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A MANUAL

EXPERIMENTS IN PHYSICS

LABORATORY INSTRUGTION
FOR COLLEGE GLAUSES

JOSEPH S. AMES, Ph.D.

UBOOUTt rSORSSOR OF FHT8IC8 IN JOHNS H0PEIH8 DKiyiBSTTT

WILLIAM J. A. BLISS, Ph.D.

ASSOCIATE IN FHT8IC8 IN JOHNS HOPKIMS CMVKR8ITT

KEW YOBK •:• GINC3NNATI •:• CHICAGO

AMERICAN BOOK COMPANY

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AUri^renrnd.
W. P. 5

'V_

PREFACE

Thebe are two reasons why the study of Physics shonld
be included in every college conrso : one is, becanse it
teaches certain intellectual methods, certain modes of ex-
act thought which are not required by other sciences in
their elementary stages ; the second is, because it teaches
methods of accurate observation and measurement. Phys-
ics, as well as any science, may be studied entirely in the
class-room; and profit may be derived from seeing per-
formed by the instructor demonstrations of the funda-
mental experiments, and from following out the logical
processes and methods based upon these; but it should
not be thought that this is the entire aim of Physics.
Every student should be taught in the laboratory how to
measure those quantities which are involved in the state-
ments of the laws of nature, and should be given an op-
portunity of verifying as many of these laws as possible.

One can divide into three classes those students who
undertake laboratory work in Physics; and the require-
ments of these classes are by no means the same. At the
present time the largest proportion of elementary students
of Physics are taking the courses as part of their prepara-
tion for other sciences, in particular medicine and engineer-
ing ; an increasing number are taking them simply in the
course of their liberal education; while a comparatively
small number look forward to continuing their work either
as investigators or as teachers. It is obvious that there
are certain laboratory exercises and methods which, while

I? PREFACE

abiolntelj neoessary to a fnture physiologist or chemist,
might be omitted in a system of general education, and
might not be so important from purely physical consider-
ations as some experiment of a similar but more funda-
mental character.

In preparing this text-book for use in Physical Labora-
tories, the needs of all three of these classes of students
have been borne in mind, how successfully it is not possi-
ble to say. The only experiments described are quantita-
tive, because it is assumed that purely qualitative ones
are demonstrated in the lecture*room. Those experiments
which are suited to a definite student or to a definite class
must be selected by the instructor ; and it is impossible to
give any precise statement as to which are best adapted for
any particular purpose. It has been impossible, of course,
to include all the experiments which might be desired ; but
it is hoped that no important principle or piece of appara-
tus has been slighted.

The object of an experiment in Physics is not simply to
teach a student to measure quantities and to verify the
laws of nature ; it should also lead him to look closely into
the methods made use of, the theory of the instruments,
the various sources of error, the possible deductions and
applications of the principles involved. The importance,
too, cannot be overstated of insisting upon the student
learning neat and systematic methods of making, record-
ing, and reporting observations. With these ends in view,
each experiment, as described in this book, is divided into
seven parts :

1. Object of Experiment, — This is simply a single para*
graph stating the chief object of the exercise.

2. General Theory. — In this section is given a brief state-
ment of the general theory of the physical laws involved in
the experiment, and the general principles made use of in
the methods of measurement. No particular forms of ap-
paratus are described, but the essential details of the neces-
sary processes are given.

PBSFAOS T

8. 8ource$ of Error. — Under this head are given the most
important difficnlties in the experiment, the most frequent
causes of error, and the essential precautions.

4. Apparatus, — This is a list of the instruments and ap- '
pliances required for the exercise, together with, in certain
cases, a brief description of the instruments themselves.

5. Manipulation. — This is a full statement of the details
of the experiment, with explicit directions as to quantities
and methods.

6. Illustration. — There is given in nearly every case the
result of an actual experiment performed in the manner de-
scribed in the body of the exercise. These illustrations are
meant to serve as guides to students in making their re-
ports, as well as to show practically how accurate the ex-
perimental methods are.

7. Questions and Problems, — These are questions suggest-
ed by the experiment, and problems serving to illustrate the
principles involved in it.

The object of this particular division and arrangement of
each experiment has been twofold. The main one was the
hope that the student might be induced to prepare himself
for performing the experiment by a preliminary reading of
the principles and methods involved. The second object
was to avoid the danger, so far as possible, of making the
descriptions apply to one particular set of apparatus.

In the use of this book an experiment should be assign-
ed a student some days before he is to perform it ; and he
ought not to be allowed to take time from the regular labo-
ratory hours for the preliminary study, which should be
done elsewhere. He should get the necessary apparatus
from the stock-room, which should be in charge of a cus-
todian of apparatus, and should set up the apparatus him-
self. Records should be made in a systematic, permanent
form; and the results should be deduced, not in the labo*
ratory, but at other times, and should be reported to an
instructor regularly (once a week or fortnight) in a suit-
able book. Those of the questions and problems which

Vi PREFACE

an inBtmctor desires answered should also receive atten-
tion in this report -book. It is .often advisable for two
students to work together while performing an experi-
ment, and in some cases it is absolutely necessary ; if this
is done, each student should take an independent set of
records and should hand in a separate report.

There is great difficulty in assigning the credit for any
particular experiment or form of apparatus ; but in every
case where it is possible suitable acknowledgment has been
made in a foot-note. Special thanks are, however, due to
four former assistants in the Physical Laboratory of the
Johns Hopkins University : Dr. W. S. Day and Mr. 11. S.
Uhler, who have taken great pains at various times in
working out the details of many of the experiments ; Mr.
0. W. Waidner, for the substance of Appendix III. on gal-
vanometers; and Dr. N. E. Dorsey, for a description of the
clock-circuit contact devised by him, which is given in Ap-
pendix II. The drawings have all been made by one of our
students, Mr. W. S. Gorsuch, Jr., to whom we are greatly
indebted for his skill and promptness.

J. S. Akes.

W. J. A. Bliss.

Johns Mopkins UNiVERsrrr, January, 1898.

TABLE OP CONTENTS

GENERAL INTRODUCTION

PAGK

0. G. 8. System 2

Physical Measurement 2

Accunicy of Hesiili 8

Graphicnl Meihods 6

Qeneral lustructions 9

PRELIMINARY EXPERIMENTS

1. A rough determination of each of the three fundamental

quantities— length, mass, and time 15

2. To determine the internal volume, or ''capacity," of a bulb. . 22

NoTB.— To measure aroax and regular volumes.

3. To determine the number of centimetres in one inch 26

^. To learn the metliod of using a vernier 30

5. Use of vernier caliper 34

6. Use of micrometer caliper 39

7. Use of the spherometer 43

8. Use of the dividing-engine 49

9. To measure the pitch of the screw of a micrometer microscope 55

10. Use of comparator or cathetometer 58

U. To delermino by the method of vilirat'ons the position of

equilibrium of the pointer of a balance 61

Tiii TABLE OF CONTENTS

EXPERIMENTS IN MECHANICS AND PROPERTIES
OF MATTER

PAOiC

Introduction to Mechanics and Propebtieb of Matter. . 67
Units aud deflnitions 67

12. To determine the linear velocity and ncceleration of a rapidly

moving body 70

13. To determine angular velocity and acceleration 79

14. To determine the mass of a body by inertia 85

15. To verify the Principle of the Conservation of Linear Mo-

mentum 90

16. To show:

1. That if different forces act upon the same body, the ac-

celeration, is directly proportional to the force.

2. That if the same force acts upon bodies of different masses*

the acceleration is inversely proportional to the mass . . 98

17. To verify tlie law of centrifugal motion, that a force mrta^ is

required to make a mass m move in a circle of radius r wiili

a constant angular velocity ta 105

18. To verify the laws of harmonic motion Ill

19. To verify the law of moments 118

20. To verify the laws of equilibrium of three forces acting jii

one point. Notr.— On the nse or spring-balances 122

21. To verify the laws of equilibrium of parallel forces in the

same plane 129

22. To verify the laws of equilibrium of an extended rigid body

under the action of three forces 133

28. To determine experimentally the centre of gravity of a

weighted bar 186

24. To determine the "mechanical advantage" and "efficiency"

of a combination of pulleys 141

25. To determine the coefficient of friction between two polished

wooden surfaces 146

26. To determine the mass of the hard rubber cylinder whose

volume was found in Experiment 5 151

The theory of a chemical balance 151

Reading a barometer 157

27. To verify Hooke's Law and to determine "Young's Modu-

lus " for a given substance by stretching a wire 168

28. To determine the coefficient of rigidity for iron 168

A method of measuring intervals of time exactly 168

29. To verify the laws of fluid pressure 179

TABLE OF CONTENTS ix

PAOK

90. To determine the density of a liquid by means of '* balancing

columns/' Nohb.— capillary corrections 188

81. To determine the density of a solid by means of a chemical

balance. Archimedes's principle 189

32. Use of Nicholson's hydrometer and detenninalion of the

density of some small solid 198

88. To determine the density of a smnll solid by Jolly's balance. 197

84. To determine the density of a floating body 201

35. To measure the surface-tension of pure and impure liquid

surfaces 204

36. To measure the density of a gas 206

37. To prove that Boyle's Law holds for air approximately 209

EXPERIMENTS IN SOUND

Iktroducjtion to Sound 217

38. A study of '* stationary " vibrations 218

1. Long flexible cord 218

2. Water tank 222

39. To verify the formula for the frequency of a stretched string

or cord when vibrating transversely 226

To study the laws of harmony 232

Nom.— To meaanre tbe rreqaency of a taning-fork.

40. To determine the velocity of sound in air by means of sta-

tionary waves in a resonance tnbe 288

41. To determine the velocity of longitudinal vibrations in a

brass rod by Eundt's method 289

42. To compare the velocity of longitudinal waves in brass and

in iron 248

48. To study the different modes of vibration of a column of gas 246

EXPERIMENTS IN HEAT

Introduction to Heat 268

Use of a mercury thermometer 254

44. To test the fixed points of a mercury thermometer 259

45. To determine the coefficient of linear expansion -of a solid

rod or wire 265

46. To measure the apparent expansion of a liquid 271

47. To determine the mean coefficient of cubical expansion of

glass between 0° and 100° C 274

Mon.~To measore tbe abaolate expanaion of a liquid.

X TABLE OF CONTENTS

PAGC

48. To nifiasure the increase of pressure of air at constant vol-

ume wlien the tempeniture is increased. Air thermometer. 278

49. To determine the specific heat of a metal~«. g,, lead or brass. 282

50. To determine the specific heat of turpentine 288

51. To determine the *' melting-point " of parafflne 291

53. To determine the latent heat of fusion of water 295

53. To determine the boiling-point of benzene 30O

54. To determine the latent heat of evaporation of water at 100^ G. 302

55. To verify tlie law of saturated vapor 306

58 To plot the ** cooling curve " of a hot body 311

EXPERIMENTS IN ELECTRICITY AND MAGNETISM

Introduction to Electuicity and Magnetism 317

Units and definitions : General directions 317

57. To plot the fields of force around various electrified bodies. . 321

58. A study of electrostatic induction by means of the gold-leaf

electroscope 325

59. To repeat Faraday's ** Ice Pail " experiment 328

60. A study of an electrical induction maciiine 332

61. 1. To sliow that the capacity of a condenser composed of

two parallel plates varies invci-sely as the distance between

i ts plates 334

2. To determine the dielectric - constant of some dielectric,

such as glass 334

62. To map a *'current sheet " 340

63. To plot the magnetic field of force 344

1. Of a magnet and the earth together.

2. Of the magnet alone.

64. To measure the magnotic inclination or dip 849

65. To compare the intensities of fields of magnetic force 356

66. To measure the horizontal intensity (II) of tlie earth's mag-

netic field 359

67. To prove that the resistiuicc of a uniform wire varies directly

as its length 365

68. To determine roughly the effect upon resistance of alterations

in length, cross - section, temperature, and material of a
conductor. 371

69. To measure a resistance by the Wheatstone wire-bridge method 876

70. To measure the resistance of a mirror - galvanometer by

Thomson's mt'tliod, using a ** Post-office liox " 381

71. To measure tiic resistance of a cell by Mance's Mclliod 385

TABLE OF COxNTKNTS xl

PAGK

72. To measure the specific resistance of solutions of copper sul-

phate by Kohlrausch's Method 388

73. To compare electromotive forces by the high - resistance

method 392

7 L To compare electromotive forces by the * ' condenser method " 396
Moth. — Use of ballistic gaWanotneler.

75. To determine the ** galvanometer constant" of a tangent gal-

vanometer. Water volUimeter 402

76. To determine O or // by the deposition of copper 409

77. To determine ihe mechanical equivalent of heat by means of

the healing effect of an electric current. Joule's law 414

78. To determine the "Magnetic Dip" by means of an "earth

inductor" 420

EXPERIMENTS IN LIGHT

ISTRODDCTION TO LIGHT. 427

79. To compare the intensities of illumination of two lights by

means of a Joly photometer 428

80. To verify the laws of reflection from a plane mirror 481

1. Plane waves.

2. Spherical waves.

81. To verify the laws of Feflection from a spherical mirror 436

1. Real image.

2. Virtual image.

83. To verify the laws of refraction at a plane surface 442

1. Plane waves.

2. Spherical waves.

83. To measure the index of refraction of a solid which is made

in a plate with plane parallel faces 446

Si To verify the laws of refraction through a spherical lens 449

85. To construct nn astronomical telescope 455

86. To construct a compound microscope. 457

87. To measure the angle between two plane faces of a solid. . . • 459

The adjustment of a spectrometer 459

88. To study the deviation produced by a prism. To measure

the angle of minimum deviation 469

89. To measure the index of refraction of a transparent solid

made in the form of a prism 474

90. To study color-sensation 476

91. To measure the wave-length of light by means of a grating. 477

zii TABLE OF OONTENTS

APPENDIX 1

LABORATORY KQUIPMENT ^^^

Aspirator pump 488

Platform air-pump 483

Drying tubes 484

Distilled water 484

Clock circuit 484

Sets of chemicals 488

Supplies 488

Books of reference 488

Glass-blower's table 489

Laboratory tables 489

Balances / 489

Giilvanometers 489

Storage-batteries 489

APPENDIX II

LABORATORY R1CEIPT8 AKD MKTHODS

Cleaning glass 491

Cleaning mercury 491

To fill a barometer iulye with mercury 494

Amalgamating zinc 495

Amalgamating copper 495

"Universal wax." 496

Cements 496

Damping keys and magnets 496

Sensitizing mixture 498

Silvering mirrors 498

Mercury cups . . . ; 498

Open iron resistance-boxes 498

Sliding resistances 499

Mercury trays 499

Simple glass-blowing 499

Standard cells 501

APPENDIX III

OALTANOMETIRS

Tangent galvanometers 504

The differential ^ulvanoraeter 604

TABLE OF CONTENTS xUi

PAOK

The ballistic galvanometer 605

D'ArsoDval galvanometers 505

Proportionality of deflection with current 506

Choice of a galvanometer 507

Controlling magnet 507

The suspended system 608

Magnets 510

The staff 511

Mirrors 511

Suspension fibres 512

Astaticism.

1. Horizontal systems 612

2. Vertical systems 518

Sensibility 514

TABLES

1. Mensuration 517

2. Mechanical units 517

8. Elastic constants of solids 518

4. Densities 518

5. Surface tension 520

6. Acceleration due to gravity 520

7. Correction for large arcs of vibration 521

8. Capillary depression of mercury in glass 521

9. Barometric corrections.

1. Correction for temperature 522

2. Correction for variation in ^^ 522

10. Frequencies of middle octave 528

11. Velocity of sound 528

12 Average coefficients of linear expansion between 0° and 100^ C. 523

13. Average coefficients of cubical expansion of liquids 524

14. Average specific heats 524

15. Specific heats of gases 524

16. Fusion constants 524

17. Vaporization constants 525

18. Vapor- pressure of water 525

19. Vapor-prcssure of mercury 526

20. Heats of combination 526

21. Thermal conductivities 527

22. Dielectric constants (electrostatic system) 527

A MANUAL

EXPERIMENTS IN PHYSICS

GENERAL INTRODUCTION

To understand properly any phenomenon implies two
things : a study of the sequence of events, the cause and
effect ; and a determination of exactly how much is in-
Yoived of each quantity which is concerned in the phe-
nomenon. We cannot understand any phenomenon unless
we can measure it.

In order to measure quantities certain standards or
units must be chosen, in terms of which to express the
numerical values. It is shown in any treatise on Physics
that every quantity which enters into the phenomena of
matter in motion can be reduced to a certain amount of
matter, a certain space, and a certain interval of time (see
" Physics,''* Art. 7). Consequently it is necessary to adopt
standards of quantity of matter, of space, of time, which
will serve as mechanical units. Similarly it is necessary to
select units in terms of which to measure electrical and
magnetic quantities.

* Here, as eleewhere in this book, this reference is to " Theory of Phys-
ics," oy J. S. Ames, and published bv Uaipei- & Biotliers: 1897.
I

2 A MANUAL OF EXPERIMENTS IN PHYSICS

0. 0. 8. SysteoL The anits of length, of matter, and of
time which have been adopted by the scientific world (see
*^ Physics," Art. 8) are the following :

Unit of Length. The centimetre, the one -hundredth
portion of the length of a metal rod which is kept in Paris
when it is at the temperature of melting ice. On this unit
are based the square and cubic centimetres, as units of
area and volume.

Unit of Qv^ntity of Matter. The gram, the one-thou-
sandth portion of the quantity of matter in a lump of
platinum which is kept in Paris. (The gram is very ap-
proximately the quantity of matter in one cubic centime-
tre of distilled water at the temperature when it is most
dense, i.e., 4° C).

Unit of Time. The mean solar second, an interval of
time such that 86,400 of them equal the mean solar
day, t.6., the average length of the solar day for one
year.

On these mechanical units are based the subsidiary
units of speed, velocity, acceleration, force, energy, etc.
The electrical and magnetic units will be defined later.

This particular system of units is called the C. G. S.
system, from the initial letters of centimetre, gram, second;
and in terms of these or of units derived from them all
physical quantities should be expressed. That is, a length,
whenever it occurs, should be measured in centimetres ;
all masses should be measured in grams ; and all intervals
of time in seconds. These units are perfectly arbitrary,
but there is no reason to suppose that the standards will
ever change; and having received the sanction of all civil-
ized countries and all scientists, they should be used in
expressing every measurement. Moreover, they are very
convenient, since they are the foundation of decimal sys-
tems, and not systems in which the smaller and larger
measures are related in arbitrary ratios, as the foot and
inch.

Physical Measurement. The object of a physical experi-

GENERAL INTRODUCTION 8

ment is in general to measure a quantity either directly or
indirectly. Thns^ a length can be measured directly by
means of a centimetre rule ; but the density of a body^
that is, the number of grams in one cubic centimetre, is
measured indirectly, since to determine the density meas-
urements must be made of quantities which are connected
with density by a physical relation which can be expressed
in a mathematical formula. In every case, however, a
series of measurements must be made of certain quantities,
either the quantities which are themselves desired or those
which enter into a cevtain formula, stating some physical
definition or law.

It is obviously impossible to know whether any one of
the observed measurements gives the true value of the
quantity^ and the separate measurements will in general
differ among themselves. We are, therefore, led to inquire
how we can best use these differing determinations so as to
deduce from them as^close an approximation to the truth
as possible, and also to learn how great an error we are
liable to in the result thus obtained.

While paying special attention, however, to the more
minute portions of a measurement, care must be taken to
make no mistake in recording the numbers which express
the larger part of the measurement. Thus, in measuring
a length of 5.21 cm. the student is far more liable to
make a careless mistake in reading 4 or 6 instead of 5
than to make an inaccurate reading of the 21. The error
in the " whole number '* mnst be most carefully guarded
against.

Accuracy of Beiralt If a long series of readings of the
same quantity has been made, the same care being given
each individual measurement, the arithmetical mean of
these readings is the most probable value of the quantity ;
and by comparing this mean value with the individual
readings a great deal may be learned as to the degree of
accuracy of the method used and the observations. The
general process is to write the readings in a vertical ool-

4 A MANUAL OF EXPERIMENTS IN PHYSICS

umn, and in another colnmn write the differences between
each of these and the mean^ placing + or — before each
difference according as that measurement is greater or less
than the mean. The difference for any reading is called
the '' residual '^ for that observation ; and if the residuals
are large, it is evident that there is much more uncertainty
as to the accuracy of the mean value than if the residuals
are small. It is evident^ too^ that if only a few observa-
tions are taken, the accuracy of the mean value is not so
great as it would be if a long series were taken. Conse-
quently it should be possible from a consideration of the
magnitude of the residuals and the number of times the
measurement is repeated to form a definite idea of the
probable error to which the mean value is liable. The
theory of this determination is given in the "Method of
Least Squares/^ a mathematical process based upon the
theory of probabilities. It is sufficient here to state that
this method shows that, if we define as the '* Probable Er-
ror '^ a magnitude such that the actual error of the mean
is more likely to be less than this rather than greater,
then the probable error of the mean of n observations is

0.6745y/— Y ^<f where s is the sum of the squares of the

residuals. The probable error of any one of the n obser-
vations is 0.675\/ -, showing that the probable error

of the mean is less than that of any observation in the
ratio of 1 : \/n.

If X is the tnie value of a certain quantity, and if a is
the mean value of a series of measurements of that quan-
tity, the true meaning of the "probable error" can be ex-
pressed mathematically thus :

aH-e>a;>a — 6,

where e is written for the probable error. In words, the
true value of a quantity lies between the mean of the ob-
servations plus the probable error and the mean minus the

GENERAL INTBODUOTION 6

probable error. In stating tbe result of the series of
measurements^ it is ordinarily said that the value of x is
a±,e, with the interpretation of 0 as given above.

When the object of the experiment is to deduce the
value of a quantity from measurements of other quantities
connected with it by a formula, it is possible to calculate
the probable error of the final result if the probable errors
of the individual quantities which are substituted in the
formula are known. Thus, if it is wished to determine
the probable error* in the product of two quantities whose
true values are x and y, and which have been measured
with the result that x is found to be a =b e^, and y is found
to be ft ± tfg, where e, and ^ are the probable errors of a and
b respectively, the product of the measurements, ab, is
compared with the product {a±e^) (bzte^). The differ-
ence is d:a e^dbft ^1 db^i ^2, but e^ €2 is bo small u quantity
numerically that it may be neglected in comparison with
tbe firet two quantities. Thus the uncertainty of a is e^,
that of i is e,, and that of oft is a 63 + ft ^i* Writing this e,
we have

e s= 06, + fttf ,

aft""a"*"ft'

^ expressed in hundredths, is the '^ percentage '' that e is
ab

of oft ; - is similarly the percentage that 0, is of a, etc.

Consequently, the percentage error of a product is the sum
of the percentage errors of the factors ; and the rule can
obviously be extended to any number of factors. There-
fore, to determine the error of a calculated quantity, ex-
press the probable error of each factor entering into the
formula as so many per cent, of that factor, add the per-
centage probable errors of all the factors, and the sum is
the percentage probable error of the product. The nu-
merical value of the probable error may be at once calcu-

6 A MANUAL OF JSXP£RIMENTS IN PHTSIGS

lated from the percentage probable error; thus, if the
product of the means of the observed quantities is 10.05
with a probable percentage error of 0.5, the probable error
is =b 10.05 X 0.005 or =b 0.05.

If in the formula the sums of certain products enter,
the probable error of each product must be calculated sep-
arately, and their sum gives the probable error of the cal-
culated quantity.

Two most important facts are apparent from the above
theory of the probable error of a product'.

1. If in the formula which enters in the experiment a
factor appears to the mih power, the percentage probable
error of this factor introduces in the product a percentage
probable error m times as large as it would if it entered to
the first power only, because percentage errors are added.
Therefore a quantity which appears in the formula as
squared or cubed must be measured with much greater
care than a quantity which appears in the same product
only to the first power.

2. If, owing to special difficulty in measuring a certain
quantity, the probable error thereby introduced is liable to
be large, care must be concentrated upon this quantitv,
and its probable error must be reduced as much as possi-
ble by repeated measurements. The other quantities which
enter into the formula may often be measured compara-
tively roughly, without appreciably affecting the error in-
troduced by the one whose value is obtained with diffi-
culty.

Qraphical Methods. In many experiments the object is
either to verify a law stating the relation which exists be-
tween two quantities or to discover one if it exists. In
expressing the result of such experiments, it is always best
to have recourse to graphical methods.

Thus, suppose it is a question of the verification of
Boyle's law for gases, viz., "at constant temperature the
product of the pressure and volume of a given amount of
gas remains constant. '^

GENERAL INTRODUCTION 7

Let the measured pressares and corresponding vol-
mnea be

82.1

12.03

88.2

11.20

962

10.26

106.5

9.86

118.9

8.31

185.5

7.29

160.1

6.17

105.

Draw two lines at right angles to each other, one hori-
zontal and the other yertical. Consider distances from
the vertical line meas-
ured horizontally to
mean Yolnmes, and
distances above the
horizontal line to
mean pressures, ac-
cording to any arbi-
trary scale which we
may find convenient.
Obviously, any point
of the region be-
tween the lines rep-
resents a certain defi-
nite state of pressure
and volnme^ since it

is at a definite distance from each of the two lines. In
the experiment we observe the gas in a number of states,
in each of which we measure its pressure and volume.
For each such state there is then a coiTesponding point
on the diagram, which we mark with a cross (x). More-
over, since the gas passed continuously from one of these
states to another, we could have found any desired num-
ber of points, forming an unbroken chain between any
two of those actually measured. This we denote by draw-
ing an nnbroken curve connecting the individual points

▲ MANUAL OF EXPERIMENTS IN PHYSICS

observed. Farthermore, we know that each of oar ob-
servations is liable to error, whereas it is unlikely that
there are sudden changes in the behavior of the gas at
the points observed. We therefore draw our curve so
that it is " smooth," even though it does not exactly pass
through each observed point ; but we try to leave as many
of these points above it as below it. Finally, we mark
along the horizontal line the scale according to which
horizontal distances denote volumes and a similar scale of
pressures along the vertical line. Distances along the hor-
izontal line are sometimes called ^'abscissae," and vertical
distances '^ordinates."

Another illustration is afforded by the measurement of
the change in volume of water as its temperature is raised,
starting from such a temperature that the water is in a solid
condition, and ending at a temperature so high that the
water is vaporized. In the figure volumes are measured
by ordinates and temperatures by absciss®. The scale

of temperature is
BO chosen that
C* Centigrade
comes at P and
100^ at Q ; then
points to the left
of P correspond
to temperature
below 0° C.

It is often a
good plan in plot-
— ^ ^ ting graphically
a series of obser-
vations to draw
around the point which records a particular observation
a small circle with a radius equal to the estimated prob-
able error of that observation. Then in drawing a curve
through the various points it may be at once seen whether
the distances of points from the curve, which always arise

100°

Fnk9

OENBBAL INTRODUCTION 9

if the curve is made '^smooth/' exceed the limits of ac-
curacy of the experiment. Obviously, considerable discre-
tion is needed in drawing these observation cnrves, but
ambiguity seldom arises.

General InstructionB. It is of the utmost importance that
the student should learn to record his observations clear-
ly and systematically, and to this end the following rules
should be observed :

1. All observations should be recorded in a suitable
note -book at the time they are made — loose sheets are
often lost or mislaid. These records should be made neat-
ly and according to some scheme which has been thought
out previous to the actual experiment. It is often con-
venient to rule vertical lines, and place different measure-
ments of the same quantity in one column, so that they
may be compared or averaged.

2. The actual observations should be recorded. In no
case should a mental calculation be made, and only its
result noted; all calculations, however simple, should be
done at a later time. The laboratory note -book should
always show the original observations. Thus, if the zero-
point of an instrument is wrong, allowance should not be
made in the observations, but the actual error and the
actual observation should both be recorded.

3. A carefully prepared report of each experiment should
be written in ink in another book, and this siiould be
handed to the instructor for his inspection and comments.
In the following chapters of this manual forms will be
given under each experiment, which should be followed as
far as possible by the student in making his report.

4. Both in the actual record and in the subsequent re-
port care should be t^ken in so entering the figures that
they indicate the precise accuracy of the measurements.
For instance, if four observers measure the same length
and note it as follows, A as 5 metres, B as 5.0 metres, C
as 5.00 metres, D as 5.000 metres, the supposition is that
A is certain of the length as being 5 metres rather than 4

10 A MANUAL OF EXPERIMENTS IN PHYSICS

or 6, but that he does not know whether the length may
not be a fraction of a metre greater or less than 5 ; B is
certain that the length is not 6.1 or 4.9 metres, but he
does not know whether it may not be some hundredths of
a metre greater or less than 5 metres ; C is certain that the
length is not 5.01 or 4.99 metres, but it may vary some
thousandths of a metre from 5 ; D, however, is certain that
the length cannot differ from 5 metres by a thousandth of
a metre. If all four observers have used the same means
of measurement, and if it is known that these are accurate
enough to ascertain the length only to within one hun-
dredth of a metre, then A is a most careless observer, B is
less so, D is untrustworthy because he overstates the ac-
curacy, while C states the result correctly. It is fully as
bad to be an observer like D as to be like B.

The same rule applies to a result calculated from meas-
ured quantities ; each figure should have a definite mean-
ing, and any uncertainty as to a result should be expressed.
Thus 5.002 dbO.OOl means that the observer is uncertain of
the final 2 in 5.002 to within one figure, i, e., it may be
anywhere between 3 and 1. The accuracy of an experi-
ment cannot be increased by carrying out the result of a
division or of a multiplication to additional places of deci-
mals ; and the accuracy of any calculated result is limited
by that of the original measurements, as has been explained
in the section on '* Accuracy of Result."

It is often convenient in expressing a large or a small
quantity to use a factorial method ; thus, instead of
54600000. it is better to write 5.46x10', and instead of
0.0000018 to write 1.8 xlO"*.

5. Before any particular experiment is performed, the
student should read carefully the description of the method
and manipulation as given in the manual, and should con-
sider especially what quantities are the most difficult to
measure, which should be measured most accurately (see
p. 6), and what particular precautions must be taken. In
using any instrument the readings should be made with

GENERAL INTRODUCTION 11

the ntmost accuracy attainable^ unless careful considera-
tion has shown that this is unnecessary, owing to the uu-
ayoidable error which may enter in another measurement
in the same experiment.

6. It is a general rule that if the scale of any instrument
is divided into small divisions, the reading should be made
by estimation to one-te?Uh of one of these smallest divisions.
In such an estimation it must be remembered that the
true boundaries between the divisions are infinitely narrow
lines, and that the broad marks actually made are intended
to spread as much on one side as on the other. Wc must
therefore mentally divide into tenths the space between the
middle of the marks, and not that between the edges.

7. In the report which is handed to the instructor, the
student should give answers to the questions and problems,
and should also carefully explain how he has avoided or
considered each of the sources of errors mentioned in the
general description of the experiment. It should be need-
less to add that logarithms should always be used, and
that numerical calculations should not be recorded in the
report.

PRELIMINAKY EXPERIMENTS

EXPERIMENT 1

(PABT 3 RSqUIRBS TWO OBSBKySBS)

Olgeet. A rough determination of each of the three fun-
damental quantities — lengthy mass, and time. (See ** Phys-
ics/' Art. 7.)

1. To Measure the Lengrth of a Straight Line
General Theory. A straight graduated bar is held paral-
lel to the line, and the points on the scale which are ''op-
posite" the ends are read. By ''opposite" is meant con-
nected with them by straight lines perpendicular to the
scale and to the desired length. The difference in the
readings gives the desired length.

Sonroes of Error.

1. The scale and line may not be parallel.

2. Oire must be taken to determine the points on the scale which

are exactly "opposite " the ends of the line.
8. There may be defects in the measuring bar itself, due to
faulty graduation, warping, or wear.

Apparatus. A sheet of paper upon which a straight line
is carefully ruled, the ends of the line being sharply de-
fined ; a metre bar.

Hanipulation. Examine the metre bar; see that it is
straight, and that there are no evident defects in the
ruling. Lay the sheet of paper on a smooth table in a
good light. Place the bar' along the lino to be measured,
and turn both paper and bar until there is a good light
on the scale and also on the line. Turn the bar on its
edge, so as to bring the graduated scale as close as possi-

18 A MANUAL OF EXPERIMENTS IN PHYSICS

pound weight in one pan and a gram weight which yon
think approximately equal to it in the other. If this
turns out to be too small, double it ; if too large, halve it.
Find thus at once two masses, one greater and the other
less than a pound. Try next a weight half-way between
these two extremes, and continue similarly until a change of
yV of a gram shifts the pointer from one side to the other.
Estimate the fraction of ^ g. which will be necessary
to obtain an accurate balance, and note the mass of a
pound thus found. Now interchange the weights from
one pan to the other, placing the pound on the side in
which the grams were, and vice versa ; and note whether
any change must be made in the equivalent of a pound in
grams as found before. If there is a difFerence, take the
mean of the two quantities. This corrects the third error.
The student will have to assume his weights to be accurate.
They are tested when necessary by weighing each large
weight in turn with those smaller than itself, and finally
comparing one of the set with a standard.

ILLUSTRATION ^ , ,^

Pound in right pan, gram weights in left, 1 lb. =453.55 grams.
Pound in left pan, gram weiglits in right, 1 lb. =453.65 grams.

Mean 453.60 grams.

8. To Determine the Period of a Pendulum with an Ordi-
nary Watch

General Theory. The '* period " of a pendulum is the
time which elapses between one passage through a given
point of its swing and the next transit when it is moving
in the same direction. Since the instant it passes through
its central portion is more sharply defined than any other,
because the motion of the pendulum is fastest then, a
period is best measured between one transit through the
middle point of its swing and the next in the same direc-
tion. Since the time of one period is very short, we in-
crease the accuracy of the experiment by measuring the
time of a large number of successive periods, say fifty.

PRSUMINART EXPERIMENTS

SomxMfl of Bixor.

1. If the oeDtral point of the swing la not marked by a sharp

line back of it, and if the passage of the pendulum across
this line is not always viewed from the same direction, the
interyal of time measured may not be a period, because
there is no way of fixing the point at which transits should
be observed.

2. Care is necessary to read the watch accurately and note the

exact time of the transits.

Apparatus. A metal ball ; thread ; a metal clamp at-
tached to a firm support about two metres from the floor
and with the jaws vertical ; a watch with a second-hand.

Manipulation. A bicycle ball may be used as a pendu-
lum by attaching it to a thread about a metre long^ mak-
ing three loops tightly around it at right angles to one
another, and fastening the intersection with a little wax.
Suspend the thread from a clamp
as shown. For a line of reference
use a second vertical thread tied
to a nail or other firm support
close behind the pendulum and
stretched by hanging from it any
convenient weight, such as a knife.

With the pendulum at rest one
observer, A, places his eye in stich
a line that the two threads coin-
cide, taking care to note some dis-
tant object, e.g., A line on the op-
posite wall, or the edge of a door,
which is in the same line ; so that,
if he moves, he can return to the same position as before.
The other, B, sits at a table with the watch open before
him and a note-book convenient. A starts the pendulum
swinging in a vertical plane through a small arc — 1 cm.
each side is enough. Holding his eye in the right position

* This drawing is taken from Wortbington, '* Physical Laboratory Prao-

Fio.6-Bad*

Fio. 7— Good

A MANUAL OF EXPERIMENTS IN PHYSICS

he observes the transits of the pendulnm thread across the
fixed thread. When both observers are ready, A warns B,
and as one thread passes the other he gives a sharp tap
with a knife or pencil on the cover of his note-book. B
notes the reading of his watch at the instant of the tap,
which he should be able to do correctly to within half a
second. A counts each transit in the same direction up to
fifty. As fifty is approached he calls out the numbers 48,
49, as a warning, and taps again just as the fiftieth period
is finished. B notes the instant of the tap again.

Take five sets of fifty each in this manner. Then let A
and B interchange and take five more.

Deduce the period from each and average.

ILLUSTRATION

Tim OP M VtBiUTIOMS

start

m. &

Mr.B

:81

reads

81i

the

watch.

481

■

Finish
h. DL s.

2:88: 1}
85: 4i

87:16
89:17
40:59

Intenral
to Mconds

94.5

98.

96.

98.5

94.

Oct 6^ 1896

Period

1.89
1.86
1.92
1.87
1.88

Mean 1.884

Greatest deviation from mean is 2 per cent.

r start

FiBiah

Intenral

h. m. &

h. ID. &

in seconds

Mr. A

2:42-58i

2:44

94.75

reads

44:51

98.5

the

46:50}

281

98.

watch.

48:44

94.

50:851

94.25

Pisriod

Greatest deviation from mean is 1 per cent.
Mean of both observers: 1.88.

1.895

1.87

1.86

1.88

1.885

Mean 1.878

Note. — Refinements upon these roefthods of measuring the three funda-
mental quantities will be introduced later. The student should note, how-
ever, that the sources of error pointed out above unde^ each heading are
common to all methods, and it will be assumed hereafter that the student is

PREUMINABY EXP£fiiMENTS 91

aware of them, and their enumeration will not be repeated under the heading
** 8oarc9e8 of Error '* in future experiments. The student must never foi|;et
to gnmrd against them, though not spedfieaUy told to do so.

Questloiis and Froblenui.

1. Exactly how much error would be produced in the measure-

ment of the length of the line, if the line and scale were in-
clined to each other by l"* ?

2. What is meant by *' density"? How could you determine

the density of a drcular cylinder of wood ?
8. Can you explain why difference in the lengths of the balance
arms is corrected by ''double weighing "?

4. What modifications of the bahince would you suggest in order

to oiake one which would be more delicate ?

5. Calculate the "probable error" of the mean period of the

pendulum.

6. Why does the amplitude of the pendulum slowly decrease ?

Does the period change as this happens ?

EXPERIMENT 2

Olgect. To determine the internal volume, or " capacity/'
of a bulb. (See " Physics/' Art. 8. )

General Theory. A large glass bulb with capillary stem
is cleaned, dried, and weighed. It is then filled up to a
certain point with water or some other liquid and weighed
again. The difference in weight is that
of the mass of water or liquid necessary
to fill the tube up to the mark. Know-
ing the temperature of the water or liq-
uid, the density — i. e., the mass of a cubic
centimetre — is found in the Tables, and
the capacity of the bulb and stem up to
the scratch is deduced.

no. 8

Sources of Error.

1. The liquid —d.^., water— may contain

bubbles of air.

2. Tbe mass of the liquid is determined as

the difference of two masses. Each
of these must therefore be measured
with extreme accuracy.

Apparatus. A glass bulb with capillary
stem — about 300 cc. is a convenient size ;
platform scales and weights as in Exper-
iment 1 ; centigrade thermometer; Bun-
sen burner. Chromic acid, alcohol, and
ether are needed for cleansing purposes.

Manipulation. The bulb must be cleaned
inside and out quite carefully. The best

PRSLIMINART BXPSBIMSNTS

method is to wash it with chromic acid ; then remoye the
acid and rinse with pure water; then remoye the water
and rinse with alcohol and ether.

The small amounts of the liquids necessary to clean the
bnlb can best be introduced by heating the bulb by one's
hands (or by a flame^ if the flame is not brought near the
ether or alcohol), thus expelling some air^ and then by intro-
ducing the opening of the tube under the surface of the
liquid and cooling the bulb.

To remove the liquids the bulb may be turned with the
tube pointed downward ; and, if the bulb be heated^ the
liquid will flow out. After the alcohol and ether have
been removed^ it is necessary to dry the bulb. To do this^
join it to the drying-tube apparatus; exhaust and allow
dry air to enter^ then exhaust^ let more enter, etc., until
the bulb is entirely dry. This process may often be has-
tened by heating the bulb gently by means of a Bunsen
burner.

The bulb should now be weighed as is explained in Ex-
periment 1. This weighing must be par-
ticularly accurate.

To fill the bulb with some liquid — e. g.,
water — the following plan is best : Make
a funnel out of a clean glass tube, about
6 cm. long; 2 cm. diameter, closed by a
tight -fitting clean wooden cork at one
end ; by means of a cork - borer make a
hole in this cork just large enough to fit
tightly over the stem of the bulb. Leave
the opening of the stem just above the us^^ki
cork ; and support the bulb in a clamp- ^^ ^
stand and pour in water, from which air -^^^ ^
has been removed, if necessary, by boil-
ing.

Heat the bulb by a Bunsen burner, thus
causing air to bubble out through the
liquid in the funnel ; cease heating ; and, fio. 9

24 A MANIML OF EXFERIMBNT^ IN PHYSICS

aa the bttlb cool0> the liquid will i*tidh in (conMquently do
not heat the bttlb too high).

By repeating the heating and cooling the entire bulb and
stem may be filled with the liquid. (A last bubble of hit
is liable to stick to the mouth of the tube^ but it oan be re-
moyed by scraping it off with the point of a knife.) Place
the bulb one side, to cool and come to the temperature of
the room. (This may take an hour, and in the meantime
the student may take up some other experiment.) When
this stage is reached^ remove the funnel ; shake out enough
water to bring the level of the surface down to within a
few centimetres of the bulb { dry the exterior carefully ^
weigh as before. Note the temperature of the air near
the scale-pan and assume this to be the same as that of
the water. Make a fine scratch with a file at the top of
the column of water in the stem. Empty the bulb; dry it$
pass a cork, only bored part way through, over the end to
keep out the dust, and put the bulb away carefully for use
in future experiments*

Deduce the mass of water contained in the bulb and
stem up to the scratch, and by the aid of the tables of
densities calculate the capacity up to this point.

ILLUSTRATION

Oct 98, IM

Mate of bulb empty, 80.05 g4 ^ ^ » _ , «. ^^

* 1. iw * II eo ^i. f ^»88 of water, 25.^1 g.
Mass of bulb fuU. 56.16 g. ) ' ^

Temperature, 19.4' c.

Mass of 1 cc. of water at 19.4* is 0.998. ^ ^i

Hence, capacity of bulb and stem to scratch at 19.5^ is -^s^sr = ^-^ ^

•Wo

Since the volume of a glass vessel increases by about
0.000026 of its amount for each degree centigrade rise
in temperature, the capacity at 0"" of the bulb is %6M
(1.-0.000026x19.6)2*25.24 cc.

NoTr-^Th6 method giTen abore for the determination of Tolnmes is the
one universally used for determining the capacity of a Tessel of irregular
form, or of one whose internal dimensions cannot easily be measured. Where
greater aocnracy is desired, mercury is used iti the place of water.

PREUMINARY EXPERIMENTS 26

The Toliime of irregular solidi is found in a similar manner by the follow-
ing process t Adj oonrenient narrow glass vessel large enough to hold the
object is filled with water up to a mark bj means of a burette, so that the
exact Toluroe of water may be noted. The vessel is then emptied and dried
and the object placed in it. Water is again run in from the burette until it
reaches the same mark. The difference in the volumes of water needed in
the two cases is the volume of the object. A sinlcer must be used for light
objects ; and, if the sinker also is of irregular sliape, its volume must be
found in a similar manner, tli^ totume of objects whose dimensions are
easily measured is of course best found by calculation.

Areas wiiieh cahnot be readily oaloulated from their dimenlioiis are
measured by transferring them to smooth, uniform card-board, by pricking
the outlines with a very fine point The area is then cut out and weighed on
delicate balances, and another piece of tlie same card-board of regular di-
mettslodS is also weighed. From the area of the latter, which can be calcu-
iMed, and the relative wetghcs, the unknown area Is easily foUttd.

The average area of the cross-section of a tubto is fouttd by weighing
the amount of mercury necessary to fill an accurately measured length of the
tube. If th« mass is m, the density p, and the length /, the average cross.

section is 7— By using a short thread of mercury and determining its ex-

act length In each suooeesive portion of the tube, the tube can be "calibrat-
ed**—that 18, any VaHatiohs in the cruss-sectioh ascertained.

Qnestioiis and Problems.
1. Why, in determining the volume of an irregular solid, as Just
described, is not the solid put in the vessel while it contains
the water ?
2b Standard gold is an alloy— 11 parts gold, 1 part copper. Cal-
culate its density.
8 Oil mixing 68 cc. of sulphuric acid with 24 cc. of water, 1 cc.
is lost by mutual penetration. Calculate the detisity of
the mixture.

EXPERIMENT 8

Ol(jeot. To determine the number of centimetres in one
inch.

General Theory. This is to measure the same length by
two rules, one divided in centimetres, the other in inches,
and then to take the ratio of the numerical values of the
length in the two units.

Sources of Error.
These are practically the same aa those of the first part of Ex-
periment 1.

Apparatus. Two rules, eacb one metre long, the one
graduated in centimetres and millimetres, and the other
in inches and fractions of an inch.

Manipulation.— Method 1. By Coincidence.— Place the
rules side by side on a steady table, with their upper sur-

l|Tnipill|[JBlBi[[IJll]ii]l[Llli|llJJ[illl|Uiim

9876548S1

ilri[il|ffiliTiiL|lnnliin^^

Fio. 10

faces at the same level, and with their graduated edges in
close contact. It is best to place the rules on a table fac-
ing a window ; but in any case care must be taken to see
that they are illuminated directly, not sidewise. Slide one

PRBLIlflNABY EXFE&IMEKTS 27

scale along the other until the axis of a division on the
inch scale is exactly opposite the axis of a division on the
centimetre one. Choose clearly defined^ narrow divisions
near the ends of the two scales, but never measure from
the ends themselves. See that the divisions chosen are
well ruled, perpendicular to the edge. Holding the rules
very tightly together with their upper surfaces in the same
horizontal plane, find two other divisions, one on each
scale, the axes of which coincide exactly. This pair should
be chosen more than twenty inches from the first pair
which were placed opposite each other. In determining
when one scale division is opposite another, sight along
their axes with the eye a little above them ; do not view
the lines from the side.

The distance between the two pairs of divisions is the
same, but is expressed in different units on the two rules.
Note the numbers of the two divisions on the inch rule.
Their difference is the difference in inches. Do likewise
for the centimetre scale, and deduce the distance in centi-
metres. Thus calculate the number of centimetres in one
inch. Report the numbers of the divisions actually read,
as well as the differences, as is shown below. Bepeat four
times, using different distances and portions of the rules,
and find the mean of the results. This is the desired ratio,
as given by this experiment.

Method 2, By Estimation. — In this method an arbi-
trary number of inches is measured in centimetres. Place
two divisions of the scales opposite each other, as in the
beginning of part one ; but, instead of searching for two
more that coincide exactly, note on the centimetre scale
the exact position of a division of the inch scale, which is
any definite number of inches — say thirty — ^from the first.
In doing this, estimate the tenths of a millimetre by the
eye, if the desired inch mark falls between two millimetre
marks. Note the readings on both scales and the differ-
ences as before, and repeat four times, using the same num-
ber of inches but different parts of both rules. From the

A MANUAL OF £XP£RUI£^iTB IN PHYSICS

mean number of centimetres thus found to be equivalent
to the given number of inchesi deduce again the ratio of
an inch to a centimetre or the number of centimetres in
one inch.
Average the results by the two methods.

ILLUSTRATION
METHOD 1

Nov. 1, ltt6

l8t Mark

9dM«rk

Intenral

Cm. Id 1 in.

loch Rule
Cm. Rule

26.75
67.8

4.00
10.00

22.75
57.8

2.541

Inch Rule
Cm. Rule

81.60

77.4

8.00
6.00

28.60
72.4

2.640

Inch Rule
Cm. Rule

84.0
88.3

2.00
7.00

82.00
81.8

2.641

Inch Rule
Cm. Rule

88.1d5
06.8

1.00
1.00

87.125
94.3

2.640

..^

Mean, 2.6406 J

METHOD 2

I8t Mark

ad Mark

Intorral 1

Inch Rule
Cm. Rule

88.00
96.14

8.00
20.00

80.00

76.14

Inch Rule
Cm. Rule

84.00
98.22

4.00
17.00

80.00

76.22

Inch Rule
Cm. Rule

81.00
94.18

1.00
18.00

80.00

76.18

Inch Rule
Cm. Rule

35.00
87,19

5.00
11.00

80.00

76.19

Inch Rule
Cm. Rule

82.00
90.25

2.00
14.00

80.00

76.25

Mean. 80.00 in. = 76.20 cm.
or 1 in. = 2.540 cm.

Mean of two methodii 1 Sn. = 2.5400 cm.

PRELIMINARY EXPERIMENTS 89

QuMtions and FroUeiiui.

1. Why is it better to compaie distances as great as twenty

indies or more ?

2. Id the second method why is the given way better than to

choose an arbitrary number of centhnetres and measure

their length in inches ?
8. Why is it better not to measure from the ends of the rules ?

Why not use the same scale division over several times 1
4 Is tlierc any objection to using two rules of different materials,

6.^., iron and brass?

EXPERIMENT 4

Otyect. To learn the method of nsing a yemier.

General Theory. With many instruments measarements
are taken by means of an index which slides along a grad-
uated scale. This scale may be straight^ as in a caliper or
barometer, or circular, as in a sextant. In the simplest in-
struments of this kind there is only one mark on the slid-
ing index, and the position of the index is read from the
scale division nearest to this mark. When the mark falls
between two divisions of the scale, the fractions of a divis-
ion are estimated by the eye, as in the second part of Ex-
periment 3. To obtain 'this fraction with greater accuracy,
the index of delicate instruments is provided with addi-
tional marks, forming a series of equal divisions, on one or
both sides of the principal mark. Such a scale on the index
is called a " vernier'*; and the principal mark, the *'zero
of the vernier,'' because in a perfect instrument it comes
directly opposite the zero of the main scale when the quan-
tity which the instrument is designed to measure is zero —
e. ^., in a caliper when the jaws are closed on each other
with no object between. It will avoid confusion to remem-
ber that the zero of the vernier replaces the single mark
on simpler instruments, and that the other marks are mere-
ly to be used in determining more accurately the position
of this principal mark.

Positive Vernier is the most usual' form of instrument.
The vernier scale numbers on the sliding index increase in
the same direction from the zero of the vernier as the as-
cending numbers on the main scale. Such a vernier is

FRELDUNART EXFERIHENTS

called "poBitiye/' and for convenience of description we
will assnme this direction to be from left to right. The
vernier divisions differ in length from those of the main

]

S S « tf 6 7

. ••

1 :

1 i 6 • 7 8 1

U IS IS
10

Li 15 16 n 18 19

FiaU

scale, bnt a whole number, n, of vernier divisions is always
made to equal w±l scale divisions. We will describe the
kind of vernier in which the vernier divisions are shorter,
and will first consider the case in which 10 vernier divis-
ions eqnal 9 scale divisions. Let the length of a ver^inB^
division be represented by v, and the length of a scale di-
vision by s.
Then in this particular vernier,

/. V = 9/10 8,
A « — r = « — 9/10 s = 1/10 8.
In words, this equation says that one scale division is
longer than one vernier division by 1/10 of a scale division.
Suppose, now, that we start with the case in which the in-
dex mark, or zero of the vernier, corresponds with a cer-
tain definite division line of the scale — say the line number
2, for example. If we look at the scale to the right of this
mark, we see that the first vernier division falls short of
coinciding with a scale division by 1/10 «. The second
vernier division falls short of coinciding with a scale divis-
ion by 2/105, the third by S/lOs, etc. It is plain, then,
that as we move the vernier along the scale to the right,
when the zero of the vernier has passed 1/10 8 beyond any
line of the scale the first vernier division line will coincide
with a line of the scale. If we move it along 2/10 8 from

88 A MANUAL QF SXPEBIMBNTfi IN PHYSICS

the Btarting^point^ the second line on the vernier wlU be
in coincidenoe, and so on. Therefore, if we wish to know
how many times 1/10 s the zero of the vernier is beyond a
division line of the scale, we have only to find the number
of the first line of the vernier that is in coincidence with a
line of the scale, and that number will give the number of
tenths of a scale division desired. If no vernier division is
in exaot coincidence with a scale division, the two divisions
which lie on either side of coincidence mnst be noted, and
by estimation one can calculate what vernier division would
coincide if the vernier scale were more finely subdivided.

This theory can be generalized for this kind of a vernier
as follows *

Let ft vernier divisions equal n-^l soale divisions. That is:
nv^in — l) 9,

n n n

« — 1^ is called the ''least count," which in this ciise is

s. If n is 10, as above, and s a millimetre, the least
fi

count is A mm.

The theory of a vernier in which the vernier divisions
are longer than the scale divisions, or of a vernier whose
divisions increase numerically in a direction opposite to
those of the scale, presents no difficulties. The first step
in using an instrument with a vernier is to determine the
least count and the kind of vernier, positive or negative,

Apparatus. Wooden model of a vernier on a large scale.

Wsjiipiilation. 1. Find the least count of the vernier.

Slide the vernier along the scale until the zero of the
vernier coincides exactly with any arbitrary division on the
main scale. Find another mark on the vernier which ex-
actly coincides with one on the scale. Let it be the j)^\
Let there be q divisions of the n^in scale between the
same two marks. Then

p v = q if ,\v = qlp a = n — ' — M 8.

Therefore {p — q)lp = 1/n, least count.

PRBLIHINART EXPERIMENTS

Find the least connt of the model in this way :
2. Practice reading the vernier and also estimating tenths
by the eye. Set the vernier at random^ and^ covering all
bnt the index with a piece of paper, estimate by the eye
the exact reading of the index to tenths of a scale division.
Next find the coinciding division on the vernier, and from
it determine again the tenths of a division. Tabulate as
below and compare the readings by the eye and by the
vernier. Bepeat twenty times and report all readings.

ILLU8T&ATI0N

Zero of vernier ooioddes with 7 cm. on scale.

Oct 0, IBM

16 ••
10-9

Jt — i

Lv, y = v. xjcwa

ouu.

BmdtBg
by Bye

Whole Namber
on Scale

Coinciding mark
of Vernier

Fimotionby
Vernier

BMMtingby

Vernier

6.7

8x1/10 = .8

6.8

7.2

lXl/10«:.l

7.1

etc

etc.

Quastlons and Problems.

1. Suppose the 6th veraier diyision moet nearly coincides with a

scale division, but falls on the side nearest the seero of the
scale, and that it is estimated to be 1/4 as far from coinci-
dence as the 5th division. What is the exact fractional read-
ing, the least count being 1/10 cm. ?

2. Give a formula applicable to such cases in general.

8. What effect would a slight irregularity in the position of the
marks on the main scale have on the vernier reading, and
how can error from this source be best avoided ?

EXPERIMENT 5

Olject. — XJsB OF Vebkieb Caliper.^- To measure the
linear dimensions of some small object and to calculate its
Yolume.

General Theory. The volume of a circular cylinder is
irr'l, if r is the radius and I the height ; consequently, in
order to determine the volume, r and I must be measured.
As r enters to the second power, however, and I to the first
only, it is evident that special attention must be paid to
the measurement of r.

There are various instruments which can be used to
measure lengths accurately, but none is so generally used

)}

0 SlOKtOM

t|lipppi|l|ll|llipi |iii[)(

Fio. 19

as a vernier caliper. It consists of two parts— « graduated
'Mimb,^' with one fixed jaw at right angles to it, and a slid-
ing index which carries a second jaw accurately parallel to
the first, and which is free to move along the limb unless it
is clamped. The index carries a vernier, and the position
of the zero of the vernier on the scale indicates the dis-

PSBLIMINARY EXPSRIMENTS S6

tftQce between the two jaws. (The caliper in the figure
has an attachment to the sliding vernier^ which is designed
for making finer adjustments. In nsing it unclamp both
screws and slide the yemier jaw nearly into contact. Then
clamp the screw in the sliding attachment, but not the ver-
nier screw^ and make the final adjustment by the *' tangent
screw " between the two parts. Then clamp the yemier
screw.)

Sources of Bnor.

1. The two Jaws may DOt be exactly parallel.

2. The linear dimeDsions to be measured may not be exactly

parallel to tbe scale on tbe limb.

8. Too great pressure may strain tbe caliper and bend tbe jaws
80 that they are no longer parallel, tbough they spring back
when the object is removed, and the. error is not noticed
in suiisequent zero readings. Pressure may also change tbe
dimensions measured, as, for instance, in measuring the di-
ameter of a hollow cylinder or of a tube.

4. Tbe same dimensions may be greater or smaller at different
parts of the object.

Apparatus. A hard rubber cylinder to measure ; a yer-
nier caliper ; reading lens ; metre bar.

Manipulation. Compare the scale on the limb with the
metre bar and learn what the unit is. If there is an inch
scale on one side and a centimetre scale on the other, use
both. Deduce the least count of each yernier. Clean all
dust or dirt from the inner faces of the jaws.

In all measurements it is as important to know accurate-
ly the reading for the beginning of the length measured
as it is to know that for the end. In this case the begin-
ning is the position of the zero of the yernier on the scale
when the moyable jaw is in contact with^ and parallel to,
the fixed jaw, and this must be determined quite as accu-
rately as the position of the index when the object to be
measured is placed between the jaws. - This first reading
is called the determination of the zero of the instrument,
or simply the ''zero reading/' and a similar set o| ^ero

86 A MANUAL OF EXPERIMENTS IN PHYSICS

readings must be taken on every instrament throughout
the course.

To Obtain the Zero Reading. Loosen the clamping screws
so that the moyable jaw is free to slide along the limb^
but not to rock on it ; slide the jaws gently into contact ;
hold the caliper between the eye and the light, and, looking
between the jaws, bring them lightly but completely into
contact along their whole length, if possible ; clamp the
jaw by means of its screw ; see that the jaws remain in
contact when the screw is clamped. Bead and note the
exact position of the yemier on the scale of the limb, de-
termining the fractions of a scale division by the vernier,
as learned in Experiment 4. (Bead both the inch and
centimetre scales, if both are given on the caliper, and re-
duce all fractions to decimals.)

