuminated. In many cases the surface may be inaccessible or at a temperature which renders close inspection impossible. In such cases a telescope showing an image formed by light reflected from the surface in question affords the means for such examination. A short focus lens, as a common reading glass, held between the telescope and the eye and properly focused, gives a well denned image of the prism surface, and enables the operator to examine it at leisure. This is especially useful in work with prismatic and interferometer surfaces, as the exact condition of the reflecting surface and the point from which the reflected light proceeds can be instantly and accurately determined. In adjusting the faces of a prism for work on the spectrometer, one face is set at right angles to the line joining two of the three levelling screws of the table, in order that its adjustment may be undisturbed by the motion of the third screw. In the case of a prism whose angles are each about sixty degrees, and on a table whose levelling screws are placed symmetrically about the axis, this arrangement may be realized for all three faces. In some prisms one face may not be exactly perpendicular to the base of the prism, and the image of the slit as reflected from this side is slightly inclined to the vertical cross-hair. In such cases it is impossible to have all the faces parallel to the axis of revolution at the same time. This fault is slightly noticeable in most prisms, but by bringing the image of the slit to the middle of the field in each case and setting the cross-hair upon the middle of the image, the error may be rendered practically negligible. 228 PHYSICAL MEASUREMENTS 167. Exercise 88. To Measure the Angle of a Prism. In measuring the angle of a prism with the spectrometer shown in Fig. 125, three methods of procedure are possible. Each method will be described in turn. (a) Prism fixed and telescope rotated. The spectrometer hav- ing been put in adjustment the prism is placed upon the table as directed in Article 166, and its faces set approximately vertical by the eye. The collimator slit is opened rather wide and illumi- nated by a small lamp. The angle to be measured is turned toward the collimator and so placed as to divide the opening of the ob- jective about equally. In the case of telescopes of small aperture it is important also that the angle be placed very near the axis of rotation, as otherwise the reflected rays may be thrown out of the field of view. Turn the telescope to position T, (Fig. 128). A black cloth placed loosely over the col- limator, prism and telescope aids materially in finding the reflected image of the slit in the telescope. It is best to catch the reflected image first in the eye placed close up to the prism and then, keeping the image in view slowly bring the telescope into position. Hav- ing brought the right hand image into the telescope ob- serve whether or not it lies in the center of the field I28 - and parallel to the vertical hair. If not, the prism must be ad- justed for level. Next turn the telescope to position T' , and see whether the left hand image is also visible, whether it lies prop- erly in the field, and whether the telescope is in such a position that readings may be made in each case. If the prism need adjustment for level, observe carefully which face is most out, and note the effect upon the position of the THE SPECTROMETER 22Q images when the prism is slightly rocked about each edge forming the angle in question. If the prism has been properly placed upon the table a few minutes trial should bring it into adjustment. In case the faces are not both normal to the base, the images can- not both be rendered vertical. In such case set the intersection of the cross-wires on the center of the image. The slit image is next drawn down* to a narrow line by means of the adjusting screw on the collimator, and brought upon the vertical hair by use of the clamp and slow motion screws. Owing to the fact that the eye can better judge of the equality of the two bright parts of the slit on either side of the dark hair, it is preferable not to make the slit narrower than from three to five times the width of the hair. The setting having been completed the position of the telescope is read and recorded. If the instru- ment have two verniers or two microscopes read both each time and combine the readings as indicated in Article 28. Unclamp the telescope and turn to position T f and repeat the operations just described. Then return to position T, set and read, to make sure that nothing has been changed. Combine this reading with the first made. The difference between this mean and the reading at T' is twice the angle A, of the prism. Prove this. Displace the prism slightly and repeat the measurements twice. Take the mean of the three as the final result. (&) Telescope fixed, prism rotated. Turn the telescope (Fig. 129), so as to make an angle of 20 to 30 with the collimator and clamp it. Rotate the prism table and adjust until the reflected image from face AC enters the telescope and lies properly in the field. Then continue the rotation until the image from face AB also enters the field. If both images are properly located for meas- urement, clamp the table to its radial arm, after making sure that the vernier can be read in each position. Bring the image from the first face once more into the field and clamp the pjg I2 g arm, making the final setting with the slow motion screw. Record the reading. Unclamp the arm, rotate 230 PHYSICAL MEASUREMENTS prism to second position, set and read. The angle through which the prism has been rotated is 180 A. Take three sets of read- ings as under (a), unclamping the table from the arm and displac- ing slightly so as to change the readings in each case. Take the mean of the three results. (c} By autocollimation. The prism is placed centrally over the axis % of the table and the telescope, fitted with either of the collimating eye-pieces de- scribed in Article 166, is clamped in position. The illumination from the side is adjusted and tested by the small mirror placed over the ob- jective. The prism is then turned so that face AC (Fig. 130), stands approximately normal to the axis of the telescope and slowly rotated until the bright image either of the diaphragm or slit enters the field. The prism is then ad- justed until the reflected images from the two sides register accurately upon the cross-hairs. The table is then clamped to the arm and read- ings made upon the two faces in succession. The angle through which the prism has been displaced is again 180- A. Make three independent sets of readings, displacing the table relatively to the arm after each set as under (b). FORM OF RECORD. Exercise 88. To measure the angle of a prism by three differ- ent methods. (a) Vernier I " '.' II istFace Date. 2nd Face Difference Mean ........ (b) Similar record. (O " Angle by method (a) = " " " (&) = Angle, Mean = 1 68. Angles by Method of Repetition. In work of precision it is frequently of advantage to employ the principle of repetition THE: SPECTROMETER 231 in the measurement of angles. With a spectrometer in which the movement of telescope and of prism table may be read separately, or in which the prism table may move either with, or independ- ently of, the graduated circle, the principle may be employed very readily in either method b or c, as described above. For example suppose that in method (b), the prism has been rotated from the first to the second position by means of the radial arm, and the readings determined in each case. If now leaving the arm fixed, the table be released and the prism rotated back to its first position, the setting made, and the table again clamped to the arm, and arm and table again rotated in the same direction as at first until the slit image from the second face again falls upon the cross-hairs, it is clear that the difference between the initial and final reading of the arm vernier will correspond to a total rotation of 2 (180 A) or 360 2A. If the operation be repeated once more the displacement becomes 540 $A, or, in case A is about 60, the total displacement is about 360 and the initial and final readings are made upon the same part of the circle. The method therefore consists in stepping off the angle a suffi- cient number of times to bring the index back to the neighbor- hood of the starting point, where "the graduations are assumed to differ little from each other. It is designed to eliminate errors of graduation in the circle, and assumes that these errors are greater than those of individual settings. The method is em- ployed to the greatest advantage when the angle to be measured is some aliquot part of 360. In some instruments the graduated circle is read from fixed verniers or microscopes, and may move either in conjunction with the telescope or each may move independently. In such instru- ments the principle may be applied to method (a). Thus sup- pose the reading has been made in position T; the telescope is clamped to the circle and the two rotated into position T' ', the prism remaining fixed. The angle thus stepped off is 2 A. The circle is now clamped and the telescope released 1 , returned to posi- tion T, accurately set and clamped to the circle, which is now re- 232 PHYSICAL MEASUREMENTS leased and telescope and circle again moved forward throug'h the angle 2 A. 1 169. Exercise 89. Index of Refraction of a Glass Prism. The effect of a prism upon light passing through it is two-fold. The direction of the light is changed, the light being bent toward the base of the prism both on entering and leaving the prism, and secondly, the light is dispersed or broken up into its constituent colors. If the prism used in the previous exercise be now placed centrally over the center of the table and turned into the position indicated by the full line (Fig. 131), an eye, placed in the position indicated by the emergent light, will perceive no longer a bright image of the slit, but a broad band of color, the spectrum of the light furnished by the lamp. This spectrum may now be received into the telescope and its parts examined. The best effect is obtained by excluding all stray light from the telescope by means of the dark cloth as in Exer- cise 88. By rotating the prism slowly and following the spectrum Avith the telescope, a position is soon found in which, no matter which way the prism is rotated, the spectrum conies to a certain point nearest the direct line from the collimator, stops and then recedes. This is the position of minimum deviation. The small lamp is now removed and a Bunsen burner substituted. The burner is so arranged that the colorless flame plays against the tip of a piece of asbestos paper saturated with sodium nitrate. An intense yellow light results. On examining the image in the telescope it is seen that the spectrum of this light consists of a single bright line, a yellow image of the slit, the sodium spectrum, for this temperature. In spectroscopes of high resolving power this line is readily seen to consist of two lines, D : and ZX. 1 See Louis Bell, The Absolute Wave Length of Light, Am. Jour. Sci, XXXV, pp. 350-352. THE SPECTROMETER 233, By closing the openings of the Bunsen burner so as to give the luminous flame, the continuous spectrum returns and we see superposed upon it the bright line due to the vapor of