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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



<J (




PHYSICAL MEASUREMENTS



INTRODUCTION

1. Benefits of Laboratory Work. The benefits to be derived
by the student from work in the physical laboratory are two-fold.
In the first place he is to become acquainted with delicate instru-
ments, to be trained to make systematic, accurate and independent
observations, and to compute from the data so obtained, the val-
ues of many of the more important physical constants. Secondly,
he is to make a searching review of the fundamental principles of
the science, and to be brought to a more lively realization of the
meaning and importance of formulae and laws deduced in the
text-books upon general physics. To this end it is urgently
advised that the student familiarize himself with all the de-
tails of the theory of the experiment, and be able to sketch from
memory the apparatus to be employed before beginning any ex-
periment. An attempt to follow directions but dimly understood,
and to manipulate apparatus whose construction and purpose are
alike unknown, can only result in loss of time, and laboratory
work under such circumstances is practically worthless as a means
of discipline.

2. Instruments. The instruments used in the physical lab-
oratory are usually of delicate construction; many of them are
costly and liable to injury from rough or careless usage. It is of
the highest importance that all apparatus should be handled with
care, and returned to its proper place after use. If any piece is
found to be out of adjustment or in need of repair, report the fact



2 PHYSICAL MEASUREMENTS

before beginning work. If any screws or other parts of an instru-
ment do not move readily, do not apply force but report the
matter to the instructor in charge. The ability to use delicate
apparatus without injuring or destroying it is an important part
of a liberal education.

3. Record of Observations. All observations, data and nec-
essary formulae, such as the time and place of the exercise, the
specific instruments used, the objects measured, etc., are to be
recorded in a note book provided for that purpose. Such a book
should have fixed leaves, and be made of paper suitable for
writing with ink. The record is to be made at the time of the
experiment. It must contain the detailed information necessary,
must be clearly written and arranged in a neat, methodical man-
ner, so that one familiar with the experiment may readily compre-
hend what has been done. It should be sufficiently specific to be
intelligible after the circumstances of the experiment are entirely
forgotten.

A suitable form of record has been appended to each exercise
in the manual, and the student will do well to follow these, at
least until he is able to arrange the material for himself. Tabula-
tion of results in columns adds much to the neatness of a note
book and materially assists in the detection of errors, either of
record or of observation. It is suggested that the note books
be casually inspected at each exercise if possible, by the instructor,
in his rounds in the laboratory.

The computation of results should generally be done at home,
and a record of the complete experiment be handed to the in-
structor for approval. In this report the date, number, name and
object of the exercise should precede a short discussion, deriva-
tion of formulae, the record of observations and the computed
results. If these reports be written on loose sheets to be returned
to the student after approval by the instructor, these sheets must
be kept, ready for inspection, in a suitable binder. A separate
page should always be begun for each new experiment.

4. Graphical Methods. It is frequently of interest to verify
a law. stating the relation that is known to exist between two



INTRODUCTION 3

quantities, or to detect and determine such a relation where it is
not known. In all such cases the results obtained by observation
are most clearly presented to the eye when plotted as a curve. -
In the application of the method it is customary to plot values of
the independent variable as abscissae and those of the dependent
variable as ordinates. In all cases where the phenomenon under
investigation is continuous, a smooth curve sketched through the
points obtained, may be assumed to represent the facts better than
any individual observation. The graphical method has the addi-
tional advantage that an accidental error is at once made evident
by the fact that the point so obtained departs markedly from
the curve.

The curve most readily plotted and tested is the straight line,
and it is advisable so to transform the assumed relation as to
render the plotting of a straight line practicable.

For example, suppose it were desired to investigate the relation
between the time of vibration of a pendulum and its length. If
we assume that this relation may be expressed by an algebraic
function of the form

T = ar (i)

we may determine the constants a and m from a series of obser-
vations. Passing to logarithms we have

log T = m log / + log a (2)

This is clearly of the form

y = m x + c (3)

and is therefore the equation of a straight line. If now we plot
values of log / as abscissae and the corresponding values of log T
as ordinates, we may decide at once whether such a relation as we



4 PHYSICAL MEASUREMENTS

have assumed exists, and we may obtain values of a and m direct-
ly from the curve.

It is not necessary that the same numerical value should be
assigned to a scale division on the horizontal and vertical axes.
In general it is best to make the value of a scale division cor-
respond, as nearly as possible, to the least quantity which we can
measure. If this be impossible, such values should be chosen for
a scale division on each axis as will cause the curve most nearly
to fill the page with the observations to be plotted. It is only in
case we wish to determine from the curve a quantity which is
represented by the tangent of an angle, i. e., which represents the
ratio between the coordinates, that it is necessary to assign to
them their proper relative values.

A table of the data from which the curve has been plotted
should in all cases accompany the curve and the values as-
signed to a scale division on the horizontal and vertical axes
must be clearly stated upon these axes. In practice it is well
to prick with a needle the exact position of each point on the
curve and then draw around each point a small circle in colored
ink. All curves should be plotted upon special cross-section paper,
drawn in ink, and the points clearly marked as indicated above.
In case special accuracy is desired it is well to use paper printed
from engraved plates. The curves are to be inserted in the note
book after the record of the experiment. This is most readily
done by cutting away about two-thirds of a sheet lengthwise, and
pasting the stub to the back of the cross-section paper.

The following data may be used as exercises in the plotting of
curves. In case any set of data does not represent continuous
phenomena how should the curve be drawn?

I. POPULATION OF THE UNITED STATES.



Year

1810
1820
1830
1840
1850


Population
3 929 214
5 308 483
7 239 881
9 633 822
12 866 020
17 069 453
23 191 876


Year
1860
1870
1880
1890
1900
1910


Population
31 443 321
38 558 371
50 155 783
62 622 250
76 303 387
93 402 151



INTRODUCTION 5

II. ATTENDANCE AT THE UNIVERSITY OF MICHIGAN.

Collegiate No. of Collegiate No. of

Year Students Year Students

1881-82 1534 1897-98 3223

1882-83 1440 1898-99 3192

1883-84 1377 1899-1900 3441

1884-85 1295 1900-1901 3712

1885-86 1401 1901-1902 3709

1886-87 15/2 1902-1903 3792

1887-88 1667 1903-1904 3957

1888-89 1882 1904-1905 4136

1889-90 2153 1905-1906 4571

1890-91 2420 1906-1907 4746

1891-92 2692 1907-1908 5010

1892-93 2778 1908-1909 5223

1893-94 2659 1909-1910 5383

1894-95 2864 1910-1911 5381

1895-96 3014 1911-1912 5582

1896-97 2975

III. HYSTERESIS CURVE FOR SWEDISH IRON.

The values are given for half a cycle only. To obtain the com-
plete curve, reverse the signs of both columns.

Intensity of Field Magnetic Induction


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