A lens should always be used in reading, and the instru-
ment should be held facing a window, if possible. It will
often be found that the divisions of the two scales shift
relatively to one another, according to the direction from
which they are viewed. Care must be taken, therefore,
to look at them in directions parallel to the marks on the
two scales, or to look down on the instrument perpendic-
ularly. This shifting with the point of view is called ''par-
allax/' and must be guarded against whenever a vernier
is used, and in many similar cases {e, g., Experiment 8).

The whole number of the reading is always given by the
scale division next below the zero of the vernier, and the
vernier reading of the fraction is always added to this.
Hence, if in the zero reading the zero of the vernier
comes below the zero of the scale, the division next below
it is — 1. If the vernier reading is then, say, .7, the whole
zero reading is — 1. -|- 0.7 or — .3, and so on.

To Measure the Length. Having obtained the zero read-
ing, loosen the screws, move the jaw, and insert the cylin-
der, so as to measure a diameter at one of its ends. Turn
the cylinder in the caliper and see whether in any direc-
tion the diameter seems to be a maximum. If it is, meas-

PR£LlMli\AKY EXPERIMENTS 87

nre in this direction. Hold the cylinder so that its axis is
at right angles to the limb^ and note carefully that the
points in contact with the jaws are diametrically opposite
and at exactly the same distance from the end. The diam-
eter to be measured should be exactly parallel to the inner
edge of the limb. If from an imperfection in the instru-
ment the jaws were not in cod tact along their whole length
in taking the zero readings see that the same points which
then touched are the ones now touching the cylinder.
Push the jaws together until the cylinder is just loosely
held between them ; then clamp the screw, and read and
note the position of the zero of the vernier as before (on
both scales if there are two). Measure similarly the diam-
eter at the other end, taking in each case the maximum
diameter. Try and close the jaws with the same pressure
in each reading.

Take a new zero reading and repeat the measurement of
the same two diameters. Repeat the readings twice more
on diameters at right angles to those already measured.
Ayerage the zero readings, and also the four diameters
measured at each place, and, subtracting the former, get
the mean diameter of each section. Average these to get
the mean diameter of the cylinder. (Do the same for the
readings taken in inches.)

Proceed in the same way to get the length of the cylin-
der, measuring it three times, taking a zero reading each
time, and again using both scales, if there are two. Note
the number of the cylinder and description or number
of the caliper. Finally calculate the volume of the cyl-
inder.

Kon.— If the cylinder to hoUow it will be necessary to measare the in-
tenial diameter also. Bring the outer edges of the tips of the caliper's jaws
into contact with the inner surface of the cylinder, the line joining the edges
being accurately at right angles to the axis of the cylinder. The scale read-
ing Is then the internal diameter minus the thickness of both jaws. The
Vatter thickness may be found by measuring it with another caliper. Hake
Sour readings at each end, and record.

A MANUAL OF EXPJB&IMENTS IN PHT8IGS

ILLUSTRATION.— Ctliitdbr No. 1.

ZUO BMdfOff

DIAMETER
MarkMKod

Calipbr No. 8

Oct 7,1

OthOT Bad

Cb.
0.07
0.06
0.08
0.07

Mean, 0.07

Id.
0.081
0.088
0.086
0.088

Cm.
4.49
446
4.51
4.50

In.
1.775
1.78
1.77
1.78

Cm.
4.60
4.61
4.52
4.49

In.
1.79
1.785
1.77
1.775

Hence, diameter j

0.088 4.49 1.776 4.60 1.780

marked end, 4.42 cm., 1.748 in. ; mean, 4.42 cm.

4.48 cm., 1.747 in. ; mean, 4.48 cm.

\ other end,

Thickness of jaws, 0.6 cm.

Tlie cylinder was hollow, and of internal diam. = 8.29 + 0.6 cm.
= 8.79 cm.

(Tiie stadent mast giye the details of this measurement, just as they
are given above for the external diameter.)

Zero RMdiog

Cm.
0.08
0.07
0.06

In.
0.086
0.088
0.088

Mean, 0.07 0.064

LENGTH
Reading on LengUi

Cm.
5.79
5.80
6.78

Id.
2.285
2.280
2.290

2.285
2.261 =

Cm.
6.72
5.72

In.
2.251

Mean of results by two scales, 6.72
/. VoL = 8.1416 X 6.72 | (^V- ^^'| = ^-^ ^

Questions and Problems.

1. What advantage is there, other than mere convenience, in hav-

ing different units of length on the same instrument ?

2. Would a theoretically perfect caliper give the same result for

the same dimensioD of the same object at all times ? Why ?
8. Why was the diameter of the cylinder measured eight times

on each scale and the length only three ?
4. In which dimension do you think you have made the least

proportional error ?

EXPERIMENT 6

Otgeet — ^Usb of Micrometeb Calipeb. — To measure the
linear dimersions of some small object — e.g., the thickness
of a wire^ a piece of glass^ a sphere, etc.

General Theoiy. A micrometer, or screw caliper, consists
essentially of a screw whose pitch is uniform and whose
motion in its nut may be accurately noted. In the ordi-
nary form of instrument the screw is rigidly attached at
one end to the inner end of a hollow cylinder or "barrel/*
so that as the barrel is turned the screw is turned in its
nut. There is a scale running lengthwise on the nut.

Fio. 18

which is BO divided as to correspond to the distance which
the screw advances each whole turn — i. e., the scale divis-
ions equal the pitch of the screw. The edge of the barrel
moves backward and forward along the edge of this scale as
the screw is turned, and so whole turns of the screw may
be noted. Fractional turns may be noted by the divisions
which are engraved on the edge of the barrel, and which
pass across the fixed line running lengthwise of the nut.

40 A MANUAL OF EXPERIMENTS IN PHYSICS

The screw ends in a flat face^ called the tooth, and as the
screw is tnmed this tooth may be made to approach or re-
cede from another flat face, which is rigidly attached to
the nut by the framework of the instrument. Hence, if
the two teeth are flrst placed in contact, and then separated
so as to just contain some solid, the number of turns of
a screw and the fraction of a turn may be observed by not-
ing the readings on the edge of the barrel and along the
nut. The pitch of th^ screw is always a standard one — e. g.,
1 or 1/a mm., 1/16 or 1/32 in., etc., and which particular
on^ it is can be determined by a rough comparison with a
standard rule or with a plate of known thickness.

Sources df XSrror.

1. The faces of the teeth may not be parallel, and (nay not be
perpendicular to the axis of the screw.

3. The linear dimensions to be measured may not be exactly par-

allel to the screw.
a The pitch of the screw may not be ezaotly the same In all
parts.

4. Pressure changes the shape of the caliper, and may strain the

jaws apart without the screw being turned. Too great
pressure may also change the dimension to be measured.

Apparatus. A micrometer caliper; a piece of plane,
parallel glass ; a reading-lens.

Manipulation. Compare the scale on the nut with milli-
metres by marking on the straight edge of a piece of paper
the exact length of ten divisions (not half-divisions, if such
are marked) of the scale along the nut, and laying it off on
the metre bar. Record the equivalent in millimetres of
one division of the scale. Each such division marks one
or more complete turns of the barrel ; to determine how
many, start with any chosen mark of the barrel opposite
the line of reference drawn lengthwise on the nut, and turn
the barrel any number of complete turns, counting the
number of whole division marks on the nut which have
been passed over. From this calculate the fraction of a
division of the instrument passed over by the edge of the

PRBLIMINARY EXPERIMENTS 41

barrel, when the latter is turned throngh one diyision of
the scale npon it. (If the scale on the edge of the barrel
has two sets of numbers npon it — e. g., 1.6, 2.7^ 3.8^ etc. —
one tnm wHl be found to correspond to one«half a division
of the nnt, and the lower of the two numbers is to be used
when the edge of the barrel is in the first half of a diyision,
and the higher when it is in the second half.) Use the
same precautions against dirt^ parallax^ and non-parallelism
of the surfaces of the teeth as in Experiment 5. The best
caliper has a ratchet at the end^ which enables the con*
tact to be made always with approximately the same press-
ure^ which is indicated by the same number of clicks of
the ratchet, turned yery slowly and counted as soon as the
surfaces touch. (Five clicks mark a. safe and convenient
pressure.)

Make a zero readings bringing the teeth into contact by
means of the ratchet and counting the number of clicks
after they touch. The integer of the reading is given by
that division of the nut which is nearest to the edge of the
barrel, but not covered by it ; and the decimal fraction of
a tnm, by that point of the scale on the edge of the barrel
which is opposite the line of reference on the nut. Bead
to tenths of the division on the edge. (If the edge is be-
low the zero mark, give a negative sign to the integral part
of the reading, keeping the fraction always positive ; see
illustration.) Next, insert the object to be measured, clos-
ing the teeth on it with the same number of clicks of the
ratchet after contact as in the zero readings and taking
care that the dimension to be measured lies in a line per-
pendicular to the faces of the teeth. Take separate read-
ings, with a zero reading for each.

If the object of the experiment is to measure a given
thickness, make all the measurements in the same place :
but if the object is to find the average thickness of various
points — e. g., the diameter of a wire — measure in as many
directions and places as possible.

Note and report the number of the instrument used, and

▲ MANUAL OF BXPIRIMINTS IN PHTSIG8

any mark or number aerring to identify the object meas-
ured,

ILLUSTRATION •

Oct 13, 1899

Plate or Glass No. 1. HiCRomriE Calipik "M 60"

Zero Reading

Reading
on 61*88

Thlckneaa

Zero Reading

Reading
on Glass

Thlckaeea

-1 .978

8.151

8.178

_ 1

.968

8.149

8.181

-1 .977

8.151

8.174

— 1

.969

8.151

8.182 -

-1 .972

8.161

8.179

— 1

.965

8.149

8.184

-1 .974

8.152

8.178

— 1

.968

8.149

8.181

-1 .971

8.151

8.180

— 1

.967

8.148

8.181

-1 .978

8.158

8.175

— 1

.968

8.147

8.179

-1 .969

8.160

8.181

— 1

.966

8.146

.8.180

-1 .972

8.160

8.178

-1

.967

8.150

8.188

Mean, 8.1796 mm.

Probable error of one observation = ± 0.0024 muL
'* " thei«8uU =±0.0006 mm.

.-. Probable thickness is 8. 1796 ± .0006 mm.

Qnestlona and Problems.
1. To how many significant figures are you entitled to carry out
your result, and why ?
What error is your measurement liable to which you could
not compensate for by taking more observations? How
could you detect and diminish it ?
Does the scale on the nut of the micrometer caliper have to
be as accurate as that on the Ihnb of the vernier caliper ?
Why?

EXPERIMENT 7

Olgeel — ^Use of the Sphebometer. — To measure the
thickness of some small object^ such as a plate of glass.

Oeneral Theory. A spherometer is an instrument made
essentially on the same principle as a micrometer caliper.
It consists of a screw which turns in a nut supported by
three legs^ as shown.
There is a scale rigid-
ly fastened to the nut
and parallel to it ; and
the screw carries a
disk perpendicular to
itself^ whose edge is
divided into equal in-
tervals. The fixed
scale is so divided as
to correspond to whole
turns of the screw.
The three legs, the
screw, and the fixed
scale are all parallel,
and perpendicular to
the plane which is
fixed by the extremi-
ties of the three legs.
By placing the instrument on a plane surface the position
of the screw can be recorded when its extremity is in that
plane, and then its position can also be noted when it i»
raised so as to allow some solid to be introduced under it.

FkkU

44 A MANUAL OF EXPERIMENTS IN PHTSIGS

In this way the thickness of a plate can be measured^ if the
pitch of the screw is known. It may be assumed in gen-
eral that the pitch is a standard one — e. g., 1 mm.^ 1/16 in.^
etc. — and the particular one which it is can be determined
by a rough comparison with a centimetre or an inch rule.
If the pitch, however, is arbitrary, its value must be deter-
mined by the use of a plate of known thickness. The in-
strument may also be used to measure the curvature of a
spherical surface, as is explained in any larger manual.
(See Stewart and Gee, vol. i.; Glazebrook and Shaw.)

Booroea of Brror.

1. The screw may not be perpendicular to the plane of the ex-

tremities of the legs.

2. The fixed scale may not be parallel to the screw.

8. The pitch of the screw may not be the same in all parts.

4. The disk may not be exactly perpendicular to the axis of the

screw.
\ 5. The divisions on the fixed scale may not agree exactly with

whole turns of the screw.

Apparatus. Spherometer; large glass plane surface; a
piece of plane, parallel glass ; a reading-lens.

Kanipnlation. — Note. — In using the spherometer raise or
lower the screw by turning the milled head. Do not turn
it by means of the disk.

Compare the vertical fixed scale with a millimetre one,
and note the value of one of its divisions. Note the num-
ber of divisions on the disk, and see if, when the disk is op-
posite a division of the vertical scale, its reading is zero, as
it should be. Next, by actual trial, find wliat divisions of
the vertical scale correspond to whole turns and what to
half turns or several turns ; that is, determine the pitch of
the screw. Now turn the disk slowly, and observe care-
fully whether it rises or falls as it is turned in such a way
that successively greater numbers on the disk pass the edge
of the vertical scale. If it rises as increasing numbers
come to the edge, that number on the disk which has just

PRELIHIXART EXPfiRIMENTS 45

passed the edge indicates the fraction of a tnm which has
heen made since the disk passed the mark on the vertical
scale next below (helow in position^ not numerically). This
fraction is therefore to be added to the scale division in or-
der to get the exact reading.

If, on the other hand> the disk falls as the nnmbers in-
crease, the fraction indicated by it is to be subtracted from
the whole number, as given by the scale division next alnwe,
for it means that the. disk has fallen that fraction since pass-
ing the mark next above. In other words, the fraction
marked by the scale on the disk is given a positive sign in
the first case and a negative in the second. The sign of
the integer of the index is positive, of course, when the disk
is above the zero of the fixed scale, and negative when it is
below. The number of the integer is to be taken, as stated,
from the mark next below the disk in the first kind of in-
strument, from the mark next above in the last. In deter-
mining the position of the disk on the scale always sight
along the upper surface of the former, and in like manner
sight along the graduated surface of the scale to determine
the fraction of a turn indicated by the disk. Bead the
scale on the latter to the tenth of its smallest divisions.

Having become thoroughly familiar with the instrument,
measure with it the thickness of some thin object, as a piece
of plate-glass. For the sake of comparison, it is well to
use the same piece of glass as was measured in Experi-
ment 6.

In the case of the spherometer the point measured from —
f. e.y the zero of the measurement — is the plane of the three
outer feet. To get the zero reading, it is necessary there-
fore to set the extremity of the screw precisely in this
plane, and to record the position of the disk when it is so
set. Place the instrument on the plane surface which
comes with it, and raise or lower the screw until the in-
strument rests squarely on it as well as on the feet. If
the central foot is in the least below the plane of the other
three, the instrument can be made to rock or to spin on

46 A MANUAL OF £XP£&Ili£NTS IN PHT8ICS

the screw-point. The screw must then be so adjusted that
it just does not rock. To do this exactly, begin with the
screw a little too high, and lower very slowly, trying the
stability continually, nntil yon notice the first trace of
rocking ; then note the disk reading. Now torn back very
slowly and stop when the rocking just stops; note the
reading again and repeat, turning backward and forward
until you have reduced the uncertainty as to the exact
point to the smallest range possible ; then take the aver-
age of the two extremes between which you are in doubt.
Finally, complete the reading of the zero by noting the in-
teger on the vertical scale. Be very careful to give both
the integer and the fraction their right signs in your mem-
oranda. Repeat the measurement ten times, moving the
screw each time considerably out of position in order to
secure ten entirely independent determinations. It may
be well to take five of the zero readings at the beginning
and five at the end of the experiment.

Now raise the screw and insert the object, taking care
that it is clean/ so that its lower surface lies in the plane
of the fixed feet — t. 6., flat on the glass plane. Lower the
screw so that its point is just in contact with the upper
surface. Make the exact adjustment as before by rocking.
Note the reading on both scales/ and give each its right
sign. Make three independent settings, then turn the ob-
ject the other side up and make three more. Note which
position gives apparently the smaller thickness for the ob-
ject, and complete the measurement by taking enough
readings to make a set of ten observations with the object
in that position. (Do not include the observations on the
opposite side in calculating the mean.)

Becord as below. Give the number of the instrument,
and a mark or number to identify the object.

If the reading on the disk is not zero when it is exactly
at a scale division of the fixed scale, care must be taken
not to make an error of a whole division of the vertical
scale on this account. A little common-sense will avoid

PRELIMINARY EXPERIMENTS

it, jaat as one woald avoid a mistake of a whole minate
in timing with a watch whose second-hand passed the 60
a little before or after the minnte-hand passed its mark.

ILLUSTRATION
FhAn-QUM No. 1. Sphuomrir, M 188

Zero
Veftical
G.5
5.5
5 5
5.5
5.5
5.5
5.5
5.5
5.5
5.5

On Olaa

Di8k
.8261
.8260
.8245
.8255
.8279
.8264
-.8278
-.8280

Vertical
18.5
18.5
18.5

18!5

Disk
- .1415
-.1421
-.1485

-.1424

Mean, 5.5 -.8271

other Side Up

Vertieel Disk

18.5 - .0824

18.5 - .0806

18.5 - .0348

18.5 - .0826

Oct 7, IBM

ThickiiMB

8.1847 turns

8.2945 turns

This last giving the greater thicknees, the experiment was finished
with the Other dde up.

On G1M8

Vertical

Disk

18.5

-.1427

18.5

-.1428

18.5

-.1428

18.5

-.1429

18.5

-.1448

18.5

-.1454

18.5

-.1452

TbickneM

Mean of all ten, 18.5 - .1488 8.1888 tume

Graatest deviation from the mean is 1 part in 60,000.

It was found by trial that the disk fell when turned in the direc-
tion of increasiog numliers, hence the minus sign is given to the disk
reading. Also by comparison with a millimetre scale the pitch of the
■crew was found to be 1 mm.

.'. 8.1888 turns = 8.1888 mm. = thickness of plate-glass Ko. 1.

(Experiment 6 gave the thickness of this same piece of glass as
8.1796. or .0042 mm. less. This is probably due to the measurements
being taken at different parts of the glass.)

50 A MANUAL OF EXPERIMENTS IN PHYSICS

If the pitch of the screw is not known^ it can be deter-
mined by placing a standard rule under the microscope
parallel to the screw, and noting how many turns of the
screw move the microscope through a known number of
centimetres. (See Experiment 9.) If the pitch of the
screw is known, the instrument may be used to measure
the distance between two points or lines by placing the
straight line between these points (or lines) parallel to the
screw, and noting how many turns of the screw will move
the microscope through the distance.

Bonrces of Error.

1. The pitch of the screw may not be the same in all parts.

2. The scale parallel to the screw may not correspond to it exactly.

3. The screw, the scale, and the " ways" may not be parallel.

4. The line whose length is measured may not be parallel to the

screw.
6. The nut may fit the screw loosely — i. «., may have " back-lash "
— in which case for a small motion of the screw in one di-
rection there might be no motion of the nut.

Apparatus. The dividing-engine ; an inch scale seven or
eight inches long.

Manipulation. In dividing-engines, as actually used, the
screw is placed horizontal, and there is a fixed platform

under the microscope, which is
itself adjustable horizontally and
vertically. The nut is split in
halves, which can be held apart
by a lever or spring, and which
must be held together by a clamp
in order to bind the screw.
P20. 16 Assuming that the pitch of

the screw corresponds with the
scale parallel to it, and that the scale is a metric (or any
standard) one, the instrument may be used to measure a
distance along a line — e. g., the distance apart of two di-
mensions on an inch rule — and the following description
gives the details of this measurement :

PRELIMINARY EXPERIMENTS 51

First determine the pitch of the screw. To do this,
loosen the nnt by raising the lever^ and slide the carriage
out of the way to one end or the other. Then compare
the scale with a centimetre mle.

The next step is to place the edge of the inch mle on
the platform parallel to the screw. Do this^ approximate-
ly, by the eye, but more exactly by means of the microscope.
Slide the carriage until the microscope stands over the inch
rule, and ''focus'' the instrument on the surface of the
rule. In a microscope which is not properly focused the
part of the object viewed, which seems to be directly under
the cross-hairs, will vary in position if the eye is moved
laterally across the eye-piece; this is said to be due to
''parallax." To avoid this, focus the eye - piece firsty by
sliding it in and out of the tube until the cross-hairs — ^not
the image of the object — are perfectly distinct ; then focus
on the object by moving the whole microscope up and down
in its holder, being very careful not to disturb the adjust-
ment of the eye -piece in so doing. When the focus is
complete, test by moving the eye across the eye-piece, and
do not be satisfied with the adjustment until there is no
relative motion of the cross-hairs and the surface of the
rule. Now adjust the level of the rule until the surface is
in perfect focurf^ the microscope is ttt^ved over its entire
length ; then, keeping the level unchanged, turn the rule
until, as the microscope is moved lengthwise, the edge of
the rule (or some line on the rule perpendicular to the di-
visions) coincides in all points with the point of intersec-
tion of the cross-hairs. These two adjustments place the
edge of the rule parallel to the screw. Now loosen the
screw which holds the microscope sleeve, and draw the mi-
croscope across the rule a short distance, so that the divis-
ion lines are in the field of view ; finally, clamp the micro-
scope screw and readjust the focus if it is necessary.

The instrument is now ready for use. Slide the carriage
until the point of intersection of the cross-hairs appears a
little to the right of TO inch mwk near the right-hand end

61 A MANUAL OF EXPERIMENTS IK PHTSIGS

of the scale, and the index on the nnt refits exactly on a
mark of its own scale. Turn the handle so that the zero
on the disk is directly opposite the fixed mark. Now close
the nnt by releasing the lever and pushing it down. The
reason why the index must be set exactly on a mark, and
the disk turned exactly to zero before closing the nnt, is
that otherwise the threads of the screw and nut would not
fit into each other, and might be seriously damaged in be-
ing forced together. Now turn the screw very slowly until
the cross-hairs coincide with the exact middle of a chosen
mark on the inch rule, not the end. If you accidentally
pass it, turn back until well to the right of it again, and
bring the hair into coincidence from the right side. Like-
wise through the experiment always move the hair on to the
mark from the same side. The object of this is to take up
the "back-lash," or looseness of the nut on the screw, owing
to the wear, which allows the screw to be rotated part of a
turn before moving the nut. Read the whole number of small-
est divisions on the scale by means of the index attached
to the nut, and the fractional part on the scale of the disk.
Turn the screw back until the cross-hairs are well off
the mark, and make three more independent determina-
tions of the exact position of this same mark. Note all
four readings and^e number of the mffrk on the inch
scale. Throw the nut out of gear, slide the carriage down
to the other end of the inch scale, throw the nut into gear,
and take four readings, in exactly the same manner, on a
division of the inch scale near that end. It will be con-
venient to choose a division which marks a whole number
of inches from the first. (Be very careful again before
throwing the nut in gear to see that the index is on a
mark and the disk turned to zero.) A given number of
inches have thus been measured in terms of turns of the
screw. Repeat the same measurements four times, using
the same number of inches, but measuring between differ-
ent marks so as to neutralize any error in the graduation
of the inch scale used. Report as below.

PRELIMINARY KXF£RIM£NT8

ILLUSTRATION

Oct as, 1809
Dnrmiio-iiNiuci ** M 7 '*

By comparison with a millimetre scale 100 divisions of the scale of
the instrument are found to equal 100 mm.; therefore the pitch of
the screw is 1 mm. to the turn. The larger-numbered divisions cor-
respond to ten turns, or a centimetre each. The disk is divided into
100 divisions, each of which = .01 turn, or .001 cm.

AN INCH MKASUBBD IN GENTIMBTRB8 BY OIVIOINO-ENOINE

Engine Soale In Cm.
liMtoz Disk
144 .0744
144 .0745
14.4 .0742

0.125
0.125
0.125
0.125

Mean, 0.125

0.5
0.5
0.5
0.5

Mean, 0.5

0.75
0.75
0.75
0.75

Mean, 0.75

0.25
0.25
0.25
0.25

1.025

14.4 .0748

Engine Scale in Cm

Inch Scale

Index Disk

6.125

29.7 .0202

6.125

29.7 .0196

6.125

29.7 .0194

6.125

29.7 .0201

14.4 .0744 6.125 . 29.7

6 in. = 29.7198 - 14.4744 cm. = 15.2454 cm.

.0196

15.7 .0455

15.7 .0458

15.7 .0455

6.5
6.5
6.5
6.5

6.5

80.9
80.9
80.9
80.9

.0886
.0885
.0884

80.9 .0885

. 6 in. = 80.9885 - 15.7446 cm. = 15.2430 cm.

147 .0277

147 .0276

147 .0278

147 .0277

6.75
6.75
6.76
6.75

6.76

29.9
29.9
29.9
29.9

.0787
.0786
.0786
.0788

29.9 .0787

.-. 6 in. = 29.9787 - 147277 cm. = 16.2460 cm.

80.2
80.2
80.2

149 .0881

149 .0882

149 .0880

149 .0881

6.25
6.25
6.25
6.25

6.25

.0278

80.2
80.2

.0280
.0279

. 6 in. = 80.2278 - 149881 cm. = 15.2448 cm.

Mean, 6 in. = 15.2448 cm.
Hence, 1 in. = 2.5408 cm.
Greatest deviation of an individual ** setting '* is 1 part in
Greatest deviation of a measurement from final mean is 1 part

100.000.
in 10,000.

64 A MANUAL OF EXPERIMENTS IN PHYSICS

QnefttioiiB and Problems.

1. Meution three or more advantages possessed by the preceding

method of tiikiug measurements of relatively long distances
over that which keeps the nut in gear while passing from
one extreme division to the other.

2. If the disk scale and the inch scale are perfect in all respects,

would the expansion of the nut and screw, consequent upon
a rapid, long-continued motion of the one along the other,
cause the results of this experiment to be too large or too
small ? Explain theoretically in detail.

3. For what purpose was the dividing-engine designed? Show

briefly how it works when used for this purpose — i,e,, oat-
line the principles involved.

EXPERIMENT 9

Olgect. To measure the pitch of the screw of a microm-
eter microscope.

General Theory. Micrometer microscopes are of two
types : in one the frame holding the cross -hairs is moved
perpendicularly
across the axis of
the microscope by
means of a screw ;
in the other, the
whole microscope is
moved at right an-
gles to its length by
a screw. The screw

in both instrnments is provided with an index marking the

whole number of turns and a
finely divided disk, or "head,*'
marking the fractions of a
turn ; and the method of meas-
uring the pitch is exactly that
made use of in the measure-
ment of a length with the
dividing-engine. (See Experi-
ment 8.) A standard rule is
adjusted parallel to the screw,
the microscope is focused on
its divisions, and the number
of turns is measured which is
required to carry the cross-
hairs a known number of scale
Fia. 18 divisions.

66 A MANUAL OF EXPERIMENTS IN PHYSICS

Bourcea of Brror.

1. The length of the standard rule may not be parallel to the

screw.

2. The dimensions of the standard rule may not be all equal.
8. The pitch of the screw may be different in different parts.

Apparatus. A micrometer microscope and stand ; a
standard rule.

Manipulation. The particular instrument used must be
carefully examined^ and the different clamps or screws
which stop or permit particular motions must be thorough-
ly mastered. The standard rule must be adjusted accu-
rately parallel to the micrometer screw^ exactly as in the
previous experiment. (The longer the standard rule or
the longer the screw^ so much the more accurately can this
be done.) Then a known number of scale divisions must
be measured in terms of the pitch of the screw, in doing
which each setting must be repeated twice and care must
be taken to avoid the '^ back-lash^' of the nut.

Repeat the observations, using different portions of the
screw and different portions of the standard rule.

It is sometimes difficult to identify the particular divis-
ion of the rule on which the cross-hairs are focused ; but,
by slipping a pin or a pointed piece of paper along the
rule until it is also in the field of view, this difficulty may
be obviated. Note the temperature, and correct for the
expansion of the standard rule, because its length is given
for a particular temperature.

PRELIMINART EXPERIMENTS 67

ILLUSTRATION

Oct 90, 1896
MlCROMICTRR EtLPIICB ON COMPARATOR " M 4 "

TompenUure 28° C. B. d: D. Metre Bar,
Dutanee mea»\ired on Standard Rule, 8 DittMans,

Screw Settings

1st Mark 2d tfiiric

0.608 5.101

Hence, 4.4947 turns = 2 dIvislonB

0.801 5.099

Iturn =0.4450 division

0.607 5.100

0.6068 5.1000

(The details of otber obserTationB are omitted here, but must be sup-
plied by the student in the report of his experiment.)

Summar}% 1 turn = 0.4450 divisioii
1 •• =0.4486
1 •* =0.4442
1 *' =0.4457
Mean result, 1 turn = 0.4444 diTision

Greatest deviation from mean is 1 part in 550. The B. ft D. metre
bar has a length 100.0180 at 17^ C. and it expands 0.000018 of its
length for a rise in temperature of 1° C. The divisions measured
were between millimetre lines.

Henoe, 1 division = 0.100018 (1 + 5 x 0.000018) cm.

= 0.10008 cm.
Henoe, 1 turn of screw at 22^ G = 0.4445 mm.
(It is evident that in this case the probable error exceeds the tem-
perature correction ; but it is useful to understand the process of cor-
rection.)

Questloiui and Problems.

1. State and explain several advantages possessed by a micro-

scope that is moved bodily by a screw which are not pos-
sessed by a microscope whose cross-hairs are moved by a
screw.

2. Would changes in temperature affect the readings on the disk

scale if it expanded uniformly in all directions ? Give some
advantages and some disadvantages of a disk of large radius.

8. State some defects of micrometer screws.

4. W^at is meant by " calibrating " a screw ?

& Would the. value of the pitch as obtained by the foregoing
method vary with the focal length of the microscope? Why?

EXPERIMENT 10

Olgeot. To measure by comparison with a standard mle
a horizontal or vertical length. Use of comparator or
cathetometer.

General Theory. A comparator or cathetometer consists

Tn.W

essentially of a massive framework provided with one or
two micrometer microscopes or telescopes. The frame-
work is so made that these instruments can slide along it
in a straight line. The two microscopes or telescopes are
carefully adjusted parallel to each other, and perpendicu-
lar to this line of motion. The length to be measured is
placed exactly parallel to this line of motion ; the two mi-
croscopes or telescopes are focused on the extremities of
the length ; the standard rule, which has also been ad-
justed parallel to the line of motion, is now placed in the
field of view of the microscopes or telescopes so that its
divided surface is exactly in focus ; the positions of the
two sets of cross-hairs are noted, and their distance apart

PRELIMINARY EXPERIMENTS

gives the length desired in terms of the standard rnle. If
the cross-hairs do not coincide with diyisions on the rnle,
their distances from the near-
est divisions may he measured
hy means of the micrometer
eye-pieces, whose pitches must
of course be previously meas-
ured.

In some types of instruments
there is but one microscope or
telescope, and the standard
rule is engraved along the line
of motion of the microscope or
telescope. With these instru-
ments the reading is made in
turn on the extremities of the
length, and the difference is
noted.

In another type of cathe-
tometer there are two tele-
scopes on the vertical stand,
and the scale is engraved on
this along the line of motion.

Booroes of Error.
L The microscopes or lele-

scopes must be exactly

perpendicular to the line

of motion.
& The line of motion may not

be a straight line.

Hanipulation. The adjust- ^^^

ments of the comparator are practically the same as those
described in the last two experiments, but one additional
adjustment is necessary : the length to be measured and
the standard rule must be placed side by side, exactly par-
allel and with their surfaces in a plane perpendicular to the
axes of the microscopes, so that each may be in focus when

•0 A MANUAL OF BXPKRIMSNTS IN PHYSICS

the platform carrying them is rolled aidewise, bringing the
two lengths sacceflgively nnder the microsoopes.

This adjustment mnst be made before the measurements
are begun, and can easily be secured by proper screws on
the instrument or by using suitable wedges or blocks. In
some instruments the platform carrying the two lengths
is not movable, but the shaft carrying the two microscopes
can be turned around its axis, and the microscopes be thus
brought to focus upon each in turn.

The cathetometer is, almost without exception, used to
measure vertical distances, and in order to adjust the
line of motion of the telescopes exactly vertical, the in-
struments are provided with levels and levelling screws.
For a complete discussion of these adjustments, see Stew-
art and Oee, vol. i. The length to be measured and
the standard rule can be adjusted parallel to the line of
motion of the instrument, and at equal distances aw^y
from the eye-pieces of the telescopes ; and the lengths are
in general compared by causing the cathetometer to rotate
around an axis parallel to the line of motion. (In certain
instruments the standard rule is adjusted permanently by
the maker parallel to the line of motion of the telescopes
or microscopes ; and in others, as noted above, the standard
rule is engraved along the line of motion.)

The temperature of the standard rule should be noted
and correction made.

EXPERIMENT 11

Otgeet To determine by the method of yibrations the
position of equilibrium of the pointer of a balance.

Qeneral Theory. The indications of many instruments
are made by means of a marker of some kind swinging over

'^Tf^

a fixed scale. Instances of this are the pointer of a balance
and the needle of a galvanometer. In all these instruments,
when sensitively made^ the indicator swings back and forth
many times before it finally comes to rest at a definite

A MANUAL OP EXPERIMENTS IN PHYSICS

pointy which marks the position of equilibrium. Time
would be wasted in waiting for it to stop, and in any case
the indications of the moving pointer are, as will be shown,
more trustworthy than those of one which has come to
rest, because the latter may not be in the true position of
equilibrium, owing to friction. It is therefore important
to learn to tell the precise position of equilibrium, and to
practise reading with the pointer moving. We do this by
noting the extremes of its swings. In deducing the point
of equilibrium from these readings, it must be remembered

that the swings are continually
decreasing in amplitude. Con-
sequently, if X is the point of
equilibrium, and (x + a) a read-
ing of the right end of the
swing, and each swing is e di-
visions less than the last swing,
the next left swing will only
carry it to a point a — e beyond

[■ - the point of equilibrium — i. e. , to

] I a point (a? — (« — «)); and the

1 point of equilibrium is half-way

between the left swing and the
mean of the two right swings,
just before and after the left
one. A little consideration will
show that no matter how many consecutive swings are
noted, if an odd number are taken so that the last swing
is on the same side as the first, the point half-way between
the mean of all the left-hand and the mean of all the right-
hand readings is the point of equilibrium.

Sonroes of Error.

1. The scale division read must be the one directly back of the

pointer at its turning-points, a condition difficult to satisfy.

2. The vibration must not be disturbed after being once begun.
9. The "damping " of the vibration must not be too great, o(he^

wise e canpot J)e i^sfpfped to be a <x)«ptftot,

Fio. 32

PRELIMINARY EXPERIMENTS 68

Appftiatns. Chemical balance in glass-case ; small mirror.

llaiupiilatioiL Lower the balance on to the knife-edges

by means of the screw in the front of the case. Set the

balance swinging, by fanning one of the scale -pans very

gently with the hand, so that the pointer moves over not

more than four or five divisions. Close the balance-case*

Note the turning-points of the swing to the tenth of a

scale division. (If the zero of the scale is marked in the

middle, disregard the figures marked and call the mark

farthest to the left of it zero.) Note as many swings as

possible before the pointer comes to rest, taking the last

on the same side as the first. Determine the point of

equilibrium from the first three swings, and also from the

firat five, the first seven, and so on. Finally, note where

the pointer actually stops, and show how closely the several

ways of determining the point of equilibrium agree in their

result:

In order not to make an error in viewing the pointer
from different directions as it swings, place a
small piece of mirror immediately back of the
pointer ; and, as it swings, move your eye so
that the pointer always covers its refiection in
the mirror. After you have once begun to read
the swings do not raise the front of the case
until the pointer has come to rest, as a draught
of air might spoil the experiment. Take care
in starting the swing you do not set the scale-
pans moving in any other way than up and
down from the beam from which they hang.

Do the experiment twice, with the same am-
plitude of swing; and twice more — once with an
amplitude of seven or eight, and once with an
amplitude over ten. When finished, raise the ^^' ^
beam off the knife-edges by means of the screw, and leave
the case closed.

A MANUAL OF KXPfiRlMfiNTS IN PHYSICS

ILLUSTRATIOlf
Balamgi ''M 826"

TaralncpoiBU

NaSwliigB

Mmo TarQiDg.poiota

Pblnta or RmI
cftlcQlaimt

L0ft

Right

LtA

Wght

8.6

18.6

8.9

12.2

8.7 +

12.6

10.6 +

9.8

11.9

8.9 +

12.4

10.6 +

9.6

11.6

9.1

•

12.2 +

10.6 +

9.9

11.8

9.8-

12.0 +

10.6 +

10.1

11.1

9.4

11.9

10.6 +

10.8

10.9

9.6 +

11.8-

10.6 +

10.4

9.6 +

11.6 +

10.6 +

The pointer came to rest at 10.6.

Beport similarly the other experiments directed.

Qaestioiui.

1. Why must the scale-pans hang quietly on the beam to get the

point of equilibrium properly ?

2. What effect does the presence of an observer have upon a sen-

sitive balance ?
8. Is there any analogy between the vibrations of a balance and
those of a pendulum?

EXPERIMENTS

MECHANICS AND PROPERTIES OP
MATTER

INTRODUCTION TO MECHANICS AND PROPERTIES OF
MATTER

Units and Definitionfl. The only nnits made nse of in
Mechanics and Properties of Matter are the centimetre,
the gram^ the second, and others deriyed immediately from
these fundamental ones. It may be useful to give the
names and definitions of these nnits. (The names and
yalue of other mechanical units which are sometimed used
are given at the end of the Yolume in the tables.)

Qouimy

Lec^

Area.

Volume

Kan

Time

Density

ADgle.

Linear speed.. . .
Linear velocity.

Linear accelera-
tion.

Angular speed..

Symbol
DefinlDg
Equation

t
arc

radius

$=x/t

«=^/<

Unit

Centimetre, cm.
S quare centimetre,

cm.'*

Cubic centimetre,

cm.» or ca

Gram, g.

Second, sec.

Radian

Deflnltkm

Seep. 2.

See p. a
See p. a

Angle such that
ratio of arc to
radius is 1.

1 cm. per sec.

Unit speed in def-
inite direction.

Change of velocity
1 unit per sec.
If direction is
unchanged,! cen-
timetre per sec.

1 radian per see.

A MANUAL OF EXPERIMENTS IN PHYSICS

Symbol

Quuititj

Deflnliig
EqusUon

Uo\J

DeflnltioB

Angular velocity..

....

Unit speed around
definite axis.

Angular accelera-

«=«/<

....

Change of angulai

tion.

velocity 1 unit
per sec.

Linear momentum

IflV

Such a motion that
nuyn x linear ve-
locity equals 1
-^.^.,lg.mov.
Ing with a ve-
locity of 1 cm.
per sec.

Force

F^fMb

Dyne

Such a force that

if "acUng" by

itself on a body

. 1

whose mass is

m would give

•

it an accelera-
tion such that
f7ia = l— 0.y., if
acting on Ig. in
its direction of
motion it would,
in 1 sec., in-
crease the speed
by 1 cm. per sec.

Moment of inertia

/=»»•*

Angular momen*

J»

turn.

Moment of force.

L^n

Force x lever-arm.

Work and energy.

W^Fx^d^

Brg

Such an amount of
work that force
X distance of
motion in direc-
tion of force
equals 1— «.y.,
1 dyne doing
work through 1
cm.

Work and energy.

• •. •

Joule

10' ergs.

Power or activity.

p=Tr/r

Watt

1 Joule per sec.

SXPEIOMENTS IN MECHANICS AND PBOPSRTIES OF MATTER 69

Otgact of Experimenta. The experiments in the following
section of the manual may be roughly classified in two
groups: riz.^ 1, those designed to teach the student how
to measure with accuracy lengths, masses, and interyals of
time ; and, 2, those designed to teach him by actual obser-
yation the properties of matter and the laws which express
mathematically the behayior of matter under various con-
ditions.

The fundamental properties of matter are inertia, weight,
aii.d elasticity ; and it is shown in treatises on physics that,
as a consequence of these properties, matter behaves in a
definite way under definite conditions. The mathematical
formulsB which express these modes of action involve mass-
es, lengths, and times ; and so, in order to verify the laws,
exact measurements of these three quantities must be made.

EXPERIMENT 12

Otyeot To determine the linear velocity and accelera-
tion of a rapidly moving body. (See " Physics/' Art. 18.)
Oeneral Theory. If a body moves very rapidly the accu-
rate determination of the distance passed over in a given
time can no longer be made by means of a watch. The
principal methods of measuring the time in such cases all
depend upon the comparison of the intervals which are to
he measured with the time in which a standard tuning-
fork of known period makes one vibration. The chief end
of the present experiment is to illustrate this method ot

measuring time.

A block of wood is arranged
at the top of suitable vertical
ways so that when released it
will fall in a straight line with-
out twisting in any way. A
heavy tuning-fork of known pe-
riod is rigidly clamped so that
as it vibrates its prongs move

Fto. 24

KXPfi&UUilMlS L\ M£0HANIC8 AND PRUPJ&BTI£d OF MATTJfiR n

to and fro horizontally. To one of theae prongs is at-
tached a suitable stylns^ and a strip of smoked glass or
other surface upon which the stylus can leave a trace is
fastened firmly on the front of the falling block. When
the tuning-fork is set in vibration and moved forward so
that the stylus touches the prepared surface^ it
leaves a horizontal trace so long as the block is at
rest. When the block is released and falls^ the
result is a wavy line^ as shown in the figure. If
the fork is now stopped and the block drawn up
with the surface still in contact with the stylus a
straight vertical line is drawn^ marking the medial
line of the vibrations of the stylus. If the points
where this straight line intersects the wavy one
are marked P^, P^y P^, etc., it is evident that the
distances P^P^, P^P^, P^Pj, and similarly P,P^,
A^e» ^^^'> ^^^ ^^^ traversed in the time taken
by the stylus to make one complete vibration.
Therefore if the period of the fork is known, it is
possible to at once determine the average velocity
between any two of these points by measuring the
distance between them along the straight line.
Or, without knowing the period of the fork, the
average velocity at different parts of the fall can
be compared. Hence :

1. Let 2; = the distance between any two points
where the curve crosses the straight line in the
same direction, and let n =r the number of vibra*
tions of the fork marked between the points ; then, "" *"
if 7 is the period of the fork, and a the average speed

with which the distance x was traversed, e = -^

2. If X and a! are spaces at different parts of the motion
traversed in an equal number of periods,

x^ '— _?L f ?.

3. If, now, x-^y X2, x^y etc., are the spaces traversed in siic-

n A MANUAL OF £XP£RIM£NTS IN PHYSICS

ceesive mtenrals of n Yibrations each, and Si, s^, s^, etc., are
the average speedg for those interraU, then ^--^i is the
increaee in speed in an interyal nT. Hence, if a„ o^are the
average accelerations in these intervals of time.

Hence, if the acceleration is constant,

«, — «, p= ^3 — «a ss «4 — «8 =: etc- ^anT;
or, rca — a?i=;jC3 — a:9 = a?4 — argspetc. zsUfi^T*.

4. Finally, if the acceleration is nniform, and if « is the
speed at a point P, and if P', P", P'", etc., are points
passed at successive intervals of n vibrations after P, and
of, ir", of", etc., the distance PP', PP", etc., then
a;' zs:snT+ian^T^,

a/" s: 3«n T+ |«n' r", and
a?" - 2a;' « an«y», a?"'- 3a;' = 3aw»r»*

•'•a;"' -3a;'""*'
Hence the student should make the following observations :
1, Obtain the period of the fork from an instructor and de-
termine the average speed for two different intervals of, say,
three vibrations each, as far apart as possible, as in 1 above.

3. Calculate the ratio of the speeds for the two intervals
without assuming the period known. •

8. Measure the spaces passed over in successive intervals
of, say, three vibrations each and show whether the acceler*
ation is uniform.

4. Select as long an interval as can be found upon the
trace which can be divided into three intervals of the
same number of complete vibrations each, and show that
the relation proved above holds. Since the spaces are long,
the irregularities in the acceleration, due to jars and local
roughness in the ways, ought not to affect the result, and
this part of the experiment may therefore be taken as a
proof that under uniform acceleration the space passed
over in ** ^ '' seconds is a; = «^ + \a^.

EXPERIMENTS LV MECHANICS AND PROPERTIES OF MATTER 78

The outline of the process is then as follows : Three or
four traces are obtained as above, and the best two are se-
lected for measurement. Each of these in turn is placed
on the dividing-engine and the medial line adjusted paral-
lel to the cross-hairs. The positions of the intersections
are then read successively the entire length of the curve.
The readings of the odd and the even intersections on the
same curve are noted separately and form two independent
series, for each of which the above relations should be de-
daced separately.

The drawing of the medial straight line may be dispensed
with in the following manner : After the plate is removed
from the block the middle point of the horizontal trace^
made before the block started, is marked with a fine cross,
and similarly a point at the farther end of the curve, which
appears to be where the curve would cross the middle line
if it were down. The curve being adjusted so that the
oross-hairs pass through each of these points, the line of
measurement will be. along the medial line to an approxi-
mation close enough for the present experiment.

Souroes of Error.

1. The stylus always has to have more or less spring, and the

motion of its free end is therefore not quite the same as
that of the fork to which it is attached.

2. Unless the ways in which the block slides are very well mads

it is difficult to draw the block back so tliat the stylus at
rest traces a line directly down the middle of the curve, and
it is no easier merely to indicate such a line by marks at
the top and bottom as described. Hence the interval be-
tween two intersections in the same direction — as P|, P,— is
often not exactly a period.
8. The actual period of the fork is not the same as it would be
without the stylus ; and, moreover, i( varies with the position
of the latter. It should, therefore, for accurate purposes be
determined with the stylus adjusted just as it is to be used.

Apparatus. A block of wood arranged to fall between
vertical guides ; a heavy tuning-fork of low pitch mounted
horizontally on a firm stand ; string ; a box of matches ; a

74 A MANUAL OF EXP£&IM£NTS IK PHYSK^

diyiding-engine^ or elge a metre-bar supported horizontally
in two clamp-stands ; and a flat block of wood to support
the plate while it is measured. To receiye the trace, may be
used : either a long strip of glass, smoked with camphor burn-
ed in a watch-crystal or other shallow pan; or, paper sensi-
tized to show a trace when an electric current passes through
it. For this second method the front of the falling-block
must be covered with metal foil conneoted to a binding-post,
and 25 cc. of sensitizing solution (see '^Laboratory Re-
ceipts''); a storage-battery, or other source of a considerable
current, and enough wire for connections are also necessary.

The block of wood should be slightly wedge-shaped, so
that it projects forward at the top, in order to insure
a continuous trace during the fall. The stylus should
be firmly attached to the prong of the fork, and should
bend down a little at the tip. A stiff bristle is often used,
but a better one may be made out of a very narrow strip of
thin glass drawn out to a point. The width of the glass
strip is vertical when it is attached to the fork, and the
point of the stylus is not carried down by the plate as much
as if it were as flexible in a vertical direction as it has to
bo in the direction in which the fork vibrates. If the elec-
tric method is used, the stylus must be soldered to the fork
and may be a reasonably flexible needle, the rounded head
resting against the paper, or copper wire (about No. 28 in
size), used double, with the bend for a drawing-point.

Manipulation. — Smoked Glass. — Fasten the plate of glass
to the face of the falling-block with "Universal." If the
face of the block itself does not tilt forward at the top, in-
cline the glass by placing a match stick between it and the
wood at the top. Set flre to the camphor in the pan pro-
vided for it, and hold the glass surface over it, moving it
to and fro until it is covered all over with a thin, uniform
layer of soot. Tie a string to the top of the block, pass
it over the peg at the top of the guides, and fasten it so as
to hold the block at a height sufficiently great to leave it a
fall of more than the length of the glass plate, and so that

EXPKRIMBKTS IN MECHANICS AKO PROPERTIES OF MATTER 75

the Btjlns on the fork will come jnst a little above the
bottom of the plate. Place the fork in position with the
stylus close to bat not touching the plate^ and with the
prongs horizontal and the direction of vibration parallel to
the plate. (If the stand provided for the fork is not high
enough to admit of a trace the entire length of the glass
plate^ it must^ of course, be raised by blocks.)

The best way to set the heavy fork into vibrations of an
amplitude great enough to make good waves the entire
length of the glass is to pass two or three turns of common
cotton string around the ends of the prongs, cross the
ends of the string, and pull on them, so as to draw the tips
of the prongs together, until the string snaps.

When the apparatus is all ready, one observer sets the
fork in vibration, as above, and pushes it up carefully so
that the stylus just leaves a slight trace against the smoked
surface. The other stands with match in hand and burns
the string supporting the block the instant the stylus is in
position. After the block has fallen, stop the fork, and, if
practicable, draw up the plate in exactly the line it fell, so
that the stylus at rest makes a straight line down the mid-
dle of the curve. It may take several trials before a good
trace is secured. Repeat the experiment until four or five
good curves have been obtained, using the other glass plates
also if necessary.

Select the best two curves, place them in turn under the
dividing-engine, and adjust the plate so that the medial
line of the curve is exactly parallel to the path of the cross-
hairs, and note the reading of each intersection of the
curve with the axis, as described in the theory of the ex-
periment. Becord the readings of alternate intersections
in different columns, so that the difference between two
successive readings in the same column is the distance
fallen in a whole and not in a half period.

Or else the metre-bar supported in the clamp-stands
may be used for measurement. Arrange the bar in the
clamps accurately parallel to the table, with the edge verti-

7« A MANUAL OF EXPERIMEIHS IN PHYSICS

cal and at such a height that the glass plate monnted on a
flat wooden block can just be slipped nnder it without the
soot being rubbed. The height of the bar should be ad-
justed very accurately, so as to avoid rubbing the trace on
tlie one hand and the danger of parallax in the reading on
the other. (The adjustment may best be made with the
help of another plate of glass of the same thickness and
the block of wood.) When the bar is properly placed^ lay
the plate flat on the block and slide it under, adjusting it
so that the medial line is exctctly under the front edge of the
bar along its entire length. (If no line was drawn down the
middle of the curve, the ends of such an imaginary line should
be marked as described in the theory of the experiment,
and both marks should be exactly under the front edge of
the bar.) Read the exact position of the intersections to
tenths of a millimetre. This is done most accurately by
standing behind the bar, looking over the top and down along
the line of the graduations on each side of the intersection.
Record the readings in two columns, as described above.

Sensitized Paper. — The method is the same as for
smoked glass, except in the following details. An electro-
motive force of ten or twelve volts is needed. One termi-
nal of the battery is connected with the fork, the other, by
means of a long, light wire, to the binding-post on the fall-
ing-block. The paper, which should be fairly smooth but
unsized, must be freshly soaked in the solution and laid,
while still wet, flat on the foil-covered face of the block.
The metre-bar need not be supported in the clamp-stands,
but can be laid directly on the paper ; and the measure-
ments should be made as described above.

After the measurements along the medial line are made,
obtain the frequency at the fork from an instructor, and
make the four following calculations, as described in the
General Theory of the Experiment. (Do not make these
calculations during laboratory hours.)

1. The average speed during the third and the next to
the last complete period.

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 77

2. The ratio of the average speeds in the above two
cases, without assuming the period of the fork known.

3. Show whether the acceleration was uniform through-
out. If it appears not to he, state whether you think there
was a real variation ;* if so^ what caused it ; and, if not,
why is the curve not uniform ?

4. Prove the relation x = st + iat*, using each series in
each trace as described.

ILLUSTRATION

LlMIAR yUX>CITT AND ACCXLIKATIOir

April 36, 1897

Ola$9 Ftate 8moktd b^ Camphor. MecuuremefUs made on Ditiding-

9.1«

9.49

p»

9.9»

10.e9

11.69

12.89

1488

16.10

P,r

18.00

i*..

ao.84

29.88

P«

26.82

P»

2ai8

Pt,

81.06

engine

DfBUiKM Fatten In Interna ST

Inoreaae in ihia Distance between
two Saooesslye Intervale

P, -P, =ar, =1.58

P,-P,=: a?, =2.20

a-, -X, =67

P„-P,=ar,=2.90

a?, -X, =70

P„-P,=a?,=3.64

ar,-a-,=74

P,.-P,=^,=4.41

•Xt -Xj =77

A7-Pii=iPu=5.20

aru-a-, =79

Af-A.= a?i.= 6 01

a'ii-a'„=81

Pn-P»=Xu=^t9

a-u-a?u=72

A.-Pit=«it=7.28

a^iT- ^15=49

P--P..= ^..= 7.79

ar„-ir„=56

A,-P., = '.. = 8.33

ar„-a'„= 48

Frequency of fork, 70.

/. period =

= 0.0148 sec

2 90

1. The average speed during the 8d interval =-^=67 cm. per sec.

7 09
The ayerage speed during the 10th interval =-^=186 cm. per sec.

7 99

2. The ratio of the speeds in these intervals is ^-^ = 2.75.

8. The acceleration as shown in the third column varied from

81 48

AT ^ A?*' '^^^ ^'^^ ^^ ^^ uniformity was probably a real varia-
tion due to the rude guides used, though part of it may be accounted
for by the swaying of the block of wood, which made its path a curve
instead of the straight line represented by the medlnl line of the plate.

76 A MANUAL OF EXPERIMENTS IN PHYSICS

. 4. Startiog from Pi the disunoes travened in 4T, ST^ and 18r,

respectiTely, are :

PfPi^S^ r= 8.88
Pj,-Pj=:J»"= 8.88

P„-P4 = «'"= 18.97

>"^ai^_ 11.88 _^

•V-ar'" 8.87"**^

which proves the relation :« = «< + ^a<> to withhi ^ = 2%, about.

The student should give similarly his readings and deductions for
the alternate points, P„ P4, etc., and for the other trace which he is
directed to measure.

Qnoattona and Froblanui.

1. Calculate linear speed of a point on the equator of the earth at

midday and ut midnight. Rudius of orbit is 82,000,000 miles.

2. A sione is dropped over a cliff into water ; the sound is heard

after 10 seconds (velocity of sound = 88.800); find the height
of the cliff.
8. A train passes a station with a speed of 60 kilometres per hour.
On passing the next station, 2 Icilometres away, iUi speed is
40 liilometres per hour. Calculate the acceleration, assum-
ing it to be constant.

4. A train has a speed of 60 kilometres per hour. A gun is fired

from the train so as to hit an object exactly opposite the
wiudow. If the velocity of the bullet is 100 metres per
second, calculate the direction of aim. Rain is falling with
the speed of 4 metres per second ; calculate the path of a
drop on the window-pane.

5. If an acceleration is 500 in yards and minutes, find its value in

centimetres and seconds.

6. A train acquires, 8 minutes after starting, a velocity of 84 kil-

ometres per hour. If the acceleration is constant, what is
the distance passed over in the 5th second f

7. Show that, when a body is thrown upward, it has, at a height

h, the same speed, whether it is rising or falling.

8. If a body falls in a vertical circle from any point of the cir-

cumference to the lowest point, along the chord joining the
two points (or along any path), it will have the same speed
at the bottom. Prove this theoretically.
0. The driving-wheel of a locomotive is 1.5 metres in diameter;
it makes 250 revolutions per minute. What is the mean
linear speed of a point on the periphery? What la the
speed for a point on top T For a point on bottom T

EXPERIMENT 18

(two OBSKRVKK8 AUB KEQUIRED)

Olgect To determine angular velocity and acceleration.
(See "Physics," Art. 22.)

Oeneral Theory. A wheel with a flat rim is rigidly attach-
ed to a horizontal axle, which is free to turn in bearings
mounted on a platform at a height of eight or ten feet above
the floor. The axle projects over the edge of the platform

Fio Q6

and has a cord wound around it, to one end of which is at-
tached a heavy weight. When the weight is released and
falls, angular acceleration is produced in the axle and
wheel. [If a be the acceleration, M the falling mass, r
the radius of the axle, and I the moment of inertia of the

80 A MANUAL OF EXPERIMENTS IN PHYSICS

wheel and axle, la = Mgr — /, where / is the friction ex-
pressed as a moment opposing the rotation of the axle.]

If the rim of the wheel be covered by a strip of paper,
either smoked or soaked in a sensitizing solation, a trace
may be obtained upon it by means of a stylns attached to a
vibrating fork, as in the last experiment. If the fork is at
rest as the axle is revolved while the weight is being wound
upon it, a straight line will be drawn on the paper. If the
fork be made to vibrate horizontally as the weight falls, it
will trace a wavy line which crosses the straight one at every
half-period of the fork. Hence the distance between two
successive points where the curve crosses in the same di-
rection will be the distance, x, a point on the rim of the
wheel has travelled in that period. If G be the angle of ro-
tation in the same interval, and R the radius of the rim
of the wheel :

If T be the period of the fork and w the average angular
velocity :

6 X

«= ;sr=-

T" RT

Similarly, if « be the length between two crossings sepa-
rated by n periods of the curve, the average angular ve-
locity during the interval is

The paper with the trace upon it is taken oflE the fly-
wheel and measured, just as in the previous experiment.
From the measurements may be deduced properties for
motion of rotation corresponding to those deduced before
for motion of translation, and in a similar manner, Le. :

1. The average angular velocity, a», for any given interval
of an integral number of periods.

2. The ratio of the angular velocities, w and J, for two
intervals at di£erent parts of the motion.

EXPERIMRKTS IN MECHANICS AND PROPERTIES OF MATTER 81

3. The mean angular acceleration for each BuccesBiYe in-
terval of any desired length containing an integral number
of periods.

4. The relation may be proved between the angle of rota-
tion 9, in a given time t, the angular velocity O at the be-
ginning of this time^ and the acceleration during the time,
to wit :

The proof is exactly similar to that for linear motion
in the previous experiment, and depends upon showing
that

where 9', 9", and 9'" are the angles through which the wheel
rotates in intervals, nT, 2nT, SnT respectively from any
chosen instant near the beginning of the motion.