incandes- cent sodium. The non-luminous flame having been restored, the prism is rotated until the position of minimum deviation for the sodium line is actually determined, the cross-hair placed upon the image of the slit, the telescope clamped and the reading taken. The prism is next rotated into the position shown by the dotted line in the figure. The light is now deviated to the left of the direct position and the position of minimum deviation is deter- mined as before. The difference between the two readings is ob- viously 2D, where D is the angle of minimum deviation for sodium light. It is shown in works on physics, 1 that when the prism is put in the position of minimum deviation, the refractive index /*, is defined by the equation where A is the angle of the prism. Derive this formula. From the measured values of D and A as obtained above, com- pute the value of ^ for sodium light for the prism under experi- ment. Repeat the experiment using lithium carbonate in place of sodium nitrate. FORM OF RECORD. Exercise 89. To determine the index of refraction of a prism for the lines Naa and Lia. Prism.. Date.. r r 2D D Computation log sin y 2 (A + D)... log sin y 2 A l / 2 (A-\-D) log /* ... /* 1 College Physics, Article 450. 234 PHYSICAL MEASUREMENTS DIFFRACTION. 170. Exercise go. Wave-length of Sodium Light by Dif- fraction Grating. A Bunsen burner (Fig. 132), is provided with a sheet iron hood in -which is cut a small triangular slit, s, about 1.5 cm long. Immediately in front and below the slit is fixed a meter rod held horizonally with the slit at the center of the rod. At a distance of some three or four meters in front of the slit is placed the grating, held in a suitable clamp, with its surface ver- tical and parallel to the meter rod. If the burner be adjusted for the luminous flame the slit appears white to the naked eye but when viewed through the grating the eye perceives in addition Fig. 132. to the white slit, a number of spectra symmetrically placed with reference to the central image. These are diffraction spectra and are characterized (a) by the relative positions of the various colors with respect to the slit, the violet being the least diffracted and the red the most; (b) by the uniformity of the dispersion of the various spectra, each color being seen at a distance from the slit directly proportional to the wave-length of the light in ques- tion. In practice the luminous flame is replaced by the sodium light and the colored spectra become a series of yellow images of the slit which, to an eye placed behind the grating, are seen projected upon the meter rod at spaces equidistant from the central image. Beginning at the inner spectra measure carefully the distances M'H between the first two images on either side of the slit, s.,s' 2 , the distance between the next two, and so on. Take half the measured distance as the distance of each image from the central slit, ss lf ss. 2 , and so on. DIFFRACTION 235 If d be the grating space, n the order of the spectrum observed, and n the angle subtended at the eye by the distance ss n , then 1 X d sin (267) where A is the wave-length of sodium light. Hence for the first three or four spectra, d sin 6- d sin 5 X = d sin 0i = = , etc. (268) In the experiment described the distances ssa. divided by a, the distance from the rod to the grating, gave directly tan n , in each case, from which the value of sin n is readily found. Determine by this method the wave-length of sodium light, using spectra of at least four different orders. The value of d for the grating used will be given by the instructor. Calculate A in millimeters. FORM OF RECORD. Exercise 90. To measure the wave-length of sodium light by diffraction grating. _> o d .... s .... a .... Date Su S'n ' SnS'n SS a sin0 n \ 2 a n 171. Exercise 91. Constant of a Diffraction Grating. If in equation (267) the wave-length of light be assumed as known, the quantities n and sin n may be determined and d the grating constant may be computed. Place a transmission diffraction grat- ing, whose constant is to be determined, centrally upon the table of the spectrometer and set its surface normal to the axis of the collimator. This is most readily done by setting the telescope upon the image of the illuminated collimator slit, and then set- 1 College Physics, Article 494. 236 PHYSICAL MEASUREMENTS ting the grating normal to the axis of the telescope, by means of an auto-collimating eye-piece. Illuminate the slit with sodium light, and determine for the first two spectra on either side the central image. Assume for A. its mean value 0.0005893 mm, and compute the value of d. FORM OF RECORD. Exercise pi. To determine the constant of a diffraction grat- ing by the spectrometer. Grating used Readings First spectrum Second Right Left Date, 2 0,= *i= d-. 2 2 = 2 = di Mean 172. Dispersion, Normal and Prismatic. We have seen in Exercise 90, that the spectrum of a diffraction grating is char- acterized by the uniformity of the dispersion produced, the de- viation of each color being absolutely fixed by the equation con- necting the wave-length of the color in question and the constant of the grating. Spectra formed by two diffraction gratings are directly comparable, the ratio of their lengths being inversely as the grating constants. Such spectra are termed normal spectra, and such dispersion, normal dispersion. With prisms however the case is very different. If we ex- amine the spectra from a number of prisms of different materials, but all having the same refracting angle, we shall find that tlv lengths of the spectra differ enormously. This is said to be due to the different dispersive powers of the different prismatic