[The work done against friction in one turn may also be
found from another trace obtained in the following manner :

The fork is set in vibration as before, but the stylus is not
pressed against the paper until the instant that the cord is
loosed from the axle, so that the falling weight imparts no
further acceleration to the wheel. If the moment of in*
ertia of the wheel be considerable and the friction small,
the angular velocity will remain practically constant for
one or two turns. This maximum angular velocity, O, may
be determined upon removing the paper by counting the
number of complete periods in one revolution of the wheel.
Let this number be m. Then

.. 2if
" = ^-
The energy is then ^/O* ; and, if the wheel makes N
turns before coming to rest and the work done against fric-
tion in each turn be W:

N may be found by timing the interval, iy between the

82 A MANUAL OF EXPERIMENTS IN FHY&I08

moment of pressing the stylus against the fly-wheel and
the moment the wheel stops. For the average angular
velocity during this period is \Q :

From this value of W the moment of friction, /, may be
calculated. For /2r = W. .-. f = W\%k.\

Sources of Brror.

Same as in previous experimeDt.

[The moment, Mg^^ of the falling weight canDot be determined
very accurately, as the cord often slips upon the axle, and the
radius, r, of the axle is not exactly the arm of the moment, un-
less the place where the cord is wound is so long that only one
layer is necessary.]

Apparatus. Fly-wheel mounted firmly in horizontal bear-
ings on a platform two or three metres high^ with the axle
projecting ; weight of about five kilograms ; strong cord
two or three metres long ; four or five strips of paper, just
long enough to go around the wheel, and lap enough to hold ;
camphor, and pan for burning it, or beaker of sensitizing
solution, as in last experiment ; dividing-engine or metre-
bar and supports, as in last experiment; heavy tuning-
fork, with proper stylus and cotton string, as in last experi-
ment ; universal ; flat blocj^ of wood.

Haaipulation. Smoke the paper and mount it on the fly-
wheel with a very little universal. Be careful to see that
the paper is tight and smooth and lapped in the proper
direction, so that as the stylus crosses the joint it will pass
from the upper to the lower layer of the lap. Adjust the
fork so that its prongs vibrate in a horizontal line and at a
height such that, when the stylus is brought into contact
with the revolving wheel, the portion of the rim which it
touches will be running away from and not towards the sty-
lus. Attach the weight to the cord and wind it on the axle.
Do not tie it on, but leave the loose end of the cord so that

£XP£filM£NTS IN MECHANICS AND PROPERTISS OF MATTER 88

it will fall off when the rest has anwound. One observer
then holds the weight still at the top and lets go as soon as
the other has started the fork and placed it in contact with
the paper on which the trace is to be made. Stop the fork
without moving it the instant the loose end of the cord
leaves tl^e axle^ and stop the wheel also as soon as it has
made one or two more tarns, daring which the fork leaves
a straight trace down the middle of the previons wavy
one.

It will be fonnd that the wheel makes a number of com-
plete revolutions daring the fall of the weight ; and, con-
sequently, the vibrating stylus leaves its wavy trace several
times over the same part of the paper. There need be no
confusion, however, in measuring the intersections, since
each separate curve can readily be followed and disentangled
from the others.

[When four or five good traces have been secured in this
manner, vary the experiment so as to measure the work
done against friction. Do not place the stylus against the
paper until just as the loose end of the cord drops off, and
leave it in vibration for one or two revolutions only. Note
with a watch the time it takes the wheel to come to rest
after the stylus touches it. Measure the diameter of the
wheel with the metre -bar and that of the axle with the
vernier calipers.]

Select the best two traces obtained in the first part of
the experiment ; stretch them carefully on a block of wood,
and measure the points of intersection of the curve along
the medial line with a metre-rod, as described in the previ-
ous experiment.

[Next count the number of periods of the fork in one
revolution of the wheel on the traces made in the second
manner. Obtain the period of the fork and the moment
of inertia of the wheel from an instructor, and weigh the
falling weight on the platform scales.]

From the data thus obtained make the deductions indi-
cated in the theory of the experiment.

84 A MANUAL OF EXPERIMENTS IN PH7SICS

ILLUSTRATION

CTbe mode of recordiDg the first part of the experiment Is exactly
similar to the iUastratioD in the preceding experiment.)

Jan. 7, 1897
The work done against friction in one turn was found as follows :
r= period of fork = .0006 seconds.
/= moment of inertia of wheel = d.00 x 10*.
Seyeral traces were made as directed, by pressing the stylus against
the paper just as the end of the string became loose. The number of
waves in one reyolution of the wheel were counted in three of these,
and found to be 82.6, 82.6, 88.0 respectively ; mean. 82.7 = m.
Hence the angular velocity while the trace was made was :
^ 2ir 2x8.1416 oo a ^. ^

The time in which the wheel came to rest after the stylus was

pressed against it was 61, 68, and 62 seconds ; mean, 62 seconds.

Hence the work done against friction in-one turn is :

„ 2irO- 2x8.1416x88 «^.^ „ ^^ .^.
Tr=— T-/= js •^ ^ 10» = 7.68 X 10* ergs.

Qaastions and Problems.

1. Explain why a common hoop does not fall over as it runs along

the ground.

2. As a wheel of a carriage turns, what is the connection between

the linear velocities of its axle and its upper and lower tire?
Let the radius be 50 centimetres and the speed of the carriage
5 miles per second.

8. A man can bicycle 12 miles an hour on a smooth road; down-
ward force with each foot in turn is 20 pounds; the length
of stroke is 1 foot; driving-wheel has circumference 12 feet.
How much work is done per second?

4. Calculate the horse-power transmitted by a rope passing over a
wheel 16 feet in diameter, which makes 1 revolution in 2 sec-
onds, the tension in the rope being 100 pounds.

EXPERIMENT 14

(TWO 0B8ERYEB8 ARE BBQUIBED)

Olifflct. To determine the mass of a body by inertia. A
direct comparison of its inertia with that of a number of
standard masses. (See "Physics/' Art. 26.)

Oeneral Theory. Two bodies are said to have the same
maaSj it, when acted upon by the same impnlse, each is
giye^ the same velocity. A simple method of testing this

LJ*"^''LJ

V-.*

I^O. 27

T5qaality of mass is to suspend the two bodies from cords,
place a compressed spring between them, release the spring
and compare their velocities. As is proved theoretically
(see Experiment 15), the velocity with which such a sus-
pended body will start off is proportional to the distance
through which it swings before coming to rest and return-
ing on its path, provided the radius of the circle in which
it swings — u e,, the length of the supporting cord — is long.

86 A MANUAL OF EXPERIMENTS IN PHYSICS

The method^ then, is to suspend by a long string the
body whose mass is desired, and by its side suspend by a
string of equal length a body whose mass is known (e,g,,
any one of a set of gram weights, or a can containing
such weights) ; compress a spring between them, release it,
and measure the two swings. If they are equal, the two
masses are equal. If they are not equal, the known mass
may be replaced by another, or it may be altered by a known
amount, until the corresponding swings are equal.

8oiizo6s of Bivor.

1. Parallax is very difficult to ayoid, siDce a considerable clear-

ance has to be allowed between the pointer and the scale.

2. A correction must be made for the difference in mass of the

vessels themselves.
8. It is didcalt to apply the impulse to each can in a line paaaing
through its centre of mass; and a part of the impulse goes to
make the cans rotate. The linear momentum could, there-
fore, only be the same for each can in case the rotational
momentums were the same.
Care is necessary to make the error from this source negligibly
small.

Apparatus. Two tin cans just large enough to contain
the cylinders measured in Experiment 5 ; a spool of strong
thread; a box of gram weights, 100 g. to 0.01 g.; a spring-
clamp, such as is constantly used to pinch a rubber tube ;
a clamp-stand and short metal rod ; a metre-bar, or, better,
a scale curved into a circular arc of a radius equal to the
length of the threads suspending the cans. In order to
bring the centre of mass in each can at the same level, the
one intended for weights should have a false bottom at the
proper height. Each can should have a vertical pointer,
and three holes pierced at equal distances around the top,
so that it can be suspended by three threads and will hang
even. Each thread may conveniently have a wire hook on
the end to hook into the holes in the can.

The three threads are tied together at the top and hang
from a thread of suitable length provided with a wire hook

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 87

at the other end. The three BUBpending threads for each
can haye to be of exactly the same length or the can will
not hang eyen ; and the saBpensions for both cans mnst
also be of equal length.

Mampnlation. By means of the hooks on the suspensions
bang the cans empty from little loops of thread placed in a
couTenient position in the same horizontal line. Adjust
the height of the loops so that the pointers on the cans are
about at the level of the eye^ the observer either stand-
ing or sitting, as may be convenient. Place the loops far
enough apart so that there is just space enough between
the cans for the spring when compressed. Compress the
spring-clamp, tie it with thread, and hang it loosely on the
bar held in the clamp-stand.
The figure shows the spring compressed and released.
Adjust the whole apparatus so
that the cans rest firmly against
the flat faces of the spring in a
line joining their centres, and be
very careful that neither can is
pushed out of the position in
which it would naturally hang
if the spring were not there. Ad-
just the clamp-stand so that the
impulse to each can is delivered
a little below its centre of figure.

Place the scale horizontally directly behind the pointers
of the cans and as close to them as may be, without their
grazing it in swinging.

When the apparatus is set up, note the positions of the
pointers on the scale, being careful to avoid parallax as far
as possible. One observer places himself in readiness to
note the turning-point of the swing of one can, and the
other observer does the same for the other can. The
thread is then burned and the spring released. A few pre-
liminary trials are usually needed to adjust the scale and
the amount the spring should be compressed so as to send

Fio. :

88 A MANUAL OF EXPERIMENTS IN PHYSICS

the cans as far as possible without sending them off the
scale. The readings should be made to millimetres each
time, and tenths, if possible. Note and record the point
of rest and the extremity of the swing of the pointer on
each can, and deduce the arc over which the can is driven
by the impulse. Add weights to the can which is driven
farthest, until the initial velocities become the same.
When believed to be the same, a trial should be made
with exchanged observers.

Note the difference of mass in the cans as found in this
manner and correct the result of the final experiment.

Place the cylinder in one can and weights in the other,
changing the weights until the swing is the same for each.
As before, when the swings are nearly the same, the ob-
servers should change places, so as to lessen the effect of
individual errors. Note the weights in the can and cor-
rect for the difference in mass of the two cans.

ILLUSTRATION
COMPABISON OF TWO MASSES BT INERTIA

April U, 188T

PoiotsofRMt

Turalng-polDta

Swings

Weights in
UftOsn

Weight too QntX
or too Small

Right

Lea

Rigbt

lAiti

Right

Left

Orams

45.0

55.0

4.0

08.2

40.1

42.8

Too small

44.8

55.8

1.8

04.4

48.0

80.1

Too great

44.0

56.1

8.2,

08.2

41.7

42.1

44.6

55.8

2.6

07.7

42.0

41.0

In doubt

44.7

55.0

2.0

07.8

41.8

41.0

44.0

56.2

8.0

07.0

41.0

41.7

45.1

56.8

2.8

08.4

42.3

42.1

■

44.0

44,7

56.0
55.5

2.0
2.4

08.4
07.7

42.0
42.3

42.4
42.2

• *

In doubt

44.0

54.8

2.0

07.3

42.0

42.5

44.7

55.2

2.6

07.4

42.1

42.2

45.0
445

56.2
65.6

2.7
2.8

08.8
07.5

42.S
42.2

42.1
41.0

Nearest

44.8

55.0

2.0

07.7

41.0

41.8

Hence the mass of the right can is .7 gram greater than that of the
left, as close as one can tell.

EXPERIMBNTS IN MECHANICS AND PROPERTIES OF MATTER 89

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<S p $g S ^ S .S ^ S S ^ SS S S £2 1 .s

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-I

EXPERIMENT 16

(two OB8RRVBRB ARE RBQUTRRD)

Ol^eet To yerify the Principle of the Consenration of
Linear Momentum. (See "Physics," Arts. 27, 28, 29.)

General Theory. There are two simple experiments which
serve as illustrations of this principle : one is, when two
bodies are allowed to impinge on each other; the other,
when two bodies separated by a compressed spring are
thrown apart by allowing the spring to extend, care being
taken in each case to avoid the action of any external in-
fluence, such as gravity.

1. Impact. Two Bodies.

The law of the conservation of linear momentum states
that if two bodies impinge directly — t. e., when moving
along the line connecting their centres so that there is no
spinning after they strike — the sum of their momentums
must be the same immediately before and after impact —
i.e.,mv+ MV= mv'+ MV, where the small letters are the
mass and the velocity of one ball and the large letters those
of the other, v' and F' being the velocities immediately after
impact,!; and F those immediately before. (The coefficient
of restitution is defined as :

_ Velocity with which the balls move away from each other_ tf— V'\
"" Velocity with which the balls approach each other "~ K— « /

The simplest method of producing definite velocities is
to suspend the body by means of a long string and allow it

£XPBK1M£NTS IK MECHANICS AND PROPERTIES OP MATTER 91

Fio. 29

to swing in a vertical circle. (See '' Physics," Art. 69. ) If

the body is suspended from 0 by a cord of length (JF, so

as to be free to move in a vertical circle of this radius,

and if it is then allowed to drop from a point A

of this circle, it will have at the bottom of its path,

P, the same speed that a body would have if it fell

through the same vertical distance

¥P, That is, F'=2^ ET, But by

XP^

geometry, BP=:—z=z.' Therefore
^ ^ 20P

the velocity of the falling ball at the
bottom of its path is always propor-
tional to AP, the chord connecting P
with the point from which the ball is
dropped. Since only comparatively
small arcs of the circle are used in
this experiment, it is not necessary to
distinguish between the arc and the
chord; and it can with sufficient accuracy be said that
the velocity of the falling ball at the bottom of its path is
proportional to the number of divisions of the arc it sweeps
over in falling ; and this can be varied at will. Similarly,
if a ball is started by a blow from its lowest position it will
rise over an arc whose length is proportional to its starting
velocity.

The simplest mode, then, of verifying the law is to sus-
pend two small spherical bodies side by side by strings
of equal length ; and, leaving one hanging freely, to draw
the other one side in the plane of the strings, and then let
it fall and strike the other. The velocities are in the same
straight line — t. e., of the centres ; they can be measured,
and so may the masses. The line of centres is horizontal,
and gravity plays no part during impact.

A more complicated apparatus is shown in Fig. 30,
which may be used for the purposes of this experiment,
or in place of that described in Experiment 14.

(^^ OF THE \

UNIVERSITY I
OF

•2

A MANUAL OF EXPERIMENTS IN PHTSIGS

nokso
2. Compressed Spriko. Three Bodies.
■I" ■■ ' ■ ^ In this method^ two bodies
— ^^ suspended by strings, as in
Part 1, are kept from tonch-
ing by a compressed spring.
This spring may be held com-
pressed by a thread, and should
itself be suspended by a long
string.

If the thread is now burned,

the spring will expand and

throw the two bodies apart

with a definite Telocity, which

may be measured as in Part 1.

^-s^ I If the spring moves one way

r A^m/'^ or another, its velocity should

V^^^/^v^ also be calculated. Then, since

Fio. 31 the three bodies are at rest be-

EXPERIMENTS IN MBCHANI08 AND PROPERTIES OF MATTER 98

fore the thread is burned^ the ezpreBBion of the law of the
conservation of linear momentum is that
mi Vi + ^3^3 + WI3 V3 = 0,
if mj, m^ m^ are the three masses^ and Vi, v^, v^ the cor-
responding velocities at the instant the spring expands.
(The same law would apply to later instants if gravity had
no influence.)

In general^ the momentum of the spring may be omitted ;
and, in any case, the spring can be fastened to one of the
bodies so as to move off with it, if it is desired.

DOQioaa of Bnof.

1. Tlie line of motion of the impact may not be along the line of

centres.

2. The line of motion may not be perfectly horizontal
8. The radii of the chxsles may not be the same.

4. Care moat be taken to adjust the path of both bodies so that
they do not rub against the scale at any point.

Apparatus. — Method 1. — A support from which two
ivory balls or two cylinders of lead may be suspended so

as just to touch when
hanging at rest. Each
ball or cylinder has a
brass pointer screwed
into its lowest point,
which moves over the
divisions of a graduated
circular arc when the
ball is swinging. Each
ball is suspended by two
strings, whose length may
be so regulated that both
balls hang just above the
graduated arc, and that
their pointers move as
close as possible to this
arc over its whole length

94 A ^LANUAL OF EXPERIMENTS IN FUYSICS

without touching at any point. Further, by regulating
the length of the strings the balls may be placed closer
together or farther apart. The whole apparatus is sup-
ported on levelling-screws.

A small and a large ivory ball are needed for the experi-
ment, and also two lead cylinders or balls. The balls are
attached by tips which screw into their tops. Other arti-
cles needed are two stiff cards, about four centimetres
square ; a balance and weights accurate enough to weigh
0.1 gram.

Method 2. — ^As in Method 1, with the addition of a
spring, either a spring pinch -cock, such as is used for
rubber tubing, or a small coiled spiral spring; thread;
matches.

Manipulation. — Method 1. Case A. — Two ivory balls;
the larger at rest, the smaller dropped.

Level the stand so that the balls hang as close to the zero
of the scale as possible. By means of the screws at the
top regulate the length of the strings so that

1. The balls touch very lightly, and the centres of both
are on the same arc, parallel to the graduated circle.

2. The brass pointers move close to the circular arc with-
out rubbing as the balls swing.

3. These pointers, when at rest, are at equal distances
from the zero, and on opposite sides.

Both balls should then hang vertically at the lowest point
of their respective arcs of motion.

When both are perfectly still, read their positions on the
scale. Gall readings to left of zero minus. One observer
raises the small ball, and with one of the cards holds it so
that the pointer lies precisely on a division of the scale.
Be careful that the ball is not twisted in any way, so that
when freed it will not be set spinning. It may take sev-
eral trials before the student learns to do this skilfully,
and also before the other observer learns at exactly what
point to watch for the end of the swing of the struck ball.
To get this point precisely, he should hold a piece of card

RXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 96

edgewise against the scale, so that the pointer of the ball
can just clear the top of it. By repeated trials the card is
set BO that the ball reverses its motion almost exactly over
it. Make ten good readings of the turning-point of the
struck ball, not counting the preliminary trials. Then, in
a precisely similar way, make ten readings also of the point
to which, the dropped ball goes after impact. Always drop
the ball from exactly the same point.

Report as shown in the illustration.

C(ue B. — Proceed as in Case A, but drop the large ball.

Que O. — Use the lead cylinders, dropping either ; but
let it be the same throughout.

Weigh the balls and cylinders to within 0.1 gram. Add
the masses of the brass caps on the ends of the cords to the
respective balls. These may be learned from an assistant.
Be very careful about the algebraic signs throughout. Veri-
fy the relation that

Calculate «, the " coeflScient of restitution,^' for the ivory
and lead balls.

Method 2. — The adjustments and precautions are exact-
ly as in Method 1, and so need not be described. It is evi-
dent, however, that it is impossible to compress the spring
each time to the same amount, and so successive readings
cannot be averaged.

ILLUSTRATION

Not. 16, 1896

Masses. Ivory baUfr--No. 1, 68.42 g.; No. 8. 82.57 g.

Lead cylinders— No. 1. 400 g.; No. 2, 882g.

Right cap, 2.75 g. ; left cap, 2.50 g.
Case A. No. 8 hung on left pord and dropped from - 20.

Masses and caps— No. 1, 71.17 g.; No. 8, 85.07 g.

Zeros- No. 1. 1.20 ; No. 8. - 1.26.

.-. Arc through which No. 8 falls is 18.75.

No. 1 at rest.

Hence, e = 18.75.
K=0.

A MANUAL OF SXPERIMGNTS IN PUYSiGS

Na3

No. 1

-4625

12.00

-4.625

12.25

-4.76

12.185

-4.75

12.25

-6.00

12.00

-4.76

12.00

-4.75

12.125

-4.75

12.00

-5.00

12.00

-4.75

12.00

Mean. -4.70

12.10

•. Arc made by No. 8 after impact— t. «., F = - (4.70 - 1.25) = - 8.46
Arc made by No. 1 after impact— ».e., F = 12.10 -1.20 = 10.90.
Momentum before impact = 18.75 x 86.07 = 657.5.
Momentum after impact=(10.90x71.17)-(8.46x35.07)g=654.7.

Differences 2.8e:.4^
10.90 + 8.45

18.75

■ = 0.76.

CaseB.
CaseC.

Report similarly.
Report similarly.

Questloiis and Problems.

1. The energy of a body whose mass is m and which is moving

with a Telocity v is 1/2 me*. It is a law of nature that energy
is conseryative. Calculate the energy before and after im-
pact and account for the difference.

2. If a man is placed on a horizontal, perfectly smooth table, how

could he move himself in a horizontal direction ? .
8. Why is there no "external influence" in this experiment?

4. If density of earth is 5.56, calculate its momentum.

5. A base-ball, whose mass is 800 grams, when moving 10 metres

per second, is struck squarely by a bat and then has a speed
of 20 metres per second; calculate the impulse and the aver-
age force if the contact lasts 1/50 second.

6. Two equal masses are at rest side by side. One moves from

rest under a constant force F, the other receives at the same
instant an impulse / in the same direction. Prove that they
will again be side by side at time 21/ y»

7. When a horse drags a cart or a canal -boat, if action equals

reaction, why is not tne horse held fast 7

EXPERUIKNTS IN MECHANICS AND PROPERTIES OF MATTER 97

8. The mass of a gun is 4 tons, that of the shot 20 pounds, the

ioitial velocity of the shot is 1000 feet per second, what is
the initial velocity of the gun ? What is the effect of the
gaaes so far as momentum is concerned ?

9. A 80-gram rifle-bullet is flred into a suspended block of wood

weighing 15 kilograms. If the block is suspended by a string
of length 2 metres, and is moved through an angle of 20°, cal-
culate the velocity of the bullet,

EXPERIMENT 16

Otgeot. To show : 1. That if different forces act upon
the same body, the acceleration is directly proportional to
the force.

2. That if the same force acts upon bodies of different
masses, the acceleration is inversely proportional to the
mass. (Sec *' Physics,'' Arts. 30, 31.)

General Theory. If a body whose mass is m is moving
under the action of any external influence with an acceler-
ation a, the product ma is called the "external force," and
is taken as the measure of the external influence ; because
if, under this same influence, a mass m! is moving with an
acceleration a\

m' a*z=zina.

This fact is to be tested by experiment. The simplest
means at our command of producing forces is to make use
of the fact that a body whose mass is rw. falls freely towards
the earth with an acceleration g, a constant at any one place
for all bodies. That is, a body of mass m is always acted
upon by a force downward fw^, which is called its weight {g,
in Baltimore, is nearly 980 ; and its value in other places is
given in the Tables).

The general method is to apply different weights to the
same body and measure its acceleration, and to apply the
same weight to different bodies and to measure their accel-
erations. The instrument used is called Atwood's machine.
It consists essentially of a very light wheel, with a grooved
rim arranged to turn in a vertical plane with as little fric-
tion as possible, and set upon a tall column. A long cord

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 99

passes over the wheel and carries at its ends two cylinders
of equal mass. Neglecting the weight of the cord, the re-
snltant force acting upon either cylinder is, therefore, zero,
since the downward force of gravity is
exactly counterbalanced by an equal
upward tension in the cord, due to the
weight of the other equal mass. The
cylinders will, therefore, remain at rest
unless some additional force is applied
to one or the other ; or they will con-
tinue to move with uniform velocity
when such a velocity has once been im-
parted to them, neglecting the effect of
friction.

Forces are applied to the system in
the following manner : One of the cyl-
inders is drawn to the top of the pillar,
and a hinged platform, arranged for the
purpose, is adjusted beneath it and held
by a catch. While the cylinder is thus
supported, a small bar of known mass,
called a "rider," which projects on each
side considerably beyond the cylinder,
is placed upon it. The catch is now
pulled away, the platform drops, and
■~"^ the mass on this side moves down while
the other rises. At a distance beneath
the platform, which may be varied at pleasure, is placed a
ring, through which the cylinder can pass freely, but not
the rider. There is a second platform which may be ad-
justed at any desired distance below the ring, and which
stops the motion.

The force which imparts motion to the system when the
hinged platform is released is evidently only the weight of
the rider, while the mass moved is the entire mass of the
system — that is, the sum of the two equal masses plus the
rider. (Allowance must also be made for the fact that the

Fio. 33

100 A MANUAL OF EXPERIMENTS IN PHYSICS

wheel itself is turned. It is extremely difficult to make
this allowance unless the slipping of the cord is accurately
known.)

The system will then move under a uniform acceleration
until the rider is removed ; and after that, since the ex-
ternal force is removed, the acceleration is zero, and the
velocity remains constant until the motion is stopped by
the lower platform. In the present experiment the ring
is placed immediately above the lower platform at the
very end of the motion. If t is the time taken by the
system in moving from the starting platform to the ring,
the whole motion being under the uniform acceleration a,
and the distance from the platform to ring being x, it is
known that x= 112 at* or a =2x/t*. Therefore, by meas-
uring the distance x, and observing /, the acceleration a can
be determined. Hence^

1. To show that the acceleration varies as the force, the
mass being constant : Place two riders, one much heavier
than the other, upon the cylinder that is to move down,
and determine the a<;cel oration. Let it be a. The ex-
ternal force is the sum of the weights of the two riders.
Then leave only the heavier rider on the cylinder that
moves down, placing the lighter one upon the other cylin-
der. Determine the acceleration again and call it a'. The
external force is in this case the difference of the weights
of the two riders, while the total mass moved is the same.

Then, if mj and w, are the masses of the riders,

-7 should = -^ -, if the acceleration varies directly as

a trii — rwa

the force.

2. To show that the acceleration varies inversely as the
mass moved, the force being constant : Determine the ac-
celeration of the equal cylinders with any suitable rider.
Let it be a. Replace the equal cylinders by two other
equal cylinders, but of a mass different from the first, and
determine the acceleration again with the same rider. Let
it be a'. Then, if M is the mass of each of the first pair,

EXPERIMEKTS IN MECHANICS AND PROPERTIES OK MATTER 101

and M' the mass of each of the second, m being the mass
of the rider,

—7 should = -^TTT -y if the acceleration yaries inversely

as the mass moved.

Sources of Error.

1. Friction and ibe resistance of the air are forces opposing the

motion in each case. The actual resultant force upon the
system is, therefore, the difference between the weight of
the rider and the sum of these forces. The true statement in

the first experiment would therefore be-7 = 7— ^ \^ — ^'

where /and/' are the opposing forces in the two portions
of the experiment. But since / and/' are very difficult to
determine, tlie experiment is so devised as to make them
small, and therefore they can be omitted from the formula.

2. The resultant force has to set in motion the wheel as well as

the weights and cord. Hence, if / is the moment of ibertia
of tlie fly-wheel, a its angular acceleration, and r its radius,
a fuller statement of the equation of motion is

foroe = (2if + TO)a + — ,

la
= (2 If + to) a + -5. if there is no slipping.

Hence, in the second part of the experiment,

^ 21f' + TO + ~

=: _ . rpjjg correction in this case can be made

if /and r are known, and if there is no slipping.
S. The time enters to the square in the formula, and is, moreover,
very hard to determine, as it is quite short ; and care must
therefore be concentrated on it

Apparatus. An Atwood's machine ; two riders of differ-
ent weights, and two different pairs of cylinders ; strong
thread, or very light cord ; a stop-watch.

Mampulation. Adj ust the cord to e^^actly the right length,
so that one mass will rest upon the top platform while the

102 A makoal of experiments in physics

other just clears the floor or the base of the machine. Hang
the heavier pair of cylinders on the cord ; pass the latter over
the wheel in the groove, and replace the bell-jar covering
the wheel and its supports, if there is one. Place the lower
platform as far down as possible, and the ring at such a
height over it that the rider has just time to be lifted be-
fore the motion stops ; thus the motion under acceleration
is as long as possible. Adjust the whole apparatus so that
one cylinder rests squarely on the top platform, and, when
it falls, passes through the ring without touching. This
is best done by le veiling-screws, with which the base of the
apparatus should be provided.

1. To show that the acceleration varies directly as the
force, the mass moved being constant :

liaise the cylinder on the side of the platforms and ring,
and support it on the upper platform. Place both riders
upon it. Release the catch and start the watch the instant
the platform drops. Stop it at the sound of the click when
the rider strikes the ring. Repeat ten times.

Repeat the experiment, placing the heavier rider on this
cylinder and the lighter upon the other. Weigh the two
riders on a platform-balance.

Let the mean durations of fall be t and t'. Then, since

the distance fallen is the same, ~=-r^. This should equal

a t ^

viy -I- Wa

2. To show that the acceleration varies inversely as the
mass, the force being constant :

Use the lighter rider and tlie same cylinders, and repeat

the observations as before ten times. Repeat again with

the same rider and the pair of smaller cylinders. Weigh

the two' pairs of cylinders on a platform-balance.

a /"
Then, if t and t' are the intervals of time, -7 = rr, and,

a V

therefore, -s^ should equal -— ,^ - ^ .

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 103

ILLUSTRATION

Aprtl 16, 1897

1. To prove that acceleration varies directly as the force.

Riders Sepwated

Ridera IVigether *

Force = 11 g — l0g = 7 g dynes

Force = 17 ^+10 ^ = 37^ dynes

sio

6.0

3.0

6.0

2.-8

6.0

Hence, /« = 35.8

2.8

Hence, <'* = 8.76

5.8

3.0

6.0
5.8
6.0

*°^^ = 85.3

2.8
8.0
8.2

*°^^=8.76

6.0

8.0

6.0

8.0

5.94

2.96

•••

a ~

85.3
8.76 ~

4.08.

The ratio of the forces is

. 27

• 7

= 8.86.

jw :> X.

- J.1-^ _t_l.A. Jl 4.1 A.^

The discrepancy, which equals ^ = 4$, is in the right direction to

be accounted for by friction, whicli would tend to diminish both forces
equally, and would therefore show its effect most with the smaller forca

Futhermore, the probable error of t is about \%, and of ^' nearly
l)j(, which makes the probal)le error of a'ja about 4^, since t and t
both enter as squares.

2. To prove that acceleration is inversely proportional to mass:

Larger CylinderB
t

sio

Smaller GyliDders
t'

8.6

5.0

4.0

5.4

8.6

5.0 Hence, <• = 26.0

3.6 Hence, ^• = 13.7

5.2

3 6

^^ 2.

^•^ ^^^ = 26.0
5.0

3.8

4.0 and a- ^3^

o.o

5.4

8.6

5.0

8.6

5.10 Mean.

8.70 Mean.

104 A MAxNUAL OF EXPERIMENTS IN PHYSICS

Mass of larger cylinders is 275 grams each.

Mass of smaller cylinders is 187.5 grams each.

Rider in eacli case 10 gnims.

2Jf + f» 285 - 5«0 ^ ^ , , ", ,

^-^rr-, — = =s7v- Inverse rutio = -— = 1.96, which shows an agree-

ment fully as close as would be expected.

QueationB and Problema.

1. Calculate the tension in the rope which draws a carriage weigh

ing 1 kilogram up an incline of 80^ with an acceleration of 1
metre per second.

2. If a nation uses 40 metres as unit length, 8 seconds as unit time,

and 100 pounds as unit mass, what is the value of the unit
force in this system in tenns of dynes ?
8. A IxHly is moving with a speed of 4 kilometres per hour, what
force in dynes will bring it to rest in 5 seconds ?

4. A particle is projected upward at an angle of 80° to the hnri-

zontal, with a speed of 70 metres per second. Find the time
before the particle again reaches the horizontal. Find the
horizontal distance.

5. What pressure will a man who weighs 70 kilograms exert on

the floor of an elevator which is descending with an acceler-
ation of 100 centimetres per second ? Discuss the tension in
an elevator. rope when rising; when falling. Discuss stress
on car-couplings when the train is starting, and when it is
in uniform motion.

6. The ram of a pile-driver weighs 250 kilograms. It falls 7

metres and drives a pile 20 centimetres. Calculate reust-
ance (if uniform).

EXPERIMENT 17

Olgecl To verify the law of centrifugal motion, that a
force mria is required to make a mass m move in a circle
of radins r with a constant angular velocity ta. (See
" Physics/' Art. 35.)

General Theory. Two spheres of different masses, con-
nected hy a cord, are placed free to slide along a horizontal
rod which pierces their centres. This rod is rotated rapid-
ly aronnd a vertical axis, and the position of the spheres is
sought, in which they will remain in equilibrium, and will
not fly to one end or the other of the rod. If Wi and Wj
are the masses of the two spheres, and r, and r^ the radii
of the circular paths of their centres when there is equi-
librium, then

m^ r, w' should equal rwj r^ w',

because they both have the same angular velocity, and each
exerts on the other the force necessary to make it move in
a circle.

Therefore, since r, and r^ can be measured, mjm^ = rjrx
can be determined, and the result compared with that ob-
tained by the use of a balance.

Scrarcas of Brror.

1. Friction of the balls on the rod can never be entirely gotten

rid of.

2. The hmI must be accurately horizontal and the axis of rotation

vertical, or else gravity will tend to move the balls one way
or the other.

106 A MANUAL OF EXPERIMENTS IN PHYSICS

8. Care must be taken not to rock the apparatus while rotating it.
4. The wire or cord connecting the masses has inertia also, and

an excess of length of it on either side of the axis aids ilie

tendency to move in that diroction.

*^^^^

Fio. 34

Apparatus. A whirling -tabic; two steel L-sqnares; two
wooden clamps ; a metre-bar ; a large yemier caliper is also
convenient in measuring the diameter of the spheres, though
the metre -:bar and L-squares can be used. The fly-wheel
of the whirling- table is rotated by means of a hand driv-
ing-wheel about an accurately vertical axis. A frame is
clamped rigidly to the axle of the fly-wheel, and carries a
stiff, straight, horizontal brass rod, on which two wooden
spheres are free to slide. The two spheres are connected
by a fine brass wire carrying a pointer, which moves over
a scale fixed to the frame, parallel to the brass rod. The
apparatus is so adjusted that the brass rod is accurately at
right angles to the axis of rotation ; and, therefore, if the
axis is exactly vertical, the force of gravity has no effect
upon the motion of the two spheres.

Hanipulation. Clamp the apparatus firmly to a table iu
a good light, taking care to make the axis of rotation ver-
tical. If the point where the axis of rotation meets the
scale is not already marked on tlie scale, determine it by
rotating the frame rapidly, and noting what point remains
perfectly at rest. A good way to test this is by making a
small pencil dot at the point which seems to the unaided
eye to be at rest. If, on rotation, the point makes a lit-
tle circle, rub it out and try again, until the true posi-

EXPERIMENTS IN MfiCllAKlCS AND i^kOP^KtlfcS OF MAtTEft 107

tion of the axis has been determined to the tenth of a
millimetre.

Pull the spheres apart, so that the wire connecting them
is stretched and both are free to move either way. Note
the position of the pointer on the scale to the tenth of a
millimetre. Rotate the apparatus rapid ly, and note which
sphere flies out. Call one sphere mj, the other t/ij, and
record the reading of the pointer just made under a col-
umn marked vi^ or TWg, according as it is mj or wjj which
flies out ; this is shown in the illustration. Make another
trial with the spheres moved so that the one that flew out
moves in a smaller circle, and record again in the appro-
priate column the reading of the pointer before rotation.
By similar successive trials the position of equilibrium is
soon found. When it has been apparently reached, test by
finding how much either way the spheres may be moved
without affecting the equilibrium. If the friction of the
rod is small, the place of equilibrium ought to be very well
defined. If it is not, note the point where w, just moves
out, and the point where mj just moves out, and call the
trnejwsition a point half-way between.
Repeat the determination five times.
By noting the reading of the pointer, its distance from
the axis may be at once calculated. It then remains to
measure the distances along the wire from the centre of
each sphere to the pointer. This may be done by measur-
ing the diameter of each sphere and the distances from the
pointer to the farther sides of the two spheres.

^Q-

Fio. 86

In the diagram P is the pointer ; Li and L^ are the outer
sides of the sphere ; 0, and Og* their centres. The quan-
tities to be measured are PL^ and PL2 (and Z, Z/g as a

108 A MANUAL OF EXPERIMENTS IN PHYSICS

check)^ and the two diameters. From these measure-
ments and a knowledge of the distance from the axis to
the pointer P, if P is the reading of the index when there
is equilibrium, r, and r^ may be calculated.

To measure the distances PL^, etc., a caliper may be
formed of the metre-rod and the two L-squares, and used
as follows :

Place the L-squares on the metre-bar so that the thin
sides of the Z's form two parallel jaws at right angles to
the bar, the distance between which is easily varied by
sliding one or the other along the bar. While one ob-
server holds the spheres apart so that the wire is stretched
tight, the other closes the caliper made as above upon the
outer extremities of the diameters of the spheres which con-
tinue the line of the wire — i. e., L^ and L^ in the figure. Be
very careful that the metre-bar is accurately parallel to the
desired length, and that the edges of the L-squares are
accurately perpendicular to the bar. Note the distance,
ZiZ/2, thus found; and, as the measurement is extremely
difficult, repeat at least live times. Similarly measure
Pi, and PZ/2, taking five readings of each ; an^ if
PL^ + Pi/2 is not equal to Z, Zg, take the mean as the
correct length of PZ, + PL^ and divide the error evenly
between the two terms. This gives PL^ and PXj. L^ 0,
and Z2 O29 the radii of the spheres, may most easily be
found by means of a large vernier caliper, great care being
taken to measure the diameters as close to the brass rod as
possible. Five readings of each should be taken, and there
is no object in reading the vernier closer than 1/10 mm.
If no caliper large enough is available, determine the dis-
tance from P to the near points of the spheres just as
PLi and PZ2 were determined, and thus find the diame-
ters. Be careful to measure all the lengths in the same
unit. Having determined r, and rj, calculate the ratio of
nil to m2, and get from an instructor the true value as meas-
ured on a balance.

EXPERIMENTS IN MKCHANICS AND PROPEKTIES OF MATTER 109
ILLUSTRATION — Vkrification of Law op Crntrifugal Motion
AxU found to be 51.8. ^'^' *• "^

Sphere mi Moved Out

4th

5ih

27.0

30.5

80.0

31.0

81.2

28.0

80 8

81.0

31.3 (?)

81.2 (?)

80.0

80.9

81.1

31.2 (?)

81.1 (?)

81.0 (?)

DO motion

81.2

81.1

80.0 (?)

80 5

81.3 (?)

31.2 (?)

80.0

30.8

....

31.2

81.2

no motion

80.9

....

81.3

no motion

....

81.0

....

no motion

....

81.1

....

no motion

....

Sphere m^ Moved Oat

I8t

4tb

Stb

82.0

81.5

31.8 (?)

32.0

31.5

1 ....

81.2

« • • •

81.6

31.4

1 ....

81.1

81.4

31.3

81.6

81.0

....

81.8 (?)

81.2

31.8 (?)

80.9

81.6

31.4 (?)

81.1 (?)

81.4

no motion

31.5 (?)

81.4 (?)

81.1
no motion

....

81.5

81.4

....

818

....

no motion

....

no motion

....

Readings for Eqailibrium

Hence, Distance of Pointer
fh>in Axis

1H...81.2
2d... 80.9
8ci...314
4ili ..81 3
6th... 81.1

20.1
20.4
19.9
20.0
20.2

Mean :

20.1

A MANUAL OF EKPERIMENTS IN PHYSICS

Dlmonsions of Apparatoa

LxL^

PLx

/•La

PL,^PI^

PL.

PL,

DUunetera

mi «,

471.2
471.5
471.3
471.4
471.1

871.1
371.3
371.2
871.0
370.9

371.1

99.8
99.7
99.9
99.9
99.6

99.8

meaD
value,

mean
value,

61.3 103.4

61.4 103.2
61.3 108.3

61.5 103.5
61.2 103.1

471.8

470.9

99.9

371.2

61.3 103.8

Wlience, radii are
TJ), = 30.6.

_ A, 0, = 51.7.
Tt=OP-\-PL^- /., 07=20.1+99.9-80.6 = 89.4.
ri = Pln- MA - a/' = 37M - 51.7 - 20.1 = 299.3.
/.^ = ^ = 3.82.
By the balance mjmi = 3.29.

QuestionB and Problems.

1. Why is it uot desirable to read the Ternier on the vernier cali-

per to the utmost accuracy in measuring the diameters of
the spheres?

2. If, in the experiment, the frame were accidentally tilted so

that the rod sloped downward from the larger to the smaU-
er sphere, would the ratio found be the true ratio of the
mksses, or smaller or larger? Why?

3. Is there any such force as " centrifugal force "?

4. If, while the frame is revolving, the cord or wire were cut,

what would happen? Why?

5. Deduce the tension of the wire just as the balls begin to move.

What is the linear speed of each ball at this moment ?

6. Prove that when the spheres are in the position of equilibrium,

?W|tJi+ w,r,= 0, where v^ and v^ are the linear velocities.
Can you give any reason why this should be so ?

7. A pail of water, whose mass is 1 kilogram, is swung in a ver-

tical circle r = 10 centimetres. What is the tension at top
and bottom of path, if the angular velocity at top is 5 ? How
many turns por second suffice to keep the water in the pail ?

8. A skater descril)es a circle of radius 10 metres, with a speed 5

metres per second. At what angle must he be inclined to
tlie vertical ?

EXPERIMENT 18
(two obserybbs are rbquihbd)

Olgect. To verify the laws of harmonic motion. (See
"Physics," Arts. 21, 25, 51.)

General Theory. Harmonic motion is defined as being
such that the acceleration is always towards a fixed point,
and varies directly as the displacement from that point.
Thus, the longitudinal vibration of a spiral spring is har-
monic motion, because the acceleration varies directly as
the elongation of the spring, and is always in such a direc-
tipn as to tend to bring the spring back to the position it
would have if not vibrating — i, e., to the position of equi-
librium.

Again, the rotational vibrations of a flat -coiled spring,
such as a watchspring, are harmonic, because the angular
acceleration varies directly as the angle of twist, and is al-
ways in such a direction as to tend to bring the spring
towards its position of equilibrium.

If the displacement from the position of equilibrium is
called X or 0, according as it is a distance or an angle; and
if the acceleration is called a or a, according as it is linear
or angular, the condition for harmonic motion is

linear a=z—cx)
angular a= —cO)

where c is some constant depending upon the inertia and
stiffness of the vibrating system. It is easily proved also
that the vibrations of a system in harmonic motion have a
constant period, ,

112 A MANUAL OF EXPERIMENTS IN PHYSICS

T=2,^L.

which is independent of the amplitude if it is small.

I. Harmonic Motion of Translation. — Since the acceler-
ation at any instant is equal to a constant times the dis-
placement, az= —ex, the force of restitution must be pro-
portional to the displacement also, because force varies as
the acceleration. In particular, consider the longitudinal
vibrations of a spiral spring under the influence of gravity.
Let the spring carry a body whose mass is M, and let its
own mass be m; let it be suspended vertically. If the
reading of a pointer on the spring is 0 when the spring
is at rest, then, when in its vibrations the pointer is at a
point X below 0, the force of restitution upward is propor-
tional to Xy F=z Kx. Consequently, if instead of allowing
the spring to vibrate, a force downward is applied so as to
produce the displacement x, the force applied must be Kx.
Therefore, if any force F produces a displacement a;, F
should equal Kx, where K is a constant for all displace-
ments, and measures the ^ ^ stiffness " of the spring. In other
words, the displacement is proportional to the stretching
force ; this is called *' Hookers Law"; and, conversely, any
system which obeys Hooke's law will perform harmonic vi-
brations if it is disturbed from its position of equilibrium.

It may be proved by actual experiment that the dis-
placements of a spiral spring, a pendulum, a tuning-fork,
a stretched string like a violin string, etc., are proportional
to the force, and hence their free motions are harmonic.

Since, then, F=z Kx, the acceleration may be determined
if the mass moved is known. In the above case of the spiral
spring of mass m carrying a body of mass M, it may be proved
by theory that the effect of the inertia of the spring is exact-
ly as if the mass if were increased by 1/3 m, if the sprixig is
evenly wound.

Hence, F^ {M+ 1/3 m) a,

&Qd. a = X.

M+AI'Sm

EXPERIMENTS IN HBOHANIOS AND PB0PBRTIS8 Of UATTEB 118

Henoe, the constants, as defined above (a zs — ex), equals

J/-H/3W
Henoe the period^

T=2t^

Jf-hX/3w

K may be f onnd by directly meaauring the displacement
prod need by a given force, as described above ; T and M
and m may also be measured, and this law may be verified.
The important facts are :

1. A system obeying Hooke's law makes harmonic vi-
brations.

2. The period varies as the square root of the mass
moved.

II. Harmonic Motion of Rotation. — The angular accel-
eration a £= — e;0; consequently, in order to turn the system
through an angle 6, a moment L must be applied, such
that

where k is the same for all angles and measures the stiff-
ness of the spring. It may be easily measured by noting
the angular displacement produced by a given moment.
Conversely, any system satisfying this condition will make
harmonic vibrations — 6.^., a.watchspring, a compass-needle,
a twisting wire, a vibrating balance, etc.

The angular acceleration is equal, to the moment of the
force divided by the moment of inertia — t. e., Z=: /o.

kB k /T

Hence, a = -j- ; and, therefore, c = -j. But Tz=. 2iri/ - j

hence, r=2ir\/4--

The important facts in this case are :

1. A system obeying Hookers law makes harmonic vi-
bration.

2. The period varies as the square root of the moment
of inertia. Therefore, we have a convenient method of
comparing moments of inertia.

114 A MANUAL OF EXPBRIMENTS IN PHYSICS

In the following ezerciBe the linear vibrations of a spiral
spring and the rotational vibrations of a fiat-coiled spring
will be studied.

The following facts will be verified by experiment :

1. The linear displacement of the spring is proportional
to the stretching force.

2. The vibrations have a period which is independent of
the amplitude, provided it is small.

3. The period of vibration varies as the square root of
the mass moved.

4. The angular displacement of the flat spring is propor-
tional to the moment.

5. The vibrations of the flat spring also have a period
which is independent of the amplitude, provided it is small.

6. The period of vibration varies as the square root of
the moment of inertia /=2mr'; and so it can be varied
by placing different masses at the same distance from the
axis, or by placing the same mass at different distances.

Souroes of Error.

1. If the wire is stretched too much, Hooke's law is not obeyed,

and hence the theory does not apply.

2. If there is too much frictioQ, either external or iniemal, the

vibrations die down rapidly and cease lo be isochronous.
8. Displacements must be talsen great enough to make the prob-
able error of one setting small in comparison with the entire
displacement.

Apparatus. For translation : An evenly wound spiral
spring; a metre-rod; two heavy weights; two light ones;
a watch. (In the laboratory of the Johns Hopkins Uni-
versity the spring used is one which is commonly used to
close heavy doors. Its mass is 450 grams ; and the heavy
weights used are 5 and 6 kilograms, the light ones 400 and
600 grams. The stiffness is such that 300 grams produces
an elongation of about 1 centimetre.)

For rotation : Special apparatus, as shown in Fig. 36. It
consists of a fiat coiled spring in a horizontal plane, one end

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 115

Flo. 36

fastened to a fixed support, the other to a vertical axle which

is free to rotate, and to which are
fastened a horizontal wheel and
a horizontal rod carrying sliding
weights ; cord ; weights ; pulley.
Manipulation. 1. Translation. —
Weigh the spiral spring and sus-
pend it vertically from a fixed sup-
port ; place back of it the metre-
rod, and fasten a fine wire to the
lower end of the spring at right
angles to it, so that it may serve
as an index along the scale. Hang
a heavy weight from the spring, taking care to see that
the weight is great enough to separate the spirals; note
the reading ; add a small weight, m^\ note the difference in
reading, Aj ; add another weight, m^ ; again note the differ-
ence, Aj. Then, if Pz=: Kx,

^^ should equal ^r^-

Calculate JT.

Do the same, using the second heavy
weight in place of the first. Calculate K
again. It should be constant.

2. With either heavy weight hanging
from the spring, set it vibrating vertically y
and measure the period, as in Experiment
1, at intervals of 50 vibrations, while the
amplitude slowly dies down.

Measure the weight of the hanging mass
on a platform balance.
Call it M^, Call the period Ty

3. Do the same with the second heavy
weight. Call its mass M^ and the period
Ti- Then, if m is the mass of the spring

itself, r,/7i should equal y^^^±l!l

1/3 in
1/3 m

Fia87

11 A A MAIOJAL OF EXPERIHEKTS IN PHTSIGS

4. Rotation. — By means of the string, pulley, and peg in
the fixed horiaontal wheel of the rotation apparatus, apply
a small weight to the wheel, thus producing a moment
around the axis. By means of a circular divided scale
measure the angle of displacement. Call the weight 971,
and the angle 6| ; then, if r is the radius of the fixed wheel,
7n^gr = *6i, if L = ke. mj should be great enough to make
0, large in comparison with the error of setting.

Apply a difiFerent small weight, ms, and measure the total
angular displacement, 63. Then, (m,+ Wj) gr^zkB^.

Calculate k. It should remain constant.

Add other wei);hts, and measure the displacements.

Plot in a curve the angular displacements and corre-
sponding weights.

5. Clamp equal sliding weights on the horizontal rod at
equal distances from the axis. Set the system in small vi-
brations, and measure the period as the amplitude dies
down. It should remain very nearly constant. Let the
masses be M^ each^ and their distances from the axis r,.
Call the period of vibration 7\.

6. Add equal sliding weights, M2, to each side of the rod^
and make the average distance of the whole mass, M^-^M^,
from the axis the same as before, r,. Measure the period
of vibration ; call it T^. The moment of inertia in this vi-
bration is greater than before by the amount 2Jfa^i*; con-
sequently, the period T", should be less than 7\. (If the
moment of inertia of the fixed cylinder and of the spring

itself is /', then T^/Ta should equal \J ^^^^ _^j^^^ ^, _^ j,.

An instructor should know the value of /'.)

Again, remove the two weights Jf,' ^^^ clamp the two
weights Ml at a difi!erent distance from the axis, r,. Meas-
ure the period T3. If rj > r^, T^ > T,, because the moment
of inertia has increased.

(The quantities Jf, and M2 in this and the previous sec-
tion have no connection with those denoted by the same
symbols in sections 2 and 3.)

EXPERIMENTS IN MECHANICS AND PROPERTIES OF JUTTER 117

ILLUSTRATION
I. — Hahmonio Motion of TRiHSLAnoN

1. Mass Position of Pointer

5869 g. 85.75

+ 200 85.11

+ aOO 84.47

T
3. First 50 vibrations, 0.85 sec.
Second 50 vibrations, 0.84 "
Third 50 vibrations, 0.85 "
Fourth 50 vibrations, 0.84 **
Mean, OsS "
8. Mass of spring, 468 = m .*. Jm =1548.
J/i = 5869; ir,+im = 5528.8; ^1 = 0.845.
M^ = 5951 ; jtf,+ im = 6105.8 ; T, = 0.885.

Elongation

0.64
1.28

\TJ M,+im

IL — Hahmonio Motion

Rotation.

Grams attached to Pulley

Angular readings

86.6'

+ 20

+ 4.6

+ 20

+ 4.65

+ 20

+ 4.5

+ 20

+ 4.75

etc.

First 10 vibrations. 1.68 sec. r^ =

:10

cm.

Next 15 vibrations, 1.70 "

Next 20 vibrations, 1.69 "

Mean, 1.690 "

Mass

M, + M,

2.005 t

1.690

2.182

Qnestioiia and Problems.

1. How would you prove experimentally that the vibrations of

a pendulum are harmonic ?

2. Draw analogy between mass and moment of inertia.

8. How could you determine the moment of inertia of the ap-
paratus itself in the rotation experiiVient if the moment of
inertia of the two masses m^ at distance r^ is 2miri' ?

EXPERIMENT 19

Olgect. To verify the law of moments — yiz., that the
proper definition of a moment around an axis is the prod-
uct of the force by the perpendicular distance from the
axis to the line of action of the force. (See " Physics/*
Art. 43.)

General Theory. The simplest method of verifying this
law is to secure equilibrium of an extended body by three
forces, and measure the moments as defined above. If
' the definition is correct, the algebraic sum of the mo-
ments should equal zero.

Thus, if a board is pivoted at P, and if, by means of two
strings attached to nails at N^ and JV^ forces F^ and F^ in

the plane of the board are
applied so as to tend to
turn the board in opposite
directions around the pivot,
there are only three forces
acting on the board (if the-
board lies in a horizontal
plane, or if the peg passes
through the centre of grav-
ity of the board in case it is
vertical) — viz., the two, jP,
and F29 and the reaction of
the pivot-peg. Taking mo-
ments around P, by the
above definition the mo-
ment of jPj equals F^ l^\ that

Fio. 38

EXPERIMENTS IX MECHANICS AN d^ PROPERTIES OF MATTER 119

otF^^F^lz; ^^^ ^^ ^h^ forces Fi and F2 are such that the
board is in equilibrium, then Fi 7| and F^ J, should be nu-
merically eqaal. This may be verified by direct, experi-
ment. The moment of the reaction of the pivot -peg
around the point P is of course zero.

Souroes of Brror.

1. It is quite difficult to measure the perpendicular distances

{| and l^.

2. In whatever way the board is supported or suspended, friction

always enters as an indeterminate force, though by proper
care it can he made small.

Apparatiu. Two spring balances ; cord ; two metre-rods ;
an L-sqnare ; a nail^ or a heavy weight with hook.

lUnipiilation. Pierce two small holes through the metre-
bar at points near its ends ; pass strings through each and
make loops ; support the metre-rod on edge at its middle point

FiO. 39

Phj means of a pivot of some kind^ a knife-edge resting on
a stool, or a string joined to a high support above. Join the
two loops at the ends Ifi and N2, to some nail below, 0, by
means of* strings, putting a spring -balance between each
loop and the nail, so as to measure the forces. (Another
convenient method is to fasten the strings OiV\ and ON2
to a heavy weight, which may be moved along the table

ISO

A MANUAL OF EXPERIMENTS IN PHYSIOS

below, thus altering the forces.) This may, perhaps, be
best done by joining Ni to 0 (through the balance), and
then twisting the other cord 0 iV, around the nail at 0
until the rod has a suitable position. It is best to make
the forces as large as possible, and to put in the balances
with their hooks down.

Measure the forces Fi and F^ as recorded on the bal-
ances, taking care to avoid friction of the scale -pointer
against the scale, and to place the balances as accurately
as possible in the lines ONi and OiV^.

By means of the L-squares and the other metre-rod, meas-
ure the perpendicular distances li and 2, from P to NiOsxid
N^O. Do this by placing one arm of the square along the
metre-rod, the other along the string ON^ (or ON^i and slid-
ing the two along the string until the metre -^ rod passes
through P. Be sure that the strings are straight between
0 and N^ and N^, and that the edge of the metre-rod passes
accurately through P.

Change the length of the string ON2, thus altering the
forces and lever -arms, and measure the quantities again.
Do this three times in all, turning the metre -rod over in
some experiments.

Show that i^i/i = /; /g-

ILLUSTRATION

Law of Mombmts

FiT9i Position

1
Foroea i

Lever-anng

Moments'

/•.

r.ix

^.J.

8A
8A

43.2
42.6

46.7
45.2

189.7

189.1

Me8ii,8T«,

SMr

42 35

45.46

DifFereoce, .6 or ^ of 1%,

Second Pi^iition, etc.

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 121

Qnastioiui and Problems.

1. Wby were moments taken around P, and not around Ni or N^,

orO?

2. Wbat effect does any bending of tbe rod bave t

3. Wby is it advisable to use large forces ?

4. Wby does friction affect tbe result when moments are taken

around a pivot ?

5. If it is wisbed to upset a tali oolumn by means of a rope of

given lengtb, pulled from tbe ground, where sbould it be
applied ?

6. A uniform pendulum-rod is pulled aside by a force applied

horizontally at its lower end equal in amount to one -half
tlie weight of tbe wbole rod. Calculate the angle which
the pendulum makes with the vertical when there is equi-
librium.

EXPERIMENT 20

(TWO OBSBRVEBS ABB BBQUIBfiD)

Olijeot To verify the laws of equilibrium of three forces
acting at one point. (See '' Physics," Art. 60.)

General Theory. If a point P is held in equilibrium by
three forces; the conditions are that if the three forces are
added geometrically they form a closed triangle ; or, ex-

Fio. 40

pressed in other words, the sum of the components of the
forces resolved in any direction must equal zero. In par-
ticular, if two of them are at right angles to each other,
and if the third makes an angle ^ with the line of one of
the others, as is shown in the figure,

Tsine ^--Wz^O,

r cosine ^- 5=0.
This particular arrangement of forces may be secured
easily in the laboratory by hanging a weight from a point

£XP£RI]IBNt8 IK MfiCHANiCS AND PR0P£&T1£S OF MATTER Ith

and balancing it by two forces in the directions T and B.
These may be measured and so may ^ ; and^ consequently,
the Uw may be verified.

A second perfectly general method is to tie three strings
together at a point ; fasten a spring-balance to each string ;
pall them in different directions, and register the forces,
both in amount and direction, when the point is in equi-
librium. A good way of doing this last is to place a sheet
of paper behind the strings, lay off the directions of the
forces on this, and construct their sum graphically.

Another method (see ^^ Physics,'' Art. 36) is to pass a
cord over two pulleys which have horizontal axes and are

in the same plane ; suspend
a weight from each end, and
looping a third weight oyer
the string between the two
pulleys, note the directions
and the amounts of the forces
which hold in equilibrium the
point where the third weight
is fastened. This may best be
done, as above, by a graphical method.
A full description of the first method is given below.