sub- stances. Again if the angles of the prisms be so adjusted that the resulting spectra are all of the same length, we shall still find that the separation of the colors in different parts of the spectrum is very different in the different substances. There is no definite relation therefore between the change of index of refraction and change of wave-length, the dispersion in each case depend- ing upon the nature of the refracting substance. This peculiarity is termed irrationality of dispersion. DIFFRACTION 237 173. Exercise 92. Dispersion Curve for a Prism. Owing to the irrationality of dispersion, the spectra from prisms of differ- ent material are not directly comparable with each other and it becomes necessary to investigate experimentally the relation exist- ing between wave-length and index of refraction for any given prism before it can be used as an instrument for study of un- known lines in the spectrum. This is most readily done by determining the indices of refrac- tion of the prism for a number of prominent well known lines in different parts of the spectrum, and plotting on co-ordinate paper the indices as ordinates and the corresponding wave-lengths as abscissae. Through the points thus determined a smooth curve is drawn which is termed the dispersion curve, for the prism in question. By means of this curve the wave-length of an unknown line may be determined as soon as its index of refraction with the given prism is known, or the index for a line of given wave- length may be predicted. The curve may be represented with a fair degree of accuracy by Cauchy's dispersion formula (269) where A and B are constants depending upon the nature of the prism substance. From a series of related values of A and p, we may determine the constants A and B. Determine according to Exercise 89, the indices of refraction of a given prism for the bright lines K a , K^ Li^ Na a , Tl, Sr 6 ^ and plot a curve with the indices thus obtained as ordinates, and the corresponding wave-lengths, (Table XIII), as abscissae. Take off from the curve the indices of the prism for the two hydrogen lines H a and 11^ corresponding to the C and F lines in the solar spectrum using the values for A given in Table XIII. Using sun- light verify the indices by actual measurement. Determine indices to three decimals. 2 3 8 PHYSICAL MKASUREMENTS FORM OF RECORD. Exercise 92. To construct dispersion curve for Prism No and determine from it the indices, of refraction for two lines of given wave-length. Date, Line 77 KR D P> X /* for Ha " H/3 From curve Observed Difference TABLES. TABLE I. Atomic Weights of Some Elements. Hydrogen = 1 Oxygen = 16.0 Hydrogen Oxygen = 16.0 Aluminum 26 90 27 1 Nitrogen 13 93 14 04 Cadmium .... 111 55 112 4 Platinum 193 30 194 80 Chlorine 35 18 35 4 Potassium 38 85 39 15 Copper 63.12 63 6 Silver 107 11 107 93 Gold. 195.70 197 2 Sodium 22 88 23 05 Lead 205 35 206 9 Sulpliur 31 82 32 06 Mercury 198.50 200.0 Tin 118 10 119 00 Nickel 58.30 58.7 Zinc 64 91 65 40 TABLE II. Density of Water at Different Temperatures. t d / d / d 0.99988 11 0.99965 21 0.99806 1 0.99993 12 0.99955 22 0.99784 2 0.99997 13 0.99943 2;J 0.99761 3 0.99999 14 0.99930 24 0.99738 4 1.00000 15 0.99915 25 0.99713 5 0.99999 16 0.99900 26 0.99688 6 0.99997 17 0.99884 27 0.99661 7 0.99993 18 0.99866 28 0.99634 8 0.99988 19 0.99847 29 0.99606 9 0.99982 20 0.99827 30 0.99577 10 0.99974 239 24O PHYSICAL MEASUREMENTS TABLE III. Density of Mercury at Different Temperatures. t t tf i * d 15.5950 11 13.5679 21 13.5433 1 .5925 12 .5654 2-2 .5408 2 .5901 13 .5629 23 .5384 .5876 14 .5605 24 .5359 4 .5851 15 .5581 25 .5335 - 5 .5827 16 .5556 26 .5310 6 .5802 17 .5531 27 .52^6 7 .5777 18 .5507 28 .5261 8 .5753 19 .5482 29 .5237 9 .5728 20 .5457 30 .5212 10 .5703 IV. Densities of Various Bodies. Alcohol at 20 C 0.789 .Aluminium 2.58 Brass (about) 8.5 Brick 2.1 Copper 8 . 92 Cork 0.24 Diamond 3.52 Glass, common 2.6 " heavy flint 3.7 Gold 19.3 IceatOC 0.916 Iron, cast 7.4 Iron, wrought 7.86 Lead 11.3 Mercury at C 13.595 Nickel 8.9 Oak 0.8 Paraffin 0.90 Pine 0.5 Platinum 21.50 Quartz 2.65 Silver 10.53' Tin 7.29 Zinc.. 7.15 TABLES 2 4 I TABLE: V. Reduction of Barometer Readings to oC. When the height of a mercury column has been measured with a brass scale, the length of which is correct at oC., the tempera- ture of the barometer being C., the following quantity, corre- sponding to temperature and height, has to be subtracted from the reading. (See equation 35.) OBSERVED HEIGHT OF BAROMETER. t 72.0 cm. 73.0 cm. 74.0cm. 75.0 cm. 76.0 cm. 77.0 cm. 10 0.117 cm. 0.118 cm. 0.120 cm. 0.122 cm. 0.123 cm. 0.125 cm. 11 .128 .130 .132 .134 .135 .137 12 .140 .142 .144 .146 .148 .150 13 .152 .154 .156 .159 .160 .162 14 .163 .166 .168 .170 .172 .175 15 .175 .177 .180 .182 .185 .187 16 .187 .189 .192 .194 .197 .200 17 .198 .201 .204 .207 .209 .212 18 .210 .213 .216 .219 .222 .225 19 .222 .225 .228 .231 .234 .237 20 .233 .237 .240 .243 .246 .249 21 .245 .248 .252 .255 .259 .262 22 .257 .260 .264 .267 .271 .274 23 .268 .272 .276 .279 .283 .287 24 .280 .284 .288 .292 .295 .299 25 .292 .296 .300 .304 .308 .312 26 .303 .307 .312 .316 .320 .324 27 .315 .319 .324 .328 .332 .337 28 .327 .331 .336 .340 .345 .349 29 .338 .343 .348 .352 .357 .362 30 .350 .355 .360 .365 .369 .374 TABLE VI. Coefficients of Elasticity. Substance Volume Elasticity Simple Rigidity Young's Modulus Velocity 7i /r Of e n M Sound Substance Volume Elasticity e Simple Rigidity n Young's Modulus M Velocity of Sound Distilled water.. . . 10' * dynes per cm 2 0222 10 11 dynes per cms 10 11 dynes per cm? 105 cms . per sec . 