Fkail

Sonroea of Bnror.
1. It is diiBcalt to measure d, or its cosine or sine.
3. Great care must be taken to keep B perpendicular to W.
8. The angles must be kept constant during the measurements.

Apparatus. Two spring-balances ; a weight of about two
kilograms ; a stick about forty centimetres long, with a
nail-head at each end ; a ball of twine ; a steel L-square ; a
metre-rod.

Hanipulatian. Tie a piece of twine to the weight and de-
termine the weight of the latter by hanging it on a spring-
balance. Tie one end of a short piece of twine to a nail,
or through a hole in a vertical wall or frame (as shown at A
in the figure). Tie the other end to the ring of the spring-

l%i

A MANUAL OF EXPERIMENTS IN PHYSICS

balance^ leaving about ten or fifteen centimetres of string
between the balance and the nail or hole. Tie the weight
to the hook of the balance by a string nearly a metre long.
Rest one end of the stick against the side of the wall or
frame. Loop the twine^ which carries the weight, once or
twice around the nail at the other end, at such a point
that the stick will stand out almost at right angles to the
vertical wall or frame. Fi-
nally adjust the whole so that
the nail-head on w:hich the
stick rests against the wall
does not slip, and so that the
stick is exactly at right angles
to the wall, as may be shown
by the square.

There is equilibrium at the
point P between W, the weight
acting vertically, T^ the tension
of the cord acting along FA
and measured by the spring-
balance, and B, the resistance
of the prop acting out from
the wall along BP,

Bead the spring-balance accurately. Measure as accu-
rately as possible, by means of the metre-bar, the lengths
of the sides of the triangle PBAy or of a triangle similar
to it. In doing this, remember that A is the point where
the line PA (produced if necessary) meets the vertical
wall ; and B is the point in the line PB vertically below
A, while P is the part of the nail around which the twine
is wound. Care should be taken to measure a straight line
precisely between these points. The line PB should be at
right angles to the line AB.

When these measures have been made, attach a string to
the hook of the second spring -balance and loop the other
end around the nail P, taking great care not to disturb
anything in so doing. While one observer keeps his eye

Fio. 42

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 186

on the index of this balance and pnlls ont with gradually
increasing force exactly in the line BP extended, the other
watches the nail on which the stick is pivoted and calls
"Read" the moment it is pulled away from the wall.

The reading of the second scale at this moment shonld
be exactly equal to R. The stick will drop before the
pressure against the wall is quite zero unless it is sup-
ported in some manner. This may be done by holding
the steel L-square in the angle of the wall and stick, be-
low the stick, thus keeping it truly horizontal until it is
pulled away. The friction is diminished if a bicycle-ball
is placed between the stick and the L-square.

The two balances shonld then be compared to see if

they give the same read-

^ "^"^ >OrS-B!!gLZ]<? mgs for the same forces.
pjo. 18 This may be done by

hooking the two together and seeing if they agree when
pulled apart.

Repeat the experiment three times, making the propor-
tions of the triangle FBA different each time.

If the suspended weight is not very large in comparison
with the weight of the horizontal stick, a correction must
be made for the latter. If IT, is the weight of the stick, its
effect is exactly the same as if the stick had no weight, and
the weight W were increased by ^W^, This is evident by
taking moments of all the forces around the point B.

1. Remembering that AB is vertical and BP perpen-
dicular to it, calculate from the dimensions of the triangle
the vertical and horizontal components of T^ and show that
they, together with R and FT, fulfil the conditions of equi-
librium. Report as below.

2. Prove graphically that the forces are in equilibrium,
using a scale of not less than 1/4 inch to the pound.

NoTS. — On the U»t of Spring-balanca to Mtatwrt Forces other than those
Acting Vertically Downward on the Hook, — A balance is so graduated as to
give correctly the weight of an object hung upon its hoolc. If the weight
hung upon the book is W^ the reading of the balance is W\ but the fore*

126 A MANUAL OF EXPERIMENTS IK PHTSIOS

acting upon the balance ii It^-hA, where A is the weight of the hook, index,
etc., of the balance itself. If, then, the weight were suapended by a cord
orer a frictlonleaa palley, and held there by the spring-balanoe placed
horizontally, the weight of the boolc, etc., would no longer act on the
spring, and the reading would consequently be IT— A. If the spring-balance
were now carried downward until it pulled rertically down, the hoolc, etc.,
would not only not weigh themseWes on the spring, but would counterbalance
A units of the weight W besides. The reading of the balance would, there-
fore, be W—2h, If i^i, /?^and /?, are the readings of the baUince in the
three cases, W= R^r=B^ + h — R^'\-ih. A little consideration will show
that when the direction of the cord awatf from the hook makes an angle 5,
with a direction vertically downward, the tension of the cord is
ig + (l~cosO)A
Hence, in all measurements of forces with a spring-balance, turn the hook
of the balance towards the force to be measured, and add (I — cos 0) A to
the reading. To find A for a given balance, suspend the ring from a nail,
hang a weight on the hook and read. Turn the balanoe wrong side up, put
the hook over the nail, and, hanging the weight from the ring, read again.
The difference between the readings is S— 2A, where_B is the weight of the
whole balance, which should be found by weighing it on another. This can
k>e8t be done by first weighing the balance plus a weight on the second
balance, and then the weight alone, ainoe a spring-balance often doet not
measure very small forces accurately.

ILLUSTRATION

Get SQ^UM

To Ybbitt the Laws or EQUiLiBRnm or Tbbu Fobcis Acnvo at thi

Sami Poiht

Balances nsed, No. 11 in 27. No. 14 in SF.
Correction for weight of hook in No. 11 :

Weight suspended from hook = 5.126.
Weight suspended from ring = 6.25.
.•.5-2A = 0.1261bfl.; A A = 1/2(5-0.125).
Weight and No. 11 suspended from hook of No. 14 = 5.50
Weight alone suspended from hook of No. 14 = 5.125
.-. Weight of No. 11 = 5 =0.875

.-. A = 1/2 (0.875 - 0.125) = 0.125.
Hence the correction to No. 11 used in the line XP which makes
an angle W-» with the vertical =+ 0.125 (1 - sin 3).

The weight of the liook No. 14 was similarly found to be = + 0.125.
Hence the correction for the reading in No. 14 in the line SP, where
it makes an angle 0 = 90° with the vertical, is -.

+ 0.125(1 - cos 90) = + 0.126.

EXPERIMENTS IN MEGHANIGB AND PROPERTIES OF MATTER 127

Tbe two balances were compared by pulling them against each
other, and the readings on both were found to be the same exactly.

The laws to be verified are :

1. Tsin d=zW,

2. TcoB ^ = R

Fio. 44

Leofftb of Sides in Centimetres

A u. of
Kzperinieiits

.•.Bin,*

86.1

42.5

74.9

.870

71.2

42.4

57.2

.808

79.8

42.8

67.1

.846

G08 dr

.494
.595
.588

Xaof
Rxpert-

1
2
8

T(ob«.)

llML

JKoba)
lbs.

r^corr.)

B,^,

.*. r COS 5

Iba

6.875

2.76

689

2.875

2.90

6.875

8.625

6.40

8.75

8.81

6.000

8126

6.02

8.25

3.21

, T line ^
lbs.

5.12
5.14
5.09

5.117roean
By actual weighing on the spring-balances W=z 6.125 lbs.
Greatest deviation of R from 7 cos ^ i» 1^^.
Greatest deviation of W from Tsin ^ is .6^.

Qaastioiui and Problems.

1. In what unit are the forces expressed in the above illustration*

and what relation does it bear to the G. G. 8. unit ?

2. What would be the result if the cord were not fastened at P,

but could slip ? What would be the value of T in terms
ofir?

1S8

A KAKUAIi OF £!XFBIIIMSNT3 IN PHTa|[C»

8. Why ahould a apring-balanoe be hvmg from a nail or fixed
support, if possible?

4. Prove that when a kite is flying, the string cannot be perpen-

dicular to the kite.

5. Show that three forces, 5, 6, and 12 dynes, cannot be in equi-

librium.

6. Three forces, 5, 12, 15, are in equilibrium ; calculate the angles

between them.

7. In a single-span bridge made, as shown, 6 metres high and 16

metres long, what is the
vertical pressure on each
pier and the horizontal
thrust when 1000 kilo-
grams are suspended
from the top point ?

8. When a person sits in a

hammock, what is the
tension on each rope ?

9. What is the advantage in the tow-line of a canal-boat beinp

long?
10. A heavy particle suspended by a cord of length 100 centi
metres is moving uniformly in a horizontal circle of nuliw
to centimetres, what is the angular speed ?

EXPERIMBNT 81

01(j60t. To verify the laws of equilibrium of parallel
forces ia the same plane. (See " Physics/' Art. 63.)

Oeneral Theoxy. If an extended body is in equilibrium
under the action of any number of parallel forces in the
same plane, the mathematical conditions are :

1. The algebraic sum of the forces equals zero.

2. The algebraic sum of the moments around any axis
perpendicular to the plane of the forces is zero.

In Terifying these laws it is most convenient to make the
forces vertical, because vertical forces may be produced by
hanging weights. To produce forces which are vertically
upward, and so can balance weights, two methods are pos-
sible; one is to support the body from above by means of
8pring<*balances, the other is to let the body rest on plat-
form-balances.

SoDxcea of Snror.

1. It is difficult to make all the forces parallel.

2. It is sometimes exceediogly difficult to determine the exact

lines of action of the forces and to measure their distance
from the axis around which moments are measured.

8. The spring-balance, whether of the platform kind or of the
more usual extension form, does not afford a very accurate
means of measuring forces, siDce there is always consider-
able friction in the balance, and the elasticity of the spring
changes with use.

4. The student must always observe the zero-point of a spring-
bahince carefully, as it is hardly ever correct.

Apparatus. Two spring-balances ; a spirit-level ; a metre-
bar ; two weights of about five pound's each ; a single pulley ;
9

130

A MANUAL OF EXPERIMENTS IN PHYSICS

twine. (The experiment should be done at a table which
has a wooden frame over it.)

If platform-balances are to be used^ two small platform-
balances; a metre -rod; two knife-edges; thread; three
poand weights ; a spirit-level.

Manipulation. Weigh the metre-bar and find its centre of
gravity by determining accurately — i.e., to a millimetre— the
point on the scale where a supporting thread must be put
for the bar to hang perfectly level. Hang the pulley from
a hook in the horizontal
bar of the frame as close
up as possible. Pass a
long string through the
pulley, and tie one end
to the ring of one of the
spring -balances; fasten
the other end so that
there are about fifteen
centimetres of string be-
tween the balance and
pulley. Hang the other
spring -balance directly
by a string from another hook less than a metre from the
first, so as to be at about the same height as the first.

Weigh the two weights, unless they are standard ones.
Hang each on a string which has a loop at the end, just
loose enough to slide easily over the bar ; and fasten two
similar loops to the hooks of the spring -balances. Hang
the weights on the bar, and the bar from the balances,
placing its width vertical, so as to make the bending as
small as possible. Slide the weights into any desired posi-
tions on the bar, and move the spring-balance loops until
the strings of these are approximately vertical. Now, by
means of the long string passing over the pulley, raise or
lower that end of the bar until it hangs exactly level, and
make it fast in this position. Make any slight change nec-
cessary in the position of the loops to make all the forces

Fro. M

EXPERIMENTS IS MECUAXICS AND PROPERTIES OF MATTER VM

exactly yertical — i. e., all parallel to the line of action of
the weight of the bar. Finally^ read the forces indicated
by the balances, and the points on the bar where the four
forces (besides its own weight) are applied. (If gram
weights are used, it will be necessary to reduce all the
forces to the same nnits, and the most convenient is the
weight of one gram.)

Make four experiments, varying as much as possible the
positions of the two weights and the balance that is not
hung from the pulley, placing the latter between the two
weights in one case. Becord as below.

Oct aO, IBM

ILLUSTRATION
Eqvilibrium or Paeallil Fobcxs

Forces acting up are considered +.
Momenu tending to turn the bar so that the 2sero end moves up are
coniiidered + ; moments are measured around zero of bar.

EXPERIMENT 1
Foroes

Weight of bar

+ 7.76
+ 6.76

Gnuns
+ 219
+ 191
-200
-100
-112.4

Sum of forces, — 2.4

Moments
9.1cm. - 1998

94.8 cm. - 17931
81.0 cm. + 6200

81.9 cm. + 8190
60.1cm. + 6681

Sum of moments, + 97
2.4

The sum of the forces is thus shown to equal zero to within 7J7)*or
not quite .6^

The sum of the moments is zero to within

97
19900

, or about .06^.

The student should report similarly the other experiments directed.

Alternative Hethoda I. Instead of the fixed spring-
balance, a platform -balance may be used, such as those
intended for weighing parcel-mail. Support it at a suf-
ficient height to allow weights to be hung upon the bar
resting on it. Place upon it a wooden rest with a sharp
horizontal edge on top, such as is used with a sonometer.
Ifote the weight indicated and deduct from future readings.

Best the metre -bar horizontally on this edge (instead of

132 A MANUAL OF EXPERIMENTS IN PHYSICS

supporting it from the spring-balance^ as described above);
hang the two weights from the bar as before, and attach the
string passing over the pulley. Level the bar as before and
read the two balances and the points of application of the
forces. Record as above.

II. Platform -balances may similarly be substituted for
both spring-balances. The only objection is that it is dif-
ficult to level the bar if the forces on the two balances are
very different. Remember to correct both balances for the
weight of the rests placed upon them.

Qnestions and Problema.

1. Which is the more important adjustment in the above experi-

ment, that the bar be le^el or that the foioes applied by the
strings be strictly vertical ? Are both essential t

2. Has the fact that the cords that puU up press on the bottom of

the bar, and those that pull down press on the top, any effect
on the validity of the experiment ? Why T
8. If nails were driven in a wide board at random and a number of
parallel forces applied one at each nail, and all perpendicu-
lar to the edge of the board and in its plane, would the dis-
tance of the nails from either edge have any effect ?

4. A man and a boy carry a weight of 20 kilograms between

them by means of a pole 3 metres long, weighing 6 kilo-
grams. Where must the weight be placed so that the man
may bear twice as much of the whole weight as the boy ?

5. A rod, whose weight is 5 kilograms and whose length is 100

centimetres, is supported on a smooth peg at one end and by
a vertical string 15 centimetn*s from the other end. Calcu-
late the teu«>iou of the string.

EXPERIMENT 22

Olgeet^ To verify the law of equilibrium of an extended
rigid body under the action of three forces. (See " Phys-
Bic8/'Art.62.)

Oeneral Theoiy. It may be proved that the conditions of
equilibrium of an extended rigid body under the action of
three forces are :

1. The lines of action of the three forces will, if pro-
longed, all meet in the same point.

2. The forces are such that their lines of action all lie in
one plane ; and, if they are added geometrically, they will
form a closed triangle.

A simple method of verification is to suspend any body
by means of a cord whose two ends are fastened to two dif-
ferent points of the body, and which passes over a nail ; a
plumb-line dropped from the nail should pass through the
centre of gravity of the body.

Sovupoos of Brov.

It Is sometimes difficult to determine the point where the forces
meet, especially if the peg is large.

Appaiatufl. A long rod which carries two or more bobs
(see Experiment 23); cord; a plumb-line; some suitable
projecting hook or nail.

Kanipulation. Fasten the two ends of a piece of cord,
about two metres long, to the rod at points near its ends.
This may be done by tying suitable knots, or by running
the cord through the rod if it is hollow, and preventing its

A MANUAL OF BXPRRIMENTS IN PHTSIOB

slipping by means of a loop. Suspend the rod from a nail
or peg, making two or more turns of the cord aroand the
nail 80 as to prevent slipping ; if necessary, knot the cord.

Drop a plumb-line from
thenaii,andmarkthepo-
sition on the rod where
this line would intersect
its axis, if it could trav-
erse it.

Change the length of
the cord, the points of
suspension, etc., and
note the points where
the plumb-line inter-
sects the axis. They
should all be the same.
Remove the string and
determine the centre of
gravity of the rod by balancing it on a knife-edge, as ex-
plained in the succeeding experiment.

Pto 48

Questions and Problems.

1. A rod, of length 1 metre and of weight IT. is hinged at ^ to
a Tertical wall. Its upper end B is connected
by a horizontal cord to the wall, so that the
rod makes with the wall the angle d. A weight
W is suspended from B. Calculate the tension
in the string the direction and amount of the
force at the hinge. (Principle of a derrick.)

8. A rod hangs from a hinge on a vertical wall and
rests against a smooth floor. Calculate the press-
ure on the floor and the force on the hinge. ' kiu. vj

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 1S6

8. A rigid rod is acted on by
forces, as shown. Wliat is
the resQltant ?

4. Calculate the direction and
amount of the reaction at the
pivot in the last problem of
£zperiment 19.

u. 5-

•- 4«-- — 5 *

Aa«)

EXPERIMENT 98

Oliiect To determine experimentally the centre of gravi-
ty of a weighted bar. (See " Physics," Arts. 78, 38, 40, 55. )

General Theory. The centre of gravity of any body (or
system of bodies) is the point in space with reference to
it through which its weight (or that of the system) acts for
all positions. In other words, considering the action of
the earth on all the minute portions of the body, it is the
"centre" of the resultant of all these parallel forces, or
the point through which the resultant will pass, no matter
how the body is turned.

If, then, a rigid body is balanced by a supporting cord or
on a knife-edge, so that it is in equilibrium, the centre of
gravity must lie in the same vertical line as the point of sup-
port ; otherwise there would be a moment and a consequent
rotation. By turning the body so as to be balanced from
another point, another line is determined in which the
centre of gravity must lie ; and, therefore, the intersection
of these two lines fixes the point itself. In this way the
centre of gravity of any board, however irregular, may be
found by means of a string, two nails, and a plumb-line.

It may be proved that the centre of gravity of any system
of bodies coincides with its centre of mass ; and the mathe-
matical conditions for the centre of mass are that
-. iWj x^ -f 1712^2+ etc.

"" 7Wi+m2-f-etc.
■ ^iyi4-yw,y2+etc.^

- migi-hm^ga+etc,

~~ wi^ + m, 4- etc.

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 187

where mi, th,, etc., ^e the separate masses, and Xi, a;,, etc.,
are the perpendicular distances of the masses from any
fixed plane ; y^, y^^ etc., are distances from a second plane
at right angles to the first; z^, z^, etc., distances from a
third plane perpendicular to the other two ; x,y, i being
the distance of the centre of mass from these three planes.
In partioalar, consider a uniform rod carrying two or
more bobs. The centre of mass of the rod itself is its
middle point ; and, since everything is symmetrical about
the axis of the rod, the centre of mass of the rod and bobs
together must lie somewhere on this axis. Take as a plane
from which to measure distances one perpendicular to the
rod at one end ; call the masses of the two bobs and the
rod itself m^, 1712, m^, and the distances of the bobs and
the centre of the rod from the plane at the end a;,, X2, x^.
Then 5, the distance of the centre of mass from the plane
at the end, is

This may be verified by actual experiment by balancing
the rod from a cord or on a knife-edge.

fikynrces of Error.
1. There may be difficulty in determintDg the centre of mass of

each bob.
9. When balanced, the centre of gravity is in the same vertical

plane as the point of support, not necessarily in the same

plane perpendicular to the axis of the rod.
8. The supporting edge or thread must be as fine as possible.

Apparatus. A round uniform bar, graduated the greater
part of its length, and provided with three weights, which
may be clamped upon it at any desired points ; twine ;
a support from which the bar may be suspended so as to
hang free (or a knife-edge on which it may be balanced) ;
Bcales and weights for weighing the bar and its bobs, capa-
ble of weighing 2000 grams to the accuracy of one gram ;
a vernier caliper ; a level.

188

A MANUAL OF EXPERIMENTS IN PHYSICS

Manipulation. Weigh the bar and e#oh of the bobs sepa-
rately^ taking pains to identify the bobs. With the yemier
caliper determine the thickness of each bob, and hence the
position of its centre of figure with regard to the plane sur-
face at one side of it — i, e., the correction to be made to the
point where this face cuts the bar in order to get the po-
sition of the centre of figure in subsequent experiments.
(Since the weights hare a flat boss on one side only, they
are slightly unsymmetrical, but for the purposes of this ex-
periment the centre of the symmetrical part alone may be
taken as approximately the centre of figure of the whole.)

Fia51

Suspend the bar without any weights upon it from a
frame over a table or from a projecting nail by a single
tight loop of twine, and slip the loop along until the bar
hangs perfectly level ; or balance it on a knife-edge. When
the supporting string is so placed as to hold the bar exactly
level, and the loop is truly perpendicular to the axis of the
bar, the centre of gravity of the latter must lie in the same
vertical plane as the loop. Hence the position of the loop
on the scale gives the distance of the centre of gravity from
the right section of the bar marked by the zero of the scale.
Read and note this position of the loop on the bar.

Now clamp the bobs in any desired position on the bar,
noting carefully the reading of the fiat surface of one aide ;

EXPERIMENTS LV MBCHANIG8 AND PROPERTIES OF MATTER 189

and, for convenience, place this side towards the decreasing
numbers of the bar (so that the correction giving the posi-
tion of the centre of fignre of the weights will always be
positive). Suspend the bar again and determine its centre
of gravity as before. Repeat with three different positions
of the weights. In each case, in making the report, calcu-
late the position of the centre of gravity from the weights
of the separate masses and their positions, and compare this
with the experimental result. In making the calculation,
the weight of the bar should, of course, be considered as
acting at the centre of gravity found experimentally for it
alone; and the movable weights may each be considered as
acting at the centre of figure of the symmetrical portion.
Report as below.

ILLUSTRATION not. 1, ISW

Weight of bar, 208 grams. Centre of gravity of
bar. 50.8.

HeDoeiT)

1.0 cm.

1.1 cm.
1.15 cm.

Bota

Weight

1081 gr.

.2 cm.

1.6 cm.

1540 gr.

.2 cm.

1.8 cm.

1918 gr.

.2 cm.

1.9 cm.

No of

Ezperi-

moot

Edg»(l) Centre

10.0
81.2
81.2
81.2

11.0

82.2

Pmtions qf Bobs

Edge (2) Centre

87.2

47.8
78.7
78.7

88.8
48.9
79.8
79.8

Rdge (3)

Centre

Centre of

Gravity

Obi.

90.7b

91.9

54.1

6.7

7.85

40.0

6.7

7.85

50.1

95.0

96.15

85.6

Fn.89

Centre of
Or»v1tj
Calculated

54.2

40.0

50.0

85.7

Greatest deviation is ^ of \%.
Qoestioiui and Problems.

1. What effect would it liave od the experimental position of the

centre of gravity as compared witli thHt calculated, if owing
to defective casting one weight had a large hole in the side
towards the lower numbers on the bar 7

2. A uniform wire ABC is bent at B

to an angle 60**, and is suspended
from A. JnS is 10 centimetres
long. Calculate length SU, so
that when the whole is in equi-
librium, SUyf'\\\ be lioHzontal. Fio 58

140 ▲ liANUAL OF EXPBRIMENTS IN PHTSICS

8. A circular table rests on three legi attached to three points of
the circumference at equal distances apart. A weight is
placed on the table. In what position will the weight be
most likely to upset the table, and what is the least value of
the weight which when placed there will upset it 7

4. A circular hole, 10 centimetres in radius, is cut out of a chx;u-

lar disk 50 centimetres in radius, the centre of the hole
being 10 centimetres from that of the disk. Calculate the
centre of gravity of the remaining disk.

5. Two bodies ''attracting" each other with a force varying

directly as their masses and inversely as the squares of
their distances apart move towards each other. Where
will they pieet 7

6. Two bodies, whose masses are 100 and 200, are connected by

a light wire, and are thrown in such a way that the centre
of gravity has a speed 10 metres per second, and that the
system revolves arcund the centre of gnivity twenty times
per minute. Calcuhite the entire kinetic energy.

EXPERIMENT 24

(TWO 0BBEBYBB8 ABB BBQUIBBD)

Olgect. To determine the '^ mechanical advantage*' and
** efficiency '* of a combination of pulleys. (See *' Physics,*'
Art. 72.)

Qeneral Theory. The '' mechanical advantage '' of a pul-
ley, or combination of pulleys, is defined as the ratio of
the force which tends to draw the lower pulley down, to
that which must be applied to the free end of the cord
passing around the pulleys in order to exactly balance the
first force. The "efficiency'' is the ratio of work done
against the force acting down on the lower pulley to that
done by the force applied to the cord, when the pulley is
raised at a uniform rate. (If no friction were overcome,
the efficiency would be 1.)

Several cases may be considered.
Ckise 1. — The cord is fastened to a
horizontal support and passes in turn
over a movable and a fixed pulley,
the three branches of the cord being
parallel.

Ckise 2. — The cord is fastened to
the hook on the top of a free pulley,
and passes in turn over a fixed pul-
ley, the free pulley and the fixed
pulley again, the four branches of
the cord being parallel.

It 18 obvious that, if the lower pul- fiq. 64

142

A MANUAL OF EXPERIMENTS IN PHYSICS

ley is in equilibrium, the tension of the cord is the same
in ail its branches^ and that the force down on the lower

pulley must equal the prod-
<> net of the tension in the
cord by the number of
branches of the cord leav-
ing that pulley.

The difficulty in the actu-
al measurement enters from
the fact that the motion of
the pulleys may be influ-
enced by friction, and so
this effect must be done
away with. The method
of doing this is as follows :
Pull the free end of the
cord with a force sufficient
to produce uniform motion
of the free pulley upward ;
this force is equal to the
equilibrium force plus the
force of friction ; then, by
diminishing the pull on the
cord, exert just enough force to allow the pulley to fall with
the same uniform motion as that with which it rose ; this
force equals the equilibrium force minus the force of fric-
tion. Therefore, the average of the two forces is the equi-
librium force.

In measuring the efficiency, the actual force producing
uniform motion upward may be measured ; and the ratio
of the work done against the force on the lower pulley, Z^,
to that done by this upward force on the cord, i^„ is evi-
dently given by

W^^F^y^h^ 5

TTi Fy X /| 7<\ X No. of branches or coni lenvinir pulley'

^2 being the distance the weight is raised, and l^ the dis-
tance the *'free end'* of the cord moves down.

Fio. 65

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 143

Sonzoes of Brror.

1. Tbe motion may not be iinifonn (this introduces an error owing
to the acceleration).

3. Tbe line of motion may not be maintained constant.
S. Tbe friction may be different in different positions.

4. The pulleys and spring-balances themselves have weight.

5. The axles of tbe pulleys may not pass accurately through the

centres.

Apparatus. Two single and one double pulley ; stout
fishing-line ; three weights of about ten pounds each ; two
spring-balances.

Manipulation. Arrange in succession the pulleys and the
suspended weight as in Case 1 and Case 2 above, attaching
a spring-balance, hook up, to the free end of the cord. Pull
vertically down on the balance. Becord the readings nec-
essary to secure uniform motion up and down ; call them
/i and /y It is often best for one obseryer to devote his
entire attention to the balance, keeping its motion uni-
form, and for the other to make the reading of the pointer.
These are not the true forces, because readings on a spring-
balance are true only when it is used vertically with hook
down. Consequently, these readings made with the hook
up are too small by an amount 2A, twice the weight of
the hook. This quantity may be determined as follows :
Suspend a weight from the hook, the ring being hung on
a nail, make the reading ; invert the balance, hanging the
hook on a nail and suspending the weight from the ring,
make the reading ; the difference between these two read-
ings is equal to the weight of the whole balance minus
twice the weight of the hook alone. To measure/,, then,
perform the experiment just described, using any weight ;
and, in addition, weigh the whole balance on a second bal-
ance which has been compared with the first to see if the
two scales agree. Having thus determined h, the true equi-
librium force is

EXPERIMENT 26

Olject. To determine the coefficient of friction between
two polished wooden sarfaces.

General Theory. The coefficient of friction between two
given surfaces is usually denoted by /*, and is defined Jby
the ratio FJF2, where Fi is the force necessary to move
one surface over the other at a constant speedy and F2 is
the force pressing the surfaces together, fi is us^alljT dif-
ferent for different speeds, and the value found in this
experiment is that which relates to very slow motion. It
is sometimes called the coefficient of statical friction.

If two surfaces are pressed together with a force F^, it
will require a force Fi=fiF2 to produce a uniform motion;
therefore, to produce an acceleration a, a force will be re-
quired equal to ma + F^. This additional force, ^j, is
called the "force of friction,*' and it equals the product
of the "coefficient of friction*' by the force pressing the
two bodies together. For a definite speed, /i depends only
on the condition and material of the two surfaces, and not
on the area over which the pressure is distributed.

If a body of mass m rests on an inclined plane which
makes an angle 0 with the horizontal, the force pressing
its lower surface against the plane is JV = tw^ cos 6. The
force tending* to make it slide down is i2 = tw^sin 6. The
force which opposes the sliding is the friction. When the
plane is nearly horizontal, the friction will be sufficient
to bring the body to rest, if it is set in motion down the
plane. But as the plane is more and more inclined, the
force down the plane becomes greater ; and the amount

EXPEBIMEKT8 IN MECHANIOS AND PROPERTIES 0¥ MATTBK 147

of friction necessary to keep the body from moying faster
and faster, when once set in motion, is also greater. Final-
ly, for a certain Talae of 0 the friction reaches its limit ;
and the body when set in motion continnes to mo?e faster

and faster. Evidently, where 6 has snch a Talne that the
motion of the body jnst remains uniform, the force down
the plane exactly equals friction for very slow motion — i, e.,
if a is this '* slipping angle/' mg sin o = -Pj ; but -Pj i^ tbis
position equals mg cos a, and

sin a

hence, /i :

cos a

= tan a.

i.e,j the coefficient of friction for Tery slow motion is equal
to the tangent of the inclination at which one body just
slides oyer the other.

SoozoMi of Brror.

1. Owing to inequalities in the boards, the friction is not the same
in all places, and so tlie carringe will start slipping at differ-
ent inclinntions.

3. Great care and Judgment must be used in determining when
there is no acceleration.

Appaxmtna An inclined plane consisting of a smooth,
wooden board, hinged to a base which fits over the corner
of the table. At the other end of the board is hinged a
support in which are a large number of holes at different

148

A MANUAL OF EXPERIMENTS IN PHYSICS

heights. A block rests upon the table and is proriaed
with two iron prongs, one of which is fitted into a hole
of the perforated support, and thus fixes the inclination of
the plane. By varying the distance of the block from the
angle and the hole by which the support is held, the incli-
nation is adjustable with the greatest accuracy. A heavy
weight should be placed on the base, which fits over the
table, so as to steady the apparatus ; and another may be
placed on the movable block if it is found to slip out (an
elastic band will generally prevent this). Two wooden
carriages with polished under surfaces of different areas ;
a weight of over five pounds to go on the carriage ; a metre-

Fio. 59

bar ; plumb-line and steel square are also needed. (A small
ball hung by a thread makes a very good plumb-line.)

Hanipulation. Place the weight on one of the carriages
and adjust the inclination of the plane until the carriage
just slides, when gently started, without either increasing
its velocity or stopping. Then measure the angle which
the plane makes with the horizontal. To do this, clamp
the plumb-line as near the top of the plane as possible by
laying a weight upon it, letting the bob hang well down
below the top of the table, and tie in it a small knot at
a point just above the level of the table. Put the square
and the metre-bar together, so that one side of the square
is perpendicular to the bar. In this way hold the bar at
right angles to the plumb-line and find the point in the

EXPSRIMENTS IN KSCHANIOS AKD PROPEBTIES OF MATTER 149

npper snrface of the inclined plane at the same level as the
knot. Measure the distance from this point to the knot,
call it LN; measure the distance along the plumb-line from
the knot to the upper surface of the plane, call it MN.

Then, tan a = 'z=r = m> the coefficient of friction between

the two given surfaces.

Fig. 60

Having made one determination of /i, change the incli-
nation and begin again, making four experiments in all.

Make another series of four experiments with the same
weight hut with the other carriage, the area of whose base
is different ; and a third series with another weight. Re-
cord each as below.

ILLUSTRATION

Not. 10, 1804

Coefficient of friction between polished piue surface of plane and
polished oak of carriage :

Larger Carriage : 6'ld. Weight.

:.li

26.4 cm.

66.0 cm.

.400

26.8 cm.

66.8 cm.

.401

26.7 cm.

67 0 cm.

.890

26.5 cm.

65.7 cm.

.408
Mean. .401

160 A MANUAL OF EXPSBIMENTS IK PHYSICS

Quastions and Problams.

H What Ib tbe efFect upon the friction between a wheel and Its
axle, of increasing (1) the diameter of the axle, (2) the length
of the parts in contact ?

2. Prove that if a heavy body is to be drawn up an inclined phtne,
the force required to do so is lesat when the angle between
that plane and the line of force equals tbe angle of fricUon,
tan-y.

8. Would an ordinary brick be less liable to slide down an in-
clined plane when placed on one face than if placed on
another ?

4. A shaft is 4 centimetres in diameter, and is making 120

turns per minute. It requires a weight of 100 kilograms
at the end of a lever
a metre long to keep
the "Prony Brake"
from moving. Calcu-
late tbe activity of
shaft. What becomes
of the energy ?

5. A body of mass 10 is set in motion by an impulse 10,000 along

a horizontal rough table whose coefficient of friction is 0.1.
At the end of two seconds it meets a smooth inclined plane.
How high will it rise?
6 A bullet, whose mass is 100 grams, is fired from a gun whose
barrel is 75 centimetres long, with a velocity 400 metres per
second. Assuming the powder pressure to be uniform, cal-
culate the force on the bullet and time taken to traverse the
barrel. It enters a wail 200 centimetres thick with a speed
880 metres per second, and leaves it with a speed 200 metres
per second. What is the average resistance of the wall, and
how long did it take to pass through ?

Fkan

EXPERIMENT 26

Direct. To determine the mass of the hard rubber cylin-
der whose volume was found in Experiment 5. The use of
A chemical balance. Beading a barometer. (See '* Phys-
ics/'Arts. 71, 129, 175.)

Oeneral Theory of the Ohemical Balance. The analytical
or chemical balance differs from scales designed for a less
accurate comparison of masses chiefly in the care with which
it is made, and in the introduction of devices for observing
much smaller differences in the equality of the masses in
the two pans and for making more delicate changes in the
weights used. It consists essentially of three parts : 1. The
pillar, or central support. 2. The beam, a rigid framework
of metal resting upon the pillar, and so designed as to com-
bine the greatest possible lightness with the least possi-
ble bending under the weights for which it is designed.
3. The scale-pans and the metal frames by which they are
hung from the beam.

The usefulness of a balance depends on the following con-
ditions : 1. It must be true — that is, when the masses in the
two pans are equal to the degree of accuracy for which the
instrument is to be used, the pointer which indicates the
inclination of the beam must return to the position in
which it was when the pans were empty. This position
is called the *^ zero'' of the balance.

2. It must be stable — that is, the beam must have a defi-
nite position of equilibrium for a definite small difference
in the equality of the masses.

3, It must be sensitive — that is, for a small difference in
the equality of the masses the deflection must be large.

152 A MANUAL OF EXPERIMENTS IN PHYSICS

The sensitiveneBS of a fine balance is secured as follows :
The beam is snpported on the pillar by means of a knife-
edge. This is a triangular prism of steel set in the beam
with the edge down. To decrease still further the friction
as the beam tilts, the ends of the knife-edge rest upon
horizontal surfaces of glass or agate set in the top of the
pillar. The scale»pans aie similarly hung from knife-edges
placed one at each end of the beam, parallel to the centnil
knife-edge and at equal distances from it. This insures
that the weight of each scale-pan and its contents acts ver-
tically in a line whose distance from the axis about which
the beam rotates is the horizontal distance between the
central knife-edge and the one from which the pan is
hung. These distances on each side, measured when the
beam is horizontal, are called the ^' arms'' of the balance.
Let

a = length of right arm of balance.
b = length of left arm of balance.

rrir = mass in right pan of balance.

rrtt =^ mass in left pan of balance.

if = mass of beam.

j9«. = ma8s of right pan, etc., when empty.

Pi = mass of left pan, etc., when empty.
d s distance of centre of gravity of beam from central
knife-edge.

If the scale-pans are removed, the position of equilibrium
of the beam is evidently such that its centre of gravity is
vertically below the knife-edge. This is the position which
is described above as the ieam being "horizontal.*'

Let the inclination of the beam when the pans are hung
upon it empty be qq. Then Mgd sinao=(j!?/5— jt?^a)^co8ao,
if oq is considered positive when the left end tips down.
Whence,

for, Pih-pra
Add the masses m^ and mi to the pans ; let them be nearly

EXPERIMENTS IN MECHANICS AND PROPERTIBS OF MATTER 168

equal, and let at be the inclination of the beam. Then, as
above.

/. tanai^tano^:

Bat if i7 is the heiglit of the knife-edge above the hori-
zontal scale at the base of the pillar^ and if a^ and x^ are

no. <»

the readings of the pointers with pans empty and loaded,
then

tan Qj — tan a^ =

.\ a;, — a?o = JSr

H
lid

Hence it is evident that to make the balance sensitive —
i.e., to make x^^x^?a great as possible for a given differ-
ence nti-^mn the following conditions mast be fulfilled:

1. The arms a and b must be as long as possible.

2. The beam must be as light as possible.

3. The distance of the centre of gravity of the beam from
the knife-edge must be small. Bat if it becomes zero, a?, -^ o^
will be infinite, and the balance will be unstable. Even in
the best balances the beam bends when the load becomes
greater, so that d increases, and hence the sensitiveness
decreases.

154 A MANUAL OF EXPfiRIHENTS IK PHYSICS

4. ^shonld be greats and consequently the beam support-
ing the pillar should be high.

The condition that the balance be stable is, from the same
equation, that d be not zero or negative — i. e,, that the cen-
tre of gravity of the beam be not at the knife-edge or above
it. In either case the beam alone is in unstable equilibrium,
and the slightest difference in the weights hung upon it
would tip it over entirely. For this reason the knife-edge
is set in the beam very slightly above the middle of the
framework.

The condition that the balance be true is that, when
ntr^znii, x^zzzoSf^; that is, for equal masses the pointer mast
return to the position which it has with pans empty. There-
fore,

m^ — m/5 = 0; i.e, a = 5.
Hence the arms of the balance must be equal or the balance
will not be true.

In the use of a balance for accurate determinations the
arms are not assumed equal, but a correction for difference
in length is made by weighing the body first in one pan, then
in the other. Let to be the mass of the body, mt the weights
which have to be placed in the left pan, and m, those which
have to be placed in the right pan to balance to when the
body is in turn in the right and left pan.

Then,

(If m, and w, are very nearly equal, -^ — ' is a close

approximation to the quantity ^InTnif) This is known
as "Gauss's Method of Double Weighing."

With a sensitive balance it is usually impossible to bring
the pointer exactly back to zero with the weights at com-
mand. Suppose the smallest change possible to be .001

£XPKRIM£NT» IN MfiCHANICS AKD PROP£ltTl£B 0^^ liATl£& U5

gram^ and that this carries the pointer from a position x^
to the left of ssq to x^ to the right. Then^ if w is the mass
of the object^ which we will suppose in the right pan, and
mg the weights in the left pan when the pointer was at a;^

wa — WjJ Md (wi| + .001) b-^wa

. ^0 — ^1 _^a — a;,
^^wa^tnjb .001 J

wa = (m, + .001 ^ *^^ J.

If toi denotes the exact mass in the left pan which would
connterbalance the object placed in the right pan,

wa = wfi,

.MC7, = W|-f-.00l5li:^;

i.e. 9 since the weights available do not enable one to place ex-
actly Wi grams in the pan, nit are put in first, then mi + .001
and Wi is calculated by interpolation. In a similar way
Wr, the exact counter-balancing mass for the right-hand
pan^ is determined ; and the correct mass is ter = Vwi w„

In order to protect the knife-edge from wear^ a^support
for the beam may be raised by a screw in the balance-case
at the foot of the pillar. This support holds the beam on
each side of the knife-edge and lifts the latter off the agate
bearings. This must always be done when a change is be-
ing made in the contents of the pans and even when the
balance is not to be used for only a few minutes. The
sliding front of the case should not even be raised or
lowered while the beam rests on the edge. It is evident
that accurate weighing requires that this support should
always replace the knife-edge in exactly the same position
each time it is lowered on the agate surfaces. Otherwise
the zero of the balance may be changed every time the
support is used.

IM A MANUAL OF EXPfiEIMfiKTB IN PHYSIOS

The weights to be used with a chemical balance nsaally
come in sets containing weights from .01 gram up. The
fractional weights are marked either as so many miUi-
grams (denoted by m.), centigrams (c.)> or decigrams (d.).
Even if no letter is given, there is seldom any conf nsion,
since a comparison of the size of the pieces and the num-
bers upon them gives a clew to the unit in which each is
expressed. The best fractional weights are made of plati-
num, though aluminum ones are often used and have the
advantage of larger size. Never under any circumstances
handle a weight with the fingers. The value of a weight
is easily altered by several milligrams by touching it once.
Always use the pincers.

Some sets contain weights of .001 gram ; but usually a
"rider" of greater size is used instead. A " rider '* con-
sists of a fine platinum wire of a definite weight, shaped so
that it will stay astride the beam of the balance at any point
where it may be placed. The top of the beam is grooved
at various points, such that the horizontal distance between
the central knife-edge and a vertical line through the groove
is a fixed proportion of the length of the arm of the bal-
ance— usually so many tenths. Hence, if a rider weighing
.01 gram, for instance, is placed in the groove marked as
being at a distance from the knife-edge of three-tenths of
the arm of the balance, it will be equivalent to a mass of
.01 X 3/10 = .003 grams placed in the pan. The rider may
be moved from one groove to the next by a carrier operated
from the outside of the case without raising the sliding
front. With a balance-beam grooved in tenths and a cen-
tigram rider, a change of .001 of a gram can thus be made.
The value of the rider, if not known, can be found by com-
parison with the other weights in the set. It can always be
assumed that a rider furnished with gram weights is a very
simple multiple of a centigram or a milligram.

NoTi. — The mass of the cjlinder found above ( w= V^wTvi or ^ — !!l j
is its apparent mass in the air. Since any body immersed in a fluid is huojed

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 157

op bjT a force equal to the weight of the fluid it dieplaoee, the weights and the
cylinder are both apparently lighter thau they would he in a Tacuum. Since
8.4 grams of brass haye a volume of 1 cubic centimetre, the volume of air
displaced by the weights can be calculated at once. The volume displaced
by the cylinder is known by its measurements in Experiment 6. Hence,
by finding in suitable tables the mass of a cubic centimetre of air at the
temperature and pressure of that in the case, one can calculate the loss
in weight of the weights, which should be subtracted from the apparent
amount necessary to balance the cylinder; also the loss of the cylinder,
which should be added.

Barometer Beading. The pressure of the atmosphere is al-
ways measured by the height in centimetres of the column of
mercury which it will support when the mer-
cury stands in a tube closed at the top, having
a very high vacuum above the mer-
cury, while the lower end of the
tube dips in a basin of mercury.
Such an instrument is called a
" barometer.*' The height is meas-
ured vertically from the level of the
top of the mercury in the cistern
open to the air to the top of the
convex surface of the mercury in
the closed tube. It is read by
means of a scale upon the metal
case surrounding the tube. The
zero of the scale is the tip of an
ivory point dipping into the cistern ;
the position of the top of the col-
umn is given in fractions of a milli-
metre by a vernier engraved on a
sliding index, whose zero must be
made to coincide with the top of the
column by a screw admitting of a
very delicate adjustment.
The necessary conditions are :
1. The scale must be vertical. This is se-
cured by hanging the barometer from a loop ■ Fio.esa

no. cs

168 A MANUAL OF EXPERIMEXIV IN PHYSICS

at the top, the scale being engraved so as to be vertical
when the instrnment hangs freely.

2. The top of the mercnry in the cistern must be at the
zefo of the scale — i. e., it must just touch the tip of the
ivory point. The height of the mercury in the basin is
adjusted by a screw at the bottom, which compresses the
sack containing the mercury and so raises its level. The
tip of the pointer must touch the mercury so lightly that
the dimple caused by it is just not visible.

3. The zero of the vernier must be accurately in the same
horizontal plane as the top of the mercury column. The
zero of the vernier is always the lower edge of a brass ring
that slides on the barometer tube. The opposite side of the
edge of the ring may be seen through the glass tube, and is
made so as to be exactly in the same horizontal plane as the
front edge. Hence, if one sights under the front edge from
a direction such that the back edge is just hidden, and then
lowers the ring until these edges appear to just touch the
top of the column, this top will be in the same horizontal
plane with them. This may be done by lowering the ring
until light can just not be seen between it and the mer-
cury.

4. This reading, 7i, gives in scale divisions of the metal
case the height of the barometer column in terms of mer-
cury at the temperature of the thermometer fastened to the
barometer.

Hence, if

P is the density of mercury at f* (7,
I is the length of one scale division at i^ (7,
g the acceleration of gravity in the laboratory, the press-
ure of the atmosphere is j9 = pghh

If a barometer at the sea- level in latitude 45° (where
g=g^) were to contain mercury at the temperature 0°,
the height H in centimetres registered by it, which would
correspond to an equal pressure J7, is given by the equation

KXrERlMBNTS IN HBCHANIGS AND PBOPERTIBS OF MATTER 169

But po = p (1 + Pt) where fi is the coefficient of cubical ex-
pansion of mercnry.
2 = (1 + at) if a is the coefficient of linear expansion
of the metal scale^ assuming the scale
to be correct at 0°.

"'^g^ 1+pt
The quantities -2. and """^ are given in tables, so if

may be calculated from h. H is called the '^ corrected
height/' because it gives the height to which the barom-
eter would rise when measuring the same pressure under
standard conditions. THe figures for densities of gases,
etc., in tables are always given in terms of these corrected
heights; and every reading of the laboratory barometer
must be corrected.

To Weigh the Piece of Hard Bubbbr.

Oeneral Method. The general method of the present ex-
periment is as follows : The object to be weighed is placed
in the right pan of the balance, and weights to balance it in
the left. After the pointer has been brought back as close
to its zero position as possible with the weights at command,
including the rider, the exact weight is calculated by inter-
polation, as shown above, between the two nearest positions
of the pointer on either side of the zero positioD. (The read-
ings for these positions of equilibrium are of course made by
the method of vibrations. See Experiment 11.) Oall the
mass thus found for the object Wi^

The object is then transferred to the left pan, and its mass

again determined. Oall it w^ Then w = ywi w^ = — i-- — ^

Finally, the true mass must be calculated, taking into ac-
count the buoyancy of the air.

Sooxoes of Brror.

1. Same as thnee of Experiment 1, Part 8. The effect of draughts
o? air is much more important, and the balance-case must

1«0 A MANUAL OF EXPSRIICJBKTS IN PHYSICS

always be closed before the poeitioa of the Winter is flotlljr

observed.
2. FrictioQ at the knife-edge, due to its baying been blunted by a

jar or to a spot of rust. This is shown by a decreased sensi-

tlTeness, and also by the zero of the balance changing.
8. The pans must not be allowed to swing. (See Experiment 11.)

Apparatus. A chemical balance «nd box of gram weights
(50 grams to .01 gram, with *' rider''). The same cylinder
as was measured in Experiment 5.

Manipulation. See that the scale-pans are free and clean.
If they are not, call the attention of an instructor unless
they are readily cleaned with a small brush, such as comes
in the box of weights. Level the balance, if necessary.
Lift the '* rider " with the pincers, and hang it on the car-
rier inside the case. Lower the support which holds the
beam off the knife-edges. Set the balance swinging over
two or three divisions of the scale and determine the point
of equilibrium with the pans empty, as in Experiment 11.
(One set of readings of five consecutive turning-points is
sufficient.) The point of equilibrium with pans empty as
thus found is the zero of the balance. (If, owing to any
cause, one pan is much heavier than the other, pieces of
paper may be added to the lighter one.)

Baise the support so as to lift the balance off the knife-
edges ; place the cylinder in the right pan and weights to
balance it in the left, aluoays using the pincers to lift weights.
Never add or remove a weight from a balafice-pan without first
raising the balance off the knife -edges. Serious damage is
done to a balance by neglect of this rule, and also to a box
of weights when they are handled without pincers.

Proceed as in Experiment 1, Part 3, to find a mass so
close to that of the cylinder that the smallest change in the
position of the rider (.001 gram) is sufficient to move the
pointer from one side of its zero position to the other.
Then close the case, and by the method of vibrations de-
termine the point of the scale about which the pointer
now vibrates. Let the reading on this point of the scale

£XP£RIM£NTS IN MEGUAXiCS AND PROPERTIES OF MATTER 161

be x^. Now make the change of .001 gram so as to carry
the poiuter over to the other side of the zero. Find the
new point of rest by vibrations. Let it be a:,. Let the
original zero of the pointer be Xq, Then if nii is the mass
of weights in the left pan^ the apparent mass of the cylin-
der is «7/ = f»/+'.001-^^— ^. In reading the weights com-
iCj — a?j

posing f»i, it is well to do it in two independent ways: (1)
Read the weights on the balance- pan^ (2) read the weights
absent from the box of weights. When weights are removed
from a 8cale*pau, always replace them in their proper posi-
tions in the box.

Interchange the cylinder and the weights^ and find the
balancing mass in reversed pans. Let it be Wr. Redeter-
mine the zero of the balance to see that it has not changed.

Finally, note the reading of the barometer in the labora-
tory and the temperature in the balance case. Galcalate w.

ILLUSTRATION

Nov. 3, 1806
DetenninatioD of mass of cylinder No. 1.

Balance used, M. S24. Box of weights. M. 81.

The beam of this balance is graduated to tenths of the arm. The

rider in the set of weights is .01 gram.

Zero of balance at start, 8.9 ; at end, 8.95 ; mean, 8.92 = x^.

Cylinder in Bight Pan,

•

Balancing mass = mi.

ghUinFfto

Rkl«r Toul

Pointer

81.06

At .6 mark, 81.066

8.6 = ari

31.06

At .7 mark, 31.067

9.1 =aj.

•'. Apparent mass of cylinder in this pan is :

u>i = 31.066 + .001-^ = 81.0666 grams.

Cylinder in Left Pan.

Balancing mass = rrir.

W'eigfata in Pan

Rider Total

Pointer

31.06

At .7 mark, 81.067

9.2 = a?,

31.06

At .8 mark, 31.068

8.7 = ^,

28
.-. Wr = 31.067 -f .001--^ = 31.0676 grams.

11 -^^

16S A MANOAL OF EXPRRIMKNTS IN I'HYSICS

Barometer, 763.8 mm. Temperature in balance-case = 18.5° C.

/. to = y/lciWr= —-5 — "" approxi mutely = 81.0671 grams.

Weight of rubber cylinder No. 1 in air of temperHturc 18.5^ at

768.8 mm. pressure = 81.0671 grams, as measured Uy brass weights.

81
The brass weights displace ^ cc. of air, because 8.4 is the density

of brass. Since the air at 768.8 mm. and 18.5'* wtfighs .001217 gram
per cc, the real weight counterbalancing ihe cyliuflcr is only

81.0671 - |i X 0.001317 = 81.0636 grams.
0.4

But the volume of the cylinder was found to lie 33.54 cc. in Experi-
ment 5. It has lost 33.54 x 0.U01317 = 0.0386 gram by the baoyancy
of ihe air. Its true mnss is, thererore,

81.0636 + 0 0386 = 81.0913 grams.

Tlie correct manner of recording a weighing is as follows:

L (left pao)

ao;- 10 +1+0.05 +001

+ rtder at 6= +0.006

£ (right pui)
Cylinder

6.3

11.3

6.0

11.1

5.8
6.0

11.3

Mean, i

J.6, etc.

QuestionB and Problems.

I 1. Wbat is the ratio of the urins of the balance in this expeif ment?

I 3. Discuss' advantages of long and short arm balances.

j 8. Where should the centre of gravity of the balance be ?

4. How can a balance be made more sensitive? How more stable ?

5. Discuss the effect of an increase in the valjuic of g, and also
an increase in the temperature upon the sensitiveness and
stability of a balance.

6. The arms of a false balance are in the ratio of 30 to 21. Wliat
will be the gain or loss to a salesman if he asks $1 00 per
pound for goods which apparently weigh 5 pounds?

EXPERIMENT 27

Object. To verify Hooke's Law and to determine ''Young's
Modulus" for a given substance by stretching a wire. (See
"Physics," Arts. 79,83.)

General Theory. 1. Hookers Law states that, within cer-
tain limits, the strain produced by any force is propor-
tional to that force. In particular, if a wire is stretched
in turn by different forcfes, the elongations produced vary
directly as the forces. If i^is the force which produces an
elongation A2, then

Fl^l is a constant.

2. For a wire of any given material — e.g.y brass — the

elongation depends upon the cross-section and the original

length in such a way that, if <r is the cross-section, and I the

1 Fl
original length, AZ = -^ — -, where^isaconstantforagiven

F I
material: or, F=z—. -•

E\% called "Young's Modulus for Stretching," and can be
determined by measurements of the four quantities involved.
Fj I, 9 can be measured directly by ordinary means, but £d
is a small quantity and requires special accuracy. There
are two methods available : one, to magnify the elongation
by means of a lever and scale ; the other, to use some deli-
cate means of measuring lengths, such as a vernier.

The simplest method is to suspend two wires of the same
material side by side from the same support ; let one carry
the scale, the other the vernier ; add weights to one and
measure the relative elongation. This method has the ad-
vantage of avoiding any error due to changes of tempera-

ir>4

A MANUAL OF EXPERIMENTS IN PHYSICS

ture in the room^ which, of themselves, would change the
length of the wire ; it also avoids any possible error due to
giving away of the supporting clamp.

SouroeB of Error

1 . Tliere are changes of temperature in the wire wLich is stretched,

owing to its suddeu elongation. (See *• Physics," Art. 170.)

2. The scale and verniar may he accidentally displaced over each

other.

3. The limit of elasticity must not be exceeded.

4. All kinks must be removed from the wires.

Apparatus. Two long wires of the given
substance, hung from a fixed support on
the wall near the ceiling, at such a dis-
tance apart that two vertical scales, one
attached to each wire, have their gradu-
ated edges in close contact. Supports for
weights are attached to the bottom of each
scale, and a five -pound weight is placed
permanently on each to keep the wires
straight.

Extra five -pound weights are also pro-
vided to be hung upon one
wire so as to produce the
stretching force. fio. 64A-SBonmr o^

One of the scales is a
millimetre 8cale> and the wire to which
it is attached carries the same weight
throughout the experiment, and so re-
mains fixed ; the other scale is a vernier,
sliding over the first. A guide, soldered
to the fixed scale, moves in a groove in
the vernier scale and keeps them parallel;
and an elastic band passed over both keeps
them in the same plane. A metre-bar, a
steel L-square, and a micrometer caliper
are also needed at the end of the experi-
ment.

mr'Ub

=-10

EXPERIMENTS IX MECHANICS AND PROPKKTIES OF MATTER 165^

Hanipnlation. 1. To show that — = constant. See that

one weight is hung on each wire, and that the scales hang
parallel. If any change has to be made in the weights allow
three minutes to elapse before taking a reading, so that the
wires may assume their normal length under this stretch-
ing force. With a magnifying - glass read very carefully
the position of the zero of the vernier scale on the fixed
scale. Estimate the tenths of a vernier division. Mark
the weights and place No. 1 on the carrier of the wire to
be stretched — the one with the vernier scale. After three
minutes, note the new reading of the vernier. Now add
weight No. 2, and after three minutes more read the ver-
nier again. Continue by adding No. 3 and noting the
stretch again. Next, remove No. 3, and after three min-
utes read. Remove similarly in turn No. 2 and No. 1.
Repeat the above cycle three times. Weigh each weight
to within a gram on the platform scales. A/ being the
change in length when a weight F is added or removed,

show that —7 is a constant for all elongations observed.
A/

2. To determine " Young's Modulus." With the L-square
and metre-bar measure the length of the wire to the point
where it is attached to the scale. (The height of the sup-
port above a mark on the wall behind the two scales may
be mieasured once for all by an instructor and marked on
the wall.) Measure the diameter at as many different
places as can be reached and in different directions, ten
times in all. In this way determine I and a, the original
length and the cross-section.

ILLUSTRATION
''Yovvo'8 Modulus" ior Bbabs otc 16,1806

Specimen experimented upoD : Brass wire of "Young's Modulus
Apparatus.'*

WeighU: 1 = 2889. 2 =: 3278, 8 =^851 grams.

166 A MANUAL OF EXPERIMENTS IN PHYSICS

Rsctdings with Different Weights on Carrier,

Initial Welgbt

+ 1

+ 1 + 2

+ 1+8+3

+ 1 + 2

+ 1

Initial I

8.90
8.91
8.91

4.71
4.71
4.70

5.41
5.42
5.48

6.14
6.17
6.19

5.88
5.42
5.40

4.71
4.70
4.70

8.91
8.91
8.90

Mean, 8.91

4.71

5.42

6.17

5.40

4.70

8.91

Part 1.— Hooke'8 Law:

fit

cm.

P
Dynes

1 added

1 removed . .

2added

2 removed . .

8 added

8 removed . .

.080
-.079

.071
-.070

.075
-.077

2389x980

- 2389 X 980

2278 X 980

- 2278 X 980

2851 X 980
-2851x980

298 X 100 X 980
802 X 100 X 980

820 X 100 X 980
825x100x980

818.x 100 X 980
805 X 100 X 980

Mean...

....

810 X 100 X 980

Qrealest deviation from Hooke's Luw. 4<£. This may be due to the
straightening of the wire, as the first elongation is so great.

Part 2. — Verlical distance of lower end of wire above mark on
wall = 0.3 cm. Distance from mark to upper end of wire = 272 cm.
.-. Length of wire = 272.0 - 0.8 =271.7 cm.= J.
Diameter of Wire,

ir?» = 8.1416 X (.0590)«= .6086,

ZeroorMterometer

wire

-.0041

0.1000 cm

-.0041

.1002 •'

-.0042

.1004 **

-.0042

.1008 "

-.0041

.1000 '•

-.0042

.1000 "

-.0042

.0988 •*

-.0040

.1002 ••

-.0040

.1000 •'

-.0040

.1000 "

Mean. - .0041 1 Mean, 0 09999 cm.
.'. DijiniftiT = 0 10410 cm. (0 04
is as Hcciirate as m-ed be, owing to
uncertainty in FjM,)

t. «., 9 :

.0065 sq. cm.

= 810x10^x980.

= 81x980xl0«x

= 9.7x10".

271 7
.0085'

BXPERIMENTS IN MECHAXIC3 AND PROPERTIES OF MATTER 167

Qaestions and Problems.

1. Is 2^ constant for uU forces of any njagnilude ? Does it apply
to compression as well as stretching ?

9. What kind uf vibrations will a stretched wire make if set in
motion longitudinally ? Why ?

8. What happens if a great compressing force is applied longi-
tudinally to a steel cylinder. (1), of small radius? (2), of lar^e?

4 A wire is elongated 12 millimetres by a force of 4 kilograms.
If a different length of the same wire is elongated 20 milli-
metres by the same force, what is the ratio of the lengths of
the two wires ? What change in cross-section, instead of in
length, would have produced the same eifect ?

5. How can a uniform rectangular beam of iron, 10 metres long,

mass 1000 kilograms, be made as stiff as possible ?

6. A hone is hitched to a loaded wagon by a long extensible

spring. How does the work done in Just starting the
wagon depend upon the elasticity of the spring?

7. A rubber band is stretched between two points, J. and B. If

A is kept fixed, and the end B moved to a position B, the
band being kept straight, prove that the work done depends
only on the distances of B and B from A, not on the path
followed by the end. {B may be farther from A than B,
or nearer.) Explain the exceptional case, when, during the
motion, the band meete a smooth peg and so is bent around
it, the line from AioB becoming thus a broken line.

EXPERIMENT 28

(two OB8BRVBRS ARB ADYISABLB)

Otjjeot. To determine the coefficient of rigidity for iron.
To learn a method of measuring intervals of time exact-
ly. (See " Physics/' Art. 82. )

Oeneral Theory. If there are two series of events in each
of which the same phenomena recur at regular intervals,
and it is desired to compare these intervals^ the most deli-
cate means known to science is the "Method of Coinci-
dences."

Suppose, for instance, that it is desired to find the period
of a pendulum by means of a clock which beats seconds,
and which is, as usual, not provided with any means for
indicating a fraction of a second, the natural method is to
count the number of seconds taken by the pendulum in
making a given number of vibrations. But it is evident
that one cannot ascertain the entire interval closer than
one whole second, which at once sets a low limit to the
accuracy of the method unless a great number of observa-
tions are timed, or unless one selects such a number of
periods of the pendulum that the interval to be timed is
exactly an integral number of seconds. The latter method
is that of coincidences, which it is proposedlo describe here.

By making an arbitrary mark near the centre of the swing
of the pendulum and observing its transits by the mark, and
listening to the clock, one soon notices that, while the pen-
dulum usually passes the mark between the ticks of the
clock, at regular intervals the two events occur exactly
simultaneously so far as the eye can tell. By starting at

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 169

such a '^ coincidence " and counting the number of ticks
and number of periods of the pendulum made until the
next exact coincidence, one would time a number of
periods which occupied very closely an integral number of
seconds. The determination would be accurate to within
the very small fraction of a second below which the eye and
car could not detect a departure from coincidence.

The method, as described above, can evidently be used
for comparing the periods of any periodic phenomena, no
matter what their relative length may be. The refine-
ments which are usually associated with the name of
"Method of Coincidences" apply, however, only when
the periods are very nearly commensurate — that is, such
that if r be one of the periods and T the other, r = NT±. £,
where N is a whole number and « a very small fraction of a
second.

After a coincidence, one of the systems whose periods are
being compared will evidently fall behind the other, since
NT seconds after coincidence they will be separated by an
iuterval c ; %NT seconds after, by an interval 2c ; and so
on. Finally they will be separated by an interval very ap-
proximately equal to T— «.«., the vibrations of shorter period
will have gained one whole period nearly on the other. An-
other coincidence will then occur which will appear to the
senses exact, unless, when T is divided by e, a fraction re-
mains which is greater than the shortest interval of time
the senses can detect. That is to say, for the coincidence to
appear exact T must = me, where m is a whole number, to
within a very small fraction of a second. We will then have
mr = mNT± nu = (mN±L 1) T.

m
Hence, by finding the whole number JV^ by a compara-
tively rough trial, and then by counting the number of
vibrations between coincidences made by the system whose
period is r, one can dispense entirely with noting the num-
t)er of vibrations made by the other system.

170 A MANUAL OF EXPERIMENTS IN PHYSICS

If it is more convenient to coant the number of vibi
tions of the other system between coincidences, one can do
so just as well. Let it hep. Then

p^mN^i;ovfn=:E^

p±l

This equation and the one on the preceding page simply
express the fact that, in the time observed, one body has
made one more vibration than the other.

If € is much smaller than the senses can detect there will
evidently be a number of apparent coincidences successive-
ly, for one vibration will have to gain several intervals c on
the other before the difference can be detected. In sach
a case the first and last of the successive coincidences is
noted, and the exact coincidence is taken to be half-way
between.

If, on the other hand, c is greater than the smallest in-
terval tho senses can detect, there will still be approxi-
mate coincidences whenever the more rapid vibration has
gained one whole period ; but these coincidences will no
longer appear exact, though the method can still be used
unless € is quite large.

To apply this method to a concrete case, let it be nsed in
determining the coefiBcient of rigidity of a certain material,
either brass or iron, by means of the torsional vibrations of
a wire made of the substance. (See " Physics,'' Art. 82.)

If a wire of a certain substance be held fast at one end it
will take a certain definite moment L to twist the other end
through an angle ^. Theory, as well as experiment, shows
that, if r is the radius of the wire and / its length,

unless d is too great, n being a constant which depends on
the material and condition of the wire. It is called the
"coefficient of rigidity" of the given substance.
Let the wire be clamped to a fixed support at its upper

%XPKRUI£NTS IN MECHANICS AND PROPERTIES OE MATTER 171

end so as to be saspeuded vertically. To its lower end let
there be clamped a flat disk, whose moment of inertia around
the axis of the wire is /. This forms a 'Version pendu-
lum." If the disk is twisted through a certain angle and
then let go, it will oscillate about its position of equilibri-
um. When, during the vibration, it is displaced through
the angle d> the moment with which the wire tends to un-

twist will be Z r= -gj-S.

Hence^ if a is the angular acceleration,

a/=i,ora = -^d,

which shows at once that the motion of the pendulum will
be a simple harmonic one of period

V vr*H

Whence, «=y;^,,

I and r can be measured directly. The moment of inertia,
/, of the torsion pendulum should be calculated from its
dimensions and mass, if possible, or else obtained of the
instructor; 7^ is determined by comparison with a seconds
clock by the method of coincidences as follows:

The wire of the torsion pendulum is made fairly long —
oyer a metre — and the approximate period found by count-
ing loth the number of periods of the pendulum and the
ticks of the clock between several coincidences, one after
the other. The period will in general be found not to be
very nearly an integral number of seconds.

In order to be able to use the method of coincidences to
advantage, it is then necessary to make the period as close-
ly as possible a whole number of seconds by changing the
length of the wire. It will be seen by the formula above
that the period varies as the square root of the length of
wire. Bearing this in mind, calculate the change in
length of the wire necessary to make the period an inte-
pal number of seconds, assuming the approximate period

172 A MANUAL OK EXPERlMKxNTS IN PHYSIOS

to be exactly right. Shorten the wire to the calculated
length ; then find the exact period by the metliod of coin-
cidences^ noting on the dial of the clock the number of
seconds p between coincidences. By the formula above :

T^ — ^ Ti= — ^, since 2J=:1 second. iV, that whole num-

ber which expresses the period most closely^ is easily found
by timing a few swings. Whether the + or — sign is to be
used depends upon which gains on the other, the clock or
the pendulum. If, after a coincidence, the pendulum drops
behind, its period is evidently longer than the whole num-
ber of seconds, and hence the — sign is used, and vice versa.
It must be noted that the coincidences are to be taken
when the pendulum passes its mark in one direction only —
not on the return swing.

Souroes of Brror.

1. The same precautions are necessary as in Experiment 1, Part 3.

* in order to insure that tlie transits observed are always those
at the same point of tlie swing exactly.

2. Unless the wliole apparatus is made on a large scale, the wire

will have to be very fine or the vibrations will be too rapid.
A fine wire must be very carefully treated to avoid kinks.
Moreover, the radius enters to the fourth power, and must,
therefore, be determined with the greatest possible care by
repeated measurements all along the wire. Further, the
wire should always be shortened when being adjusted, not
lengthened ; because, where it has been clamped a kink has
been made.
8. Care is necessary to start the torsional vibration so that the
pendulum does not swing as well as twist

Apparatus. A torsion pendulum ; wire at least 100 centi-
metres long; brass ring or bar ; micrometer caliper; metre-
bar.

Manipulation. Straighten the wire> remove all kinks, and
clamp it firmly at the upper end in the support near the
seconds clock. Attach the carrier with whatever load may
be furnished tightly to the other end, making the length of
the wire between the points where it is held considerably

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 17&

oyer a metre-^-iay 120 centimetres. Attach, a lo&g pointer
of flue wire to the bottom of the peudalnm, bend the end
of the pointer yertically down, and arrange in line with it
a Tertical thread and some other mark of reference as a
line of sight in observing the transits, jast as
ill Experiment 1. (Ask the instructor to start
the clock.)

Start the torsional vibrations through about
forty degrees. Several trials may be necessary
before this can be done without setting tlie
pendulum swinging. The best way is to start
with a much larger torsion than is finally de-
sired, and then to stop the swings with light
touches of the finger, which also decrease the
torsional vibrations.

One observer notes the transits of the pendu-
lum pointer past the thread, being careful to
keep the line of sight always the same; he
also listens to the ticks of the clock, and gives
a sharp tap the instant they coincide. The
other observer watches the clock -dial care-
fully (or the hands of his own watch, if the
clock has no dial), and notes the second of the tap, and
also, after that is made sure of, the minute and hour. Ob-
server No. 1 meanwhile counts the transits of the pendulum
in the same direction, still listening to the clock. When the
next exact coincidence occurs he taps sharply again, notes
quickly on paper the number of periods of the pendulum
made between coincidences, and begins counting the tran-
sits again for the next interval. Observer No. 2 again notes
the second, minute, and hour of the tap. Continue similar-
ly until five successive coincidences are noted.

Four intervals between coincidences have thus been found,
and the number of swings of the torsion pendulum in each
noted. Calculate the period as thus determined. Let it
he T, Measure I', the length of the wire between the fixed
points. .. Then calculate I, the length the wire should have

Fio. 65

174 A MANUAL OF EXPERIMENTS IK PHYSICS

in order that the period may be exactly the integral num-
ber of seconds next below T—i.e., if r'= 8.24, let T=zS,

for example. Bemember that /s^^* Shorten the wire

as closely as possible to this length. A good way is to mark
on the wire under the support with a pencil — not a scratch —
the difference of length necessary, and raise the wire through
the support until it clamps it at the mark exactly.

Readjust the pointer and line of sight if necessary. Pre-
pare to note the exact second of coincidences just as before,
thoQgh it is no longer necessary to count the swings of the
pendulum. Note thus a series of consecutive coincidences —
say eleven ; then allow some definite number — say nine— to
pass unnoticed ; then note eleven more, or whatever may
have been the number noted in the first series. If there is
doubt as to the number of coincidences which were not
noted, it can be readily calculated from the interval of time
between the first and last coincidence of the first series.
It may be used to determine N, the nearest whole number
to the period, and to note which is gaining on the other.

To find the exact interval between coincidences from the

data, proceed as follows : Let ^|, ^2> ^3' ^^'9 ^® ^^^ times of

the first set of coincidences; t'l^ ^'s, ^'3 those of the last set.

Then the approximate interval of the coincidences is the

t —t f ^ f

average of " ^ and ■ " ' • Call it p\ It is known,

furthermore, that between ^1 and t\ there are a whole num-
ber of such intervals, which is evidently the nearest whole

number to ^"7 ^' [ ^ , ^ would itself be exactly a whole

P \ P

number if there were no slight errors anywhere in the
observations. The object here is to eliminate these slight
errors.) This whole number represents the exact num-
ber of intervals between the times ^| and t'y There will
evidently be the same number between ^2 ^^d ^3, ^3 and t'^
and so on. From each of these pairs, therefore, a determi-
nation of the length of the interval can be obtained, and

SXPSRIMEXTS IN MECHANICS AND PROPERTIES OK MATTKK 170

the mean of these determinations is the most probable valne
of tho interval between coincidences to be found from the
data of the experiment. Call it p.

Calculate the period from the formula ^= ^ ..'

Measure at least twice, by means of the metre-bar, the
length of the wire between the points where it is held by the
carrier and by the support. Measure the diameter of the
wire at twenty different places with the micrometer caliper.

Obtain from the instructor the moment of inertia of the .
carrier, and add to it the moment of inertia of the load
placed upon it calculated from the mass and dimensions
of the parts.

[It is a very useful exercise, for students who have the
time, to determine the moment of inertia of the carrier by
ex}>eriment. To do this, remove the load and measure
the period exactly as above with the carrier alone. If the
length of the wire has to be changed again, to make the
period approximately commensurate with seconds, calcu-
late what the period with the carrier alone would have
been with the length of wire unchanged. Call it T.
Then, if /' is the moment of the carrier alone,

r IT

But if /'' is the moment of inertia of the load, which can
be calculated, /= /'+ /".

The experimeut is written for two observers, but one can
do it alone. He can find the approximate period with a
stop-watch, and, when the period is adjusted to be nearly
an exact number of seconds, he can find the time between
coincidences by making a mark on a sheet of paper at the
exact moment of coincidence and another at each suc-
ceeding tick until he can catch the time from the dial of
the clock. He can then count back by means of his marks
to the second when the coincidence was observed.

176

A MANUAL OF EXPERIMENTS IN PHYSICS

ILLUSTRATION
RxoiDiTT OF Iron Wibi

Deo. 10, 1896

To Find ApproxinuUe Period of Torsion Pendulum for a Length of
128 Centimetrst

Times of
Coincidences

Intenrals
(Seconds)

Na of Periods of
Pendalum In
these Intervals

Period

H. M. S.
2 14 82

15 14

16 21

17 08
17 45

42
67
42
42

5
8
6
6

8.40
8.88
8.40
8.40

Mean...

....

.... , 8.40 1

Hence, to make the period 8 seconds exactly, it should have a

leiiglh = (^*x 128 = 1115.

Length measured, after clamp was adjusted, and found to be exact
\y 111.8 centimetres.

7o Find Ema Period

Times of 0

oinridenoes

Setn.

lolmrvftlteiweeDCor-
responding Coinci-
dences in eaoh Set

Setl.

H. M. S.
2 44 6

H. M. S.
3 52 46

4120 sees.

47 85

56 16

4121 '

51 11

59 51

4190 '

54 10

4 8 11

4141 *

57 59

626

4107 *

8 1 81

958

4107 *

469

18 14

4095 *

828

16 24

4076 •

11 48

19 59

4091 '

14 58

28 80

4118 *

18 28

26 58

4110 "

Mean

4109 sees.

In Set T., 10 intervals between coincidences = 2064 seconds.
m Set II. , 10 intervals between coincidences = 2052 "

Mean, 2058 f^ccon

EXrEKIMENTS IN MEGUAXIGS AND PROPERTIES OF MATTEK 177

Hence the interval between coincidences is about 206 BeoondB.

Dividing 4t09 by 306. tlie result is 19.9. Tlie nearest whole num-
ber is 20, which must be tlie number of intervals between correspond-
ing coincidenoes in Set I. and Set IL

/. 20 intervals = 4109. /. 1 interval = p = 206.5.

In the time between llie two sets of coincidenoes it was found that
the pendulum made 10 swings in 79 seconds. Hence, i\r= 8. After
each coincidence the pendulum was found to get gradually ahead of
tlie clock. Hence,

^=m5TI=®-2065 = ^-^^«^^^«-

Tlie length of the wire was measured twice more and found to be
1 11.2, 111.4 centimetres. Mean, 111.8 centimetres, as before.

The "bob" of the pendulum consisted of a carrier loaded with a
rectangular bar and a ring of brass. The moment of inertia of the
Ciirrier alone was found by a separate experiment to be 90.68 = /(.

The dimensions of the ring were : External diameter, 6.85 centi-
metres ; internal, 6.50 centimetres ; mass, 40.1 grams.

.-. Its moment of inertia = /,= ^^^^ (3- 18)' 4- (2.75)* ^ ^^^

The dimensions of the bar were : Length, 5.16 centimetres ; breadth,
0.62 centimetre; mass, 16.5 grams.

.-. Its moment of inertia = /, = 16.5^-— ^'"*l-^5:^* = 87.1.

1«

.'. The entire moment of inertia of the pendulum is:

/=/, + /,+ /, = 481.4.
Diameter of wire:

0.2500 mm.

0.2501

0.2500

0.2498

0.2499

0.2496

0.2498

0.2499

0.2498

radius = 0.01249 cm.

Mean, 0.2499 mm.
•The coefficient of rigidity of the given specimen of iron wire is:
_8ir//_ 8irx 111.8x481.4
^" 7" r*~(. 01249)* X (7.961)" ■" '

178 A MANUAL OF EXPERIMENTS IN PHYSICS

Quostiona and Problems.

1. If the carrier io your experiment weighs 41 grams, what woakl

te llie radius of the hollow cylinder of the same mass which
could be substituted for it without changing the period?
(Neglect the thickness of the cyliuder.)

2. What is this radius called ?

3. What would be the radius of a very thin hollow sphere contain-

ing llie same mass of bniss which could be similarly substi-
tuted if ouc cubic centiiiieire of brass weighs 8.4 grams and
the moment of inert la of a sphere nbout a diameter is f Mr^l

•L A piece of shafting, 10 metres long, 5 centimetres radius, is
twisted through 1° by a curtain moment. How may Uie
shaft be changed so that the twist will be 80'?

^. if this same shaft is made three times ns long, and of twice
the (liameier, what twist will be produced by the same fon$
applied tangentially?

EXPERIMENT 29

Otgeet To verify the laws of fluid pressure^ p = pgh and
force F^pA. (See " Physics/' Art. 89.)

Oeneral Theory. It is proved from theoretical consider-
ations that the pressure due to a vertical height A of a fluid
of density p is j9 = ggh^ \ig is the acceleration due to gravity.

The general method of verification is to lower a hollow
cylinder, closed at one end, into a liquid, and to measure
the force required to sink it a given depth. The force up-
ward equals the pressure on the closed end multiplied by
the area; and if a weight mg makes the cylinder sink a
distance h^ the force A{>gh must equal mg^
or wi = Aph,

This may be verified by causing the cylinder to sink to
different depths, and measuring the corresponding weights,
by using cylinders of different sections, and by using liquids
of different densities.

Bonxoea of Brror.

1. Capillary action makes it difficult to read the exact depth of

the cylinder.
8. The metal cans may not be exact cylinders.
8. It is difficult to make the cylinders float exactly vertical, and

80 the scale marked on the cylinder may not occupy the'

same position during the experimenr.
4. The scales may not be accurate, and may not agree on the two

cylinders.

Apparatus. Two hollow metal (or wooden) cylinders,
closed at one end, of different cross -sections, each about
20 ceutimetres long, with equally spaced horizontal divis-

180

A MANUAL OF EXPERIMENTS IN PHYSICS

ioQB ruled on them ; a deep battery-jar ; one pound of lead
shot, of two or three millimetres diameter; a yemier cali-
per; kerosene.

Manipulation. Measure the diameters of the two cylin-
ders, and test the accuracy of their construction. Test
also the accuracy of the scales. Nearly fill the jar \%dtJi
tap water, and by means of shot so load the two cylinders
that they will just float upright in the water. It may be

necessary to redistribute the
shot by means of a long wire,
so as to make the cylinders
jjpi f LJ float exactly vertical. Add

r^ "^ shot, grain by grain, un-

til each cylinder floats so
that a division mark comes
exactly at the surface of the
water. (This may often be
best tested by looking up at
the mark from below the sur-
face.) When making this ad-
justment, press the cylinders
from time to time deeper into the water with the finger and
allow them to rise slowly, thus keeping the surfaces wet.

Then add enough shot to each cylinder to make it sink
one more division ; add grain by grain, and count the num-
ber added, estimating, if necessary, the fraction of a shot
which would make an exact- adjustment.

Add enough more to sink each cylinder another division,
counting the number added ; continue for as many divi-
sions as possible.

Record the number of shot in each case ; they should be.
the same for each one of the cylinders, it p=ipgh, for the
mass of each shot is approximately the same, and so the
weights are in the same ratio as the number of the shot ;
and it should, from the formula, require the same weight
to sink the cylinders each additional distance h. Take the
mean for each cylinder. Assuming the divisions to be

Fig. 66

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 181

eqnaly the pressures which correspond to equal depths of
the same liqnid are measured by the weight of these shot>
and miim^ should equal ^1:^3. Or if tii and n^ are the
number of the shot^ rii : n, should equal Ai : A^.

Float either cylinder in a similar way in a ditferent liqnid
of density p — e.g., kerosene oil — and measure the mean
number of shot necessary to lower it through one division.
If the mean number is n\, and Ui is the mean number in
the first liqnid, the foUowiag relation should hold :
ni:n'i=p:p'.

p and p' may be found in the tables, or may be measured
directly in the following experiment :

ILLUSTRATION
VXBinCATIOH OP P^pgh AVO ow pAssF

In Water, p = l

Decs, 1896

lUCylliidflr,
B»dliua86-f *o«l>

MCjlinder,
Radios 0.77+ inch

85 shot

86 "

86 -

87 •*
86 *

73 shot
78 "
78 •*
72 "

78 ••

Mean, 86 shot

72.6 8bot

240
Ratio of areas, ^ = 1.98.

Ratioofforce8,^ = 2.02

But the diyisioos of the
second cylinder were slight-
ly longer than those of the
first.

Qreatest deviation from mean less than 8^.

In KeromM OU, p = 0.788
lit Cylinder,
IUmIIub a60+ inch

28.0 shot
28.5 •*
28.0 *•
28.5 ••
28.0 •*

Ratio of densities, 0.788.

qAn

Ratio of pressures, ^^ = 0.786.

Mean, 28.8 shot

Qiieatloiis and Problems.

1. What error will be made in the above experiment if the float-

ing bodies are slightly conical 7 What if they do not float
Tertical ?

2. What is the effect of the atmospheric pressure ?

182 A MANUAL OF EXPERIMENTS IN PHYSICS

8. How was any work dooe ? Show bow it may be calculated.

4. The pressure at the bottom of a lake is three times that at a
depth of 2 metres. What is the depth of the lake? (Atmos-
pheric pressure = 76 ceutimetres of mercury.)

6. A sphere, 1 metre radius, is just immersed under water. What
is the total force upward ?

6. Show that the total thrust on the five faces of a cube filled with

a liquid equals three times the weight of the liquid, omittiog
atmospheric pressure.

7. A cube, 10 centimetres on each edge, is filled half with water,

half with mercury. Calculate the force on the bottom and
on each of the sid^.

8. A vertical tank having its base in a horixontal plane is to be

filled with water from a source in that plane. The area of
the croBs-section is 5 square metres, the height is 10 metres.
Calculate the work requ ired to fill it. Does this depend upon
the poBitioo of the inlet pipe T

EXPERIMENT 30

Olgect. To determine the density of a liquid by means of
" balancing columns/' (See " Physics/' Art. 91.)

Gteneral Theory. The height to which a liquid will rise
in a tube varies directly as the pressure and inversely as
the density, but does not depend upon the area of the tube.
This fact can be made use of to compare the density of the
two liquids. Two methods will be described : one for use
with liquids which do not mix or act chemically on each
other — e.g.y mercury and water; the other, for use with
any two liquids.

1. If the two liquids are contained, as shown, in a U-
tabe, and if the liquid of density
Pi stands at a height h^ above the
surface of separation of the two
liquids, and the liquid of density
P2 stands at a height ^3 above the
same level, then, since the press-
ure is the same in both arms of
the tube at this level.

na67

If, now, the level at which the liquid stands in each tube
be changed considerably by pouring in an additional amount
of one of the liquids, and h\ and A'j are the new heights of
the free surfaces above the new surface of separation,

9% *'i

184

A MA^'UAL 0¥ EXPERIMENTS IN PHYSIOS

(By taking the second set of observations the correction
for capilhirity is eliminated.)

2. If by suction the two liquids
are drawn up into two glass tubes
joined by a T-tube at the top, and
if the heights of the liquids abore
their respective free surfaces are
hi and h^, then, since there is the
same difference of pressure act-
ing on each,

and also, as in (1), if a second set
of readings be taken at changed
levels.

Consequently, if the density of one of the liquids is known,
that of the other can be calculated.

Sources of Error.

1. CorrectioDs may be necessary owing to capillarity.

2. The scale and tubes must be parallel and approximately vertlcaL
8. The pincli-cock in Method 2 must not leak, otherwise the press-
ure will change.

Method 1. — For Liquids which Do Not Mix

Apparatus. A wooden stand holding a glass U-tube, about
two centimetres in diameter, open at both ends ; a metre-bar
is held between the parallel branches of the U, the latter and
the bar being vertical ; a steel L-square and level ; a ther-
mometer ; mercury and water are the most convenient liq-
uids to compare by this method, and about 250 grams of
clean mercury in a lip-beaker, and clean tap water should
be used ; a wooden tray should be obtained in this and all

KXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 185

other experiments where mercury is used, and the whole ap-
paratus should be set up in it.

ManipulatioiL Pour enough mercury into the tube to rise
to a height of about four centime-
tres in each branch. With the aid
of the L-square and level, adjust the
stand so that the U-tube and metre-
bar are vertical. To do this, place
the level on one limb of the square
and set the other in turn against
the front and the edge of the metre-
bar. Pour water into one arm
until it rises to about two- thirds
the height of the tube. By a
thread lower the thermometer first
into the mercury and then into the
water, and note the temperature of
each. The thermometer should be
at least a minute in each before be-
ing read. Take two long, narrow
pieces of paper, each with straight,
smooth, parallel edges. Turn one
end of each into a paper tube, just
fitting over one of the glass tubes.
The other end should project over
the scale, so that its
upper edge marks
exactly the plane of
the top of the paper
tube. Slip one end

of the indices thus made over each branch of
the U and adjust to the level of the liquid sur-
faces in the same manner as the barometer in-
dex is adjusted. Thus determine o^, the read-
ing of the level of the top of the water column.
The edge of the paper is horizontal if it reads
fn, 70 the same at both edges of the bar. Measure

Fro. 69

186 A MANUAL OF EXPERIMENTS IN PHTSIGS

to the bottom of the hollow, or meniscus. In a similar man-
ner determine x^, the reading of the level of the bottom of
the water column, measuring to the top of the mercury me-
niscus ; and likewise x^y the reading of the level of the top
of the mercury in the other tube.

Pour in enough water to fill the tube nearly to the top.
Bead again the level of the same three surfaces. Call them
<^'v * 2> ^V Finally, redetermine the temperatures. Then,
if p and D are the densities of mercury and water at their
respective temperatures.

Note. — If only one reading is made on the height of a Uqnid in a tabe^

and if the tube id narrow enough for the surface of the liquid to be much

curved, a correction has to be applied equal to the diittanoe through which

the surface tension would raise or depress the liquid in the given tube with

reference to the level at which it would stand in a very large vessel This is

27*008 ^
equal to A = ■ for capillary tubes, where T is the surface tension of

rpg

the given surface ; ^, the angle at which it meets the walls of the tube ; r, the

radius of the tube ; p, the density of the liquid ; and y, the acceleration of gravity.

When h is positive, the liquid is raised above the level of a large vessel,
and vice vena, 2* and J^ differ very widely with the purity of the liquids and
cleanness of the tube ; and the formula is departed from as the meniscus
ceases to be spherical and becomes flat on top — t.e., in tubes of over two
millimetres in diameter.

For tubes of seven or eight millimetres the student may assume the fol-
lowing values :

18
Water-air, . . . A = -r-^ cm.

diameter

Mercury- water, > , 1.06

>A=rr- cm.

Mercury-air, ^ diameter

Measure the tube and see if the correction is large enough to affect the results

just obtained.

A uble is also given at the end of the volume for the correction of mercury

in glass tubes.

Method 2 (Hare's Method).— For Liquids which Mix
OR Affect Each Other Chemically
Apparatus. Two small glass jars or beakers ; two glass
tubes, each about a metre long and one centimetre or more

KXPERIMEXTS IN M£GfiAKlGS AND PROPfiRTIfiS OF MATTER 187

in diameter, bent at the bottom as shown in the fignre, so
that 60 to 90 centimetres of each can lie right along the
edges of a metre-bar placed between them, while the lower

ends dip in glass jars placed
one each side of tlie bar and
close to it ; a steel L-square
and level; a thermometer;
a clamp - stand ; a three-
way glass connector; two
short pieces of rubber tub-
ing, and one piece half a
metre or over in length,
with a pinch - cock on it ;
some kerosene or other liquid to con^-
pare with water.

Kanipolation. Fasten the tubes each
side of the metre -bar with string or
rubber bands. Support them in a ver-
tical position in the clamp -stand, the
ends dipping in the glass jars. Con-
nect the tubes to tWo branches of
the three-way connector, and fasten
the long rubber tube tightly over
the third. Having filled one jar with
kerosene and the other with clean tap
water (or whatever liquids are to be
compared), determine the tempera-
ture of each, leaving the thermome-
ter in each at least one minute. Now
open the pinch-cock and suck the liquids up so that they
stand at convenient heights in each tube. Close the cock
tight, and determine the level of the top of the liquid in each
jar and each tube. Use a piece of paper with a straight-edge
as described in Method 1. Bead the level of the liquid in
the centre of the tubes and jars, not at the edges.

Change the level in each tube as much as possible, and
read the levels again. Then, if pi is the density of the liq-

nan

188

A MANUAL OF EXPERIMENTS IN PHYSICS

aid whose heights have been x^-^x^ and x\^x'2, and p^ that
of the liquid whose heights have been x^^x^ and ac^'^ — as'4,

Pi («'3-«3)-(aj'4-«4)
Repeat the whole experiment four times^ chang^n^^ the
height of the liquids in the tubes each time. Finally^ note
the temperature again.

ILLUSTRATION
Dbmsitt or MxRCDBT BY Mkthod 1

Dec. 81, 1895

Temperature of mercury at beginning, 17.8^; at end, 20.2^;
mean, lO"*.

Temperature of water at beginning, 18''; at end, 20.4^; mean, 19.2®.
Density of water at 19.2°= D = 0.99855.
Diameter of tube = 0.8 centimetre.

»l

--I

s»

•'a

7.27
7.42
7 59
7.62

61.21
68.78
72.82
70.93

77.25
81.17
85.26
83.36

2 11
1.94
1.88
1.85

1.71
1.56
1.41
1.43*

6.46
6.86
7.06
600

18.56
18.55
18.53
18.52

Mean.

.... 1 ....

18.54

.-. p = 13.54 D = 13.54 x 0.99855 = 18.52.

.'. The density of mercury at 19° is found tu be 18.52.

QuestionB and Problems.

1. Why are you directed to take the temperature of the mercury

before that of water ?

2. What is the weight supported by the clamp-stand in Method 2?

3. What corrections would be necessary under each of the fol-

lowing conditions : Mercury tube much smaller bore than
water tube ; tubes and bars not verlictil but parallel to one
another ; neither vertical nor parallel ; cross-section of tubes
irregular ; insoluble particles of dirt, such as broken glass,
etc., in the liquids ; one liquid volatile in Method 2.

4. In Method 1 what would happen if more water were poured in

than the weight of all the mercury in the tube ?

EXPERIMENT 81

01(jeot. To determine the density of a solid bj means of
a chemical balance. Archimedes' principle. (See " Phys-
ics/' Art. 92.)

General Theory. When a solid is completely immersed
in a liquid (surronnded on all sides), it is buoyed up with
a force equal to the weight of the liquid displaced. This
is Archimedes' principle.

This buoyant force may be determined by weighing the
solid first in air, next when in the liquid ; for the difFer-
ence is the force desired. If the density of the liquid is
known, the Yolume of the solid can be calculated, because
it equals the mass of the displaced liquid, as just deter-
mined, divided by the density of the liquid. But as the
mass and the volume of the solid are now both known, its
density may be at once found by dividing one by the other.
Of course a liquid must be chosen whose density is
known and which does not in any way act on the solid ; if
the solid floats in the liquid it may be made to sink by
loading it with a heavy weight, due allowance being made
for this in the observations and calculations. Special
measures must be adopted for certain substances which are
granular or very porous.

Souroes of Brror.

1. There is always capillary action on the thread or wire which

supports the solid in the liquid.

2. Air-bubbles may cling to the solid.

8. Friction and capillary action between the object and the sides
of the vessel must be guarded against.

190

A MANUAL OF EXPERIMENTS IN PHYSIOS

FiaTa

Apparatus. A chemical balance and box of weights, 50
grams to .01 gram, with rider ; the cylinder 'measured
in Experiment 5 and a beaker large
enough to hold it when completely sub-
merged in water, but small enough to
go on a pan of the balance; a brass
stand designed to be placed oyer the
balance -pan and to rest entirely upon
the bottom of the case so that the beaker
may bo set upon it without its weight
acting upon the balance ; half a metre
or so of very fine wire or thread and a
thermometer ; camelVhair brush.

Manipulation. Weigh the cylinder
very carefully, as in Experiment 26.
(If the same cylinder has already been
previously weighed it need not be
weighed again.) Place the stand in
position and the empty beaker upon it. Hang the cylin-
der from the hook over the balance-pan by means of the
wire, so that it does not touch (or come very near to) the
sides or bottom of the beaker, but can be completely sub-
merged when the beaker is filled with water. Adjust the
wire so that as little as possible of it will be under water,
and, if possible, so that only one strand will pass through
the surface of the water, so as to avoid capillary effects.
In this position weigh the cylinder and the supporting
harness, and thus find the weight of the harness. Re-
move the beaker, leaving the cylinder hanging ; fill it with
enough water to completely submerge the cylinder when
it is replaced later ; place it under the receiver of an air-
pump, and exhaust the air from the water. Water from
which the air has recently been expelled by boiling may be
used to advantage and need not be exhausted of air under
the pump. See that the beaker is dry outside, replace it
on the stand, and hang the cylinder inside it again, so as
to be completely submerged in the water. Take the tern-

EXPEmiCENTS IN MECHANICS AND PROPERTIES OF MATTER 191

peratnre of the water and remove all air -bubbles which
may cling to the cylinder.

The balance will be found much less sensitive than when
the cylinder was in air; the cause is the capillary ac-
tion of the water on the wire. The sensitiveness can be
increased in the following manner, and the weight of the
cylinder in the water measured : Find the exact weight
in the pan necessary to bring the pointer to equilibrium
at a point five divisions to the left of the zero found for
the balance, and also the weight necessary to bring it to
equilibrium five divisions to the right of the zero. Both
weights should be found to tenths of a milligram, and
their mean will be the weight necessary to bring the point-
er to its zero — i. e., the correct weight of the cylinder in the
water. . Finally, determine the temperature of the water
again.
Then if Wi = weight of cylinder -f- wire in air,

fTjts '' " " in water + weight of wire.
FT,— W2 = los8 of weight of cylinder in water (neg-
lecting loss of weight of wire in water),
= weight of water displaced by cylinder.
Let F= volume of 1 gram of water at given tem-
perature,
then F( IT, — TTj) = volume of the cylinder.

From the volume of the cylinder thus found and its
weight in vacuo as previously measured, its density may
at once be calculated.

ILLUSTRATION

J«n. 13, 1897
DiMBiTT or Hard Rubber

Cylinder 1. Box of weights, M d42. Balance, M 258.

Zero of balance was 10.87.

Weight necessary to balance cylinder + wire in air :

With 81.301 grams in left pan, pointer read 10.89
•* 81.802 ** " " *' •* " 18.68

/. Wi = 81.801 + (^of O.OOA = 81.8011 grams.

192 A MANUAL OF EXPERIMENTS IN PHYSICS

To find the mass of the cyliDder suspended in water + that of the
wire:

The zero was redetermined and found to be 8.49. Hence the weights
necessary to bring pointer to 8.49 and 18.49 were found.
7.587 grams in pan, pointer read 12.8

7.688 ' ** " 14.4

.'. 7.5874 grams would bring it to 18.6 —

7.681 grams in pan, pointer read 8.2

7.682 •• ' " 4.8

.'. 7.6812 grams would bring it to 8.5 +
.-. The mean of 7.5812 and 7.5874 grams would bring it to the zero-
point 8.49— f. e„ 7.5848 grams, which is the weight sought = TT,.

Temperature of water at starting = 16.6^, at end 17.0°.

Mean temperature = 16.8^.

Vol. of 1 gram of water at 16.8° = 1.00101 cubic centimetres.

Tr, = 31.8011

Wt= 7.5812
.'. 28.7199 = weight of water disphiced.
.'. 28.7199 X 1.00101 = vol. of cylinder = 28.7489 cubic centimetres.
Mass of cylinder in vacuo = 81.091 grams.

.\ Its density is ^^^ = 1.8094.
28.744

Queationa and Problems.

1. Compare the volume of the cylinder found as above with that

obtained for the same cylinder in Experiment 5.

2. What is the total weight on ihe brass stand while the cylinder

is being weighed in the water ?

8. How could densities be determined with a platform-balance ?

4. 80 cubic centimetres of lead, 20 cubic centimetres of cork,
10 cubic centimetres of iron are fastened together and sus-
pended in water from a balance. What is the apparent
weight?

6. A solid weighs 8 grams in water, 9 grams in air. What is its
weight in vacuo ? Temperature is 0° C. and pressure 76
centimetres of mercury.

6. A brick is dropped into a vessel containiog mercury and
water. What will be its position of equilibrium ?

EXPERIMENT 32

Object. Use of Nicholson's hydrometer and determina-
tion of the density of some small solid^ such as a coin. (See
'*PbyBic8/'Art. 93.)

General Theory. A Nicholson's hydrometer is simply a
floating body which has pans designed to carry a small
body, first in the air and then in the liquid. If the weight
is known which/ will submerge the hydrometer to a def-
inite point, then thediflPerence between this and the weights
which, together with the small solid, bring the hydrometer
to the same point, when the solid is in the air or in the
liquid, give the weight of the solid in the air and its ap-
parent weight when in the liquid. Consequently, by Archi-
medes' principle its density may be calculated, if that of
the liquid is known. If the solid is one which would it-
self float on the liquid, it may be confined below the sur-
face by a wire cage permanently attached to the bottom of
the hydrometer.

Sources of Brror.
The same as in the preceding experiment.

Apparatus. A Nicholson's hydrometer and tall glass jar ;
a box of small weights, 5 grams to .01 gram ; a small brush
with a long handle, or a piece of cotton wool tied on a stiff
wire ; a five*oent piece, or any small coin or other light ob-
i^t of like size, the density of which is to be determined ;
« thermometer ; a card-board or paper cover for the jar.

A« ordinarily made, a Nicholson's hydrometer is a water-
tight, hollow cylinder of metal, usually with conical ends.

194

A MANUAL OF EXPERIMENTS IN PHYSICS

At one end there is a long^ thin stem carrying a small
platform at the top> and at the other end is hung a heavy
conical weight to hold the cylinder upright when floating
in water. There is a mark around the
stem about half-way. up. The conical
weight is flat or slightly hollowed on
top^ so that it can hold the object whose
density is to be determined.

Manipulation. Glean the stem of the
hydrometer with a little caustic soda or
potash, and rinse under a water-tap ; fill
the jar with clean water, which has been
previously boiled and allowed to cool,
and place the hydrometer in it ; remove
all air-bubbles with the brush or cotton,
and cover with the card-board, making in
the latter a slit to the centre, wide enough
to prevent the stem rubbing against
it. Take the temperature of the water.
Dip under water the entire stem of the
hydrometer to a point well above the

^ — ^ mark upon it, so as to wet it ; and try

and keep the stem wet during the ex-
periment. Let the hydrometer rise to its natural position
again, and add enough weights to bring the mark down just
below the surface of the water; then slowly remove weights
until the mark is exactly level with the surface. If too much
weight happens to be removed, add weights until the mark
is below the surface and try again. If, finally, the smallest
weight which can be removed changes the mark from be-
low to above the surface, estimate the fraction that would
have to be added to bring about an exact balance at the
surface. Let the weight thus found be Wi grams.

Take off the weights, place the coin in the pan, and then
add exactly enough weights to bring the mark again to thB
water-level, determining this weight just as in the first
case. Let it be w^ grams.

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 195

Now remove the coin (leaving the weights on the pan)
and place it on the cone at the bottom of the hydrome-
ter, being very careful to allow no air to be caught under
it^ and also to remove any air-bubbles on the instrument.
Additional weights will now have to be put in the upper
pan to bring the mark to the surface. Let the total amount
in the pan necessary to do this be w^ grams, which should
be determined in exactly the same way as «;, and t&j.

Finally, redetermine the temperature, and let F=the
volume of one gram of water at the mean temperature of
the experiment.

Then Wg— Wi= weight of coin in air.

W2—W2= weight of water displaced by coin.
(Wg— W2) ^= volume of water displaced by coin,
= volume of coin.

Then ^ — ^~ ' is the density of coin if the buoyancy

of the air is neglected.

Be very careful throughout not to allow the hydrometer
to rub against the glass or card-board cover, not to let air-
bubbles collect on the stem, and not to wet the upper pan
of the instrument or the weights.

ILLUSTRATION

„ Jan. U, 1897

Htdromkter

Dennty of a Nickel Five-cent Piece,

Weight in pan without coin: 9.52 grams did not quite raise mark
to surface, 9.51 grams were too little. .'. Wi = 9.516 grams.

Nickel in upper pan: 4.82 grams too little, 4.33 grams too much.
.'.«?,= 4.825 grams.

Nickel in lower pan: 4.900 grams jiist brouglit mark to surface.
.*. w,= 4.900 grams.

.'. Weight of nickel = 9.515 - 4.825 = 5.190 grams.

Loss of weight of nickel in waier = 4.900- 4.825=0.675 grams.
Temperature of water at starting = 18.4°; at end, 16°; mean = 147°.

Volume of 1 gram of water at 14.7°= 1.00069 cubic centimetres.

.'. Density of the nickel five-cent piece = a 575 x~l"Q0069 ~ ^'^^'

Kan. — ^This should not be mistaken for the density of nickel itself, since
the five-cent piece is an alloy of nickel with copper.

196 A MANUAL OF EXPERIMENTS IN PHV9IGS

Questtoiift and Probl«nui.

1. What would be the effect of a cooBiderable change {n the tem-

perature of the water during the experiment?

2. What effect hag capillarity on this method of determining

density?

8. If the surface tension of water is 74 dynes per centimetre, and
the slem of the hydrometer 2 millimetres in ciiaiueler, how
much less weight in the pan is necessary to sink the hy-
drometer to the mark than if there were no surface tension?

4. What are the adTantages of the slim stem ? Of the conical ends?

EXPERIMENT 38

Otgeot. To determine the density of a coin or other small
solid by means of Jolly's balance. (See " Physics/' Art. 93. )

General Theory. A Jolly's balance
is, essentially^ a long, fine spiral
spring, suspended from a fixed arm
so as to hang in front of a vertical
scale graduated on a long strip of
mirror. The spring carries at its
lower end two small weight -pans,
the lower of which is always im-
mersed in water in a glass vessel
placed on a small platform provided *
for it. I

A white bead on the wire sup-
porting the top pan serves to mark
the position of the bottom of the
spring relatively to the scale di-
vision— i.e., the extension of the
spring. The mirror on which the
scale is engraved is on the front of
a hollow vertical column, and the
arm which supports the spring is
carried by a rod which slides inside
the column and may be clamped
at any desired height. The height
of the platform can also be ad-
justed. 11 \ nw
The principle of its use is essen- q§a.^^^O>--lft^^
tially that of Nicholson's hydrom- T^ ? ^^
eter ; a certain weight extends the '"'^ ''^

198 A Manual of experiments in paysics

spring a definite amount^ the solid in air pins a measared
weight extends it the same amount^ the solid in the liquid
pins another measured weight also extends it the same
amount ; hence the weight of the solid in air and in the
liquid may be determined and the density calculated.

There is, however, one point of superiority of Jolly^s
balance over the hydrometer : a spiral spring obeys Hooke's
Law quite closely for small elongations — i. e., the elongation
is proportional to the change in the stretching force, and so,
if no weight in the box of weights is small enough to make an
exact adjustment of the spring, the fraction of the weight
which would have done so may be calculated from a meas-
urement of how far the smallest weight extends the spring.

Sources of Error.

The same as io tbe two precediDg experimeDtB.

Apparatus. A Jolly^s balance ; a box of weights, 10 to
.01 grams; a small beaker ; a thermometer ; a small brush;
a silver coin or other small solid.

Hanipnlation. Set up the apparatus where there is a good
light on the scale, and level it so that the spring hangs
parallel to the scale and the image of the bead is on the
scale when the eye is held in such a position that the bead
just covers its image. Pill the beaker nearly full bf water
which has been boiled, and put it on the platform. Place
about six grams on the upper pan (the weight of the coin
being less than six) and adjust the height of the top of
the spring and the platform, so that the lower pan hangs
well under water about the middle of the beaker, and so
that the white bead does not come below the engraved
scale. Take the temperature of the water. Placing the
eye so that the bead just covers its image, note the po-
sition of the top of the latter on the scale to within a tenth
of the smallest division. Note, similarly, the position of the
top of the bead for two other weights different from the
first — e.g., 6.1 and 6.2 graQis — but make no change great
enough to necessitate readjusting the height of the top of

EXPERIMENTS IN MECHANICS AND PROPkRTISS OF MATTER 199

the spring. Remove the weights, holding with a finger the
spring extended ; place the coin in the upper pan and add
weights enough to bring the bead exactly to the first of
the positions previously observed. If^ in the final adjust-
ment to this end, the smallest change possible with the
weights at hand carries the bead beyond the desired po-
sition, estimate the exact fraction which would carry it
there, from the knowledge previously obtained as to the
elongation produced by a small weight. The diflference
between the weight previously found and the one thus
found necessary, in addition to the coin, to produce the
same extension is evidently the weight of the coin. Simi-
larly, find the weight of the coin, using as the standard po-
sition each of the other positions of the bead noted before
the coin was introduced.

Remove the coin from the upper pan (keeping tlie spring
extended) and place it in the pan under water, being care-
ful not to catch a bubble of air under it. Add weights to
the upper pan until the bead is brought once more into
each of the three positions noted successively. In each
case the difference between the weights needed in addition
to the coin when the latter is in the upper pan and when
it is in water is the loss of weight of the coin in water.

Take the temperature again at the close o^the experi-
ment. Be very careful throughout to keep air-bubbles
from collecting on any part of the apparatus under water,
and to keep the upper pan and its contents dry.

ILLUSTRATION , „ ,^

Jan. 13, 1897

DnsiTT OF Com Siltxr bt Jollt*s Balance, Detcrmined from a Silyer Dime.

PMtlon of
Bead

Welghto,
wItlioaiCoin

Weights, with
Coin Id Air

Weights, with
Coin in Water

Weight of
Coin

1.088

in Water

428.1
426.3
427.4

Mean.

OraniB
6000
6100
6200

Grams
8.824
8.924
4020

Grams
4.084
4.129
4.227

Grams
2.176
2.176
2.180

Grams
0.210
0.205
0.207

....

2.177

0.207

2()0 A MANUAL OF EXPERIMENTH IN PHYSICS

Greatest deviatioa from mean is about 8ii.

Initial temperature of water, 16.6° ; fiual, 17.4° ; mean. 17.0°.

Volume of 1 gram of water at 17° 1.00106, which may be taken as 1,
within the range of error of this experiment.

.'. Volume of 0.207 grama of water = volume of coin = 0.207 cubic

centimetres.

2.177
.'. Density of coin silver = a-qq^ = 10.5.

QuesUoiia and Problems.

1. What percentage of error in the density of a coin would be

made if a bubble of air 1/10 the size of the coin were c aught
under it in this experiment ?

2. Why is the lower pan kept under water throughout ?

8. What would be the effect of a considerable change in Uie fem-
perature of the water during the experiment ?

EXPERIMENT 84

Olijjeet. To determine the density of a floating body.
(See "Physics/' Art. 94.)

General Theory. If a body is floating in any liquid^ the
weight of the liquid displaced equals^ by Archimedes' prin-
ciple, the weight of the body itself. So, if v is the volume
of the liquid displaced, p the density of the liquid, and v'
and p the volume and density of the floating body,

vpg= v'p'g,
or

p=pr/v'.
Consequently, if the floating body has a shape which ad-^
mits of accurate measurement, and if the density of the
liquid is known, that of the floating body can be at once
determined.

In this experiment a rectangular block of wood will be
floated in water.

Sooroes of Brror.

1. The main source of error is the diffloulty of measuring exactly

bow much of the block of wood is under water.

2. The weight of the block may be iDcreased by the water sonk-

ing in. .

Appamtufl. A rectangular block of wood which has been
soaked in paraffine ; a large battery* jar ; a metre-rod ; a
thermometer.

Maalpulation. Measure the dimensions of the block of
irood by means of the metre-rod. If its edges, which are
parallel, all have the same length, by means of a sharp pen-
cil mark millimetre lines along the edge which will be ver-

202

A MANUAL OF EXPERIMENTS IN PHYSICS

tical when the block is placed in water. Pill the battery- jar
with tap water ; read its temperature ; set the block float-
ing, and note the reading of the water on the marked edge
of the block ; remove the block and measure accurately the
distance along the edge from the
bottom corner to the point where
the water stood. (Estimate to
tenths of a millimetre.) The ratio
of the volume of the water displaced
to that of the block equals that of
this height just measured to the
length of the entire edge if the block
is perfectly rectangular.
Repeat, turning the block over after having carefully
dried it. Place the block again in its first position and
repeat the observations. Take the temperature of the
water again.

_ji --_._

FiO. 75

ILLUSTRATION
Dknsity by Flotation
Block measured and found to be rectangular.

Dm. ao, 1806

Length of Edge

5.24 cm.
5.23 *'
5.23 "

5.25 •*

Mean, 5.24 cm.
Mean temperature of water, 18® C.
of accuracy of the experiment.

.'. density of wood, p'= r^ = 0.607.

length of fedgQ fn Water
8.18 cm.

8.16 ••
8.20 *•

8.17 '*
Mean, 8.18 cm.

.*. density is 1 within the limits

QaestioxiB and Problems.

1. A block of brass, 10 centimetres thick, floats on mercury. How

much of its volume is above the surface, and how many centi-
metres of water must be poured above the mercury so as to
reach the top of the block ?

2. An iron spherical shell, 5 millimetres thick, floats half immersed

in water at 4° C. Calculate the diameter of the shell.

SSPfiftlHENTS IN MECHANICS AND t>ROP£RTlSS Ot* MATTER i6S

8. A block of wood weighing 1 kilogram, whose deDslty is 0.7, Is
to be loaded with lead so as to float with 0.9 of Its volume
immersed. What weight of lead is required (1) if the lead
is on top? (2) if the lead is below ?

4. A floating body projects 1/5 of its volume above water at
4** C, what proportion would project at 80*' C? What is
the density of a liquid from which 1/8 of its volume would
projectt

EXPERIMENT 36

Objeot. To measure the snrface-tension of pure and im-
pure liquid surfaces. (See " Physics/' Art. 97.)

General Theory. By definition, the surface-tension T is
the force which acts across one centimetre of a liquid snr-
face. There are many methods by which it may be deter-
mined, but only two are in the least suitable for element-
ary laboratories, and these will be merely indicated here,
not described in full.

1. Method of Capillary Tubes. — This method is suitable
for pure liquids which wet the solid forming the tube — e. g.,
water in glass. The method is to draw out a fresh capillary
tube so that its inner surface is free from all dirt, and, after
it has been well soaked in distilled water, to place it yertical,
keeping one end under the surface of the water and measur-
ing the height to which the column of water rises inside. It
is easily seen that, if

A is this vertical height,
p is the density of the liquid ;
r is the radius of the tube at the top of the liq-
uid column;
g is the acceleration duo to gravity ;

, 2T ^ pgrh
h=i — , or7'=^.

These quantities can be measured as follows :
r, by means of a micrometer eye-piece, the tube
being carefully broken off at the proper point.
A, by means of a scale, or by a hook shaped as
^^^^/ shown. This can be made of a metal wire

Fio. 76 or of a piece of glass. It is fastened to the

EXPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 205

capillary tube so that the lower point just comes up
from below to the free surface of the liquid in the
basin, and the distance between the upper point and
the surface of the liquid in the capillary tube is meas-
ured by a cathetometer ; the distance between the two
points can be determined once for all by a cathetome-
ter or dividing-engine.
p and g may be found in tables ; and hence T may be cal-
culated.
2. Method of Ring and Balance. — If a metal ring is sus-
pended horizontally from one arm of a balance so that the
plane of its lower side is exactly parallel to a surface of a
liquid, the ring may be lowered into the liquid, then raised,
and the additional force necessary to tear it away from the
liquid measured by means of the balance.

If r is the radius of this ring, the force required is pro-
portional to rTj and so the surface-tension of many sur-
faces may be compared. This method is particularly suit-
able for the study of the effect of the presence of grease,
dirt, etc., upon the surface-tension of water.

ILLUSTRATION

SUBVAOK-TKNfilON OV DiSTILLRD WaTRR

Jul 15, 1897

TQbeNal

Tube No. a

80.47
80.47
80.48
80.48

0.0793
0.0798
0.0798

80.48
80.49
80.50
8061

0.0788
00799
0.0800

Me>ui. 80.475

0.0793

Menu. 80.495

0.0796

Temperature, 16°. p = 0.999

Byflnttube. T=71M
By second tube, 7=72.28

EXPERIMENT 30

Oiyect To measure the density of a gas.

General Theory. The most obvious method of measuring
the density of a gas is to weigh a known volume of it on a
balance — i.e., to weigh a hollow sphere empty and then
when filled with gas, and to measure the volume of the
sphere. The difficulty is in securing and keeping a vacuum
in the sphere. It is, however, the method which will be
described in detail.

Other methods which are more suitable for chemical
purposes, or which give comparative results, may be found
in special treatises.

SoQxces of Error.

1. It is impossible to secure a perfect vacuum, and allowance

should he made for this.

2. It is possible that the stopcocks may leak during the weighing.

3. It is difficult to measure the volume accurately.

4. The temperature of the enclosed gas is difficult to determine.

Apparatus. A hollow brass sphere with
stopcock and hook ; an air-pump, such as
a good aspirator water -pump; rubber tub-
ing ; thread ; metre-rod ; two L-squares.

Manipulation. By means of the metre-rod
and L-squares measure the external dimen-
sions of the sphere in as many directions as
possible ; weigh it, with the stopcock re-
moved, on the platform - balance ; and cal-
^^' '" culate the internal volume, assuming the

density of brass to be /o = 8.4. For, if r is the external
radius, m the mass, and v the internal volume,

jaLPERIMENTS IN MECHANICS AND PROPERTIES OF MATTER 207

(This meaanrement might, of coarse, be made more ac-
curately by filling the sphere with water at a known tem-
perature, and measuring its temperature and mass.)

Put the stopcock on the sphere, and by means of rub-
ber tubing join it to the aspirator-pump. Soft rubber
tubing sboald be used ; and all joints should be tied tight-
I3' with linen thread and then coated with shellac. Ex-
haust the air by allowing the pump to run for about twenty
minutes ; close the stopcock ; remove the sphere from the
pump ; weigh it by means of a high balance, from one of
whose pans it can hang. Gall the mass t^j. Open the
stopcock and allow the air to enter slowly into the sphere ;
when it has ceased to enter weigh again and call the weight
m^ (These weighings must be done on each arm of the
balance in turn, and with extreme accuracy.)

The water-pump used exhausts to within a pressure of a
few centimetres of mercury ; hence w, — mj is the weight
of the air inside the sphere, because the buoyancy of the
atmosphere produces no effect in this difference. Note
the barometric pressure and temperature. The density of
air, then, at this pressure and temperature is

Bepeat the measurements of m^ and m^ and again note
the temperature and pressure.

ILLUSTRATION D«x 21, IMS

Dbmsitt or Air

Pressure, 75.62 centimetres. Teoiperature, 17.6^ C.

Weight without stopcock, 001.7 grams.
.-. (iir7.6»-c)8.4 = 601.7.

.*. « =1781 cubic centimetres.
Weight empty, with stopcock,

mi = 620.106 grams.
Weight full of air, with stopcock,
«!, = 622.279 grams.
Mean, 15.200 cm. /. m, - mi =2.178 grams.

...i>=!!!iZJ!il=o.00128.