1.45 Glass (flint) . . . Brass 3.47 to 4.15 10 02 to 10 85 2.35 to 2.40 3 44 to 4 03 5.74 to 6.03 9 48 to 11 2 5.0 34 Steel Iron (wroug't) Iron (cast). Copper 18.41 14.56 9.64 16.84 8.19 7.69 5.32 4.40 to 4.47 20.2 to 24.5 19.63 13.49 11.72 to 12.34 5.1 51 5.0 3.7 242 PHYSICAL MEASUREMENTS TABLE VII. Viscosity and Surface Tension of Liquids at 20 C. Coeff. of Viscosity Surface Tension Alcohol grams per cm. sec. 0.0125 dynes per cm. 23 Ether 0025 18 Glycerine . . 8 5 Mercury 015 68 Machine Oil. 1.95 Olive Oil 0.225 35 Turpentine. 0015 27 Water 0102 73 Uniform thin length = / TABLE VIII. Moments of Inertia. Rod, axis through middle, Rectangular Lamina, axis through center and parallel to one side, a and b length of sides, a the side bisected ...... Rectangular Lamina, axis through center and perpendicular to the plane, a and b length of sides ...... Rectangular Parallelepiped, axis through center and perpendicular to a side ; a, b and c length of sides, axis perpendicular to side contained by a and b ..... Circular Plate, axis through center perpendicular to the plate, radius = r Circular Ring, axis through center perpen- dicular to plane of ring, outer radius = R, inner radius = r Right Cylinder, axis the axis of figure, r = ra- dius of section . . . Sphere, axis any diameter, r = radius I=M I=M a 1 -- Ir TABLES 243 TABLE IX. Boiling Point of Water Under Different Barometric Pressures. .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 72. 73. o 74. I 75 - I 76. 277. a a 98.49 98.88 99.26 99.63 100.00 100.37 98.53 98.92 99.29 99.67 100.04 100.40 98.57 98.95 99.33 99.70 100.07 100.44 98.61 98.99 99.37 99.74 100.11 100.48 98.65 99.03 99.41 99.78 100.15 100.52 98.69 99.07 99.44 99.82 100.18 100.55 98.72 99.10 99.48 99.85 100.22 100.59 98.76 99.14 99.52 99.89 100.26 100.63 98.80 99.18 99.55 99.93 100.29 100.67 98.84 99.22 99.59 99.96 100.33 100.70 TABLE X.Heat Constants. Substance Cubical Expansion Specific Heat Boiling Point Heat of Vaporization Alcohol 0.00110 0.58 78. 3 cal. 210 ~~ Ether 00163 54 34 9 90 Mercury 000181 0332 356 7 62 Turpentine 00094 42 159 70 Water 00018 1 100 539 Substance Linear Expansion Specific Heat Melting Point Heat of Fusion Aluminium 000023 22 658 Brass 000019 0.093 900 Copper. . 0.000017 0.093 1084 /~M KK (jrlass 000008 19 1200 cal. Ice ... 000106 504 79.5 g Iron 000012 11 1520 30 Lead. 000029 031 327 6 Nickel 000013 11 1450 Platinum 000009 17 1755 27 Ouartz, fused 00000045 0.19 2000 Silver 000019 056 961 21 Tin 000023 054 232 13 Zinc 000029 094 419 28 244 PHYSICAL MEASUREMENTS TABLE XL Vapor Tension of Liquids. Temperature Alcohol Ether Mercury Water C. cm. cm. cm. cm. 10 0.65 11.4 1.25 18.5 0.00004 0.46 10 2.40 28.8 0.00005 0.92 20 4.41 44.0 0.0001 1.75 30 7.84 64.5 0.0003 3.18 40 13.36 91.0 0.0006 5.49 50 22.00 127.5 0.0013 9.20 60 35.1 173 0.0026 14.89 70 54.1 229 0.0051 23.33 80 81.2 300 0.0092 35.54 90 119 384 0.0162 52.59 100 169 490 0.028 76.00 110 236 607 0.045 107.5 120 322 760 0.075 149.1 TABLE XII. Index of Refraction for Sodium Light. Dry Air, 0C Ice 1.0003 1 31 Glass, Light Crown. . 1 515 Calcite, Ordinary rav 1.659 Water, 15C. Water, 20C. Alcohol Benzine 1.3336 1.3332 1.362 1.501 Light Flint . . . Heavy Flint. . Carbon Bisul- phide, 20C. .. 1.609 1.75 1.628 Extraordinary ray Quartz, Ordinary ray.. . . Extraordinary ray 1.486 1.554 1.553 TABLE XIII. Wave Lengths of Lines in Solar Spectrum, in Air at 20 C, Pressure 76 cm.; ^ = 0.001 mm. Line Element Wave Length Line Element Wave Length Line Element Wave Length fi A 1 A* A Atm. O 0.7628 D, Na 0.58900 e Fe 0.43836 a Atm. O 0.7185 Ei Fe, Ca 0.52703 f H 0.43405 B O 0.68701 h Mg 0.51837 G Fe, Ca 0.43079 C H 0.65629 -c F? 0.4*576 h H 0.41018 a. O 0.62781 F H 0.48614 H H, Ca 0.39685 A Na 0.58960 d Fe 0.46682 K Ca 0.39337 TABLES^ 245 Wave Lengths of Lines in the Flame Spectrum of Various Metals. Element Wave Length Element Wave Length K Li Na 0.768 0.671 0.589 Tl Sr K 0.535 0.461 0.404 TABLE XIV. Electrical Resistance of Metals. (a) Specific conductivity, referred to mercury. Aluminium (soft) 32.35 Copper (pure) 59 Iron 9.75 Mercury 1 Nickel (soft) 3.14 Platinum 14.4 Silver (soft) 62.6 Tin... 7 (b) Specific resistance, in ohms cm. at 18 C. Substance Specific Resistance Temperature Coefficient Aluminium 0.321 17 3.0 1.2 4.2 9.58 1.0 1.08 1.4 0.16 J , XlO' 5 0.0036 0.0040 0.0004 0.0005 0.00001 0.00092 0.005 0.0036 0.0025 0.0037 Copper German Silver Iron ... Manganin .... Mercury Nickel Platinum, pure Platinum, commercial Silver TABLE XV. Electrical Conductivity of Solutions at i8C. (a) Specific conductivity, in ohms' 1 cm' 1 . Concentration NaCl CuSOi ZnSO 4 AgN0 3 5 10* 15* 0.0672 0.1211 0.1642 0.0189 0.0320 0.0421 0.0191 0.0321 0.0415 0.0256 0.0476 0.0683 246 PHYSICAL MEASUREMENTS (b) Equivalent conductivity. Gramequivalents per Liter NaCl KCl KN0 3 iH 2 SO< 0.01 0.1028 0.1224 0.1182 0.308 0.1 0.0925 0.1120 0.1048 0.225 0.5 0.0809 0.1024 0.0892 0.205 1.0 0.0743 0.0983 0.0805 0.198 TABLE XVI. Numbers Frequently Required. i cm. = 0.3937 in. i mile = 1.6093 km. IT 2 = 9.8696. I in. = 2.540 cm. i km. = 0.6214 mile, log 77 = 0.49715. log TT 2 = 0.99430. Base of natural logarithms: = 2.7183. log e = 0.43429. Factor to convert common into Naperian logs = 2.3026. Density of dry air at oC. under a barometric pressure of 76 cms., 0.001293 gm/cm 3 . Coefficient of expansion of air . . . 0.00367. Velocity of sound in air at oC. . 332.4 m/sec. i calorie = 4.181 X io 7 ergs for water at 20 C. i atmo. pressure = 1.0132 X io 6 dynes/cm 2 , g at latitude 45 and sea level = 980.63 cm/sec 2 . Specific heat of steam at constant pressure . 0.4776. Specific heat of steam at constant volume . 