Bzieraal Diameter
of Sphere

15.22 cm.
15.18 "
15.16 "
15.24 "
15.20 ••

308 A MANUAL OF KXPERIMENTS IN PHYSICS

QaesUoiui and Proble

1. ABSuming tbat the aspirator- pump producer a vacuum sucIj
that tbe pre«ture is 2 centimetres of mercury, make the nec-
essary correction in tbe calculation, assuming the approxi-
mate value of the density, 0.00122» to be accurate enough for
the purpose. (See "Physics," Art. 108.)

S. Calculate tbe mass of 1 cubic metre of dry air and 1 cubic
metre of air saturated with water vapor, the temperature
being 20^ C, and the pressure 76 centimetres of mercuiy.
(See "Physics," Art. 107.)

EXPERIMENT 37

Otgeet. To prove that Boyle^s Law holds approximately
for air. (See " Physics,'' Art. 108.)

Oeneral Theory. It has been found by experiment that
if the temperature is kept constant^ the pressure of a gas
is directly proportional to its density. This is Boyle's Law.
Expressed in an equation, it is

p = kp, if temperature is constant, or, substituting for

p its value m/v,
pv = km, if temperature is constant, where ^ is a con-
stant for a definite gas at a definite temperature.
To verify this law it is necessary simply to measure the
pressures and corresponding volumes of a given amount of
a gas under such conditions that the mass and temperature
do not change ; and the products of p and v should have
the same numerical value. A quantity of gas — e.g., air —
is enclosed in a glass tube, one end of
which is sealed, by having some liquid like
mercury (which vaporizes only slightly) fill
and close the rest of the tube. The volume
may be calculated from measurements ot
the length and cross-section of the tube ;
and so may the pressure, by determining
the difference in level of the mercury sur- J
faces in the closed tube and the one open
to the pressure of the atmosphere.

Thus, if the tube is as shown in the fig-
ure, the air being confined in the closed
branch, the pressure on it is the atmos-
pheric pressure + pgh, where h is the no. 78

no A MANUAL OF EXPfiRIMEKTB IN. PHYSICS

difference in vertical height between the two sarfaces of
the liquid, and p is the density of the liquid, the + sign
being taken if the open column is higher than the other,
the — sign if it is lower. This apparatus is designed to
study pressures greater than one atmosphere.

Souxoes of Error.

1. It is sometimes difficult to prevent bubbles of air from enter-

ing or leaving during the experiment.

2. When a gas is compressed its temperature rises greatly ; and

conversely.
8. If tb» glass tube is not perfectly clean the mercury sticks to
it« and the tube must be tapped before reading until the
surface becomes convex.

Apparatus. A barometer tube bent into the form of a
U, the closed branch being considerably the shorter, and
the space between the branches just wide enough to hold
a metre-bar. The whole is mounted on a stand so that
the bar and both branches of the U are parallel and ver-
tical. The student also needs a steel L-square and level,
a thermometer, a funnel, a mercury - tray, and enough
mercury to fill a length of about a metre of the U-tube—
I. e., 600 or 700 grams for a tube of 8 millimetres' di-
ameter.

Manipulation. Set up the apparatus in the mercury-
tray, being careful to see that the tube, funnel, and mer-
cury are clean. If the latter has a little dirt on it, it may
often be improved by making a cone-shaped bag of a piece
of glazed paper, pricking a few pinholes in the bottom, and
filtering the mercury through these, not allowing the last
few drops to escape. Set the metre -bar vertical, as de-
scribed in Experiment 30. Pour enough mercury into the
open branch of the U to close the bend and rise seven or
eight centimetres in the open branch. There is now im-
prisoned in the closed branch the mass of air to be experi-
mented on ; and great care must be taken not to change
the quantity by allowing any to escape or a bubble more

KXP£R1MSNTS IN MECHANICS AND PR0PBBTIB8 OF MATTER 211

to enter. Note the temperature of the air by lowering the
thermometer into the mercury, which is a good conductor
of heat, and also by hanging it alongside the closed branch
containing air. Bead and note the barometer, taking care
to see that the mercury in the
basin just reaches the index. (See
Experiment 26.) Do not ''cor-
rect ^* the reading. Determine the
level of the top of the interior of
the closed tube with the straight-
edge of a slip of paper, as de-
scribed in Experiment 30. Meas-
ure to a point below the apex of
that surface about 1/3 of the
height of the curved surface which
closes the top. Similarly^ by
means of the strip of paper^ de-
termine the level of the top of
each of the two mercury columns.
Make all readings to the tenth of
a millimetre. Note the precau-
tion to tap the tubes, so as to
shake the top of the mercury col-
umns a little before reading. The
pressure on the air is obviously
the barometric pressure + that due
to a column of mercury whose
height equals the vertical height
between the surfaces of the two
columns.

Change the volume and pressure of the enclosed air by
pouring in very carefully and slowly enough mercury to
raise the level in the open tube about ten centimetres. Be
careful not to let any air in or out of the closed end in
doing so. Kote the level of the mercury columns as before.
(The level of the top of the closed space should not change
unless the tube slips in its clamps.)

FlO. 7»

212

A MANUAL OF EXPERIMENTS IN PHYSICS

Repeat, similarly, five or six times more, adding aboat ten
centimetres eaoh time to the height of the mercury in the
open tube. Finally, read the barometer and take the tern*
perature of the enclosed air again.

Measure the total length of tube filled with mercury in
the last experiment. Pour all the mercury into a separate
clean vessel previously weighed and weigh it.

Plot the measurements in a curve, the volumes being ab-
BcisssB, the pressures ordinates.

ILLUSTRATION j^i. ^^ i^

To Proti that Boyli*s Law is Appbozimatblt True por Aik —

pV = kfn AT Ck)NSTANT TiMPKRATURB

Barometer at beginning, 77.026 cm. )

" end, 77.016 *' l ''uncorrected/'

Mean, 77.02 cm. )
Temperature of air, l(i.iy> at beginning ; 17.4^ at end.
Temperature is, therefore, constant approximately.
In the following table A is the cross-section of tbe tube, assumed
to be uniform, p tbe density of mercury at 17^, and g gravity.

Pt the pressure, is the barometric pressure + tbat due to the column
of mercury in tlie tube.

Top of

Cloeed

Mercury In
Closed Tube

Mercury In
Open Tube

Tube

16.29

8 26

12.08 X A

8.88

S2Mpg

987.6 X pgA

4.09

11.20 x^

16.27

08.20 pg

987.9 X pgA

6.08

10.26 X A

24.19

96.18 f^

986.8 X pgA

6.94

9.86 x^

84.61

106.69 p^

987.8 X pgA

6.98

8.81 x^

48.86

118.89 f^

988.0 X pgA

8.00

7.29 X A

66.46

195.^ pg

987.6 X pgA

9.12

6.17 X A

92.24

l^Mpg

988.1 X pgA

Mean,

....

987.6 X pgA

Since pg and A are constant, the greatest departure from the mean
is .8 or M%. The law is, therefore, shown to hold for air to tliat
degree of accuracy for pressures of from 1.08 to 2.1 atmospheres.

Non. — For studying the behavior of air at pressures less than that of the at- .
mosphere it may be confined in a tube, as shown in figure 80. To acoonipUih
this, hold the tube closed end down; fill it nearly full of mercury ; put a finger
oyer the open end ; carefully inyert and place the open end in a deep basin of

KXPERDIENTS IK MECHANICS AND FBOPERTIES OF MATTER 218

mercory. The impriioiied air bubbles up to the top of the tube, and its vol-
nine snd prtitinre may be changed at will by raising the tube. They may

be ea«l1y mGaeured^ihe
only !t»striitn<?nt Di^d-
ed being tt uietrc-roJ.
Another form of in-
»trum<^tit wlikli niav
be used to Btudjf tlie
proportjea of tbo ftir,
At press 11 re 8 both
^Gflt^r und Igau thun
that of the atmos-
pherCt is alio wn in
fipurt^ 81, It t'onaisis
of two glass tubeg,
iiboiit SO ceiitiinetreB
lorp^, connected bj a
flexible rubber tub*;
the end of one is sealed,
ftnd iiierftUT7 ia poured
mto the open end of

the other «o «i] to itopmon some air in
the clofled tube. The two g\BM tiibeg ar^
he^d ID cUmpfl ^tiicb can be tnoved
aloug a vertk^l ec«le.

In ihb iv*y the surface of the raereurj
ill the open tube may be brought either
abore or b«low ih^t In iHb closed tube.
The prewfUre und volume may be thu»
varied at wiNi&Ddfas they can bfj meas-
uredt Bojle^a I^w c^n be verified.

QtieBtiona and Problans, j^l

1. Recincc the prcs.su re in your ^

Last experitDcnt to dyncs
per square uenti metre.

2. From the length of tube filled

Fio. 81

214 A Manual of experiments in physics

with mercury (p = 18.64 at 20°) and the weight of the mercuiy,
calculate A and reduce the volume of airto cuhic centimetres.

8. From a knowledge of the density of air at a definite tempera-
ture and pressure (see tables) calculate k for air at the tem-
perature of your experiment.

4 Which is the essential condition for accurate measurements in
this experiment, that the bar be vertical or parallel to the
branches of the U-tube?

5. A barometer has a cross-section 2 square centimetres, and is so

long that, as the mercury stands at 76 centimetres, there is a
vacuum space 10 centimetres long. Some air is allowed to
enter, and the mercury falls 10 centimetres. What was the
volume of the air before it entered?

6. In a vessel whose volume is 1 cubic metre there are placed the

following amounts of gas : (1) Hydrogen, which occupies
1 cubic metre at atmospheric pressure ; (2) nitrogen, which
occupies 8 cubic metres at pressure of 2 atmospheres;
(8) oxygen, which occupies 2 cubic metres at pressure of
8 atmospheres. Calculate the pressure of the mixture.

7. A glass tube, 60 centimetres long, closed at one end, is sunk

open end down to the bottom of the ocean. When drawn up
it is found that the water has penetrated to within 5 centi-
metres of the top. Calculate the depth of ocean, assuming the
density to be constant. (Principle of Lord Kelvin's sound-
ing apparatus.)

8. A barometer contains traces of air, and when the mercury is

70 centimetres high the "vacuum space" is 20 centimetres
long ; on lowering the tube into its tank so that the column
of mercury is 67 centimetres high, the *' vacuum space" is
12.6 centimetres. What is the true barometric pressure?

EXPERIMENTS IN SOUND

INTRODUCTION TO SOUND

Unita Sonnd, being a sensation, cannot be expressed as
a certain number of nnits^ but the vibrations and waves
which produce sound are purely mechanical processes, an(i
80 they must be measured iu mechauical units. (The fre*
quencies of several musical scales may be found in the tables
at the end of this Manual.)

Olgect of Experiments. The experiments of this section
may be divided into two groups— via., the study of vibra-
tions and the measurement of the velocity of sound-waves
in certain bodies. There are no essential difficulties in the
experiments, as they involve measurements of length, mass,
mi time only. In Experiments 30 and 42 a moderate
ability to distinguish differences of pitch is necessary;
and it is desirable for a student who is wholly devoid of
this sense of pitch to perform these with one who is more
musically constituted.

EXPERIMENT 88

(TWO OBSBRVBBfl ARB RBqUIBXD)

Oljeot A study of " stationary " vibrations. (See * * Phys-
ios/'Art. 138.)

1. Transvene VibrationB of a Cord

Oeneral Theory. A cord is held fixed at its two ends
nnder a definite tension. Transyerse vibrations are im-
pressed at one end and travel to the other^ where they are
reflected ; and two trains of exactly similar waves travelling
in opposite directions are thns prodnced.

When the length of the waves is such that the distance
between the fixed points is an integral number of half
wave-lengths, "stationary vibrations*' are produced, in
which the "nodes,'' or still points, are those where the
two trains are at each instant tending to move the cord
in opposite directions. The nodes break up the cord into
an integral number of vibrating segments, each of which
is 1/2 a wave-length long. The connection between the
number of vibrations per second and the number of such
segments in the length of the cord can then be studied.
Thus, if V is the velocity of transverse waves in the cord,
X the wave-length of the particular set of waves, n the fre-
quency of the vibration, L the length of the cord, N the
number of vibrating segments,
X = 2X/J\r,
t; = wX = %nLlN.

For a defitnite value of N, n must have a certain value
which can be measured, and so v can be calculated. As long
as the tension remains constant, the velocity v should also.

£XP£RlMfiNTS IN SOUND 219

If the tension in the cord is changed, the velocity of
transverse waves changes ; and theory shows that the ve-
locity should vary directly as the sqnare root of the ten-
sion. (See " Physics/* Art. 142. )

These transverse vibrations may be produced by fasten-
ing the cord to one prong of a tuning-fork, in which case
n, the frequency, remains constant, and the tension must be
varied so as to produce different values of N; or by setting
the stretched cord in vibration by means of one^s hands, in
which case the tension remains constant and the frequency
is varied at will. The latter method is the one t<o be more
fully described below.

Sonioes of Error.

1. The teosion may not remain ooDstant

2. The vibratioiia may not be harmoDic.
B. The length may change.

ApparatiuB. A long spiral spring ; a canvas bag ; several
weights — e. g., 1 kilogram, 2 kilograms ; a clamf -stand ; a
Btop-watch ; a metre-bar ; a plumb-line.

Manipulation. The closely wound spiral spring of wire,

Bix or seven metres long, is fastened at the top so as to hang

clear of obstruction in a room with a very high ceiling, or

in a stairway. The bag in which different weights may be

placed is hung on the end. Place about one kilogram in

the bag, and, while the spring is hanging free, insert it at

a point near the bottom in a clamp, which will maintain

the tension and the length constant. Catch hold of the

spring, the thumb and forefinger being just above the

clamp, the rest of the hand helping to hold the clamp

Btill. With a sidewise movement of the finger only, send

transverse waves up the spring at regular intervals. These

waves are reflected at the upper end, and, if the motion of

the hand is so timed that an integral number of vibrations

IB made in the time it takes the wave to travel to the top

and back, stationary vibrations are set up.

^mi, time the motion so that the whole spring moves

2S0

▲ MANUAL OF EXPERIMENTS IN PHYSICS

back and forth together — t. e., there is no node between the
fixed points at each end. Keep the hand moving at exact-
ly the same rate, and time a namber of yibrations great
enough to take an interval of over
a minute. Let one student meas-
ure the time exactly with a stop-
watch, while the other student
mores the spring and counts aload
the motion of his finger as indicated
by the sense of feeling. If the mo-
tion is very rapid he should count
1, 2, 3, 4 ; 1, 2, 3, 4, etc., each set of
four being marked by the other ob-
server. Whence, find N^ the num-
ber of vibrations per second with
the spring moving in one segment.
Repeat the measurement of N^
three times more.

Now move the hand faster, so that
stationary waves are formed with a
node half-way between the fixed
points. Determine N^ in the same
manner as N^, Similarly, find the fre-
quencies of vibrations which have
three and four vibrating segments.
Loosen the clamp, put all the
weights in the bag, clamp again,
and repeat the experiment.

Measure L with a plumb-line and
metre-bar.

The tension of the spring is (in
the average for its whole length)
equal to the weight on the end plus 1/3 the weight of the
spring. Weigh the weights used and the spring.

From the two experiments show that if Tis the tension
in the first set of observations, and T' that in the second,
the two relations hold :

xxb

ticKSa

EXPERIMENTS IK SOUND 281

1. If P is constant, v is the same for all wave-lengths.

2. — = sJ -pp ; ». «., the velocity varies as the square root of
the tension.

ILLUSTRATION
Stationary Vibrations

Febi 3, 1897

Length of spring between flxed points = 708 oentimetres.
I. Mass of spring, 1215 grams.

. Naof
Safmrats
ilT

No. of

Vlbniioni

Timed

lotorval

.'.«

, 2X708

9=3 MX

67.4 >!
68.2

68.0 I
59.0

Mean, 68.15 J

74.4 ^
78.8

0.688

1416

974.2

100

78.0
78.4

Mean, 78.65 J

73.4 ^
78.2

1.858

708

961.4

160

78.0
78.6

Mean, 78.8

72.8 ^
74.0

2.046

472

966.8

200

74.2
78.6

Mean. 78.66 J

2.716

854
Mean,

961.6
966.7

The greatest deviation from the mean is less tban \i,

Tlie weiglit hung on the end of the spring was 994 grams.

. Tension =(994 + ~)^ = 1899y = T.

^2«

A MANUAL OF SXPERIMENTS IN PHYSICS

II. The weight hung od the end of the spring was now increased to
1090 grams. /. Tension = (l990 + ^)^ = 289(^ = T.

No. of

Vibmtioni

Timad

Intemtl

.'. n

axTnn

e = iU

( i 52.0 ^

58.0

50-

58.2
68.0

Mean. 62.8

68.4 >!
68.2

0.947

1
1

1416

1841

100

68.0
62.8

Mean, 68.1

64.0 .
68.8

1.888

708

1888

160

68.6
64.2

Mean. 68.9

64.4 ^
68.8

2.788

472

1814.

200

640
64.6

Mean. 64.2 J

8.690

854
Mean.

1806
j 1828

The greatest deviation from the mean is about 1^.

Further. ?; = ^ should equal \/§J=\/5; •• ^.. 1-87 should

equal 1.81, showing a departure from an accurate yeriflcation of the
law of about 4.4)(. This is probably due to the fact that the law
applies rigidly only to perfectly flexible uniform cords which are
vibrating through very small amplitudes.

2. Surfaee Vibpations In a Tank of Watep
General Theory. Vibrations are produced in a tank of
water by means of a paddle (or otherwise), and the con-
nection between the length of the vibrating segment and

EXPfiRIMSKTS IN SOUND

223

the frequency of the yibration is studied. As in Oase 1, if
V is the velocity of surface waves, \ the wave-length of the
waves which set up the vibration, n the freqnenoy of the
vibration, L the length of the tank, N the number of nodes
(for in this case each end is a loop),

x=2Z/Jv;

v = nX=:2nZ/JV;
ft, L, N can be measured, and hence v can be calculated for
a train of waves of a definite wave-length (X = 22//iV) in
water of a definite depth. The velocity for waves of dif-
ferent length may be found by varying N,

Similarly, by changing the depth of the water in the tank
the velocity under these new conditions may be measured.

Sources of Bxror.

1. Tbe vibrations may not be exactly statioDary.

2. Care is necesaary in counting tbe number of nodes when the

waves are sbort.

Apparatus. A tank with glass sides ; a paddle with a
square blade a little smaller than the inside cross -section

of the tank ; a stop-watch. (The tank at Johns Hopkins
Uniyersity is 140.6 centimetres inside length, by about 35
centimetres depth and 10 centimetres width.)

Mtnipulation. Fill the tank with water to a depth of
U centimetres. Set up stationary vibrations of various
lengths^ as described below. In each case find n, the fre-

824 A MANUAL OF EXPERIMENTS IN PHYSICS

quency, by means of the stop^watch^ counting the number
of vibrations for as long a period as the waves last in perfect
form. Do this fonr times. Note the number of nodes N.

1. Oreate vibrations in which there is one node by rais-
ing one end of the tank and setting it down very gently.
By properly timing the lowering, perfectly stationary vi-
brations can be obtained. (It will probably be necessary
to try several times before a perfect wave is obtained.) The
vibrations are stationary when there is a node at a perfectly
fixed point at the middle of the tank, which does not move
to and fro or up and down at all, while the water rises at
one end of the tank exactly at the same time it falls at the
other end.

2. Set up with the paddle stationary vibrations of a much
shorter wave-length — i. e., such that there are three or more
nodes in the length of the tank. The best way to do this
is' to hold the paddle by the top of the blade, one hand on
each side of the handle, the thumb pressing against the
side towards the body, and two or three fingers against the
other side, all pointing down the blade. Kow stand facing
the end of the tank, lower the paddle almost to the bottom
with the blade nearly upright, and rest the hands one on
each side of the tank. Move the paddle to and fro in the
direction of the length of the tank with a motion of the
fingers only. Hold the paddle so loosely that it is carried
to and fro with the water by each wave, and time the press-
ure of the fingers so as to assist this motion slightly each
time. It is therefore necessary to begin at once with a
motion of about the right frequency, which can then be
gradually adjusted to the exact frequency necessary to give
absolutely stationary vibrations. The test is as before, that
there is no motion of nodes Or loops lengthwise of the tank.
If a crest travels back and forth> or is irregular in its po-
sition, the vibrations are not quite stationary.

n can best be found by counting the motions of the
hands, not the water ; the student plying the paddle de-
termines this number by his sense of feeling; a&d| if the

EXPERIMENTS IN SOUND 225

motion is very rapid, counts only 1, 2, 3, 4 ; 1, 2, 3, 4, etc.,
and another observer makes a mark for each set of four.

Repeat the experiment with a depth of about twenty-
fonr centimetres in the tank, using waves of the same length
as before.

Calculate the velocity of the waves of each length in the
shallow and in the deeper water, and show whether the ve-
locity 18 the same for waves of different length and water
of different ^epth, ^nd, if not, what effect these conditions
have. Record the experiment as in the illustration for
Part 1, noting that different depths take the place of dif-
ferent tensions.

QoaBtioiui and Probloiiui.

1. In what respects do water-waves differ from sound-waves ?

2. What part of a soupd-wave corresponds to a crest ?

8. Can you Qnd any explanation in your experiment for the fact
that waves apprnacliing a sbore always turn so as to present
a front approximately parallel at all points to the sbore line f
lUnstrate with a diagram.

EXPERIMENT 80

(Tbi8 experiment requires in tlie student a sliglit ability to distin-
guish pitch.)

Olgeot To verify the formnla that the frequency of a
stretched string or cord when vibrating transversely is

1 fr

where I is the length, T'the tension, a the cross - section,
and p the density. (See "Physics/' Ai-t. 142.)

General Theory. 1. To prove that a cord under constant
tension has a frequency inversely proportional to its length.
It is possible to keep a cord under constant tension by
means of a weight, and to vary the length of the vibrating
portion by means of bridges or frets ; thus the string can
be brought into unison with various standard tuning-forks,
and the relation between the frequency and length deter-
mined.

2. To prove that if the length of the cord be kept con-
stant but the tension varied, the frequency varies directly
as the square root of the tension. Two methods may be
used : either to bring the pitch into unison with standard
forks by varying the tension, or putting the cord under a
definite tension to vary the length of a second cord, whose
tension is kept constant until their pitches are in unison;
then to use a different tension for the first cord and to de-
termine the new length of the second cord which will bring
the two into unison again.

In this second method the frequencies vary inyersely as
the length of the second cord, and so their ratio is known.

EXPERIMENTS IN SOUND

227

3. Since the frequency is known in (1) for a definite
length and tension, the density may be calcnlated from
the formula and compared with the known value for the
cord.

It shonid be noted that this experiment is simply a repe-
tition of the preceding experiment. Part 1, for vibkations
t-oo rapid to be counted directly.

Sonroas of Brror.

1. The tension may change the density.

2. The tension may not equal the stretching weight exactly.

8. It is necessary to hold the wires closely against the frets, other-
wise the lengths are unknown.

Apparatus. Two monochords (or sonometers) with wires
about .03 centimetre radius; four weights (3500 grams,
two of 5500 grams, 7000 grams are convenient) ; a metre-
rod; a micrometer caliper; a box of tuning-forks, fre-
quencies about 256-512.

Kanipulation. The sonometer consists of a long wooden
resonator-box, over which is stretched a wire whose length
may be altered by means of two bridges that slide on a

guide running lengthwise of the box. The wire is attached
to a brass peg at one end of the box, passes over the
bridges, and over a bent lever at the opposite end, hinged
Bo as to move freely in a vertical plane. To the end of
the wire beyond the lever is attached the stretching weight

928 A MANUAL OF EXPERIVENTS IN PHYSICS

Pari 1. Set np one sonometer as shown in the illustra-
tion. Hang the heaviest weight on the wire, and keep the
tension thus produced the same throughout this part of
the experiment. Strike the fork of lowest pitch {ui^t 256
ribrations) on the knee, rest it on the resonator-box, and
change the bridges until the wire when placked gires ex-
actly the same note as the fork. Pluck the wire half-way
between the bridges, and deaden the vibrations against the
fret by pressing down with a dnger on the wire just back
of the fret farthest from the weight. When the string is
nearly in tune, count the beats between its note and that
of the fork, and change the bridges until they are no
longer heard. When the note is exactly that of the fork,
measure the length of the vibrating part of the wire be-
tween the two bridges and note it. Repeat, having first
removed one bridge entirely and replaced it at an entirely
different position.

Repeat the experiment similarly with the octave of the
first fork, and also with one of the intermediate forks.

The arms of the lever are intended to be of equal length.
To guard against a possible error from this source (which
would make the tension on the part of the wire on the
sonometer different from that of the part from which the
weight hangs) take the lever off, reverse it, and repeat the
experiment.

Show that the frequency of the vibrations of the string is
inversely proportional to its length — i. e., if »i, n,, n^ are
the frequencies of the three forks used, and li, Zj, ^3 are

the lengths of wire in unison with them, -^ = , and — = 7*,

or, more simply, njli = Wjli = n^s*

Part 2. The heaviest weight is still hanging from the
first sonometer. Place the bridges on it as far apart as
possible, and keep the distance between them precisely
the same throughout this part of the experiment. Set up
the second sonometer with a stretching weight of about
S6W^ gnuns. Vary the distance between the bridges on

EXPERIMEKTS IN SOUND 289

this until the two wires are exactly in unison, testing
hy beats, as in Part 1. When exactly in nnison measure
the length of wire between the bridges of the second
sonometer and note it. Change the position of the bridges
on the second sonometer and repeat. Repeat, similarly,
with the two lower weights on the first sonometer. Re-
verse the lever on the first sonometer and repeat again.

Since the frequency of a wire, as proved in Part 1, varies
inversely as its length, the frequency of the first sonom-
eter is in each case inversely proportional to the length of
wire on the second sonometer when in unison with it — i. «.,
nJi = nj^ = n,2s, where n is the frequency common to
both sonometers and I is the length of the wire in the
second.

Hence, show that, if Ti, T^, T^ are the tensions of the
first wire.

I.e..

T, "" T, T/
l]T,^l\T,^l\T..

Part 3. Measure ten diameters of the wire on the first
sonometer with the micrometer caliper, and calculate from
the measurements of Part 1 the density of the wire. Com-
pare this value with the one given in the tables.

KoTK. — ^The frequency of a standard fork maj be obtained by comparing
it directly with a standard doclt, making use of tlie method of coincidences.
(See Experiinent 28.) By suitable means the clock may be made to illumi-
nate once a second a small round opening directly behind a prong of the vi-,
brating fork ; so that, as time goes on, more and more of the opening ap-
pears uncovered, corered again, etc., periodically, exactly like the gaining
of the torsioD'penduIum on the dock. The same formula may be applied,
i»d thos tlie period of the fork deduced.

daiO A MANUAL OF S1F£B1M£NT8 IN PHYSICS

ILLUSTRATION

Transvuuik ViBRinoiiB or a Stritcbbd Wieb

PUrt 1.

To prove 1*1^1= yi,/,= n,2,.

Fork

Frequdocy
n

Leuglli of Wire

•

59.8

Ut,

866

(Lever reversed)
60.0

15880

60.0

Mean; 59.9

46.66
46.75

Mi,

820

(Lever reversed)

47.40

47.85
Mean, 47.04

16050

80.1

1
1

80.2

512

(Lever reversed)
80.2

80.4

Mean, 80.2

16460
Mean. 15280

Greatest deviation from mean Is less than 2^.

SZPER1MSNT8 IN SOUND

l$l

jrBft 2.
n? n: n!

TopfOve^ = ^^ = ^.or/;2\ = /S2', = /;r,.

T— lou In DjiMS
OD Wlrv Na 1
1 y

length ofWIraKa 3

/•r

78.8
79.0

8500X// .

(Lever reversed)

79.2

79.9
Mean. 79.8

2202xl0»x^

68.8
68.6

i
5580x^ .

(Lever reverped)
68.2
68.2

2226xl0«x^

Mean, 68.45 i

56.9

66.7

! 7000 X;7 J

! (Lever reversed)

2275xl0*x^

1 . 57.1

I i

57.2

Mean. 57.0 )

.Mean, 2284 xl0«x^

».

Grentest dcvijitioii from mean is 2%,

282 A MANUAL OF EXPERIMENTS IN PHYSICS

PUH 9.

Diameter of Wire, .9H, .836. .839, .329. .828. .829, .882, .881. .880. .889
mHlimetre. Mean = .0829 ccDtimetre.

n=:512, / = 80.2. 2*= 7000 g.. cross - aectioo = » x (.0165)«. denaity
p=s8.89.

NoTi.-»A student with a good ear for pitch will find it a usefal etereise
to study the interval between the notes of the musical scale us follows :

Find, by one trial only, the length of wire in unison with each of the forks,
giving the principal notes of the scale^t. e., not sharps or flats— including the
octave of the lowest note.

Calculate and express deciroally the ratio between the lengths of wire giy*
ing successive notes. Which intervals are senjitones ?

Select from the box the three notes CT/,. if/,, and Sol^ Sound them to-
gether and note the harmony. Calculate the ratio of tlie length of wire
giving ift and &>/ to that giving Ut. Sound, similarly, Uti, Jie^, Mi^, Wbat
is the effect ? Calculate the ratio to Ut again.
• What is the combination Wj, 3fi^ 5W, (Cj, ^. Gi) ?

What other similar combinations are there in the diatonic scale f

What would be the length of wire giving Sol^ on the chromatic scale ?
(See " Physics," Art. 162.)

QnestlonB and Problems.

1. If the two arms of the lever are as 9 : 10, what would be the per-

centage difference betwecrti the true tension of the wire and
the tension calculated from the suspended weights if the
longer arm is horizontal?

2. Would an error from this seUtte a/fect the accuracy of Part 1

or of Part 2?
8. What effect has a rise of temperature upon the pitch, intensity,

quality of (1) a piano-string ? (2) An organ-pipe ?
4. Describe the effect of a sounding-board upon a piano-string.

Is there any effect upon the duration of the vibration ¥
6. The third harmonics of two notes hnye the ratio 16 : 20. What

ia the ratio of their fundamentals ?

EXPERIMENT 40

(two OB8EIIVBB8 ARB REQUIRED)

Otjeet To determine the velocity of sound in air by
means of stationary waves in a resonance tube. (See
"Physics/' Art. Ul.)

General Theory. When an organ-pipe^ closed at ohe end
and open at the other, is sounding, if V is the velocity of
sound-waves in the gas, n the frequency of the vibration,
L the length of the pljie (corrected for the open end), the
possible values of n are given by

F=«.|Z.

Oensequeatly, if a fork whose frequency is n is vibrdting^
a number of different tubes will resound to it. ii will be
the fundamental frequency of the shortest of th&se tubes^
the second partial for the next longer, the fourth partial
for the third in lengthy etc. Hence, if ^i, l^i h* etc., are

the lengths of these tubes, v = «.4Z, = «. -^ ~^~fi^'
/. /x = ^ = -^, or 4 = 3/1, h = ^hf etc.,

O 0

tod t-^=«.^^-|,

— i. 0., the difference in length between two suocessive

184 A MANUAL OF EXPERIMENTS IN FHYSIOS

lengths of tube which resound to the same note is one-
half the wave-length of the wave in the tnbe.

To yerify this fact the following method is devised : A
sonnding tuning-fork is held at the month of a tube whose
length can be varied ; the greatest length of tube for which
resonance occurs is noted, and the tube is slowly shortened
until each successive leugth for which there is resonance is
ascertained. A number of determinations of the half wave-
length are thus obtained.

The pitch of the fork being known, the velocity in air is
determined from the relation v = nX. By repeating the
experiment with other forks the effect of the frequency on
the velocity (if any) can be noted.

(In an open organ-pipe the position of the loop at the
open end is not exactly at the end, but beyond it, at a dis-
tance approximately equal to the radius of the pipe.)

Ctonroes of Brror.
1. The loudness of the fork gradually diminishes, and care is
necessary to distlnguifih this from a decrease in loudness
due to the resonance of ihe tnbe becoming less.

Apparatus. A long glass resonance -tube, with a small
side tube attached near one end. Subber tubes are joined
to this branch tube. (The tube at the Johns Hopkins Uni-
versity is 144 centimetres long, and the listening-tube is
connected about 10 centimetres from the top. The in-
ternal diameter is 28 millimetres.) The bottom of the tube
is tapered so as to join with a length of rubber-tubing con-
necting with the water-tap, near which the experiment
must be done. Rubber tubing of sufScient length for this
purpose is needed ; also a thread with a small plumb-bob ;
a metre-bar; and a set of forks containing at least two
of 250 vibrations and over. (In another form of appara-
tus the resonance -tube is lowered vertically into a deep
basin of water, thus changing the length of the column
of air.)

Caution, Be careful not to wet the tuning-forks*

EXPERIMENTS IX SOUND

2»5

Kanipnlation. Stand the tube near the sink and connect
with the tap by means of the rubber tube. Adjust the
resonance -tube by the plumb-line so that it is approxi-
mately vertical. Turn the water on in order to drive out
the air in the rubber
tube and fill about a cen-
timetre at the bottom
of the glass tube. Place
in the ears the listening-
tubes which are joined
to the small side tube.
Strike the ut^ tuning-
fork (or the lowest of
the two selected for the
experiment) against the
knee, and, holding it at
the mouth of the tube
with the plane in which
the two prongs lie ver-
tical, turn the water on
80 that it rises quite rap-
idly in the tube. The
sound in the ear will be
found to vary in inten-
sity as the water rises, and one hand must be kept on the
stopcock and the water turned off the instant the sound is
at its loudest. The approximate height of the water for
resonance having thus been found and marked by a little
strip of wet paper placed on the tube, disconnect the tube
from the tap and allow the water to flow out slowly into
the sink. The vibrating fork must be held over the tube
while the water is flowing out. Do not, however, allow
the water to flow out too slowly, for the change in inten-
sity of the sound may become so gradual that the maximum
is not readily noted by the ear. Stop the flow again the
moment the sound begins to diminish, and lay the strip of
paper again with one edge at the level of the top of the

Fio.86

286 A MANUAL OF EXPERIMKKTS IN PHYSICS

water column. Baise and lower the level of the water Biini>
iarly until yon are conyinced that yon have laid the strip
exactly at the level which corresponds to maximam reso-
nance in the tube. Measure the distance from this strip to
the upper edge of the tube or to any fixed level in the tabe.

Repeat the observation twice^ removing all marks be-
tween the trials so as to secure perfectly independent de-
terminations. Take the mean as the correct distance for
maximum resonance of the top of the water column below
the top of the tube (or the fixed level).

Starting with the water level at this pointy turn on the
water and let it again rise rapidly in the tube. The sound
of the fork will become at first weaker, reach a minimum,
and then again increase to a maximum. Turn the water
off, as before, the moment this maximum is passed ; mark
it and proceed to determine the distance from the edge of
the tube or the fixed level, as in the previous case, making
three independent determinations. Take the temperature
of the air in the tube. If the tube is long enough, find,
similarly, a third point of resonance. The distance between

successive maxima is -.

Repeat with ut^ the octave of the fork already tried, or
some other fork of a pitch considerably higher than the first.

Calculate the velocity of ^ound in air of the temperature
of that in the tube, as explained above. vz=.7i\ = n'\\

X X'

-, and are given by the above experiments ; and either

the frequency of the fork is marked upon it or else its
name ; and in the latter case the frequency may be found
in the tables.

Reduce the value of the velocity in air thus obtained to
its value at 0° by noting that the velocity diminishes ap-
proximately 60 centimetres per second per degree centi-
grade as the temperature falls. (More accurately,

273+^°

£XP£RIMENTS IK SOUND

237

ILLUSTRATION
ViLociTT OF Sound in An
Fork Ui^ 266 yibrations per second.

F0lk 1,1897

20° C.

Koi of Nod«

liMiujc« Beluw Upper
KdgtofTape

Maui Distaooe

Ist.

ad.

97.2.98.8.97.5 cm.

81.1.80.4.81.6 *

97.Tcm.
31.0 **

66.7 cm.

YariatioD is about 1%,

Fork ^i;«, 512 vibrations per second. 20* C.

Nodo

Dtetance Below Tape

Mean

A'
2

lat

4Ui

114.2.118.6.114.9 cm.
80.6. 81.2, 81.8 *•

46.6. 48.1. 47.7 "

14.7. 14.9, 16.1 *•

114.2 cm.
81.0 "
47.6 "
14.9 '

88.2 cm

88.5 "

32.6 "

Mean, 88.1 cm.

Variation is about 1%.

From W,. F=2 x 66.7 x 856 = 84, 100 cm. per sec.

From Ut^, F= 2 x 88.1 x 612 = 88.900 *

Mean. 84.000 cm. per sec., velocity at 20<> C.
.-. Velocity at zero = 82,800 ** '• "

QoastioiiB and Problems.

1. Allowing for the error of your experiment, does it indicate

any difference between the velocities of long and short waves
in the column of air?

2. In your second experiment, of what length of tube is Ut^ the

fundamental, and of what lengths is it the 2d, 4th, and 6th

partials ?
8. How much beyond the end of the tube in each case is the loop

which is usually described as being at the open end of an

oigan pipe ?
4 In calculating the length of a pipe to ^ive a certain note, would

the correction for this error at the open end be the same for

any note?

5. Calculate the change of velocity in air due to rise of tempera-

ture O^' to 20^^. Will the velocity change with the barometric
pressure?

6. Describe an experiment which proves that the velocity of sound

is greater in a solid than in air.

2B8 A MANUAL OK EXPBRIMKNTS IN PHYSICS

7. If a mass of air were confined in a ckMed Tesael of conataat

volume, would changes in the temperature affect the Teloci-
ty of sound in it ?

8. Allowing for the diameter of the tube, what must be the length

of an open tube whose diameter is 6 centimetres, and whicli
is filled with air, to respond most loudlj tea iuning-tofk ci
a80 vibrations per second ?

EXPERIMENT 41

Olgect. To determine the velocity of longitudinal yibra-
tions in a brass rod by Kundt's Method. (See '* Physics/'
Art. 158.)

Oenend Theory. A brass rod is clamped at its middle
point, and set in longitudinal vibration ; one end of the rod
is provided with a small disk, which fits in a resonance-tabe
coaxial with the rod. Consequently^ if the resonance-tube
is of suitable length, the gas in it will be set in vibration
by the vibrations of the brass rod. The frequencies are the
same for the two vibrations ; and, if the length of the vi-
brating segments (i. e., half the length of the waves which
produce the vibrations in the gas) can be measured, the
raiio of the velocity of longitudinal waves in brass and in
the gas can be determined ; for

If the gas in the tube is air, the velocity in it may be as-
sumed to be known from the preceding experiment.

In the vibrating rod there is a loop at each end and a
node at the middle point ; and so the wave-length of the
longitudinal waves in the rod is twice the length of the

^HBSiiiai

Flo. 86

^ The positions of the nodes and loops in the column
of gas may be recorded by a little fine, dry powder, such as

240 A MANUAL OF EXPERIMENTS IN PHYSIOS

cork dust, sprinkled lightly in the tube ; for, when there
are sharply defined nodes and loops, the dust gathers in
definite heaps marking those points. If the powder is very
light it will fly away from the nodes, leaving bare places in
the tube ; while, if the powder is heavier, it will remain
inert at the nodes and will collect in transverse ridges at
the loops. (These traiisversp ridges q,r^ due to differences
in pressure, caused by the air flowing to and fro between
the flne particles.) There is never any ambiguity as to
where the nodes are ; because, if the resonance - tube is
closed, both ends of the column of air are nodes, and so
it is 0a8ily seen which other points are nodes.

Eknuces of Srxor.

1. The brass rod is heated by the rubbing te a tetnperatare 90t

readily determined. The velocity foun(l by the ei^periok^D^
canuot, therefor^, be reduced to standard couditiops.

2. The friction of the gas against the sides of thf^ glass tub^

changes slightly the velocity in tlie gas.

3. The frequency of the vibration of the brass rod is not exactly

what it would be if it were vibrating freely.

4. The brass rod may not be clamped in such a way as to be bdd

at the exact node of the vibniUon.

Apparatus. A glass tube 2 or 3 centimetres in diameter;
a piston which fits the tube tightly, attached to a rod about
50 centimetres long ; a brass rod 3 or 4 millimetres in diam-
eter, with a piston on one end fitting the tube only loosely ; a
vise ; a small piece of rough cloth and some resin ; a metre-
bar ; supports for the tube ; cork dust or lycopodium powder.

Manipulation. Arrange the apparatus as shown in the fig-
ure, clamping at its middle point in the vise the rod which
is to vibrate, and which carries the smaller piston. Before
putting the tube in place scatter small amounts of the dust
as evenly as possible ins|de it. The loose piston must not
be very near the mouth of the tube. Set up longitudinal
vibrations in the clamped rod by stroking it with the resined
eleth. The note is better if the cloth is pulled ef^tirelypS
the end of the bar. Push the tight piston in or oiit oatfl

E^PEBIMENTS IN SOUND

241

the ridges of cork dust become as sharp as possible. Meas-
ure the distances between the nodes. To do this it is not ad-
visable to measure between two adjacent nodes, but between
two as far apart as possible.

Measure from the face of the fixed pistpn to a sharply de-
fined node near tl^e othef piston. The distance betiireen
two adjacent nodes is then easily calculatpd, and is equal

Tap the tube so as to destroy the ridges and repeat,
making in a similar manner four determinations of X.
Measure the entire length of the brass rod and find the
temperature of the air in the tube.

(The Telocity of sound in air at zero degrees may be
taken as 332 metres per second, to which .6 metre should
be added for each degree above zero.)

The velocity of another gas than air may now be found
by allowing it to flow through the tube slowly so as to fill

X'
it, finding — the distance between the ridges, whence the

\'
velocity in this gas V = r- ^.

ILLUSTRATION
Vklocity ov Soukd in Brass

Fel».8,iW7

Length of bar = 60.9 centimetres. .*. X = 121.8 centimetres.
Temperature inside the tube = 19°. .'. Velocity of sound in air,
843.8 metres per second.

NaofLoo|»

between Nodes

MeaMrad

DtBtanoe

x/a

58.6

5.944

11.89

68.7

6.87

11.74

64.8

6.89

11.78

58.6

6.86

11.72

Mean, 11.78

121 8
. Vx'= 848.8 X '^ = 8664 metres per second = velocity of sound

in brass.

242 A MANUAL OF EXPERIMENTS IN PHYSICS

Qnesttons and Problems.

1. Deduce the frequency of the note given out by the braas rod.

2. Given that the density of brass is 84. deduce iu elHsticity in

dynes per centimetre. (See *' Physics," Art 148.)

8. What would happen if the rod were clamped at another point
than its middle ?

4. What would happen if the clamp were too broad?

6. Gould the Telocity of sound in water be found by this method,
using a proper substitute for cork dust?

6. Is the velocity of transmission along the brass roil of tbe vi-
brations in this experiment the same as the velocity of trans-
mission of the vibrations in Experiment 89 along tlie brass
wire ? Explain fully.

EXPERIMENT 42

Olgeot. To compare the velocity of longitudinal wayes
in brass and in iron. (See " Physics/' Art. 167.)

General Theory. Two wires— one brass, the other iron —
are stretched side by side and set in longitudinal vibration.
The lengths are then adjusted until the pitches of the two
vibrations are the .same. If t^i is the velocity of longitudi-
nal waves in the brass wire> and 2| the length of the brass
wire, and v^ and ^ similar quantities for the iron wire, then,
since the frequencies are the same,

Sooxoe of Error.
The main source of error comes from the Inability of Che obfierver
to decide when the two strings are in unison.

Apparatna. A large sonometer with brass and iron ¥rires ;
clamps ; metre-rod ; two pieces of cloth, and resin. The
large sonometer is similar in principle to those already de-
scribed in Experiment 89, but can carry two or more wires.
The two wires, the brass and the iron, are placed under
tension by means of pegs similar to those used to tighten
the wires in pianos. The pegs are turned by a key which
fits them ; and the wire, being thus wound around them,
can be tightened to any degree desired. At each end of
the sonometer are stationary clamps, in which both wires
are firmly held after being tightened, so as to prevent
slipping at the pegs and loss of tension. There is also, one
movable clamp for each wire. The jaws of all the clamps,
fixed and movable, are lined with lead, so as not to cut

244 A MANUAL OF EXPERIMENTS IN PHYSICS

the wire. A fixed centimetre scale^ 150 centimetres long,
runs the entire length of the sonometer.

Hanipolation. Stretch the wires so that they are firm
and straight; hnt there is no need of having them very
tense. Place a clamp on each wire, with one edge against
the scale, and tighten the clamp-screws.

Stroke both wires with the resined cloths, holding the
clamp still ; and then change the position of the clamp on
one until the note given out by both wires is the same.
When the notes are nearly the same, move only about one
millimetre at a time until they appear to be exactly in nni-

hj.).h).J.J.LJ.J.J.LJ.LLJ.)i)JL).).).im

Vn. 87

son. Note the scale -reading of the side of the clamp on
each wire next to the part of the wire which was made to
vibrate; then continue to move the same clamp as before in
the same direction, and note where a difference in the notes
can again be distinguished. The mean of the two readings
thus made on the same wire is then taken as the correct
reading, as corresponding to unison of the two wires. Meas-
ure the lengths of the two wires. Bepeat four times, tak-
ing different lengths of wire in each case.

Oalculate the ratio of the velocity in iron to that in brass,
and determine the absolute velocity in iron from that in
brass, as found in Experiment 41.

KXPERIMENTS IN SOUND

245

ILLUSTRATION
YxLOOiTT or Souin> ur Iboh

Feb. 19, IWT

fixed End of
BoCh Wires

Movable Clamp

Lengths

Iron
Brass

Bra« Wire

Iron Wire

Brus

Iron

66.9

100.0

66.9

100.0

1.496

80.0

110.8

80.0

119.8

1.491

50.0

74.6

50.0

74.6

1.492

150

80.8

47.1

69.2

102.9

1.487

100

60.0

15.9

90.0

134.1

1.490

Mean,

....

1.491

.*• Velocity of longitudinal waves in iron = 1.491 x velocity in brass
= 1.491 X 8554 metres per second (by Experiment 41)= 5291 metres
per second.

QueatloiT and Problems.
1. Does tbe result of the experiment depend on the tension of the
^res? the size of the wires? the temperature of the wires?
•^Hiy?
A. Wlij cannot beats be heard in this experiment?

EXPERIMENT 48

(TWO OBSBBTBB8 ABB BBQUIBBD)

Olgeoi. To stndy the different modes of vibration of a
column of gas. (See " Physics/' Art. 148.)

General Theory. The column of air in a tube open at
both ends is set in vibration by a siren placed near one
end. A siren consists essentially of a circular disk in

which, at regular intervals
near the edge, small holes
are made ; and, when a blast
of air is blown through the
holes as the disk revolves,
the air is set in vibration
with a frequency equal to
the number of pulses which
come through the disk in
one second. The siren is
set up in front of the reso-
nance-tube ; and, by alter-
ing the speed of the disk,
different rates of vibration
can be noted, to each of
which the column of air re-
sponds.

A column of air open at both ends can vibrate in various
ways, such that there are successively 1, 2, 3, etc., nodes in
its length ; and the frequencies of the vibrations to which
it responds are, therefore, in the same ratio. Further, if

Fio. 88

EXPERIMENTS IN SOUND

247

the freqnencieB and the lengths of the vibrating segments
are known absolutely, the yelooity of Boand-wayes oan be
determined as in Experiment 40.

SouiOBs of Bnoir.

1. The belt connecting the disk with the wheel by which it Is

rotated may slip and the frequency be less than that indi-
cated by the speed of the handle.

2. It is difficult to keep the speed absolutely constant, and it can-

not, therefore, be determined accurately.

Apparatna. Siren ; bellows ; rubber tnbing ; short piece
ol glass tubing about 6 millimetres diameter; clamp-
stands ; resonance-tube, a glass tube abont 70 centimetres
long and 3 centimetres in diameter ; watch.

The rubber tubing should be joined to the bellows, and
in its other end the short glass tube inserted, so as to seiYe
as a mouth-piece for the air-blast.

Hampulation. Place the resonance-tube in a clamp-stand
so that it is perpendicular to the disk, with one end oppo-
site a hole. On the other side of the same hole fix the
blast -tube so that
the air is blown di-
rectly down the res-
onance-tube. Blow
the bellows and re-
▼olve the siren,
gradually increas-
ing the speed until"
the tube resounds.
Then keep the speed
SB constant as pos-
sible, and time by
the second-hand of your watch fifty turns of the handle
which revolyes the disk. Stop the siren. Begin again and
repeat the determination three times.
Next, increase the speed beyond that necessary to give

FIG.8S

24fi A MANUAL OF EXPERIMENTS IN PHYSICS

the lowest tone of the tnbe^ and find, as before, the speed
nfeoessary to produce the next higher note td which the
tnbe resounds. Make three trials again. Increase the
speed still further and find a third note ; and continue the
experiment until a speed is reached which cannot be con-
veniently kept constant long enough to count.

It is necessary to time a greater and greater number of
turns of the handle in each case — say, 100 for the second
not^, 150 for the third, and so on, or else the interval be-
comes too short to be accurately timed. Take the tem-
perature in the tube.

Stop the bellows. Reyolve the handle very slowly and
count the number of turns made by the disk while the
handle revolves ten times. Do this threb times. Deduce fi|,
the number of turns made by the disk to one of the handle.
Count the number of holes in the disk. Let it be «,.
Then, if the haiidle is turned at a speed of N turns per
second, the number of holes which pass the blast per sec-
ond— i, B.y the pitch of the note — is JV^Wjnj.

Measttre the length of the tube. Draw diagrains to il-
Itlstrate where the nodes and loops ard in each case, and
state which note is the fundamental of the tube, and which
partial correspoilds to each of the others. Show in each
case the relation of the pitch to the fundamental.

Deduce the velocity of sound in each case and average.
The loop at the open end of an organ-pipe is at a distance
beyond the end equal approximately to the radius of the
tube, and due allowance must be made for this.

Fab. 9, 1»7

.*. 1 turn of handle pro-
duces 6.67 turns of
disk, .Mil =6. 67.
There are 45 holes in disk — t. e,, n, = 45. .*. One turn of the disk
per second Corresponds to a pitch of n{n^ = 255.

ILLUSTRATION

SlRKN

Turns of Handle

Toms Of Disk

EXPERIMENTS IN SOUND

249

Frequencies Giving Bemmanee

LowMt
No. of turns timed. 60

Time in seconds.

f 58.0
54.8
53.7
62.7

Mean, 52.45

100

^ 50x255 ^,

Frequency , 52.45 =^^

58.6
58.8
58.4
54.0

68.7
475.1

4th

150

200

51.2

50.4

51.4

506

52.8

51.2

51.6

50.8

51.75

50.75

789.0

1005.0

Mean frequency corresponding to lowest pitch is -^^ta - = 242.

0«. ID

Length of tube, 68.2 cm. ; radius, 2 cm. .'. X = 140.4.
VeiocUy of sound, 242 x 140.4 = 84,000 cm.

QnastlcMis and Problems.

1. Why is the pitch shown in this experiment likely to differ more

from th<' tliooroficjil vflliie ilie liiglior it is?
8. Why does nut the tube resound for all notes f

EXPERIMENTS IN HEAT

INTRODTTCTTON TO HEAT

ITnits and Deilnitioiis. Aa is shown in treatises on Physics^
the main '^ effects of heaf are changes in Tolnme^ in tem-
peratnre^ and in state or condition ; and all these effects
are due toHhe addition of energy to the minnte portions
of matter in the body which experiences the heat-effect.
Naturally, therefore, amounts of heat -energy should be
measured in ergs ; but in almost every case this would be
impossible. Consequently, a subsidiary unit of energy is
adopted which admits of ready use, and which can be meas-
ured in terms of ergs. This subsidiary unit is the amount
of heat-energy required to raise the temperature of one gram
of water from 10** to 11^ 0. It is called the " thermal unit,''
or a " calorie '*; and its value in terms of the erg has been
determined by experiment to be

1 calorie = 4.2 x 10' ergs.

By actual experiment it is found that the amount of heat-
enei^ required to raise the temperature of one gram of
water one degree centigrade is not exactly one calorie at
all temperatures ; but the variations are so slight that in
preliminary experiments, such as those in the following
section, they may be neglected.

To measure temperature the centigrade scale is used ;
and the thermometer in universal use is the mercury-in-
glass one. The centigrade scale is one on which the tem-
perature of melting ice is called 0**, and that of the vapor
rising from boiling water 100**, provided the pressure of the
vapor is 76 centimetres of mercury under standard condi-
tions (t. e.f when the barometric reading is 76 centimetres
"corrected," as explained in Experiment 26).

S54 A MANUAL OF EXPERIMENTS IN PHYSICS

* A mercnry-in-glasB thermometer is an instmment whose
reading must be subject to many corrections if accaracy ia
to be obtained.
The chief corrections may be thns summarised:
!• The bore of the tabe may be irregular. The correo-
tion for this may be found by a process of calibration by
means of a thread of mercury. (See Experiment 2.)

2. The scale which is marked on the instrument may be
irregular, or it may be so placed as not to coincide exactly
with the position of the mercury column at definite tem-
peratures. The correction at 0° and 100** may be deter-
mined as is shown in Experiment 44; and the errors at
other points may be learned by comparing the instrument
with some standard thermometer whose errors are known.
(If the thermometer is made of a kind of glass whose prop-
erties are known, its errors may be deduced from observa-
tions on it by itself.)

3. The effect of pressure, both external and internal, on
the volume of the bulb must be noted. This effect is very
noticeable if the thermometer is used first vertically and
then horizontally, or if it dips deeply in a liquid. The ex-
act correction may be determined by subjecting the bulb
to various pressures and measuring the effects; in most
cases of laboratory thermometers this correction is neg-
ligible. This effect also causes a difference between read-
ings made when the mercury column is falling and when
it is rising, owing to differences in capillary pressure.
Owing to this fact, readings should be made when the
mercury is either always rising or always falling, prefer-
ably the former ; and in any case the stem of the ther-
mometer should be tapped gently before a reading is made.

4. The change in the volume of the bulb owing to molec-
ular changes in the glass is most important. This change
appears in two ways : there is a slow decrease in volume of
the glass, which continues for years after a thermometer
is made, and which is shown by the gradual rise of the mer-
cury column when the temperature is maintained constant

EXPERIMENTS IN HEAT 26d

—0. g,, the rise of the 0° point ; and also^ if the tempera-
tnre of the instrument is varied from 0° to any point f and
then back to 0^, the mercary colamn will always stand
lower than at first, owing to the fact that the glass lags be-
hind the change in temperature and the volume of the glass
does not return to its previous value for some days, per-
haps weeks. This "depression of the zero point'' depends
upon the temperature to which the instrument has been
exposed, upon the time of exposure, and upon the rapidity
of the return to (f 0.

Owing to this fact it has been found convenient to de-
fine temperature as follows : let Vioo be the reading at 100°
C, Vq the reading at 0° after the exposure of the thermome-
ter to the temperature r, Vg the reading at t^, then

where t^o is the reading at (f after the thermometer has
been heated to the temperature lOO'^ (it is nearly equal to

Although the mercury-in-glass thermometer is the one
in universal use, it is not the standard instrument. The
civilized countries of the world have agreed to accept as
the standard thermometer one filled with dry hydrogen at
the initial pressure of 100 centimetres of mercury. To re-
dace the readings of the mercury thermometer to the stand-
ard hydrogen thermometer, corrections must be learned by
direct comparison of the two instruments once for all, and
these can then be applied in all subsequent readings.

Ol(ject of Experiments. The experiments in the following
section are measurements of expansion and of quantities of
heat-energy. Every experiment involves a measurement
of temperature, and there is no physical measurement quite
80 difficult to make accurately. The chief difficulties in
the measurement of temperature may be thus summa-
rized:

1. Error of the Instrument, as Described in the Previous
'Article. — In the accurate use of a thermometer all the cor-

Sd6 A MANUAL OF EXPfiBI)i|SIfT3 PT PHYSICS

reotions must be mftde; but for ordinary pprpp^es it is
sufficient to standardize the 9cal3 at O'' and 100°, calling ti^0
veadingB at these temperatures Vq and v^oq, and to deftna
the temperature at any other reading Vo w

Vioo-fo

2. Stem 'Correction. — The portion of the thermometpr
whioh is not immersed in the body whose temperature is
desired is at a different temperature ip genera}, Qrud so the
mercury column does not record the correct temperature.
This error may sometimes be avoided by enclosing the en-
tire thermometer iq f^ glass tube, and filling this with water
at the same temperature ^ that of the substance in which
the bulb is placed. In general, however, some correct^ion
must be applied, assuming the average temperature of the
projecting stem to be somewhere between the tempera-
ture of the bulb aud that of the room. Thus, if T, is the
true temperature of the bulb> t^t that of the surrounding
air or room, the average temperature of the stem will be
between these two (it is practically impossible to say ex-
actly what); call it i°. Let a be the coefficient of apparent
linear expansion of mercury in glass, and let the mercury
column project h degrees^ then the stem correction is evi-
dently Aa(/°, - H-

3. Error in Reculxng. — In reading a therpiometer great
care must be exercised to look at the mercury column in a
direction perpendicular to tho scale. In some thermom*
eters it is possible to get a reflection of the scale divisions
in the mercury by looking a little from one side ; i^n4 the
divisions and their images may be brought into lii^ by
moving the eye, thus securing the proper direction. (See
Experiment 33.) In other cases the student must U9e
what care be can to make the correct reading.

4. Error Due to Radiation. — Great care must be taken tp
keep the thermometer from interchanging heat-energy by
any means (but particularly by radiation) with other bodies
than the one whose temperature is desired. To avoid tififi

EXPEKIMENTS IN H£AT

261

danger^ screens of non- transparent, non^condncting sub-
stances should be interposed between the thermometer and
neighboring bodies.