0.3637. Vibration frequency of C 3 on the scientific diatonic scale, 256. Vibration frequency of A 3 on the musical equal-tempered scale, 435. TABLES 247 TABLE XVII. Squares, Cubes, Square Roots, Circumferences and Areas of Circles. n IT ;/ 1 A T ;/ 2 n- ** v 1 3,1416 0,7854 1 1 1,0000 2 6,2832 3,1410 4 8 4142 3 9,4248 7,0686 9 27 7321 4 12,566 12,566 16 64 2,0000 5 15,708 19,635 25 125 2361 6 18,850 28,274 36 216 4495 1 21,991 38,485 49 343 6458 8 25,133 50,265 64 512 8284 9 28,274 63,617 81 729 3,0000 10 31,416 78,540 100 1000 1623 11 34,557 95,03 121 1331 3166 12 37,699 113,10 144 1728 4641 13 40,841 132,73 169 2197 6056 14 43,982 153,94 196 2744 7417 15 47,124 170,17 225 3375 8730 16 50,265 201,06 256 4096 4,0000 17 53,407 226,98 289 4913 1231 18 56,549 254,47 324 5832 2426 19 59,690 283,53 361 6859 3589 20 62,832 314,16 400 8000 4721 21 65,973 346,36 441 9261 5826 22 69,115 380,13 484 10648 6904 23 72,257 415,48 529 12167 7958 24 75,398 452,39 576 13824 8990 25 78,540 490,87 625 15625 5,0000 26 81,68 530,93 676 17576 099 27 84,82 572,55 729 19683 196 28 87,96 615,75 784 21952 291 29 91,11 660,52 841 24389 385 30 94,25 706,86 900 27000 477 31 97,39 754,77 961 29791 568 32 100,53 804,25 1024 32768 657 33 103,67 855,30 1089 35937 745 34 106,81 907,92 1156 39304 831 85 109,96 962,11 1225 42875 916 36 113,10 1017,9 1296 46656 6,000 37 116,24 1075,2 1369 50fi53 083 38 119,38 1134,1 1444 54872 164 39 122,52 1194,6 1521 59319 245 40 125,66 1256,6 1600 64000 325 41 128,81 1320,3 1681 68921 403 42 131,95 1385,4 1764 74088 481 43 135,09 1452,2 1849 79507 5o7 44 138,23 1520,5 1936 85184 633 45 141,37 1590,4 2025 91125 708 46 144,51 1661,9 2116 97336 782 47 147,65 1734,9 2209 103823 856 48 150,80 1809,6 2304 110592 928 49 153,94 1885,7 2401 117649 7,000 50 167,08 1963,5 2500 125000 071 248 PHYSICAL MEASUREMENTS TABLE XVII. Continued. Squares, Cubes, Square Roots, Cir- cumferences and Areas of Circles. n if n Xir2 2 n* v 51 160,22 2042,8 2601 132651 7,141 52 163,36 2123,7 2704 140608 211 53 166,50 2206,2 2809 148877 280 54 169,65 2290,2 2916 157464 348 55 172,79 2375,8 3025 166375 416 56 175,93 2463,0 3136 175616 483 57 179,07 2551,8 3249 185193 550 58 182,21 2642,1 3364 195112 616 59 185,35 2734,0 3481 205379 681 60 188,50 2827,4 3600 216000 746 61 191,64 2922,5 3721 226981 810 62 194,78 3019,1 3844 238328 874 63 197,92 3117,2 3969 250047 937 64 201,06 3217,0 4096 262144 8,000 65 204,20 3318,3 4225 274625 062 66 207,35 3421,2 4356 287496 124 67 210,49 3525,7 4489 300763 185 68 213,63 3631,7 4624 314432 246 69 216,77 3739,3 4761 328509 307 70 219,91 3848,5 4900 343000 367 71 223,05 3959,2 5041 357911 426 72 226,19 4071,5 5184 373248 485 73 229,34 4185,4 5329 389017 544 74 232,48 4300,8 5476 405224 602 75 235,62 4417,9 5625 421875 660 76 238,76 4536,5 5776 438976 718 77 241,90 4656,6 5929 456533 775 78 245,04 4778,4 6084 474552 832 79 248,19 4901,7 6241 493039 888 80 251,33 5026,6 6400 512000 944 81 254,47 5153,0 6561 531441 9,000 82 257,61 5281,0 6724 551368 055 83 260,75 5410,6 6889 571787 110 84 263,89 5541,8 7056 592704 165 85 267,04 5674,5 7225 614125 220 86 270,18 5808,8 7396 636056 274- 87 273,32 5944,7 7569 658503 327 88 276,46 6082,1 7744 681472 381 89 279,60 6221,1 7921 704969 434 90 282,74 6361,7 8100 729000 487 91 285,88 6503,9 8281 753571 539 92 289,03 6647,6 8464 778688 592 93 292,17 6792,9 8649 804357 644 94 295,31 6939,8 8836 830584 695 95 298,45 7088,2 9025 857375 747 96 301,59 7238,2 9216 884736 798 97 304,73 7389,8 9409 912673 849 98 307,88 7543,0 9604 941192 899 99 311,02 7697,7 9801 970299 950 100 314,16 7854,0 10000 1000000 10,000 TABLES 249 TABLE XVIII. Trigonometric Functions. Arc Sine Tangent log arc log sin log tan 1 2 0,0175 0349 (V0175 0349 0,0175 0349 "2,2419 5428 "22419 5428 "2,2419 5431 3 0524 0523 0524 7190 7188 7194 4 0698 0698 0699 8439 8436 8446 5 0873 0872 0875 _9408 _9403 _9420 6 7 1047 1222 1045 12K) 1051 1228 U)200 0870 L0192 1)859 1.0216 0891 8 1396 1392 1405 1450 1436 1478 9 1571 1564 1584 1961 1943 1997 10 1745 1736 1763 2419 2397 2463 11 1920 1908 1944 2833 2806 2887 12 2094 2079 2126 3210 3179 3275 13 2269 2250 2309 3558 3521 3634 14 2443 2419 2493 3879 3837 3968 15 2618 . 2588 2679 4180 4130 4281 16 2793 2756 2867 4461 4403 4575 17 2967 2924 30)57 4723 4659 4853 18 3142 3030 3249 4972 4900 5118 19 3316 3256 3443 5206 5126 5370 20 3491 3420 3640 5429 5341 5611 21 3665 3584 3839 5641 5543 5842 22 3840 3746 4040 5843 5736 6064 23 4014 3907 4245 6036 5919 6279 24 4189 4067 4452 6221 6093 64S6 25 4363 4226 4663 6398 6259 6687 26 4538 4384 4877 6569 6418 6882 27 4712 4540 5095 6732 6570 7072 28 4887 4695 5317 6890 6716 7257 29 5061 4848 5543 7042 6856 7438 30 5236 5000 5774 7190 6990 7614 31 5411 5150 6009 7333 7118 7788 32 5585 5299 6249 7470 7242 7958 33 5760 5446 6494 7604 7361 8125 34 5934 5592 6745 7733 7476 8290 35 6109 5736 7002 7860 7586 8452 ' 36 6283 5878 7265 7982 7692 8613 37 6458 6018 7536 8101 7795 8771 38 6632 6157 7813 8216 7893 8928 39 6807 6293 8098 8330 7989 9084 40 6981 6428 8391 8439 8081 9238 41 7156 6561 8693 8546 8169 9392 42 7330 6691 9004 8651 8255 9544 43 7505 6820 9325 8753 8338 9697 44 7679 6947 9657 8853 8418 9848 45 7854 7071 \,0000 8951 8495 0,0000 250 PHYSICAL MEASUREMENTS TABLE XVIII. Continued. Trigonometric Functions. Arc Sine Tangent log arc log sin log tan 46 0,8029 0,7193 1,0355 ~T,9047 1,8569 0,0152 47 8203 7314 0724 9140 8641 0303 48 8378 7431 1106 9231 8711 0456 49 8552 7547 1504 9321 8778 0608 50 8727 7660 1918 9409 8843 0762 51 8901 7771 2349 9494 8905 0916 52 9076 7880 2799 9579 8965 1072 53 9250 7986 3270 9661 9023 1229 54 9425 8090 3764 9743 9080 1387 55 9599 8192 4281 9822 9134 1548 56 9774 8290 4826 9901 9186 1710 57 9948 8387 5399 9977 9236 1875 58 1,0123 8480 6003 0,0053 9284 2042 59 0297 8572 6643 0127 9331 2212 60 0472 8660 7321 0200 9375 2386 61 0647 8746 8040 0272 9418 2562 62 0821 8829 8807 0343 9459 2743 