5. Capillary Error. — Before making a reading always tap
the thermometer gently.

In measuring quantities of heat -energy errors due to
thermometers enter, but the main difficulty is caused by
transfer of the energy in other ways than in the one de-
sired. Thus there is constant danger of loss of heat-
energy by radiation, conduction, and convection,^ which
can be largely prevented, however, by suitable precau-
tions. Further, allowance must always be made for the
heat-energy which is necessarily consumed in any change
of temperature of the vessels which contain the substances
mainly involved in the heat-transfer. In the description
of each experiment attention will be
directed to these possible dangers,
and methods of correction will be
given.

One direction cannot be empha-
sized too much : Stir constantly any
liquid whose temperature is desired.
The stirring should not be violent,
otherwise it may itself cause a rise of
temperature ; it should, however, be
thoroDgh and unceasing.

Bnnsen - burner. The gas-burner
which is ordinarily used in labora-
tories for heating purposes is the
Bunsen-bumer, which is so devised
as to allow the gas when it enters to
mix with suitable amounts of air,
and thus it secures violent combus-
tion. Screws or stopcocks are intro-
duced so as to regulate the flow of
both gas and air, and they should be
17

Fio. 90

258 A MANUAL OF EXPERIMENTS IN PHTSIGS

SO adjusted as to cause a bine cone about 1^ inches high
.to burn quietly above the mouth of the burner, the sur-
rounding colorless flame being about 5 inches high. The
hottest portion of the flame is just outside the tip of the
blue cpne. If the flow of gas or air is suddenly disarranged,
the flame may ^^ strike back'' and burn in the tube at the
point where the gas enters. (This is liable to so heat the
burner as to cause the rubber tubing to burn.) When this
happens it is best to put the burner out and relight

EXPERIMENT 44

Oljeot. To test the fixed points of a mercnry thermome-
ter. (See " Physics/' Art. 168.)

The thermometer, once tested, should be used in all later
experiments.

Qeneral Theory. The general discussion of a mercury ther-
mometer is given in the introduction ; and it is seen that the
two standard temperatures on the centigrade scale are those
of the equilibrium of ice and water, and of water and steam,
under standard conditions of pressure. These temperatures
are called 0° and 100° ; and the readings of the scale of a
thermometer at these temperatures must be carefully de-
termined. The obvious method is to place the thermome-
ter in turn in a mixture of ice and water and in the steam
rising from boiling water. If the pressure of the steam is
not the standard one, due correction can be made. The
temperature of equilibrium of ice and water is unafFected
by slight changes in pressure; and that of steam rising
from boiling water is changed at the rate of an increase
of 0.1° for an increase in pressure of 2.68 millimetres of
mercury.

Sonzces of Exror.

1. The ice or water may be impure.

3. Id determining the freezing-point it is very difficult to get the
entire mass of water exactly at 0^. There may be, conse-
quently, warmer water near tlie bulb of the thermometer.

3. The pressure around the thermometer may not be that recorded

on the barometer.

4. There may be loss or gain of heat in the thermometer by radi-

ation.

260 A MANUAL OF EXPERIMENTS IN PHYSICS

• 1. To Determine the 0° Point on the Centigrade Scale ]

Apparatus. A centigrade thermometer. (Note the nnm- |

ber.)

For Method A. — A long, wide test-tube; a stout wire
stirrer, bent at one end into a ring, just large enough to
move freely over the bulb and stem of the thermometer;
a cork to fit the test-tube ; a deep glass or metal vessel ; a
supply of distilled water and of coarse xsommon salt (NaCl);
ice or snow.

For Method B.— A stand, from which to hang the ther-
mometer ; one large and one small beaker, or a set of cop-
per calorimeters ; a cover and stirrer for the smaller of these
vessels ; three long corks ; sufficient cracked ice or snow to
fill the larger vessel.

Manipulation. Method A.* — Bore a hole in the centre of
the cork to fit the thermometer tightly, and a slit on one
side for the handle of the stirrer. Wash the thermometer,
the stirrer, and the inside of the test-tube carefully in dis-
tilled water. Fill the tube with distilled water to such a
height that the thermometer can be submerged well above
the zero mark, but not so high that the water in the tube is
not entirely covered by the freezing mixture when the tube
is put into the larger vessel. Insert loosely in the test-tube
the cork and the thermometer, and turn the latter so that
the stirrer does not hide the scale. Place a layer of salt
and ice at least two centimetres thick on the bottom of the
larger vessel, then insert the test-tube and pack around
the test-tube alternate layers of ice and salt. Stir the
water in the test-tube continually. When a cap of ice be-
gins to form on the inside of the tube and crystals of ice
fioat in the water about the thermometer, begin to record
the readings of the latter about once a minute, always stir-
ring. Be careful not to let the entire mass of water in the
tube freeze soKd. To read the thermometer, lift it by means

* This method is due to Professor Ostwald, qf Leipsig.

EXPERIMENTS IK HEAT

261

of the cork^ which should be loose in the tube as directed,
until the top of the mercury just shows above the water and
ice in the tube^ the test-tube itself being raised far enough
out of the freezing mixture to admit of reading the ther-
mometer in this position. Take pains to have the line of
sight perpendicular to the thermometer. Bead to one-tenth
of a division on the thermometer scale, as quickly as possi-
ble, and at once lower the thermometer in the test-tube and
the latter back into the freezing mixture. When the posi-
tion of the mercury has remained the same for five succes-
sive readings, it may be assumed that the water in the test-
tube and the mercury of the thermometer have both attain-
ed the temperature at which distilled water freezes under
natural conditions — i. e., 0° C. The mean of the last five
readings is, therefore, the true zero on the thermometer scale.
Take out the test-tube, allow the ice that has formed in it to
melt, and then repeat. Take the mean of the two results. If
this is $0^, the correction to the thermometer at 0° is = — 6o°-
Method B. — Wash the thermometer, the stirrer, the in-
side of the smaller vessel, and the ice in tap water. Support
the smaller vessel inside of the larger on three corks. Fill
the larger with ice, cracked to the size of chestnuts. Sup-
port the thermometer in the centre of the inner vessel, with
the stirrer fitting over it. Fill the inner
vessel one -half with ice, and add dis-
tilled water up to the brim. Stir con-
tinually, and occasionally raise the ther-
mometer, until you can just read it, and
lower again quickly into the ice and wa-
ter. As soon as the mercury has fallen to
r,note the reading every minute until it
has remained stationary for five minutes,
and take the final reading as the zero of
the thermometer. Repeat the experiment, having mean-
while taken out the thermometer and allowed it to warm,
80 as to start again with a reading abpve 1°. Take the
mean of the two results as the correct reading of the ther-

FiOw91

262

A MANUAL OF EXPERIMENTS IN PHTSIGS

mometer at 0** C. If this reading is Bq^, the correction to
the thermometer at 0° is = -- Oq°,

2, To Determine the 100^ Point on the Centigpade Seale

Apparatus. A copper vessel^ with a water-tight bottom,
has its feet firmly fastened in position by a plate of copper,
and from one side near its top projects a short copper tube.
Into the top of this vessel can be fitted
tightly a cone-shaped copper pipe, with
its base of the same diameter as the ves-
sel, and with its vertex placed upward.
The vertex of the cone is cut off so that
it admits a cork. Near the top of the
cone is a small projecting tube similar to
that of the vessel. This entire piece of
apparatus is called a " hypsometer.''

Manipulation. Fill the lower vessel with
water to within two centimetres of the side
tube. Close this tube by stuffing into it a
roll of paper or a piece of cork. Pit the
upper part of the hypsometer very tightly
to the lower part, but do not close the
tube near the top. Push the thermom-
eter, whose zero mark has just been tested, through the
hole in the cork at the top of the instrument until
ninety-nine and one-half scale divisions are hidden below
the upper surface of the cork. The thermometer should
fit closely into this hole to prevent its falling, and also
to hinder the escape of steam around the thermometer,
which would make observations upon it more difficult.
Adjust the cork until the thermometer is vertical, and
turn the thermometer around into that position in which
the scale divisions are most distinct and easily read. The
preceding adjustments will keep the bulb of the thermom-
eter well out of the boiling water, at the same time allow
free circulation of the steam around almost the entire col-
umn of mercury. Place a single Bunsen-bumer, attached

FlO. 92

EXPERIMENTS IN HEAT 268

by rubber tubing to a gas - pipe, under the hypsometer.
When the water has been boiling freely for some time,
and the top of the mercury column has become station-
ary, read the barometer, as explained in Experiment 26, in
Mechanics. Note carefully, estimating to tenths of the
smallest division, the position of the top of the mercury
colamn in the thermometer. After two minutes repeat
this reading, and immediately afterwards note the baro-
metric pressure again. Take the mean of these pairs of
results, keeping the barometric and thermometric readings
separate. Gall the mean thermometer reading 6,.

The variation in the temperature of the steam with the
pressure is given above ; hence, calculate the true tempera-
ture of the steam at the pressure read from the barometer,
correcting the observed height for the temperature of the
mercury in the barometer and for the latitude. The differ-
ence between this value and the observed scale-reading is
the correction for the observed mark of the thermometer.
Finally, allow the thermometer to cool to about the tem-
perature of the room. Then place it in ice and again test
the freezing-point.

ILLUSTRATION
THKRMOMirrKR No. 60 •'•"• ^' ^•^

Freezing-point— Method A
Last five readings:

Experiment 2
-0.2
.36

Experiment 1
-0.3
.86
.26
.8
_J8
MeaD,-0.3

Mean, -0.2
Mean = - 0.26 = V.
Boiling-point
TheTTOometer in steam. 09.6, 99.6. Mean, 99.66 = Q,.
Barometer corrected for temperature and latitude, 75.7, 75.6.
Mean pressure during experiment, 76.66.

Temperature of steam at 75.66 pressure, 100 - ^^^^q '^ = ^-^ = '..

S64 A MANUAL OF EXPERIMENTS TN PHYSICS

Hence

0, = -0.26°. /. Correction at zero = +0.26°.

tf. = 99.56°. t, = 99.87°. /. Correction at boiling-point = + 0.88^.

The true temperature, when the thermometer reads 100°, is :

Correction at 100 Is + 0.82°.

To find change of freezing-point after boiling (Method A) :

TIrlAll Trial 2

-0.4 -0.5

.85 .65

.4 .45

.4 .5

Mean. -0.4 Mean, -0.5

Mean = -0.46°
showing that boiling has lowered the zero point by O.flo.

Questions and Problems.

1. What is the true temperature when your thermometer reads

40°?

2. Is the calculation just made of the correction at iDtermedlate

points accurately true? If not, why is it inaccnrate?

8. A thermometer is so graduated as to read 10° in melting ice and
70° in steam at normal pressure ; what is the temperature
on the centigrade scale when the reading on this thermome-
ter is 50°?

4. Which experiment should be performed first, determining the
zero reading or the boiling-point?

EXPERIMENT 45

Olgeet. To determine the coefficient of linear expansion
of a solid rod or wire. (See " Physics/* Art. 169.)

General Theory. By definition, the average coefficient of
linear expansion of a snbstance between two temperatures
/j"* and fa° is

J»-'> -a

where l^ and l^ are the lengths of a linear dimension at
/j^ and t^, respectively. Consequently, the method is to
measure the change in the length of the rod or wire
between the two temperatures and to substitute in the
formula.

The change in the length, 1^ — /j, is the quantity which is
in general the most difficult to measure, owing to its mi-
nuteness, and particular care must be devoted to its meas-
urement. The general method is to have the rod or wire
BO fastened to one end of a lever that the least change of
length of the rod or wire produces a great motion of
the other end of the lever, and this can be read with ac-
curacy. The rod or wire is then immersed in succession
in baths of different known temperature — e, g., water or
steam — and if the original length of the rod or wire is
known the coefficient of expansion may be at once cal-
culated.

Bouroes of Brror.

1. The rod or wire may not be exactly at the temperature of the

surrounding gna or liquid,
d. There may be some slipping of the lever.

266 A MANUAL OF EXPERIMENTS IN PHYSICS

8. The rod or wire may not be at the same temperatare throuslb'

out its entire length.
4. The greatest error enters through the measurement of the

shorter lever arm.

ApparatOB. Method 1. — '* Expansion of metal-rod appa-
ratus/' consisting of a rod supported in a cylinder with
steam connections, lever, scale, etc.; kerosene -oil can,
in which to boil water; thermometer; metre-rod; rubber
tubing; Bunsen-burner.

The *' expansion of metal-rod apparatus'' consists essen-
tially of a metal rod supported horizontally in a cylindrical
jacket through which steam may be passed. One end of
the rod rests firmly against a fixed steel point ; while the

FkO. 98

other end, which is free to move, rests against one end of
a lever. Consequently, any change in length of the rod is
at once indicated by a motion of the lever arm, which
moves over a finely divided scale. If the magnifying
power of the lever is known, the actual change in length
of the rod can be calculated.

Manipulation. Join the boiler to the steam-jacket by a
short piece of rubber tubing, and provide for the escape of
steam from the jacket by another longer piece of tubing.
See that the rod rests against the steel point, and that the
lever and scale are properly adjusted.

Insert a thermometer in the steam-jacket and note the
temperature /j, and the scale reading of the lever. Fill
the boiler half full of water and set boiling. When the

EXPERIMENTS IS HEAT

467

thermometer iudicates a fixed temperature note this and
call it tz- Note also the scale reading. Bemove the Bapsen-
bomer and allow the rod to cool. When the temperature
has returned to its former value note the scale reading.
Liet the mean difference of the scale readings at the two
temperatures tz and ti be h. The actual elongation of the
rod can be calculated from a knowledge of the lengths of
the arms of the lever. Measure these with the greatest
care and let their ratio be p. Then the elongation I2 — li
equals kip. The original length of the rod may be learned
from an instructor, or in some cases it <nay be measured
directly. Then the average coeflScient of linear expansion
may be calculated from the formula

Apparatus. Method 2. — ''Expansion of wire apparatus/'
consisting of a wire supported in a long^ vertical glass tube,
with steam connections^ lever, steel scale, clamp -stand,
etc. ; kerosene can ; thermom-
eter ; rubber tubing ; Bunsen-
bumer.

The "expansion of wire ap-
paratus'^ consists essentially of
a long, vertical glass tube closed
at both ends, and so arranged
that steam may be passed in at
the top and out at the bottom.
The wire to be tested is double
the length of the tube, and
both ends are passed through
the cork at the top and fast-
ened to a bar passing across
the end of the tube. The bend
of the wire at the bottom goes
around a pulley, to which is at-
tached an arm carrying a hook
passing loosely through the fiom.

268 A MANUAL OF EXPERIMENTB IN PHYSICS

cork at the bottom. A long leyer is hung upon the book

so that the arm one side is six or seven times longer than

the other. The shorter arm is held firmly at its end nnder

a sharp edge fixed firmly to the stand supporting the tube.

This edge is the fulcrum^ and the long arm of the lever

carries a metal vernier at right angles to it at its extreme

end. A clamp -stand can be adjusted to hold the steel

scale vertical and parallel to this metal vernier and as close

to it as possible. The elongation of the wire is evidently

•is J • XT- X- Distance fulcrum to vernier.

magnified m the ratio : ^. , ■r-'^ : — ; — ; —

Distance fulcrum to hook.

Kaaipalation. Method 2. — Connect the glass tube at
the top of the instrument with the escape -pipe of the
boiler by means of a short, straight piece of rubber tubing.
Measure accurately the distances between the fulcrum and
the vernier and the fulcrum and the hook, and calculate
the ratio. Place the steel scale in a clamp-stand back of
the vernier and note how finely the vernier reads, as ex-
plained in Experiment 4, in Mechanics. It is convenient,
when the whole instrument has been adjusted, to raise the
scale until one of its divisions is coincident with the sero
division on the vernier. The wire is not accessible for
measurement, and the length may be obtained from the
instructor.

Before heating the water note very carefully the tem-
perature (^i) of the inside of the tube, as indicated by the
enclosed thermometer. Note the position on the scale of
the zero division of the vernier. Pass steam from the
boiler through the tube containing the wire ; and, when
the thermometer inside the tube ceases to indicate any
change of temperature, read it carefully, correcting for
inaccuracies in its scale, etc. Call this temperature ^.
Again note the position on the scale of the zero division
of the vernier. The difference between this reading and
the like one before made is the number of scale divisions
over which the zero division of the vernier has passed while
the wire has bad its temperature raised from ti^ to /s^-

EXPERIMENTS IN HEAT 26».

This distance shonld be expressed as a decimal fraction
of a centimetre. Remove the Bunsen flame and allow the
instrnment to cool to the temperature of the room. While
this is taking place the student should deduce the value in
centimetres (/2— ^i) of the increase in length of the wire
from the ratio of the lever-arms, combined with the diflper-
ence between the initial and final readings of the vernier.

Calculate the average coefficient of linear expansion. Cal-
culate also the coefficient as referred to 0** C.

ILLUSTRATION , ^ ,^

Jan. 80, 1890

GoimciiiiT or Linear Expansion of Brass Wire
Corrected Temperature

Vernier lUadings
At IS.eo, 6.28 cm.
At 98.4° 0.71 "
Length of shorter lever-arm = 8.11 cm.
•' whole lever =89.87 "

8.11 X 4.57
^'~^= 89.87 =0-^^'

— '•""'* - = 0.00001894.

'h(t^-t,)'

QnestioiiB and ProblemB.

1. Does the cross-section of the wire, whether it be hollow, square,

circular, large nr small, influence its coefficient of linear ex-
pansion, and, if so, how ?

2. Qive some objections to measuring l^-^li while the tempera-

ture of the wire was rmng from ti° to <,°.
8. Is there any relation between heat and energy or work, and, if
so, what is it ? How is this known ?

4. Explain in detail the correction which would have to be ap-

plied to the above formula if the glass tube rested upon a
support under its lower end instead of being firmly bound
at its upper end.

5. Why is the steam let into the tube from its upper end ?

6. A steel boiler has a surface 10 square metres at 15° C, what is

the increase in area when its temperature becomes 90° C?

S70 A MANUAL OF EXPERIMENTS IN PHYSICS

7. A clock, which has a pendulum made of brass, keeps correct

time at 2(P; if the temperature falls to (PC, how many sec-
onds per day will 'it gain or lose ?

8. If 45^ G. is the maximum temperature to which railway rails

are liable, calculate the space which should be left between
their ends if they are 16 metres long and are laid down
812090.

EXPERIMENT 46

Olgect To measnre the apparent expansion of a liquid.
(See "Physics," Art. 176.)

General Theory. Since a liquid must always be held in a
solid, if the temperature of the containing vessel is raised
the apparent increase in volume of the liquid is the true in-
crease diminished by the increase of the
solid. The simplest mode of measure-
ment is to have the liquid contained in
a large bulb which is provided with a
capillary stem, the volume of the bulb
and the bore of the stem being known.

The liquid is made to stand at a cer-
tain height in the tube when the tem-
perature is 0° 0., and the change in
height when the temperature is in-
creased to t^ is noted. If Vq is the
original volume of the liquid at 0°, v
the apparent volume at t^, the coeflS-
cient of apparent expansion /3 is given
by the equation

v:=Vo{l+lit),

Bnt r — Vo is the apparent change in
volume — that is, it is the increase in
height of the liquid in the tube multi-
plied by the cross-section of the tube.

Fig. 96

272 A MANUAL OP EXPERIMENTS IN PHYSICS

The volume of the balb and the bore of the tube may be
measured as in Experiment 2.

Soaroea of Error.

1. These are practically the same as in Experiment 2.

Apparatus. The bulb used in Experiment 2; a large
beaker of ice; a 10-centimetre rule; a Bunsen-burner, tri-
pod^ and asbestos dish; clamp-stand; thermometer; stirrer;
30 cubic centimetres of glycerine.

Hanipulation. Fill the glass bulb used in Experiment 2
with glycerine exactly as there described, and leave the
surface of the liquid in the stem at a height of about 2
centimetres above the bulb when it is placed vertical in
a beaker of ice. Support it in a clamp-stand, so as to be
surrounded on all sides by ice. When the temperature
has fallen to 0°, as is indicated by the liquid in the stem
reaching a definite position, record this position by stick-
ing to the tube a bit of paper which has a sharp edge.
Remove the bulb, and notice the apparent change in vol-
ume of the liquid as it assumes the temperature of the
room. Fill the beaker with water, and place it on the
asbestos dish which is on the tripod. Support the bulb
upright in the water, taking care not to wet or remove
the bit of paper. Baise the temperature of the beaker of
water by means of the Bunsen-burner until some temper-
ature about 25° G. is reached, if this temperature does not
raise the liquid too high in the tube. Keep a thermome-
ter in the water, and stir constantly ; keep the tempera-
ture constant at 25° for a few minutes, if possible, and
record the height of the liquid in the stem by means of
a second bit of paper; record the temperature. Remove
the burner and the bulb; and by means of the decimetre
scale measure the distance apart of the two positions de-
termined at the temperatures 0° and t°.

From a knowledge of the bore of the tube and the vol-
ume of the buy;), calculate /3.

EXPERIMENTS IN HEAT 278

ILLUSTRATION
Appamnt Expamuon of Gltcerivk ^^^ **' ^*^

Volume of bulb at 0^, 25.24 cc.

Radius of bore of stem, 0.099 cm.

Heoce, cross-section = 0.0808 sq. cm.

Temperature, f" = 25®.

Rise of surface in stem, 10.1 cm.

/. «- r, = 10. 1 X 0.0808 = 0.81 oc.

f - t?o 0.81

.'. ^= ^5^ = 25 34 ^-25 = 0.0004a

EXPERIMENT 47

Olgect. To determine the mean coefficient of cubical
expansion of glass between 0® and 100° C. (See "Pbjs-
ics/' Art. 176.)

Oenend Method. If a liquid is contained in a glass bolb,
the apparent expansion of the liquid is less tlian the abso-
lute by the expansion of the glass. Consequently, ii the
apparent expansion of a liquid whose absolute expansion
is known can be measured, the expansion of the glass
can be calculated. Such a liquid is pure mercury, whose
mean coefficient of expansion between 0® and 100** 0. is
0.0001816.

In practice a glass bulb whose mass is known is filled
with mercury at 0** C; the bulb is then heated to a known
temperature — e.g., to 100° C. — and the mercury which es-
capes is caught and weighed; the bulb is then weighed
again, and the mass of the mercury remaining in the bulb
is thus calculated.

Let M=z mass of mercury left in bulb at temperature T,
m = '' " *' expelled from 0° to i\

Call p, = density of mercury at ^%
Po= '' " " "0%
Vt = volume of glass bulb at t^,
Vo= *' *' *' *' '* 0°.

The coefficient of cubical expansion of glass = o = ' .— «

But r, = — , Vo = , andpo = f>«(l + /30>

Pi Po

where /J = 0.0001816.

EXPERIMENTS IN HEAT

276

Hence

V, — Vo

if/3-

m
J

Vot M+m

= o*

and 80 a can be calculated.

Sources of Error.
These are the same as in ExperimcDt 3. %

Apparatus. A ^' weight-thermometer/' consisting of an
elongated glass bulb, to which is joined a capillary tube;
mercury ; mercury-tray ; porcelain crucible ; hypsometer
and rubber -cork with one opening; all the ap-
paratus for cleaning and filling a bulb, as in
Experiment 2.

Manipulation. The weight-thermometer must
be cleaned, weighed, and filled with mercury ex-
actly as in Experiment 2. A mercury tray must
be used to hold the clamp -stand and bulb, in
order to catch the mercury in case any is spilled.
If there are any traces of air in the bulb the mer-
cury must be carefully boiled in such a way that
bubbles of mercury yapor run up one side of the
bulb at a time. This can be easily brought about
by heating the bulb uniformly for a time, and
then keeping the flame at one point of it until
bubbles form there and roll up. When the bulb
and stem are completely free from air let it cool
without removing the cup at the top. When its
temperature haa fallen to about that of the room,
immerse it, still suspended from the clamp-stand,
in a beaker of crushed ice or snow. Leave it
thus for at least fifteen minutes, always keeping the fun-
nel half full of clean, dry mercury, and occasionally stir-
ring the mixture of ice and water. During this time care-
fully clean a small porcelain crucible and then weigh it as
accurately as possible. Without removing the thermometer
from the mixture, quickly remove the funnel from its top^

Fio 96.

276 A MANUAL OF EXPERIMENTS IN PHTSIGS

allowing the excess of mercury to escape into the beaker.
If a globule of mercury should happen to stick to the top
of the thermometer-tube immediately scrape it off with a
knife-blade. Bemove the thermometer from the mixtnie,
holding the bulb in the hand, and catch in the crucible all
the mercury which is then driven out by the heat of the
hand. When the mercury ceases to come out from the tube
fasten the weight-thermometer yertically in a hypsometer
by a rubber cork which holds the thermometer very tightly.
Boil the water, and catch in the crucible the mercury
which overflows while the mercury in the thermometer
expands. The simplest way is to hold the crucible by the
side of the tube and scrape the mercury into it. When
the mercury ceases to overflow, remove the burner and
allow the thermometer to cool.

Note the barometric pressure, correct it for the tempera-
ture and latitude, and deduce from it the temperature of
the steam.

Weigh the thermometer with the mercury left in it as
soon as it is cool enough to be handled comfortably. Also
weigh the crucible with the escaped mercury.

Let r = the temperature of the steam, as calculated.
w = mass of empty weight-thermometer.
c= " " " crucible.
w'= " " weight-thermometer -h mercury left in

it after heating to t°.
c' = " '* crucible + collected mercury.

Hence m^c'—c, mass of mercury expelled from the ther-
mometer,
and Jf=w'— «; " " " left in thermometer,

and .'. ^ — 22 = L., the coefficient of cubical expan-

sion of the glass, can be calculated.

EXPERIMENTS IN HEAT 277

ILLUSTBATION

lUjS, 1888

Cubical Ezpamsiom or Glass

f^ = 101 .28®, calculated from barometer,
to = 27.081 grams. v>' = 289.268. /. Jr= 262.282.
e= 4588 " &= 8.600. /. m = 4.116.

if/S - ^ 962.282 X 0.000182 - ^^^

/. a= L = iriifl = 0.0000265.

M+m 266.848 u.uuwsoo.

Questiona and Problems.

1. Bzplain in brief how the above result can be used to find the
mean coefficient of ezpansioD of any liquid.

d. la there any relation between the coefficients of linear and cu-
bical expansion of tbe same substance; and, if so, what is it 7

8. Describe a method for determining tbe absolute coefficient of
expansion of mercury.

4. Describe a method by which a weight-thermometer may be
used to measure temperature.

6. A cylinder of iron 50 centimetres long floats upright in mer-
cury. If the temperature rises from 0° to 100^ C, how far
will the cylinder sink ?

ft A glass test-tube contains 50 cubic centimetres of mercury at
lO"* C. If tlie temperature is raised to 80^ C, what is the
appareni volume of the mercury f

EXPERIMENT 48

Olgect. To measare the increase of pressare of ar at
constant volume when the temperatare is increased. (See
"Physics," Arts. 177, 178, 179.)

General Theory. If the volume of a given amount of air
is kept constant, but the temperature varied, the pressure
will change according to the law,

where jPq is the pressure at 0° C, ^

at r C,

^ is a constant^ and is called the '^ coefficient of ex-
pansion of the gas at constant
volume." (It is known that
/3 is the same for all gases —
Charles's Law — and that its
value is the same as the co-
efficient of cubical expansion
at constant pressure.)

The general method is to
enclose the gas in a bulb
which has a long, bent stem^
as shown, the open end of the
stem being connected by a
rubber tube to a vessel con-
taining mercury. By raising
or lowering this vessel the
mercury may be made to rise
or fall in the stem connected
with the bulb; so, however the
Fio.07 pressure of the gas changes^

EXPERIMBNTS IK HEAT 279

the Yolame may be kept the same. Farther^ by measur-
ing the differences in height of the surfaces of the mer-
cury in the two arms^ the pressure on the gas may be
measured.

The gas is then subjected to different temperatures, and
the corresponding pressures measured, while the volume is
kept constant.

BonroeB of Brror.

1. The air may not be dry.

2. Bubbles of air may cling inside the rubber tube and cause

trouble by escaping.
8. The stem of the bulb is not at the same temperature as the bulb.

(In accurate work allowance must be made for this.)
4. Allowance sUpuld also be made for the expansion of the bulb.

Apparatus. Air thermometer-bulb, rubber tubing, mer-
cury reservoir on stand ; mercury-tray ; beaker of ice; spe-
cial boiler ; Bunsen-burner ; tripod ; thermometer.

Manipulation. The bulb has been filled by an instructor
with dry air, and the apparatus should not be disturbed
during the experiment.

Set up the "air thermometer'' in a mercury-tray ; hang
a mercury thermometer on the frame between the two mer-
cury columns, and surround the bulb with cracked ice so as
to reduce the temperature to 0°. As the gas contracts, the
pressure decreases ; and care must be taken to keep lower-
ing the vessel of mercury so as not to allow the mercury to
be drawn over into the bulb. By means of the movable
basin bring the mercury surface in the stem to a fixed
mark, either a scratch or a point inserted in the glass.
Keep the mercury at this point during the entire experi-
ment. When the temperature has fallen to 0**, as is indi-
cated by no further change in the mercury in the open
tube, record the difference in height of the two mercury
surfaces. Displace the movable vessel and readjust. Do
this three times in all. Gall the mean difference of the
two surfaces h^. If the free surface is higher than the
surface in the stem of the bulb, the pressure on the gas

SWO A MANUAL OF EXPERIMENTS IN PHYSICS

in the bulb is p^^ h^ pg + atmospheric pressure. Read the
barometer and the mercury thermometer attached to the
air thermometer. ''Correct^' both Aoand the barometric
height.

Remove the ice, and place the bnlb in a special boiler in
which water can be raised to boiling, and so steam can be
made to surround the bulb. When steam is issuing freely
from the boiler, record the difference in height of the two
mercury columns, /i^qo, the volume of the gas being kept the
same as before : make three settings. Bead the mercury
thermometer again, and also the barometer. The pressure
of the gas, ^100= pffhioo+ atmospheric pressure. ''Correct"
the readings as before. Calculate the temperature of the
steam from the barometer reading.

Therefore, if j8' is the ** apparent" coefficient in glass,
;>ioo-=J5o(l-flOO/J'),

The absolute coefficient equals /)', plus the coefficient of
expansion of the glass.

Now remove the boiler, allow the bulb to cool, and, keep-
ing the volume of the gas the same as during the rest of the
experiment, measure the pressure when the bulb has reached
the temperature of the room. From this pressure calculate
the temperature.

ILLUSTRATION

« ^ , Nov. M, 1896

ExPAifsiOH or Dry Aib

Difference in height, - 8.76 cm. Temperature of mercury, 22,2°.

Corrected. -8,74 "
Barometer height," 76.59 *'

.•.p^ = 7a.86x 18.6x980.

Boaing-poirU, 100.23°
Difference in height, + 23.86 cm. Temperature of mercury, 28.8°.

Corrected, + 32.76 '*
Barometer height, '* 76.59 "

/. Aoo= WW X 18.6 X 960.

EXPERIMENTS IN HEAT 281

Ck)mction for glass = 0. 00003

Absolute coefflcicDt of expansion = 0.00865

Qnestioiui and Problems.

1. Calculate R for oxygen, hydrogen, nitrogen.

2. Form product R x "molecular weight."

3. Calculate «/, "mechanical equivalent," from R, Cp, (7».

4. If hydrogen fills a glass tube containing 500 cubic centimetres,

open at one point, at temperature 20° C, pressure 76 centi-
metres of mercury, how much gas in grams will escape if
the temperature is raised to 100° C. ?
6. How much work is done by 1 gram of hydrogen, if it expands
from volume 500 cubic centimetres to volume 1000 cubic
centimetres, the pressure being constant, the temperature at
starting being 20°? What must be the rise in tempera-
ture to produce this expansion ? How much heat has been
furnished ?

6. Calculate on the kinetic theory the average speed of a mole-

cule of H, of 0. of N, at 0° C. and pressure 76 centimetres.

7. The formula for steam is H^O; 20 cubic centimetres of i?

and 20 cubic centimetres of 0 are mixed ; an electric spark
is passed ; what is the resulting volume, the initial and final
temperature and pressure beiug the same ?

EXPERIMENT 40

Olyeot To determine the specific beat of a metal — 0. g,y
lead or brass, cut in small pieces. (See " Pbysics/' Art.
184 a.)

General Theory. By definition, the specific heat of any
substance is the number of calories necessary to raise the
temperature of one gram of it one degree centigrade. This
is different for different temperatures of the substance,
and so the average specific heat through a certain number
of degrees is usually measured. It may be assumed within
the limits of errors of this experiment that the specific
heat of water is constant; and hence by the definition of
the calorie its value is one.

If M grams of a metal at temperature T^ are quickly
placed in m grams of water at temperature t^y the water
being contained in a calorimeter whose *' water-equivalent **
is a, the final temperature f^ reached after equilibrium is
such that, if s is the specific heat of the metal,

if no heat energy is gained from or lost to surrounding
bodies.

By " water-equivalent" is meant the number of calories
required to raise the temperature of the calorimeter, the
stirrer, and the thermometer one degree. Owing to the
small numerical value of the specific heat of the calorime-
ter, a is a small quantity; and its value can be determined
by a preliminary experiment, the method used being one
which is not so accurate as that of the main experiment.
The method is this: Pour mj grams of water in the calo-
rimeter; measure its temperature, ^,° (keeping the stirrer

EXPERIMENTS IN HEAT

288

and thermometer in it) ; add a known mass (m,) of water
at a higher temperature, T^ ; and let the final temperature
of eqnilibriam be T^. Then, since the specific heat of
water is 1,

This value may be substituted in the first formula, and
8 thus determined.

£k>iirce8 of Brror.

1. Heat energy may be lost by radiation, conduction, or evapora-

tion. If the radiation is considerable, it may be allowed for
in either of two ways. (See Experiments 52 and 54.)

2. There may be u drop in temperature as the metal is transferred

from the heater to the water.
8. There may be differences between the two thermometers.
4 The metal may not be quite dry.

Apparatus. A hypsometer without the conical top; a
copper dipper fitting on top of this ; two centigrade ther-
mometers; one large and one small
nickel-plated calorimeter; a stirrer;
a large cork to fit the larger calo-
rimeter; a pasteboard cover for the
dipper; a beaker-glass; about 200
grams of shot or 100 grams of brass
wire cut in fine sections; tripod
with asbestos plate; Bunsen-bumer,
etc.

Manipulation. 1. Preliminary, —
Determining the water- equivalent
for the smaller calorimeter.

Thoroughly dry the smaller calo-
rimeter and its stirrer, and find their combined m^ss by
weighing on a platform-balance. Call this mass m'. By
means of the large cork with a hole in its centre fit the
smaller calorimeter inside the larger, with their axes ooin-

FiO. 98

884

A MANUAL OF EXPERIMBKTS IN PHYSICS

FlO. 99

ciding and their upper edges flush, as represented in the
diagram.
Weigh the combined apparatus, and add enough water
to fill the inner calorimeter about
one-quarter full. Let the mass of
the water added be m^ grams. Keep
the stirrer in the inner calorimeter.
Put one of the thermometers int^
the inner calorimeter, with the bulb
near the bottom, and cover the open-
ing with a piece of paper or card-
board. After the mercury colunm
has been observed to remain station-
ary for several minutes, read the
temperature. Apply the proper cor-
rections to this reading, and call the temperature thus
found t^. This should be the temperature of the inner
calorimeter.

Fill with water a beaker-glass whose capacity is at least
equal to that of the smaller calorimeter, and heat it io
about /,° -h 15°. Quickly fill the calorimeter about two-
thirds full of this heated water, noting the exact tempera-
ture (7*,°) of the water just before pouring it into the
calorimeter. Before reading this temperature the water
must, of course, be well stirred to insure its being at a uni-
form temperature throughout. After the water has beeu
poured into the calorimeter, stir it gently but thoroughly,
noting the temperature (7*) after a few seconds, when equi-
librium seems to have been reached. Then remove tlic
thermometer, and weigh the smaller calorimeter and stirrer
with the water it contains. The mass thus found, dimin-
ished by m' + mj, is the mass (jtu^ of water used, and so
a, the water-equivalent, may be calculated from the for-

Dry the calorimeter and stirrer, and determine a twice

EXPERIMENTS IN HEAT 285

more, correcting all the observed temperatures for the
errors of the thermometer. Average the three values
of a.

2. Experiment Proper. — Determining the specific heat of
lead or brass. Pour enough water into the boiler to rise to
within a short distance of the bottom of the dipper^ and
commence heating it. Weigh out on the platform-balance
about 200 grams of small dry shot or 100 grams of pieces
of brass wire. Let the exact mass be ^!f gcams. Pour the
metal pieces into the dipper on top of the boiler, and cover
it with a cap made of a piece of card -board. Plug up
the escape-tube in the side of the boiler, so that the steam
will be forced out only between the flange of the dipper
and the upper edge of the boiler. The second thermom-
eter, which has not yet been used, must be set vertical,
with its bulb surrounded by the metal and its stem through
the cap of the dipper. While the water in the boiler is
being heated and vaporized, weigh out about 50 grams of
water in the smaller calorimeter. This may be done by
placing weights equal to about (m'+50) grams on one
scale -pan of a balance, and the calorimeter and stirrer on
the other pan, and then pouring water into the calorimeter
until the scale balances. Let the exact mass of the water
be m grams.

Adjust the smaller calorimeter in the larger one, as ex-
plained above. Insert in the water the thermometer used
in finding the water -equivalent, with its bulb near the
bottom, and cover the vessel with a cap ; note the temper-
ature of the water. Note the reading (T^) of the ther-
mometer in the dipper when it has reached its highest
temperature, which it will retain as long as the boiler
works properly. Next, quickly remove the cap and ther-
mometer from the dipper, raise the cap of the calorimeter,
and very rapidly, but carefully, pour the hot metal into the
calorimeter. Instantly replace the cap on the calorimeter,
and thoroughly stir the water with the stirrer. Let the
temperature of the water Just before the metal is trans-

286

A MAKUAL OF EXPERIMENTS IN PHYSIGS

ferred be t^, and let the highest temperature reached
after the hot metal has been poured in be f*,

Bepeat the experiment proper three times in all, correct-
ing all temperatures^ and always using perfectly dry metaL
Bead the thermometer to at least tenths of a degree.

Specific heat = « =

M{T,^t)

ILLUSTRATION
Spkcifig Hbat or Lbao
1.

March 10, 1881

mi=80.0 grams
f?i,= e0.16 *•
r, = 81.66*' C.
r =26.8^0.
*, = 18.4<> C.
a = 4.08°

Mean, a =4.

lit
if=313.18gr8.
51.18 "
«8.7*> C.
28.7° C.
20.8* C.
0.0314"

t =

i. =
» =

Waier-EkiuitalorU

2d Sd

80.0 grama 80.0 grams

62.25 *• 68.75 "

85.r C. 88.1 « C.

29.(r C. 27.9" C.

19.6" C. 18.9" C.

4.06" 8.98"
02" = water-equivalent of calorimeter.
2. Speeific Heat of Lead
(m'= 42.24 grams)

9d 3d 4th ■ 601

50.28 grs.
99.40" C.
28.0" C.
19.1" C.
0.0319"

49.88 grs.
99.1" C.
27.4" C.
18.6" C.
0.0818"

4th ■

68.21 grs.
98.8" C.
29.7" C.
21.6" C.
0.0317"

52.45 grs.
98.6" G

27.rc.

19.4" C.
0.0812"

Mean, « = 0.0816"

Questloiis and Problema

1. Explain in full the derivation of the above formula for i,

2. In finding the water-equivalent of the calorimeter, why was it

necessary to fill the beaker-glass /W/ of water?
8. Give a reason for keeping the nms thermometer in the calo*

rimeter throughout the experiment.
4 Is the highest temperature reached by the metal the same as

the true boiling-point of water under the existing oonditioDsY

5. Could this method be used if there were chemical action be-

tween the lead and the water ? Qive reasons.

6. Calculate the error introduced in • if an error of 0.1^ ireiv

made in residing t, f„ or T^

EXPERIMENTS IN HEAT 287

7. If 1 gram of water and 1 gram of mercury are heated In turn

over the same burner, which will boil in the least time, if
their initial temperatures are 0^ C. ?

8. A litre of water whose temperature is (P C. is mixed with a

litre of water whose temperature is 100°; what will be tlie
final tempentme?

290

A MANUAL OF EXPERIMENTS IN PHYSICS

ILLUSTRATION

N<nr. 13.1SST
Stanmo Hbat or TuRPuiTiinB

a = 19.97

M- 54.72 grams.
«= 28.18 "
U = 97.20 C.
T = 49.8° C.
r, = 20.4°C.

* .*. Si =0.465

Jf= 50.21 grams. ^
fii= 27.46 *•
U = 97.5° C.

r=5i.o°c.

7; = 20.8°C.

> .\ *, = 0.466

Jf = 57.21 grams. '
i» = 80.02 *•
U =97.8°C.
T = 56.5° C.
2;= 80.1° c.

- /. «, = 0.472

1^=62.24 grams. ^
i»= 26.88 ••
t^ = 97.4° C.
T = 52.0° C.
li = 24.6°C.

- /.^ = 0.451

i/ = 58.88 grams. ^
w= 82.56 ••
U =98.3°C.
r = 62.9°C.
r.= 19.8°C.

/.«. = 0.460

Mean, 0.461

Qaestions and Problems.

1. Explain in detail the derivation of the above formuia tot.

2. Describe a method for the determination of specific heats in

which there is no need for a correction due to the water-
equivalent of the calorimeter or to radiation.

EXPERIMENT 61

Olgeot. To determine the '^ melting-point^ of paraffine.
(See '' Physics/' Art. 187.)

OenenQ Theory. When a solid is heated, its temperature
rises gradually nntil it begins to melt (or vaporize), and
while the solid is changing into the liquid state the tem-
perature either remains constant or changes at an abnor-
mal rate. All crystals and most pure substances keep their
temperature unchanged while the process of fusion is in
progress, provided the mixiure of solid and liquid is well
stirred. For such substances the fusion- or melting-point
is the temperature at which the solid and liquid are in
equilibrium together. (See Experiment 44.) Obviously,
such substances, as they pass from the liquid into the solid
state, begin to solidify at the '' fusion temperature.''

However, when waxes and certain other bodies which be-
come "pasty" — such as plumbers' bolder — begin to melt,
the temperature does not remain constant, but continues
to change during the entire process until they are liquefied
completely ; and if they are cooled when in the liquid
state the temperature at which they begin to solidify is not
that at which they previously began to melt. The average
of these two temperatures is definite for any one substance,
however ; and this is called the fusion-point.

To determine this temperature for paraffine, therefore, it
is simply necessary to observe the temperature at which it
begins to melt and that at which it begins to solidify after
having been melted. (Naphthaline has been recommended
as a sai table substance to use in this experiment in place
of paraffine. It has a definite melting-point, but its odor

292 A MANUAL OF EXPERIMENTS IN PHYSICS

when melting is most disagreeable, and so it should be
melted under a hood.)

Souxoes of Brror.

1. The true temperature of the paraffine may not be recorded by

the thermometer.

2. Errors may be introduce<l by radiation, if the temperature

differs much from that of the room, or if the apparatus is
exposed to ttic radiation from very hot bodies.

Apparatus. Some small pieces of paraffine wax ; a centi-
grade thermometer ; a large beaker-glass ; a large test-tnbe ;
an iron tripod ; an asbestos plate ; a piece of glass tnbing
about fifteen centimetres long and six millimetres in diam-
eter; a Bunsen-burner, with rubber tubing; wire stirrers,
etc.

Maniptxlation. Heat the glass tube uniformly on all sides,
by twirling it around in a Bunsen flame until it becomes
quite soft ; draw it out of the flame and pull it out into a
long capillary tube about one millimetre in internal diameter.
Break off the capillary tube so as to leave about four centime-
tres beyond the point where it widens out into the tube prop-
er, and bend the capillary point into a hook, as shown.
Fill the capillary tube entirely with paraffine, either by
placing fragments of the wax in the upper wide part of
the tube, and heating the tube in hot water, allowing
the wax to run down into the capillary part, or by liq-
uefying the wax in any suitable vessel and drawing it
up into the capillary tube by suction. Fill the beaker-
glass with tap water and place it on the asbestos plate
upon the tripod. Bind the thermometer to the wide
part of the glass tube by means of thread or a rubber
band, so that its bulb is below the middle of th^ capil-
f'lt lary tube. Place them in a test-tube, together with a

(wire bent so as to form a stirrer. Suspend the test-
tube vertically in a beaker of water, submerging the
1 greater part of the capillary tube. Heat the water
|i^ gradually, and note carefully, estimating to tenths of
Flo. 100 the smallest scale division the temperature at which

EXPERIMENTS IN HEAT

' 298

the solid parafBne commences to liquefy^ which will be
shown by its losing the opaque, whitish color and becoming
transparent and colorless. Stir the water constantly in both
test-tube and beaker* If babbles of air obscure the capillary
tube by sticking to
it, remove them by
stroking the tube
with the wire stirrer.
When all of the sub-
merged wax has melt-
ed, remove the flame
and allow the water
to cool. Continue to
stir, and note at what
temperature the whit-
ish color reappears in
theparaffine. Thear-
ithmetical mean of
the two temperatures
thus found is called
the melting-point of
theparaffine. Repeat
this process of h eating
andcoolingfour times
in all, taking the av-
erage each time, and finally taking the mean of all four
results.

If the thermometer has large corrections for its fixed
points, their influence upon the above -noted temperature
must be calculated, and the resulting corrections applied
to the observed temperature to get its true value.

Fiaioi

294

▲ MANUAL OF EXPERIMENTS IN PHYSICS

HMtlDg

62.8*'

68.2''
68.8"

ILLUSTRATION

MbLTING-POINT op PARAFnNB
Cooling

64.8°

64.4*'

64. 6**

64.5°

68.8°
58. r
58.9°
68.9°

Sept 4, 1891

Average. 68.88°.

Thermometer No. 48

^ 0° mark too low by 0.4°.
nOO° •• *• high** 0.6°.

100

lUU

Correctea lemperatare = ^^_^ ^(53.88 - 0.4) = 68.96°

Qnestions and Problems.
1. Metitiou two or more objections to performing tbe above ex-
periment by simply immerBing the bulb of the thermometer
in liquid parafflne and noting tbe temperature when it solid'
ifies and again when it fuses by reheating.

EXPERIMENT 62

(two OB8KBVBB8 ARE &BQUIHBD)

Olgeot. To determine the latent heat of fusion of water.
(See " Physics," Art. 189.)

General Theory. The latent heat of fusion is defined as
the number of units of heat required to convert one gram
of a substance from the solid into the liquid state, without
change of temperature. A known mass of ice is melted by
putting it in water whose mass and temperature are also
known ; and the consequent fall in temperature of the wa-
ter is noted. The energy required to melt the ice and raise
the temperature of the water thus formed to that of the
mixture is given out by the calorimeter and the water it con-
tains, provided there is no external radiation or conduction.
Let a = water-equivalent of calorimeter and stirrer.
m = mass of water in the calorimeter before the ice is

put in.
if = mass of ice melted.
L =z latent heat of ice.
T^ = temperature of water and calorimeter before the

ice is put in.
e^ = temperature of water and calorimeter just after
the ice is all melted.
Then, if there are no extraneous losses or gains of energy,
(e + L)M=z {m + a)(r- e).
. r_(m + a)(y^e)

The above deduction of the formula is based on the as-

296 A MANUAL OF EXPERIMENTS IN PHYSICS

samptioii that there are uo other exchanges of heat than
those between the ice, the water, and the calorimeter. This
assumption is not justified unless special precautions are
taken to avoid heat being added to the calorimeter and its
contents^ or taken from them, by the air and surrounding
objects. If the initial temperature of the water is equal to
t])at of the surrounding air, then, when the ice is added, the
temperature of the water will be lowered, and consequently
heat will flow into the mixture from the air, and the final
temperature will be too high. This difficulty may be
avoided by making the initial temperature of the water
greater than that of the room, and so choosing the amounts
of water and ice used that the final temperature will be
just as much below that of the air as the initial tempera-
ture was above it. Thus the amount of heat radiated by
the water and the calorimeter, while cooling to the tem-
perature of the air and surrounding bodies, is counterbal-
anced by the amount of heat absorbed while cooling from
the temperature of the air to that finally reached, pro-
vided the rate of fall be the same, above and below the tem-
perature of the air. (Experience shows that 20° C. is a
good range through which to cool the water by adding ice.)
Time will be saved by knowing beforehand the amount
of ice necessary to bring about the fall in temperature of
20® C. This may bo calculated from the above formula by
assuming the approximate latent heat of ice to be known.
(If the approximate value of L were not known, it could be
found by a preliminary experiment similar to this one, but
leaving out the radiation correction.) Assume Z = 80 ap-
proximately ; and if ^° is the temperature of the air where
the experiment is performed, and m the number of grams
of water to be used, the number of grams M' of ice neces-
sary to cool it from T° = ^°-f 10 to e** = r-10 is given
by the formula.

(e+80)3f'=20(m+a) ; whence Jf'=2o(^^j=20^,-
An approximate value of a may be found by multiplying

EXPERIMENTS IN HEAT

297

the mass of the calorimeter and stirrer by 0.095, which is
the specific heat of copper. Consequently^ M' may be cal-
culated. Therefore, approximately, the above masses of
ice and water should be used so as to avoid having to take
account of radiation when the cooling takes place from
(/° + 10^) to(/°-10°).

Sources of Brror.

1. There still may be losses or gains of heat energy by ntdiation

or coDduction.

2. The ice may not be diy.

3. Care must be taken not to lose wat«r by splashing while stir-

ring or putting in the ice.

4. The ice must be kept below the surface of the water.

ApparatoB. A large-size nickel'plated calorimeter ; a cir-
cular stirrer covered with wire gauze; a vessel large enough
to enclose the calorimeter ; a thermometer ; some ice ; dry-
ing-paper ; cotton-wool ; a large beaker, in which to heat
water; a tripod; asbestos dish ; Bunsen-bumer, etc.

Manipulation. The water- equivalent of the calorimeter
and its stirrer must be either determined as in the two pre-

vious experiments, or calculated from a knowledge of their
mass and material, in which case their specific heats may
be found in tables.

Then proceed with the experiment in the following man-
ner: Carefully weigh the calorimeter and stirrer together
on the platform-balance. Gall the mass m\ Fill the calo-
rimeter about two-thirds full of water, leaving the stirrer

298 A MANUAL OF EXPERIMENTS IN PHTSIGS

in, and weigh again. This mass, less m\ is the mass of water
needed to fill the calorimeter two-thirds full. (If there is
a fraction of a gram, only the nearest whole number may
be noted.) This whole number is m, the mass of water
which is to be used in the final experiment. Note the tem-
perature f* of the room near the balances with the ther-
mometer to be used in the rest of the experiment. From
m and V^ calculate M\ as explained above. Empty and
thoroughly dry the calorimeter and stirrer. Rest the calo-
rimeter on three corks of equal height inside the beaker.
Fill the space between the calorimeter and glass with cotton-
wool. Make a level pad of cotton-wool on the scale-pan
and place on it the beaker, with the calorimeter, thermom-
eter, and stirrer inside. Counterpoise this whole mass {K)
with weights. Add weights equal to m grams to the coun-
terpoise already in the scale-pan. Ileat more than m grams
of water to about (i -h 20) degrees in any suitable vessel.

While the water is being warmed break into pieces, the
size of small chestnuts, a good deal more than Jf' grams
of ice. The cracked ice may be kept from melting by put-
ting it in a covered beaker, which is floated deeply in a
larger vessel full of ice and water. The quantities of ice
and water just mentioned need not be weighed, but roughly
estimated. When everything is ready, pour the heated
water into the calorimeter until the mass K-\' m is exactly
balanced, using a pipette to make the final adjustment
Add weights equal to the whole number nearest M' grams
to the K+m grams already on the scale -pan. Stir the
water continually but gently ; and, when its temperature
is a little above (/ + 10) degrees begin to add dry pieces
of ice, always placing them under the gauze of the stirrer.
Keep on stirring and add the ice slowly, but never let all
the ice in the calorimeter melt before more is put in. Note
T°, the temperature of the calorimeter, at the very instant
before the first piece of ice enters. The ice should be thor-
oughly dried by wiping it with cold drying-paper; and it
should never be allowed to touch any warm object, such as

&XP£RIMfiNTS IN HEAf

^90

the hand. Continne to add dry pieces of ice until the scales
are very nearly balanced, Test the balance often by press-
ing down with a finger the pan holding the calorimeter^ so
as to feel how much more ice is needed. Stir the mixture
geutly and continuously^ noting from time to time the fall
in temperature. When the mass in the calorimeter has
become only a very little too small add one piece of dry
ice, and note the temperature (6°) of the mixture at the
very instant the last piece of ice disappears. Now make
whatever slight change may be necessary in the weights in
the other pan to bring about an exact balance. Call the
total weight thus found TT. Then the exact mass of ice
put in and melted is W^{K+in) grams = Jf. Substitute
m, MyQ, and Tin the formula and calculate the latent heat
of ice. Repeat the experiment and tabulate the results as

below.

ILLUSTRATION

Latcnt Hbat of Fusion of Igb
CcUculation far M'
m' = 270.96 grams ^

V4I r 111X1 •4- II imrk V ^'/i

-=228 grams.

Oct 24, 18M

m =1021.03 '

t = 2s.rc.

jy_ 20 (1021 +0.095x271)

28.7+70

Bzpertment 3
26.7
1045 grams.
222.61 "
88.6° 0.
140° 0.
80.01

KzperinMDt 1
a = 25.7
m = 1021 grams
M= 224.07 "
T= 84.6° C.
e = 14.4° 0.
X= 80.08

Mean =80.02
QuestioiiB and Problema
1. Why is ii necessary to have the stirrer covered with gauze and

to place the ice beneath it?
2 If in a copper calorimeter of mass 100 grams, which contains
1000 grams of water at 80° C, there be dropped 10 grams
of ice at QP C, what is the final temperature?
8. If 10 grams of lead at 100° G. be put into a Bunsen ice calorime-
ter, what change in volume will ihe mercury index indicate ?
4. How many ergs are required to make 1 gram of ice pass from
—10° 0. to 60° 0. , the atmospheric pressure being 76 centime-
tres of mercury ?

OF THE

UNIVERSITY

EXPERIMENT 68

Olgect To determine the boiling-point of benzene. (See
" Physics/' Art. 194.)

General Theory. The boiling-point of a liquid is the teoi-
perature at which ebullition takes place freely under a
given pressure; or^ what is the same things it is the tem-
perature of equilibrium of the vapor and liquid at the given
pressure. To determine it^ a liquid is made to boil freely.,
and a thermometer is immersed in the vapor (not in the
liquid)^ and its temperature is read.

Bouroes of Brror.

1. TLe most serious source of error is loss of heat by radiation.

2. The pressure on the vapor must remain constant.

8. Irregutur boiling should be avoided as far as possible by pat-
ting in the liquid a few pieces of sharp-pointed gUss, and
by heating gradually.

Apparatus. A boiling apparatus^ consisting^ as shown, of
one test-tube held inside another, the inner having two
holes in it— one in the side near the top, the other at the
bottom ; a condenser, consisting of a tube or flask sur-
rounded by a water-jacket; a large beaker; stirrer; Bun-
^en-burner, tripod, etc.; a roll of asbestos; 20 cubic centi-
metres of benzene; thermometer; clamp-stand.

Manipulation. Caution. —The vapor of benzene is most
inflammable, therefore be sure that all connections are
tight.

Arrange the apparatus near a water tap and sink.

Insert the roll of asbestos in the inner test-tube, so as to
make a lining to it; pat a thermometer through the cork,

EXPERIMENTS IN HEAT

SOI

and also a glass tnbe leading to the condenser. Place the
benzene in the bottom of the outer test-tube, and lower
into place the inner tube, holding it by means of a tight-
fitting cork so that it does not touch the liquid.
Support the larger test-tube in a clamp-stand, ^"^^
and place around it a water -bath — e,g^, a y

beaker filled with water, supported on an asbes-
tos dish and tripod. Join the inner test-tube
to a Liebig^s condenser, and attach to the other
end of the condenser a small Florence flask to
catch the condensed liquid.. It may be well to
have this flask surrounded as far as possible by
cold water. Connect the water-jacket of the
condenser to the water tap and sink, and turn
the water slowly on.

By means of a Bunsen-bumer raise the tem-
perature of the water-bath until the benzene is boiling free-
ly; stir the water constantly by means of a stirrer. When
the thermometer reaches a steady state, read it carefully.
Turn out the Bunsen-bumer; read the barometer; then
relight the burner, and make another reading.

Remove the asbestos lining from the inner test-tube, and
measure the boiling-point again, and account for the ob-
served change.

Fio. 108

April 38, 18M

ILLUSTRATION

BOILINO-POINT OF BCNZKNB
a. with Asbestot 6. Wlthoot

80.8<>±.2 81.0°

80.4° ±.1 81.0°

Mean, 80.85° Mean, 81.0°

Barometer, 75.76 ccntiroetre.9.

Questionfl and Problems.

1. Expliiin in full the influence of radiation in this experiment

2. What is the effect of dissolved substances upon the boiling-

point ? Upon the vapor- pressure ? Upon the temperature
of the vapor?
8. What is the influence of the material and smoothness of the
walls upon boiling and upon oondens;ition?

EXPERIMENT 64

(TWO 0BBEBVBB8 ABB REQUIRED;

Olgect To determine the latent heat of evaporation of
water at 100° C. (See " Physics," Art. 196.)