63 0996 8910 9626 0412 9499 2928 64 1170 8988 2,0503 0480 9537 3118 65 1345 9063 1445 0548 9573 3313 66 1519 9135 2460 0614 9607 3514 67 1694 9205 3559 0680 9640 3721 68 1868 9272 4751 0744 9672 3936 69 2043 9336 6051 0807 9702 4158 70 2217 9397 7475 0870 9730 4389 71 2392 9455 9042 0931 9757 4630 72 2566 9511 3,0777 0992 9782 4882 73 2741 9563 2709 1052 9806 5147 74 2915 9613 4874 1111 9828 5425 75 3090 9659 7321 1169 9849 5719 76 3265 9703 4,0108 1227 9869 6032 77 3439 9744 3315 1284 9887 6366 78 3614 " 9781 7046 1340 9904 6725 79 3788 9816 5,1446 1395 9919 7113 80 3963 9848 6713 1450 9934 7537 81 4137 9877 6,3138 1504 9946 8003 82 4312 9903 7,1154 1557 9958 8522 83 4486 9925 8,1443 1609 9968 9109 84 4661 9945 9,5144 1662 9976 9784 85 4835 9962 11,4301 1713 9983 1,0580 86 5010 9976 14,3007 1764 9989 1554 87 5184 9986 19,0811 1814 9994 2806 88 5359 9994 28,6363 1864 9997 4569 89 5533 9998 57,2900 1913 9999 7581 90 5708 1,0000 00 1961 0,0000 oo TABLES 251 XIX. Logarithms. N 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 OS99 0934 0969 1004 1038 1072 1106 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405, 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6(518 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7C67 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 /574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 PHYSICAL, MEASUREMENTS TABLE XIX. Continued. Logarithms. N 1 2 3 4 5 6 7 8 9 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8000 8007 8.014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 67 8261 8267 8274 8280 8287 8293 8299 8306- 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 ' 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 100 00000 0043 0087 0130 0173 0217 0260 0303 0346 0389 101 00432 0475 0518 0561 0604 0647 0689 0732 0775 0817 102 00860 0903 0945 0988 1030 1072 1115 1157 1199 1242 103 01284 1326 1368 1410 1452 1494 1536 1578 1620 1662 104 01703 1745 1787 1828 1870 1912 1953 1995 2036 2078 105 02119 2160 2202 2243 2284 2325 2366 ^407 2449 2490 106 02531 2572 2612 2653 2694 2735 2776 2816 2857 2S98 107 02938 2979 3U19 3060 3100 3141 3181 3222 3262 3302 108 03342 3383 3423 3463 3503 3543 3583 3623 3663 3703 109 03743 3782 3822 3862 3902 3941 3981 4021 4060 4100 110 04139 4179 4218 4258 4297 4336 4376 4415 4454 4493 1 !! 1 1 ! INDEX Numbers refer to Articles Abbe eye-piece 164 Absorption, electric 145 After effect, elastic 47 Air, expansion of Table XVI Air, free water 86 Air thermometer 87 Ammeter 116 Ammeter, calibration of 144 Ampere, definition of 100 Angle, measurement of 24-35 Angle of prism, measurement of... 167 Angles by repetition 1 68 Angular deflection of mirror 29 Astatic pair of magnets 107 Atomic weights Table I Autocollimation, angle by 167 Balance 37-42 Balance, Jolly's 54, 71 Ballistic galvanometer 113-115 Ballistic method for magnetic meas- urements 153 Barometer 46 Barometer readings, corrections of . . 46 Batteries, E. M. F. and resistance of 135-139 Batteries of constant E. M. F 101 Batteries, storage 101 B. A. unit of resistance 99 Bending, laws of 56 Boiling point, determination of 80 Boiling point of water at different pressures Table VIII Boyle's law 63 Buoyancy of air, correction for .... 41 Cadmium standard cells 101 Calibration of galvanometer 1 1 1 Calibration of thermometers 80 Caliper, micrometer 14 Caliper, vernier 18 Calorie 88 Calorimeter 90 Calorimetry 88-95 Capacity, comparison of 145, 146 Capacity, definition of electrical . . . 103 Capacity, resistance 128 Capacity, standards of . . . 103 Capacity, thermal 88 Cadmium standard cell 101 Cathetometer 20 Cells, Daniell 101 Cells, standard 101 Cells, storage 101 Cells, in series and in parallel .... 117 Charge and discharge key 115 Circular measure 25 Clock circuit for time measurement. 44 Coefficient of cubical expansion ... 84 Coefficients of elasticity, Table VI 49-52 Coefficients of inductance 104 Coefficient of rigidity 52, 58, 59 Coercive force 154 Coincidences, method of 44 Collimator 164, 165 Commutation curve, magnetic 153 Commutator, Pohl's 105 Computation, hints on 9 Condensers, electric 103 Conductivity, electric 128 Constant of ballistic galvanometer.. 114, Contact measurements 1 1 Controlling magnet 107 Copper coulometer 142 Coulomb, definition of 102 Coulometer 142 Current, measurement of electric 140-144 Curvature of lenses I55-I57 Curves, plotting of 4 Damping of galvanometers ...". .108, 112 Daniell cell, treatment of 101 D'Arsonyal galvanometer ..106,108,112 Declination, magnetic 150 Density, measurements of 67-69,54 Density ....Tables II, III, IV Deviation, position of minimum . . . 169 Difference of potential, of cell 135,136 Diffraction, wavelength by 176 Diffraction grating 170, 171 Diffraction grating, constant of .... 171 Dilatometer 85 Dip, magnetic 150 Dispersion curve 173 Dividing engine 22 Double weighing 42 Earth's magnetic field 150 Effective length of bar magnet .... 150 Elasticity, Coefficients of, Table VI 47-59 Electricity, quantity of 102 Electrochemical equivalent 141 Electrolytic resistance 128 E. M. F., measurement of 129-138 E. M. F. of standard cells 101 Equivalent of heat, mechanical, Table XVI Errors of observation, influence of . . 7 Error, probable of result 6 Eccentricity, correction for 28 Expansion, coefficient of linear ..82,83 Eye-piece in telescope 160 Farad, definition of 103 Figure of merit of galvanometer 112 Filar micrometer, angles by 33 Fixed points of thermometer 80 Flexure, Young's modulus by 57 Flux, magnetic 147,153 Focal length of lenses, 158 Freezing point, determination of 80 Frequency of tuning fork 77 Fusion, heat of, of water 93 g, measurement of 62 Galvanometers 106-108 Galvanometers, calibration of in Galvanometer, ballistic, constant of PHYSICAL MEASUREMENTS Galvanometers, damping of 108-112 Galvanometers, figur of merit of ... 112 Galvanometers, sensitiveness of 106, 112 Graphical methods 4 Graphical method for tuning fork.. 77 Grating, diffraction 170,171 Gauss eye-piece 164, 165 H, determination of 150, 151 Heat constants, Table X Heat, mechanical equivalent of, . Table XVI Heat of vaporization 94 High resistance, measurement of... 122 Homogeneous light 169 Hooke's law 48 Hysteresis curve 154 Images, optical 158, 160 Inclination, magnetic 150 Index of refraction Table XII 162, 163, 169 Induction, magnetic 152 Inertia, moments of Table VI Instruments, treatment of 2 Insulation resistance, measurement of 122 Interference method, sphereometer 15 Internal resistance of cells ....135-139 Interpolation 8 Iron and steel, magnetic properties of . .. 152-154 Jolly's balance 54, 71 Keys, electrical 105 KirchhofFs laws 1 18 Kohlrausch's method for electrolytic resistance 128 Kundt's method for velocity of sound 75 Laboratory work, benefits of i Laws of bending 56 Legal ohm 99 Length, measurement of 11-23 Lens curves 159 Lenses, curvature of 1 55-157 Lenses, focal lengths of 158 Level, constants of 32 Level tester 31 Lever, optical 30 Limit of elasticity 47 Line measurements 19 Lines of magnetic induction 149 Magnet, equivalent length of 150 Magnetic dip 150 Megnetic field 149 Magnetic hysteresis 154 Magnetic moment 150 Magnetometer 150 Magnetisation curve 153 Magnifying power of telescope ... 160 Mass, measurement of 37-42 Maxwell's rule 123 Magnetizing field 152 Mechanical equivalent of heat, Table XVI Melting point of tin 95 Melting points Table X Mercury, density of Table III Mercury contacts 105 Microfarad 103 Micrometer, filar 21 Micrometers, optical 21 Micrometer cathetometer 21 Micrometer gauge 14 Micrometer screw 13 Microscope, index of refraction by.. 163 Microscope, magnifying power of... 161 Mixtures, method of (heat) 89 Mixtures, method of (electricity)... 146 Modulus of torsion 52 Modulus, Young's. .. .51, 52, 55, 57, 76 Mohr's balance 69 Molecular conductivity 128 Moment of couples 52 Moments of inertia Table VI Moment of inertia, determination of 63, 64 Mutual inductance 148 Nernst and Haagn's method for bat- tery resistance 139 Noninductive winding 99 Normal dispersion 172 Numerical tables XV-XVII Objective 1 60 Ohm. definition of 99 Ohms law, calibration by in Optical lever 30 Parallax 1 65 Pendulum, simple 60 Pendulum, torsional 45, 59 Permeability, magnetic 158 Pohl's commutator 105 Polarization of cells 137 Post office box 124 Potential difference of cells ...135, 136 Potentiometer 132 Potentiometer method for E. M. F.. 131 Prism, angle of 167 Prism, refraction through 169 Prism, index of refraction of 169 Prismatic dispersion 172, 173 Protractor 68 Quantity of electricity 102 Radian 25 Radiation, correction for 92 Radius of curvature iSS^S? Record of observation 3 Reflecting surfaces 166 Reflection, radius of curvature by.. 157 Refraction, index of, Table XII 162, 163, 169 Remanence, definition of 154 Repetition, angle by 168 Reports, final 3 Resistance boxes 99 Resistance capacity 128 Resistance, measurements of ...120-128 Resistance, standards of 99 Resistances, in series and in parallel 119 Resistivity 124 Resting point of balance 38 Rheostat 99 Rigidity, coefficient of 52, 58, 59 Sagitta, definition of is 6 Secohmeter 147 Selfinductance, comparison of 147 Sclfinductance, standards of 104 Sensitiveness of a balance 39 Sensitiveness of a galvanometer 106, 112 Sextant 35 Shear 5^ INDEX 255 Shunts 1 10 Simple rigidity 52, 58, 59 Slide wire bridge 125 Sodium light 169 Solenoid, magnetic field inside of .. 151 Solids, expansion of Table X Sound, velocity of, Table VI Specific heat Table X 88, 89, 91 Spectrometer, The 164, 165 Spectrometer, adjustment of 165 Spherical surfaces, curvatures of ... 155 Spherometer 15, *S& Standard cells 101 Standard resistances 99 Stem correction for thermometers.. 81 Strain and Stress 5 * Stretching, Young's modulus by ... 55 Subdivision, method of 12 Surface tension 7* 7 2 Surfaces, reflecting 166 Telephone for electrical measure- ments 128, 139 Telescope and scale 29 Telescope, magnifying power of... 160 Temperature, definition of 7 Thermal capacity 88 Thermo-couple, E- M. F. of 134 Thermometer, calibration of 80 Thermometer, fixed points of 80 Thompson's method for galvanometer resistance 126 Thompson's method for comparing capacities 146 Time, measurement of 43. 44 Torsional vibrations, coefficient of simple rigidity by 59 Trigonometric functions, angle by 27 Tuning fork, frequency of 44, 77 Units, fundamental 10 Units, electrical 99-104 Vacuum, reduction to weight in ... 41 Vaporization, heat of 94 Vapor pressure Table XI Vapor tension, measurement of 96-98 Velocity of sound in metals Table VI 75 Verniers 16 Vernier caliper 18 Vibration, determining rate of .... 44 Viscosity 73, 74 Volt, definition of 101 Voltmeter 116 Voltmeter, calibration of 133 Water, air free 86 Water, boiling point of Table IX Water, vapor pressure of 89 Water equivalent of calorimeter.... 90 Wavelength of light, determination of 170 Wavelength of lines in solar spec- trum Table XIII Weighing, to make a single 40 Weighing, double , . 42 Weston standard cell 101 Wheatstone bridge 123 Wheatstone bridge box 124 Young's modulus 5 1 Young's modulus by stretching .... 55 Young's modulus by flexure 57 Young's modulus, from velocity of sound 76 Zeiss-Abbe eye-piece 164 Zero point of thermometers 80 UNIVERSITY OF CALIFORNIA Medical Center Library THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to fines according to the Library Lending Code. 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