General Theory. The latent heat of evaporation is by
definition the number of calories required to make one
gram of a liquid pass into the state of vapor at a definite
temperature ; conversely, this number of calories is given
up by the vapor if one gram condenses to liquid at the
same temperature. Consequently, if m grams of steam
are condensed at /q^ by being passed into M grams of water,
whose initial temperature is T'q®, the temperature of the
water will rise to a temperature t°, where, if a is the water-
equivalent of the calorimeter, and L the latent heat of
steam at (q^, t satisfies the equation

m{L + to- 1) = (Jif + a){t - To).

The temperatures and masses may all be measured, and so
L can be determined, for

r^ = ~^{t-T,)-{t,-t).

Particular precautions must be taken to guard against
interchange of heat energy with surrounding bodies.

Souroes of Error.

1. If the steam is not quite dry as it enters the water, a great error

is introduced.

2. There are always losses due to radiation and conduction.
9, Tt)9 pressure on the steam should be kept constant.

EXPERIMENTS IN HEAT

808

Fio. IM

Apparatus.* Two calorimeters — large and small — one
fastened in the other by a cork ; a cover for the inner
one ; a condensing vessel to go in the smaller calorimeter ;
stirrer; thermometer; rubber tubing ; boiler; glass water-
trap ; Bunsen-bumer. The condensing
vessel^ into which the steam is to be ad-
mitted^ consists of a metal can on legs^
which rests inside the smaller calorim-
eter.

Hanipnlation. Determine the water-
equivalent^ a, of the inner calorimeter,
condenser^ and stirrer, either by actual
experiment or calculation. Weigh the
calorimeter and its appliances empty and
dry ; then fill it about two-thirds full of
water, allowing none to enter the condens-
er, and weigh again. Eecord both weigh-
ings and deduce the mass, if, of water in the calorimeter.
Place the smaller calorimeter inside the larger, and put on
the cover, leaving both tubes of the condenser projecting —
the one to connect with the boiler, the other as an opening

into the air of the room —

so that the pressure inside
the condenser may be as
closely as possible that de-
noted by the barometer.
Set up the boiler as near the calorimeter as possible, and
connect it to the condenser with rubber tubing. Insert in
this connecting tube a glass trap, made as in the figure, and
designed to catch any water that may be condensed. The
trap must be vertical, and placed so that a short piece
of rubber joined to it fits directly on to the condenser.
Wrap the connecting tube and trap in cotton-wool. Place
a board between the calorimeter and the boiler and
burner. When the apparatus is set up as described, light

* This form of apparatus was suggested by Professor Schuster, of Man-
chester, England.

KiO. 105

804 A MANUAL OF £XP£RIMEKTS IN PHYSICS

the burner^ disconnect the trap from the condenser, and
tarn the trap away from the calorimeter, so that the steam
may not etrike the latter until it is desired to make the
connection again. When steam has been issuing freely
from the tube long enough to heat it thoroughly, measure
the temperature, jT**, of the water in the calorimeter, note
the time on a watch or clock, and at once make the con-
nection with the condenser. Stir constantly, and read the
temperature from time to time, recording the minutes and
seconds. When the temperature has risen to about 50°
or 60° C, deftly remove the steam-tube and note the time.
Note the exact temperature at the time of removal of the
steam-tube.

Allow the water to stand for five minutes, stirring con-
stantly and noting the change of temperature at intervals of
thirty seconds. Remove the calorimeter from its outer ves-
'sel, weigh, and thus calculate the amount of steam which has
condensed, 7n, The temperature of the steam as it enters,
/o°, may be calculated from the reading of the barometer.
It will be noticed that the temperature of the water con-
tinues to rise for some time after the steam-pipe is discon-
nected, owing to the time taken for the temperature within
and without the condenser to become the same. Then the
temperature falls gradually. The rate of- fall — t. c, the de-
crease in temperature per second — is approximately a meas-
ure of the rate of loss of heat by radiation, while the tem-
perature of the water was rising. Consequently, the product
of the time taken by the temperature of the water to rise
from T^o to its highest value by the rate of fall is a correc-
tion which must be added to the highest temperature reach-
ed, in order to give the temperature which would have been
reached if there had been no loss by radiation during this
rise. Let f" be this corrected highest temperature, then

Bepeat the experiment, using different amounts of water
and different temperatures.

EXPERIMENTS IN HEAT

806

NoTX. — A graphical method of making the radiation correction just de-
scribed is as follows :

Plot the times from the instant steam is admitted as abscisss, and the
temperatures as ordinates. The curve oonnecdng the points will be as shown.
The highest recorded tern- S
neratnre ia given bjr the
highest point of the curre; g
and the radiation correotion j^
id found by producing back-
ward the line of cooling,
which 18 approximately
straight. The pdnt where
this line intersects the tem>
peratnre axis gives (be lein-
perature which woald have
been reached if there bad
been no radiation.

Expertmeni I
Jf = 70.e65
a =18.10
JH= 8.180

U =100°

t =68° + 4o = 72o
./:=586

FiG.^Ofl
ILLUSTRATION
Latint Hxat of Stiam

Experiment II

jf=70.oao

a =18.10
m= 5.800
T.= 1.5°
t. =100°
t =40.6°
:.X=686

Hay 8, 1897

8o chosen as to
avoid radiation
correction.

Questions and Problems.

1. Explain the energy changes — kinetic and potential — when 1

gram of ice is melted and then boiled.

2. Explain upder what conditions ice can be made to pasa direct-

ly into the form of vapor.
8. Bxplain mathematically why the radiation curve gives the
same result as the calculation.

4. Describe and explain the action of an ice-making machine.

5. If the sphere and balance- pan of a Joly steam calorimeter be

of platinum and weigh 10 grams, what will be the increase
in weight if, before steam is admitted, the sphere contains
10 grams of ice at 0^ CT
0. A cryophorus contains 50 grams of water at (P C. ; bow much
ice is formed when 1 gram of water is evaporated ?

€€

EXPERIMENT 55

Object. To verify the law of saturated vapor. (See

Physics/'Arts. 192, 193.)

General Theory. If a liquid and its vapor are in equilib-
rium at a definite temperature, the pressure of the vapor
is independent of the volume occupied. In other words,
for a definite temperature of saturated vapor there is a def-
inite pressure, and vice versa.

There are two methods by which this law may be veri-
fied : 1. By measuring the boiling-point of the liquid; for
it may be shown to be constant at any definite pressure.
(See Experiment 44.) 2. By actually measuring the press-
ure produced by saturated vapor at any definite tempera-
ture, the volume being varied.

In carrying out this second method, the direct plan is to
force some of the liquid into a barometer so that it floats
on the top of the mercury column, and to measure the
pressure by the difference between the heights of a true
barometer and the mercury column on which the liquid
rests. The temperature may be kept constant by suitable
baths, and the volume of the vapor may be varied by caus-
ing the mercury column to rise and fall relatively to the
top of the tube. The formula for the pressure is given in
full below.

Sources of Error.

1. There may be traces of gas, particularly air, mixed with the

vapor.

2. The vapor may not be at the temperature of the bath.

8. Expansion and compression of a gas always change its tem-
perature unless they are done very slowly.

EXPERIMENTS IX HEAT

807

Apparatus. There are two forms of the apparatus :
1. A long clean glass tube closed at one end is filled al-
most entirely with mercury. Air and all other gases were
expelled when the apparatus was made, and afterwards a
little water was introduced which
remains in the space above the mer-
cury, partly as liquid and partly as
water vapor, the latter being al-
ways saturated, since more water
would evaporate were it not. The
other end of the tube is bent up
and left open. A metre scale is
put between the tubes so that
the level of the mercury in each
may be measured. The open end
is connected with a siphon and
pinch-cock, so that mercury may
be siphoned in or out of the tube,
and the volume of the vapor in-
creased or decreased at will by
the consequent change of level of
the mercury surfaces. The upper
end of the closed tube is sur-
rounded by a wider tube, to be
filled with water of any desired
temperature. The wide tube con-
tains a thermometer and stirrer to
measure the temperature and make '
it uniform throughout.

2. A long, straight, clean barometer tube is supported
vertically with its lower end immersed in a reservoir of
mercury. Air, etc., were expelled as in the other appa-
ratus, and ether was introduced, which now fills the tube
above the mercury, partly as liquid and partly as saturated
ether vapor. There is a water reservoir with thermometer
and stirrer, as in the other apparatus, about the closed end
of the tube to regulate its temperatpre. ^he volume oc«

Fio. lOT

808

A MANUAL OF EXPERIMENTS IN PHYSICS

capied by the ether and its yapor is increased or dimin-
ished by raising the tube out of the mercury or lowering
it into it. A metre -rod is used to measure the heights,

and a clamp -stand
to hold the tube.

Manipulation. Ar-
range the apparatus
over a mercury4rayy
so that the closed
tube and millimetre
scale are truly ver-
tical. Fill the water
reservoir around the
upper end of the tube
with tap water, and
hang the thermom-
eter in it. Hang
another thermome-
ter in the air near
the lower end of the
closed tube. Fill the
open tube in Appa-
ratus 1 (push down
the tube in Appa-
ratus 2) until most
of the vapor is con-
densed. Do this
slowly, so as to avoid
great changes of
temperature. Stir
the water in the res-
ervoir and watch the
thermometer until
it becomes stationary. Call the temperature T°. Read
the barometer, and, after stirring again and noting that
the temperature is still the same, note the difference in
height of the mercury columns in the tWQ tubes (in Appar

Fio. 108

fiXPERIl[£NTS IN HEAT 809

rains 2 the height in the tube above the free snrface), and
also the height of the liquid above the mercury inside the
closed tube. Use a piece of paper as described in Experi-
ment 29 to make these measurements accurately. Then^ if
h = height of mercury in closed tube above that in the

open tube,
h' = height of liquid above mercury,
/^= ** corrected '' barometer reading,
T=z temperature in water reservoir,
t = temperature of lower thermometer,
P =5 density of mercury at ^°,
p = density of the liquid at jT®,
the pressure of the vapor will be given by the formula
pz=. (13.6/^— Ap— AV)980 dynes per square centimetre.
Stir the water in the reservoir constantly, and siphon out
enough mercury to increase the volume above the mer-
cury by 20 centimetres. (In Apparatus 2 raise the tube to
this extent.) Wait ten minutes at least before reading the
new difference of level, stirring meanwhile ; and if the
temi)erature of the water in the reservoir falls, add enough
hot water to bring it back exactly to T^ again. When this
temperature has been permanently obtained with the new
volume, make the same series of readings as before — L e.,
the temperatures on the two thermometers, and the heights,
h and h\

Increase the volume again in the same way as much as
can be done without evaporating all the liquid on top of
the mercury. (There must be some liquid present as such,
or the vapor will no longer be saturated. ) Wait ten minutes
or more again, restore the temperature T^ by adding hot
water if necessary, and repeat the readings as before.

Bepeat the whole of the above process with the reservoir
at the temperatures 60® and 80® ; but at 80° one increase of
volume will be sufficient. Take the barometric reading
again at the close of the experiment. Correct for any
great errors in the thermometer and calculate the vapor-
preesnres.

810

A MANUAL OF EXPERIMENTS IN PHYSICS

Feb. S, 1086

ILLUSTRATION

Law of Saturated Vapor

p is constant for different volumes if tlie temperature is constant

Water-vapor

p = {\9.QH- pk - p'A')980.

Corrected, 11= 75.88 at beginning. ) . . tt r,^ ^
ri- IVK01 . ^ > Mean, JBr= 75.8a.

J7= 75.81 at end. S

Case {A). 7'*» = 19.8** C. (corrected).

^^ = 19.8^

p' at 19.8*' =

=0.998; p = 18.56.

*P+*V

Difll from lf«aB

74.83

1.26

+ 1008.32

- .81

74.28

8.72

1010.18

+ 1.00

74.20

2.59

1009.15

+ .02

Mean, 1009.18
Hence it is evident Uiat the pressure has remained constant to witiJic
1 part in 1000, its value being

p = (75.82 -74.2)18.6x980 = 1.62x13.6x980.

se(B).

' = 60.2^

= 19.4^

p -0.988.

/9 = 18.54,

hp-\-hy

Differences

60.98

1.88

827.58

-.41

60.95

2.41

828.21

+ .27

60.89

8.15

828.09

+ .15

Mean, 827.94
Again, the pressure has remained constant, its value being

l>=(75.82-60.9)18.6x980=14. 92x18.6x980.
Ckue{C). 2"'=80.1°. ^^ = 19.4''.

=0.972.

= 18.54.

ft'

*p+*y

DlffereDoet

40.26

1.15

546.50

+ .09

40.18

2.63

546.82

-.09

Mean, 546.41
Hence p=(75.82 - 40.2)18.6 x 980 = 85.62 x 18.6 x 980.
(Plot the observations, and draw a curve through the pointB.)

Qoestiona and Problems.

1. Wliy does the temperature fall below T° when the volume i«

increased, and rise when it is decreased ?

2. Would the pressure stay the same if the volume were in-

creased after all the liquid had evaporated ?
8. What change would take place ?

EXPERIMENT 66

(two OBSBRVBBS abb BBQUnUBD)

Oldeot. To plot the '* cooling curve " of a hot body in a
space sarronnded by walls at a constant lower temperatare :
1. When the surface of the body and the walls are of pol-
ished metal. 2. When they are blackened. (See " Physics,''
Art. 206.)

General Theory. The rate of radiation of a body depends
npon the nature of its surface, and upon the temperature
and nature of surrounding bodies. The simplest method
of comparing the radiation of different bodies under differ-
ent conditions is to compare the rates at which their tem-
peratures change — i,e,yio study their "cooling curves.''
Such a curve is obtained by observing the temperature of
the body at small intervals of time, and plotting the observa-
tions— ^the intervals of time reckoned from any fixed instant
as abscissaB, the corresponding temperatures as ordinates.

The two most interesting cases are a polished body sur-
rounded by another polished surface, and a blackened body
surrounded by another blackened one. To make the re-
sults comparable, the temperatures of the outer bodies
should be kept constant and the same ; and the initial tem-
peratures of the two inner bodies should be the same.

Bonroes of Error.

1. Tbe surfaces may not be at the temperatures of the thermome-

tere used.

2. Loflses of heat may occur by radiation otherwise than to the

outer bodies — by conduction or by evaporation.

Apparatus. Three thermometers ; two large and two
small calorimeters with corks to hold the latter in the

812 A MANUAL OF EXPERIMENTS IN PHYSICS

former^ one set being bright and clean and the other blackeD-
ed ; covers and stirrers for the smaller calorimeters; a glass
jar large enough to hold both large calorimeters^ a stirrer
for it^ and flat corks on which to stand the calorimeters; bal-
last to pat in the bottom of the large calorimeters to keep
them steady ; a vessel, etc. , for heating water ; a watch.

Manipulation. Heat plenty of water. Place the ballast
in the large calorimeters, and then the corks in which are
fitted the small calorimeters. Stand the calorimeters on
the corks in the bottom of the glass jar. Take the tem-
perature of the room, and fill the glass jar to three-fourths
the height of the calorimeters with water of that tempera-
ture, mixing some of the heated water with tap water for
that purpose. Be careful not to splash water so that it
might run down between the small and large calorimeters.
Fit the covers of the small calorimeters with thermometers
and stirrers, and see that they go on easily. Place the
watch open on the table.

When all else is ready, and the water is boiling, ponr
enough into each small calorimeter to fill them to the same
height — two-thirds full is enough. Fill the polished one
first, and then, immediately after, the other ; and let another
observer put the cover on each the moment it is filled, turn*
ing the thermometers so that one man can observe both.

As quickly as possible, when both are filled and covered,
Observer 1 reads the thermometer, and taps on the table
with his pencil when the mercury in one passes the first
degree mark as it falls, then turns to the other thermome-
ter and taps similarly for that, calling "black" or "bright"
each time, according to which calorimeter it is in. Obsen-
er 2 reads the watch, and puts down the minute and sec-
ond under the name of the proper thermometer. Observer
1 stirs both vessels, and continues to tap for each thermom-
eter every two degrees just as the mercury crosses the mark,
always naming the thermometer ; and Observer 2 notes the
time of each tap in the proper column. As soon as the fall
in temperature becomes slow enough, Observer 1 should

EXPBRIllKNTS IN HEAT

318

give the temperature-reading itself^ as well as the name of
the thermometer; and^ though this may not be possible
with the first one or two readings^ it will be easy to coant
back and see what temperatares the times noted correspond
to. Continue the experiment until both have cooled down
to about 40° C. Stir the water in the outer jar occasionally,
and take its temperature every five minutes at least ; and
if it gets warmer than the room, pour in a little cold water.
Report as below, and plot the ** cooling curves '' on co-
ordinate paper, using the same axes for both, and dotting
one curve to distinguish it.

ILLUSTRATION
Law op Radiation

Jan. 34, 1896

Totisbed Calortneter

Temperature

Black Calorimeter

Water d Battery -Jar

1 Time

Time

a M. &

oc.

H. H. S.

OC.

8 21 05

22.5

8 21 20

8 22 40

8 21 45

8 24 25

8 22 55

8 26 05

8 24 20

8 27 40

8 25 55

8 29 85

8 27 25

8 81 40

8 29 00

8 88 55

8 80 45

8 86 20

8 82 28

8 89 00

8 84 15

8 41 85

8 86 05

28.6

8 44 10

8 88 05

8 47 05

3 40 25

8 50 18

8 42 80

8 58 40

8 45 15

28.76

8 57 15

8 48 08

4 1 80

8 51 20

24.0

4 6 22

8 54 45

4 11 65

8 58 85

24.4

4 17 40

4 2 50

24.5

4 24 25

4 7 45

4 82 28

4 18 80

J4^ _

When the black calorimeter was at 40® the other one was at 45.5'' C.

814

A MANUAL OF EXPERIMENTS IN PHTSIGS

Cooling Curvet

— Blackened calorimeter.

— Polished calorimeter.

80
75

\ >

50
45

, , X >.

80 40

Fio. 109

Abscissae. Minutes after 8 h. 20 m.

Qaeatioiis and Problems.

1. Which cooled the faster, the black or polished calorimeter ?

2. How would the experiment have been affected if the inner sar-

face of the large blackened calorimeter had been polished?
8. Would the cooling have been faster or slower Jlfthesipatt calo-
rimeters had been put out in the open air of the room with-
out the outer Tessels, water, etc. ? Why ?

EXPEEIMENTS
ELECTRICITY AND MAGIIETISM

UTTRODUCTION TO ELECTRICITY AND MAGNETISM

Units and DefinitioiiB. The aotaal meaanrement of all
electric and magnetic qaan titles finally resolves itself into
meaaurements of mass, time, and distance ; and, therefore, in
their expression the G. O. S. system should be used. The def-^
inition of the unit of electrification and of a unit magnetic
pole, however, must be based upon electrical or magnetic
properties. There are three systems of such units in use ;
but, €ts the relations between them are known, it is simpl; a
matter of convenience as to which is used in any case. The
definition which serves as the basis of the first system, which
is called the ''electrostatic system,'^ is this: If two equal
electrical charges are at a distance apart of one centimetre
in air, and if their quantity is such that the force between
them is one dyne, each of these charges shall be called the
unit charge. Upon this definition are based that of unit
difference of potential — viz,, that existing between two
points such that one erg of work is required to carry a
unit charge from one to the other ; also, that of unit capac-
ity— viz., a conductor has a capacity one if when charged
with a quantity one its potential is one, or if charged with
quantity z, its potential is x; for the capacity G is by
definition such that the charge e=:CV, where V is the
potential.

The definitions which serve as the basis of the second
system, which is called the ''electromagnetic system," are
these : If two equal magnetic poles are at a distance apart
of one centimetre in airy and if their strengths are such
that the force between them is one dyne, each pole shall be

818 A MANUAL OF EXPERIMENTS IN PHTSIGS

called a unit pole ; also, if a steady electric current is flow-
ing in a conductor bent in the form of a circle of radiiu r
centimetres, and if the force on a unit magnetic pole at
the centre of the circle is 2x/r dynes, then this current
shall be called a unit current — t. e., a unit quantity of
electricity "flows by'* in one second. This evidently takes
a different quantity of electricity as the unit from that
used in the electrostatic system ; and upon it are based
definitions of unit difference of potential, unit capacity,
unit resistance, etc. — e. g., on this system a unit resistance
is that of a conductor such that if an electromagnetic unit
difference of potential be maintained at its two ends, an
electromagnetic unit of current will flow through it.

It is found by actual experiment that one electromagnetic
unit of quantity = v electrostatic units, where t^ = 3 x 10'*
very nearly.

It follows, then, at once, that
one electromagnetic unit of potential = 1/v electrostatic

units of potential,
one electromagnetic unit of capacity = v* electrostatic units
' of capacity,

I one electromagnetic unit of resistance r=l/t;* electrostatic

I units of resistance,

I It is found by actual experience that neither of these sys-

, tems is convenient for daily use, and so a system has been

i adopted called the " Practical System." The units are de-

fined as follows : The unit quantity, called the "coulomb,'*
is such that if it is passed through a solution of silver nitrate
(prepared in a definite way) an amount of silver will be de-
posited equal to 0.0011180 grams. The unit difference of
potential, called the "volt,'' is such that the potential differ-
ence of a staiidard Clark cell at 15"* C. is 1.434 volts. The
unit resistance, called the "ohm," is that of a column
of mercury at 0° C, 106.3 centimetres long, of uniform
cross - section, and containing 14.4521 grams. The unit
■ current, called the "ampere," is such that, if it flows for
one second, one coulomb passes. The unit capacity, called

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 819

the ^' farad/' is that of a conductor which, when charged
' to 1 Yolt, contains 1 coulomb.

It is foand by direct comparison that within the limits
of accuracy of experiment

1 coulomb = 10~' electromagnetic units.
/. 1 ampere = 10"' *' "

1 volt = 10" " '*

1 ohm = 10* " ''

(Hence, a potential difference of 1 yoJt at the ends of a
conductor, whose resistance is 1 ohm, will produce a flow
of 1 ampere.)

1 farad = 10~' electromagnetic units.
The "" micro-farad" is 0.0000001 of a farad— t. c,

= 10""" electromagnetic units.
Since the energy of a charge is \ quantity x potential,
the energy of 1 coulomb at potential 1 volt is 10^ times the
energy of 1 electromagnetic unit of charge at 1 electro-
magnetic unit of potential, but by definition of this last
quantity the energy of an electromagnetic unit quantity
at that potential is 1 erg. Hence the energy of 1 ooulomb
at 1 volt is 10' ergs, or 1 joule (see p. 68).

The energy furnished by a current % in one second is T-B;
hence, the energy furnished in one second by 1 ampere flow-
ing through 1 ohm is 10* ergs, or 1 joule — i. e,y the activity
of such a current through such a conductor is 1 watt.

Olyect of Experiments. All the experiments in this sec-
tion may be divided into two classes : one consists in the
general study of electric and magnetic phenomena; the
other, in the accurate determination of various quantities,
such as magnetic moments, electric currents, resistances,
etc. As stated before, in all these experiments the only
quantities directly measured are lengths, masses, and in-
tervals of time ; but owing to the fact that magnetic and
electric phenomena are involved, special precautions are
necessary.

1. In electrostatic experiments, moisture must be carefully
guarded against.

880 A MANUAL OF EXPERIMENTS IN PHYSICS

2. In order to discharge a body thoroughly, remove it from

the neighborhood of other charged bodies, and paM*
the flame of a Bunsen-bumer rapidly over it.

3. In magnetic experiments, care must be taken to avoid

the influence of bodies containing iron — e,g.y window-
weights, brackets, beams, common red bricks, etc.

4. In the study of electric currents the need of makiug

good contact everywhere cannot be too much empha-
sized. Wires should never be twisted together at the
ends, but should be either soldered or joined by a
metal connector or mercury-cup.

All plugs of a resistance-box must always be pushed
in again after any one has been removed.

5. All wires leading to and from instruments should be so

wrapped around each other that there is no apprecia-
ble area between the current going up one wire and
down the other.

G. Reference is made to Appendix III., '' Galvanometer,'^
for remarks on the proper use of the instrument.

7. The resistance of every conductor changes when a car*
rent is passed through it, owing to rise in tempera-
ture ; therefore currents should be kept on for as
short times as possible, and all other heating effects
should be avoided.

3. To make less current pass through a given instrument,
three methods are open : (a) To use fewer cells in the
battery, and so have a smaller £. M. F. {b) To insert
resistance in the circuit, thus making the current
smaller although the total E. M. F. is the same.
{c) To put a *' shunt" around the instrument — i. a., to
insert in parallel with it a certain resistance. .In this
case, if r^ is the resistance of the instrument and r%
that of the shunt, the current through the instru-
ment equals — r— times the total current flowing.

ELEOTKOSTATICS

EXPERIMENT 67

Olfjeot. To plot the fields of force around various elec-
trified bodies. (See "Physics/' Arts. 222, 228, 230.)

Oeneral Theory. A line of electrostatic force is defined
as a line snch that at each point its direction is that in
which wonid move a mobile, positive charge placed at that
point. In other words, it is the path a positive charge
would follow if the body carrying the charge had no in-
ertia. Again, if a small elongated body which is charged
+ at one end and — at the other is placed at any point of the
electric field, free to rotate, it will turn and place itself
tangential to the line of force at its centre.

To map the lines of force, then, at any point of the field,
the simplest method is to hold by an insulating support a
small elongated conductor — e.g., a bit of moistened thread,
or even a non-conductor, if it is pointed, e. g., Sk bit of
paper in the form of a needle, so that it is free to turn —
for under induction the small body becomes charged + at
one end, — at the other, and so will take a position tilong the
line of force. If the pointer is free only to turn around
an axis, it will set itself along the component of the force
which lies in the plane perpendicular to this axis.

An equipotential surface is perpendicular to lines of
force ; and so, if the field of force has been drawn, the sur-
faces may be easily constructed.

Bowces of Brror.
1. If the charges on neighboring bodies change, the direction of

the field of force will change.
SI

aa2

A MANUAL OF EXPERIMENTS IN PHTSIG8

2. If the pointer receives too great a charge it will modify the
field.

Apparatus. Two tin cans or other metallic bodies of
large surface ; two fiat wooden boxes
filled with paraffine for use as insulating
stands ; a pointer consisting of an insTi-
lating handle carrying a light paper vane
strong on a stretched silk fibre, as shown.
The fibre is passed through a hole at the
centre of the vane, which is about 1.5
centimetres long, and is attached to the
prongs of the handle, which is easily made
of glass rod or tubing. A drop of wax
on the fibre keeps the vane from sliding
along it. The yane can then rotate
freely in a plane at right angles to the
fibre.

Three sheets of paper are needed, the
size used at Johns Hopkins XJniversitj
being about 60 x 45 centimetres. An
Fia.110 electric machine must be near by to

charge the can.
Manipulation. Field about a SbigU 0/iarged Condudor.
— If the surface of the paraffine in the stands is very dirty,
scrape it clean with the back of a knife. Place a can on
one of the stands in a position to be assigned by the in-
structor. ' Divide one of the sheets into two halves by a
pencil line, and in the middle of one half draw in oatline
a horizontal section of the can and stand ; in the other, a
vertical section. The drawings should be about half-size.
Keeping the can in the assigned position, hold it against
one of the knobs of the machine until it is highly charged.
Place it on the table near the drawing of the horizontal
section. Hold the pointer so that the fibre is vertical.
Bring it close to the can, so that one tip of the vane almost
touches the side. Mark the point on the diagram and the
direction of the vane. Draw the pointer slowly away in

EXPERIMENTS IN ELECTRICITY AND MAGNETISM SSd

snch a direction as to move the vane always in the direc-
tion in which it points — i. e.y allow the vane to move as it
would if free to follow the line of force, but keep it in the
same horizontal plane. Watch the motion very carefnlly,
and repeat several times, nntil the path of the line of force
is definitely located and fixed in the mind. Then draw the
line on the diagram.

Take another point on the surface of the can in the same
horizontal plane about half a centimetre away, and draw in
a similar manner the line starting from that point. Con-
tinue similarly around the can until the starting-point is
reached. In this way a complete horizontal section of the
field of force will be obtained. Becharge the can whenever
necessary.

To study the vertical section, hold the pointer with the
fibre horizontal, and bring the vane as closely as possible
to a point of the can where it touches the stand. Mark
the position, and draw the line of force as in the case of
the horizontal lines. Next take a point on the surface of
the can half a centimetre above the first and in the same
vertical plane, draw the line of force, and repeat similarly
until lines have been drawn leaving the same vertical sec-
tion of the can at points half a centimetre apart, all the
way around, to the line where the surface of the can and the
stand again meet. This completes the vertical section.

Two Bodies Charged Alike. — Place two cans on their
stands and bring them near each other, about 5 centimetres
apart. Draw a sketch of the horizontal section of the two
in the middle of one side of a sheet of paper, and a vertical
section in the middle of the other side of the same sheet.
Charge both at the same knob of the machine and replace
in the positions drawn.

Starting with the pointer near one can, draw the horizon-
tal line of force from that point until it can be traced no
farther, or until it ends on the other can. In the latter case,
mark very carefully the position where it ends. Draw other
pnes in the horizontal section from points half a centimetre

8i4 ▲ MANUAL OF EXPEBIMKNTS IX PHYSIOB

apart all around the first can, and also fill in lines for the
second can from portions of the same horizontal section at
which no lines from the first end.

When the horizontal section is complete, draw the Terti-
cal section in a similar manner.

Two Bodies Oppositely Charged. — Draw the two sections
of the two cans again in the same positions on the third
sheet of paper. Charge them at different knobs of the ma-
chine, and, placing them in the positions drawn, trace again
as before the horizontal and vertical sections of the field of
force.

Finally, label each sheet and draw in ink — ^red, if avail-
able— five equipotential surfaces for each case.

An interesting variation is to plot the field about the
knobs of the machine itself.

Questions and Problems.

1. Why does tbe pointer set itself along the line of force ? Is tbe

paper pointer a couductor ?

2. Wbat further observations, if any, would be necessary to de-

termine from your diagrams the actual direction of tbe
force at any point of tbe field ?
8. Show by a diagram in your book how the field in each esse of
two cans would differ if one of tlie cans was unchnrged.

4. Do any lines of force start from the mside of the chd ? and, If

so, wha^t is their direction ?

5. How does recharging the can affect the lines of force and the

equipotential surfaces ?

6. Do lines of force pass through a conductor ?

EXPERIMENT 68

Ol{jeotb A stady of electrostatic induction by means of
the gold-leaf electroscope.
General Theory. Read " Physics/' pp. 283-300.

Soiuroe of Bnor.
Leakage must be carefully guarded against

ApparatOB. Two gold-leaf electroscopes (" Physics/' p.
276) ; a piece of copper wire about 30 centimetres long ;
a rod of ebonite^ glass, or sealing-wax ; a piece of fur or silk.

Manipnlation. Clean carefully the metal plates of the
electroscopes. Note carefully in each of the following cases
all movements of the gold leaves, describe them fully, and
explain. Whenever the gold leaves are described as being
charged^ determine and state what has become of the elec-
tricity of opposite sign. The charge produced on an ebon-
ite rod when it is rubbed with silk or fur is by definition
called ''negative.''

(1) fiub the rod very slightly. Touch it to the knob
and then remove it. Describe and explain.

(2) Bub the knob itself, remove the rubber, describe and
explain.

(3) Discharge by joining the knob to the earth with the
hand. Bring down the charged rod near enough to make
the leaves diverge well, but not so much that either touches
the side of the bottle. If one does touch, discharge the
electroscope by touching it with the finger and begin
again. Touch the knob with the finger while holding the
rod still. Bemove first the rod, then the finger. Describe
and explain.

826 A MANUAL OF EXPERIMENTS IN PHYSICS

(4) Repeat, bat remove first the finger, then the rod.
Describe and explain. Does it make any difference whether
the rod is held nearer to the leaves or to the knob ?

(6) Leaving the electroscope charged, bring in turn the
rod, the rubber, and the palm of the hand down towards
the knob from above. Do not touch the knob in any case.
Describe and explain.

(6) Repeat, but approach the objects to the leaves.

(7) Discharge and recharge as before, but note carefully
the exact distance between the knob and the nearest point
of the rod when the finger is removed. Bring the rod
down slowly and describe and explain the motion of the
leaves while the rod is farther from the knob than the
charging distance, when it reaches this distance, and when
it is brought nearer.

(8) Connect the knobs of the electroscopes by means
of the copper wire, and separate them as far as possible.
Call the left electroscope A, the right B. Approach the
rod to the knob of A from the left. Describe and explain.

(9) Holding it near, but to the left of A, touch A with
the finger, and remove first the finger, then the rod. Show
by a diagram the charges on leaves and knob of both A and
B and on the wire. Explain.

(10) Discharge. Bring up the rod to the same position
near A as before, but charge by touching B. Give dia-
gram of charges again. Explain. Would the chlirges left
on the whole system differ in any way if, in charging, the
rod were brought near different points? if the electro-
scope were touched at different points ? Why ?

(11) Charge the system with the rod held 2 centimetres
to the left of A. Remove the rod and describe and ex-
plain what happens when it is again approached to within
1 centimetre. Give diagrams of charges when rod is 3 cen-
timetres, 2 centimetres, and 1 centimetre to the left of A

(12) Repeat, bringing the rod down towards the middle
point of the wire, and give diagrams where it is 3, 2, and 1
centimetres above it.

£XPBElli£NTS IN ELECTRICITY AND MAGNETISM 827

In each diagram draw three eqaipotential surfaces (in-
cluding that ofo, + l and — 1^ if they oocar) and five lines
of force from each knob.

Note. — In thia and all similar experimeDts the student should be oarefol
in answering each section to give a brief description of what was done, and
not merely gire the number of the sections and a description of what hap-
pened. The instructor has not tinie to refer to the directions to find the proc-
ess that gave the result described.

QootioM and Problems.

1. A sphere of radius 5 centimetres is at potential — 6 ; it is then
Joined to another sphere of radius 10 centimetres, while its
potential is kept constant. What are the final charges on
the two spiieres ? How much work has been done, and how
has it been done ?

8. If the potential had not been kept constant, what would have
been the final potential and charges r Discuss the initial
and final energy.

EXPERIMENT 60

Object To show, after the method of Farada/s "loJ
Pail " experiment (see " Physics/' Art. 232), that :

1. If a charged body is placed inside a closed conductor,
a charge of opposite sign is induced on the inner surface of
the conductor and one of like sign on the outer, the two
induced charges being equal in amount to each other and
to the charge on the body.

2. That an electric charge at rest within or withont a
closed conducting surface can produce no force or induced
charge on the other side, unless the equal and opposite
charge which must always exist somewhere is itself on that
other side.

Apparatus. A small tin can and one large enough to con-
tain it, the bottom of the latter being covered with a layer
of paraffine ; a cylinder of wire gauze ; two tin plates; an
insulating stand; two gold-leaf electroscopes; 60 centi-
metres of copper wire ; a small sphere, or other conductor

without points, about the
size of a 50 -gram weight,
hung upon a silk thread
about 25 centimetres long ;
access to an electrical ma-
chine; a rod and rubber,
as in Experiment 58.

Manipulation. Place the

smaller tin can on the insa-

lating stand, and connect it

with one electroscope by

Fio. HI means of the wire. Do

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 829

not let the wire touch the table or any other conductor.
Charge the other electroscope as in Experiment 58. Ex-
cite the electrical machine, separate the knobs, and test the
sign of the electricity upon one of them by charging from
it the sphere and bringing it near the free electroscope^ the
kind of whose charge is known. Note the sign of the
charge, and always charge the sphere the same way by
turning the machine in the same direction and using the
same knob.

1. Completely discharge the can and electroscope at-
tached by *' earthing '' them with the finger. Charge the
ball, and lower it into the can, being very caref nl not to
touch the can, wire, or electroscope with the hand or the
ball.

(a) What is the sign of the charge of the electroscope
leaves ? (Test with the rubbed rod.)

(b) Bemove the ball without touching the can, and test
the sign of its charge by means of the free electroscope.
Has the charge of the ball been changed ?

(d) Explain logically what conclusion you can draw as to
the charge indnced on the can when the ball was inside it.

(0) Describe exactly what would have happened had the
can, etc., been '^ earthed '^ for a moment while the ball was
in it, and what would have been their condition after the
ball was removed. If you do not know, try the experiment.

(/) Start again with the can and electroscope entirely
discharged. Charge the sphere again, lower it slowly into
the can, touch it against the bottom, and let it roll around
so as to insure contact. Finally, remove the ball. Describe
and explain the motions of the gold leaves throughout the
prooess.

{g) Is there any charge left upon the ball ?

(A) What further conclusions besides those of (d) can
yon now draw as to the signs and amounts of the induced
charges P

S80 A MANUAL OP EXPERIMENTS IN PHYSICS

(i) Place the smaller can inside the larger^ and the larger
on an insulated stand. Connect the larger with the elec-
troscope. The paraffine insulates
the cans at the bottom, and they
must not be allowed to toncli
elsewhere. Discharge the cans
and electroscope. Lower the
charged ball slowly into the in-
ner can, and finally touch it and
let it roll around the bottom.
Fio. 112 Describe and explain the mo-

tion of the gold leayes.
{j ) What would have happened if the inner can had been
'^ earthed '^ for a moment while the ball was inside?

(k) What would have happened had the inner can been
''earthed" for a moment, as above, but the ball remoYed
without touching it to the can? Try the experiment, if
necessary.
2. Place a tin plate upon the insulating stand.
(/) Discharge the electroscope, put it on the plate, sur-
round it with the wire screen, and cover with the other tin
plate. Connect the closed conducting surface thus found
with one knob of the electrical machine, and charge it un-
til sparks can be drawn from it. Is any effect produced on
the electroscope ?

(m) Discharge the cage, take off the top, and lower into
it the sphere highly charged. Be careful not to let the
sphere touch anything. Earth the cage with the finger,
then remove the finger, and finally touch the ball to the
cage. Describe and explain the indications of the gold
leaves at every step of the above process, showing where the
opposite charge to that upon the ball is situated.

(n) Place the electroscope outside the cage, as close to it
as possible. Discharge it completely, and carefully remove
or discharge any bodies in its neighborhood that might be
charged. Charge the ball highly, and lower it into the
cage. Repeat, touching the outside of the cage with the

£}CP£RIMENTS IN ELECTRICITY AND MAONETISk 881

finger while the ball is lowered. Describe the indications
at each step and explain, showing, as before, the positions of
the opposite charge.

QaestionB and Problems.

1. A sphere 8 ceDtimetres itidius is placed in a space where the
potentiHl is 6. It is joined to the earth for a moment, and
then removed from the region. What is its charge ?

3. Two insulated spheres, each 5 centimetres radius, are connected
by a long wire. One is in a space whose potential is 6, the
other in a space at potential 8. What is the potential and
charge of each sphere ?

8. Why does a charged body inside a closed conductor induce a
charge on a body outside unless the closed comluctor is
"earthed/* while a charged body outside can produce no
effect inside whether the cage is * * earthed " or not t

EXPERIMENT 60

Olyeet A stndy of an electrical induction machine.

Apparatus. A Voss machine (a Holtz, WimshurBt, or
any induction machine will answer) ; an electroscope^ with
rod and rubber for charging it; a " proof -plane/* which
may readily be made of a small coin fastened with wax on
the end of a glass rod ; a high-resistance galvanometer^
with leads of copper wire sufficiently long to reach to the
machine ; two metres of cotton string.

ManipHlation. Draw a diagram of the machine in year
note-book. Charge the electroscope by means of the rod.
Excite the machine ; stop it without allowing it to turn
backward when it stops. Charge the "proof- plane" by
induction from one of the knobs— i. «., by bringing it near
the knob and touching it with the finger for a moment
Do not allow a spark to pass to it. Test the sign of the
charge of the " proof -plane *' with the electroscope, and
note on your diagram the sign of the charge. Test eyerj
metal part of the machine, including both knobs, the
brushes, condensers, cross-bar, buttons on the revolTing
plate, and the tin-foil pasted on the back plate. Keep the
revolving plate in the same position exactly throughout;
and if it becomes necessary to renew the charge on the
machine, stop turning when the plate is again exactly at
the right point.

Next, test the sign of a button just before and just after
it passes each of the brushes. From these observations
explain briefly the operation of the machine.

What is the function of the tin-foil on the stationary
wheel ? Of the cross-bar ? Of the condensers ?

BXPERIMRNTS IN ELEGTRICITT AND MAGNETISM 888

Place the electroscope 50 centimetres from one of the
knobs of the machine, and push the two knobs of the ma-
chine together until sparks pass as the wheel is slowly
revolved. Note the behavior of the electroscope as the
wheel is tarned> and state briefly what inference you can
draw as to what goes on in the ether around the machine.
What is the greatest distance at which you can notice
any effect on the electroscope ? What do you infer as to
the state of a mass of metal, such as the plumbing of a
honse, jast before, during, and just after a flash of light-
ning in the neighborhood ?

Separate the knobs and connect each to one terminal of
the galvanometer by means of the wire. Turn the machine
at a speed just high enough to get a small deflection, say 1
centimetre, on the galvanometer. Count the turns of the
wheel made in a minute. Bepeat, using three times the
speed, and note the deflection again. Is the current, as
shown by the galvanometer, proportional to the speed ?
Wet the string and connect to the galvanometer through
it instead of through the wire. Turn the machine at
either of the two speeds already observed. Is the deflec-
tion changed by the insertion of the wet string ? Reverse
the direction of turning the machine and note the effect
on the deflection.

Note in particular that the machine produces a current
which can deflect a galvanometer needle, and that this
current increases with the speed — i, e,, the difference of
potential maintained between the knobs, and decreases as
the resistance through which it must pass increases.

Questions and Problems.
Draw ihe lines of force for an electropliorns at each of the four
steps : charged, cover oo, joined to earth, cover removed.

EXPERIMENT 61

(two OBSBBTSB8 ABB BBqUIIUED)

Otgeet 1. To show that the capacity of a condenser com-
posed of two parallel plates varies inversely as the distance
between its plates. 2. To determine the dielectric -con-
stant of some dielectric^ snch as glass. (See '' Physics,"
Arts. 237, 239.)

General Theory. The "capacity*' of a condenser is de-
fined as the ratio of the charge upon one of its surfaces to
the difference of potential between them. In symbols

If we have a means of charging a condenser to the same
potential difference under various conditions, and can in
each case measure e, we can show how the capacity varies
under these conditions. For instance, if we have a con-
denser consisting of two parallel plates, and vary the dis-
tance between them, while t^i— V3 is kept the same, we can
show that the capacity is inversely as the distance by meas-
uring e in each case.

The best method of comparing the charges is to make
use of the fact that, if a portion of one of the plates near
its centre is made movable, the force pulling it towards the
other plate is

2rcrM

where v is the surface density of the charge,

A is the area of the movable portion or disk,
£ is the dielectric-constant of the medium between
the two plates,

EXPERIMBNTS IN ELECTRICITY AND MAGNETISM 886

Hence, if this force is measured for different distances apart
of the plates and is found to hare the valaes Fi and P^,

and

As nsnally arranged, the disk is cnt out of the upper
plate, and the force on the movable disk is measured by
attaching it to the arm of a balance and noting the weight
on the other pan necessary to just prevent the disk from
being pulled down. The difference of potential between
the plates is made the same in each case by connecting
each plate to the corresponding plate of a second trap-
door electrometer. By keeping the distance apart of the
plates in this second electrometer the same, and its counter-
poise the same throughout, its disk will drop for exactly
the same difference of potential in every trial. If we con-
nect the plates of the condenser to the poles of an electric
machine and excite the machine, when the disk on the sub-
sidiary electrometer drops it indicates that the given po-
tential difference is reached. By changing the weights in
the pan of the balance of the variable electrometer we can
adjust it so that the plates of the two electrometers fall
together. In this manner the force is measured at the in-
stant the plates reach the given difference of potential.
Hence,

!• If the distance between the plates of the condenser is

C e

varied, the ratio 7^, which equals -^ aii<l is measured by

Y -J, may be proved to equal ^—t. e.^ the square root of

the force, being proportional to the capacity, should be in-
versely proportional to the distance between the plates.

2. By placing a thick plate of glass or other dielectric
hetween the plates— the total distance being the same as

886

A MANUAL OF EXPERIMENTS IN PHYSICS

in one of the experiments with air — the yariation in the
capacity (and consequently in the force) can be noted.

Note. — The constant of the dielectric composing the plate may be de-
termined in tliis wny. For theory shows that if i^^is the force on the roora-
ble disk, when there is a thickness d of air between the plates, and F* the
force when there is a thickness d" of air, and d' of anoUier dielectric, then,
if K' is the dlelectric-ooustaut of the dielectric, and if of air,

^±.^'

y -^ = J ; whence K =■

"-"■^

since jr= 1. C^he experiment requires too much care for the ordinaiy
student, howerer.)

Bonrcea of Brror.

1. If the plates of the "guard-ring" of the electrometer are not
parallel, tlie formulae do not hold.

3. The distance between the plates enters to the square, and, be-
ing a small quantity, is difficult to measure accurately.

Apparatus. Two '^ guard- ring ^' electrometers: the up-
per plate of each may be made out of a metal disk set
upon a metal tripod. The centre of the disk is cat oat
carefully and attached by light wires to one arm of a tall

EXPERIMENTS IN ELECTRICITY AND MAGNETISM ft87

balance^ the clearance between the cat-oat piece and the
fixed '' guard-ring '' being made as small as possible. A
second disk of aboat the same diameter as the gaard-ring
is fastened firmly to the end of a shellacked glass tabe^
which is held in a clamp-stand. For the electrometer in
which the distance is to be varied, the lower plate should
preferably be provided with an insulating -stand, which
allows it to be readily raised or lowered while its plane is
kept horizontal. The tripod and the stand of the lower
plate should both be provided with levelling screws. A
"trap-door" electrometer, in which the movable part of
the upper plate is hinged at one side and kept from fall-
ing by a spring or sliding counterpoise, is convenient for
the second instrument. A le^l ; vernier caliper ; box of
weights^ 100 grams to 0.01 gram; a thick glass plate,
wider than those of the electrometer ; and an electric in-
duction-machine and wires are also necessary.

Fio. 114

Kanipulation. Set up the apparatus as shown in the
diagram. Adjust the plates of the condenser so that they
are one centimetre apart. Bemove the tripod which holds
the upper plate, or "guard-ring"; level the lower plate ;
replace the tripod, and level the guard -ring. Make the
wire connections as shown. Adjust the movable disk so
that its under surface is accurately in the plane of the
under surface of the guard-ring when the beam is horizon-
tal. This may be done approximately by means of the
levelling screws on the balance and by blocks if necessary.
Such a balance should be provided with screw -stops to
limit the tilt of the beam, and the final adjustment may

888 A MANUAL OF EXPERIMENTS IN PHYSICS

be made by placing a weight in the pan greater than that
necessary to counterpoise the disk, and then supporting the
weight-arm by the screw-stop, so that the disk is exactly in
position. If the disk does not hang level, it mij be ad-
justed by bending slightly the wires by which it is sus-
pended. See that the disk is free to moye without fric-
tion. Find the weight which, when placed in the pan, will
balance the disk while the apparatus is uncharged. Pkce
a small number of grams — e, g., 6— in the pan in addition
to this balancing weight and excite the machine. Adjust
the counterpoise of the secondary electrometer so that tbe
disk falls about simultaneously with that of the other when
thus weighted. This adjustment is then left unchanged
during the remainder of th^ experiment.

Adjust the weight in the weight-pan of the balance until
the disks fall exactly together. If one disk has a greater
mass than the other, care must be taken to note the first
trace of motion in the heavier disk. When adjusted, note
the weight in the pan, and, deducting that necessary to
balance the weight of the disk, deduce the force. Measnre
with the caliper the distance between the plates at five
equidistant points around the circumference, and average,
to get the distance of the disk above the lower plate.

Repeat with distances of 2 centimetres and 3 centimetres
approximately between the plates. Level the lower plate
and guard-ring each time, but on no account touch tbe
counterpoise of the second electrometer.

Pass the flame of a Bunsen-burner over both surfaces
and all around the edge of the glass plate, being carefnl
not to crack it by keeping the flame too long on one spot.
Place the glass plate on the lower plate of the condenser,
level as before, and determine the force on the disk. Ex-
plain fully why it is greater than when the plates were sep-
arated by a thickness of air equal to the combined thick-
ness of the glass and air in this experiment.

EXPERIMENTS IN ELEGTRICITIT AND MAGNETISM 889

ILLUSTRATION

Variation of tbe capacity of a condenser with the diBtanoe apart of
its plates. (The dielectric is air.) ^

Wei£^ht necessary to balance the disk when uncharged = 128.60
grams.

Dtetanoe between PJatee In Centimetres Weight in Pan Force, F

1.10,1.12,1.22. Mean, 1.16 = d| 18400 6.40 grams.

2.02,2.02.2.04. Mean. 2.08 = rf, 180.60 1.90 "

2.99,2.98,2 06. Mean, 2.96 = flf, 129.66 .96 <*

^=*/r^=1.41. ^ = 1.46. Difference, 4 jr.

i?*, V .95 d.

This difference is well within the limit of accuracy of the experi-
ment.

Questioiis and Problems.

1. A condenser of two circular disks 20 centimetres in diameter,

and separated by a sheet of mica 0.1 centimetre thick, is
charged to potential 2. What sized sphere would have the
same capacity ?

2. Calculate the capacity of a Leyden Jar whose capacity is 16

centimetres, and the height of whose coatings is 20 ceuti-
metres, the thickness of the glass being 0.1 centimetre. (Ap-
ply formula for parallel plates.)

8. A licyden jar of capacity 1000 is charged to potential 10,
another of capacity 600 is charged to potential 6 ; the outer
coatings are put to earth and the knobs are connected. Cal-
culate initial and final energy, and explain their difference.

4. Two condensers are made exactly alitce, each consisting of an
inner and outer concentric sphere, radii 10 and 12 centime-
tres. One has sulphur as the dielectric ; the other, air. Tbe
former is charged with 100 units ; and then the two are con-
nected, loner sphere to inner, outer to outer. What are the
charges on each, and the potential ?

ELECTRIC CURRENTS AND MAGNETISM

EXPERIMENT 62

Olfjeot To map a "current sheet/' (See "Physics,''
Art. 296.)

General Theoiy. If an electric current enters a condnct-
ing sheet — e. g., a layer of conducting liquid spread over a
glass plate, a piece of tin-foil — at one point, and leaves it at
an opposite one, the flow through the sheet is spread out
and may be said to follow certain lines, called "lines of
flow.'' At right angles to these lines there will be lines
of constant potential, because in a conductor there is always
flow from high potential to low, and a line along which
there is no flow — t. e., a line at right angles to a line of
flow — must be a line of constant potential. These lines of
constant potential may be easily mapped by either of two
methods to be described below ; and so, if they are known,
the lines of flow may be drawn at right angles to them.

One method of mapping the equ {potential lines is to join
two wires to a galvanoscope, one to each binding- post;
then, keeping the terminal of one wire fixed at some point
on the current sheet, to moye the terminal of the other wire
over the sheet, tracing out points for which no deflection is
observed in the galvanoscope.

This method cannot be used if the current sheet is not
steady but varying (such a current as is obtained from an
induction-coil); but a difference of potential between two

EXPE&IMKNTS IN ELECTRICITY AND MAGNETISM 841

points in such a current may be detected by a telephone,
used in place of the galvanoscope in the former method.*
In either of these ways lines of constant potential and
the lines of flow may be mapped at diflerent points of the
current sheet.

SoQice of BxTor.

The lines of flow may change during the experiment, owing to
changes in concentration, etc.

Apparatus. A small induction-coil; a shallow^ water-
tight box, with a plane glass bottom, to the under side of
which a sheet of co-ordinate paper is pasted ; a telephone ;
a storage circuit; wires; a loose sheet of co-ordinate paper
of the same size as that on the box, and a very small quan-
tity of common salt.

FKkUB

Kampolation. Arrange the apparatus as shown. I is an
induction-coil ; W, W, the primary or storage circuit ; W",
W", the secondary circuit, which should be made of very
long wires ; T, the telephone.

Pour tap water into the box until its bottom is cov-
ered to a uniform depth of a little less than one centi-
metre. Dissolve a pinch of salt in the water in the box,
thoroughly stirring the solution. The positions of the
telephone terminals are determined with reference to the
co-ordinate paper which is pasted to the under side of the
box, and whose rulings are clearly visible through the glass
* This form of the exp«rimeDt is due to Professor Crew, of Eranston, 111.

842 A HANUAL OF EXPERIMENTS IN PHTSIGS

bottom. The simplest way of recording the snccessiye
positions of the telephone terminals is to transfer their
co-ordinates directly to the sheet of paper similar to that
on the bottom of the box. Start with one telephone terminal
fixed at an arbitrary but definite point by clamping it to the
side of the tank, not far from either one of the electrodes,
where the circuit enters or leaves. Note its position on
the loose sheet. Place the diaphragm of the telephone
near the ear, and at the same time move the other tele-
phone terminal through the liquid oyer the bottom of the
box. Always keep the end of the terminal perfectly straight
and perpendicular to the bottom of the box. A drumming
sound will be audible, whose intensity varies as the termi-
nal is moved from point to point. Try to concentrate the
attention upon the sound in the telephone by ignoring the
buzz of the interrupter ; and for this reason it is very de-
sirable to have the induction-coil as far off, and its noise as
much mufQed, as possible. Note the successive positions
occupied by the movable terminal when no sound can be
heard in the telephone, and record them with reference to
the axes on the loose sheet of co-ordinate paper.

To find these positions systematically, move the free tele-
phone terminal along a co-ordinate line near and parallel
to either side of the box. Approach the point of no sound,
or of minimum vibration, from both directions, in turn,
along this line, until its exact position is ascertained and
recorded. Now find, in like manner, a point of no sound
on a co-ordinate line one centimetre farther away from the
same side of the box and parallel to the line first chosen.
Continue in this way until an edge of the box is enconn-
tered. Draw a curve through the points thus found, which
is the equipotential through the point marked by the fixed
terminal.

Next, place the fixed telephone terminal at a point one
centimetre (or more) from its initial position. Again find
and record the position of points of no sound on equidistant
parallel straight lines, and draw the equipotential through

EXPERIMENTS IN ELECTRICITT AND MAGNETISM 848

them. Finally^ repeat the processes just described until the
whole surface of the bottom of the box has been traversed —
i.e., until the fixed terminal has been moyed around half of
the box, from one electrode to the other. Then draw, at
short inteirals {e.g., one centimetre apart at the middle),
curyes cutting the equipotentials at right angles. Indicate
the "lines of flow" by arrow-heads.

As a variation of this -experiment, place a symmetrical
piece of metal in the middle of the box, and plot the equi-
potential curves. Make the depth of the water just suffi-
cient not to cover the metal.

ILLUSTRATION

Feb. 18, 1896

Fio. 116

QoestioAs and Problems.

1. Explain by the aid of a diagram what would happen if the cur-

rent were intercepted by placing a plate of glass aoross a
part of the liquid.

2. Could this experiment be performed with pure water In the

box? Why?
8. Why is it that only a minimum sound can be found in some

places?
4. If equal quantities of electricity did not pass every section of

the electrolytic conductor in the same time, what would be

the result? Show the analogy between electricity and an

incompressible fluid.

EXPERIMENT 08

Oljeot. To plot the magnetic field of force :

1. Of a magnet and the earth together.

2. Of the magnet alone.

Oeneral Theoiy. If a magnetized needle is supported so
as to be free to tnm, and is placed in a magnetic field, it
will set itself tangent to the line of force at its centre. If
the needle is short enoagh, the difference between the
straight line joining its extremities and the enrye of the
line of force may be neglected. Hence^ if a large magnet
be placed in the centre of a sheet of paper, and a small
pocket-compass be placed at any point of the sheet, the
position of the ends of the needle may be marked ; and the
line joining the ends is then a small part of the local line
of force. By moving the compass on, so that one end of
the needle again falls upon the point just marked for the
other, another section of the line of force may be plotted;
and so on until the line meets the magnet or the edge of
the paper. Another line may then be drawn similarly.

If the magnet and the piece of paper are stationary, the
field plotted will be that due to the earth and magnet com-
bined, since the force at any point is always the resultant
of the force due to the two. We can, however, eliminate
the effect of the earth in influencing the direction of the
line of force at any point in the following manner: The
magnet is secured to the sheet of paper, and the latter is
placed free to move upon a table, so that it can be turned
in any direction as required, carrying with it the magnet
upon it. The compass is placed where it is desired to be-

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 846

gin the plotting of the field. The needle will not lie in a
north -and -south line unless it so happens that the local
line of force due to the magnet alone is north and south.
For, if the force due to the magnet alone is in any other
direction, there will be a component turning the needle
ont of the line in which the earth alone would hold it ;
consequently, if the needle happens to lie north and south,
it is known at once that the field of the magnet alone at
that point is north and south; and therefore the extremi«
ties of the compass-needle are points on the line of force
of the m^net alone, as well as of the earth alone. If, as
is nsnally the case, the needle does not point north and
south, the paper, together with the magnet and compass
upon it, can be turned until the needle does so point. In
other words, the magnet and its field are rotated until the
line of force of the magnet at the desired place is north
and south. The extremities of the needle are then marked
on the paper, the compass is moved on, and the next sec-
tion of the line is plotted similarly. Since the paper with
the magnet fixed upon it is always turned so that the di-
rection of the line of force on the paper is the same as that
of the earth, the direction of each little section is the same
as if the earth were not there ; and the whole field as plot-
ted is that of the magnet alone.

Soaroes of Snror.

1. Owing to the size of the compass-case it is impoasible to place
the marks exactly at the end of the needle as they should be.

3. Owing also to the size of the case the difference between the
curTe of the line of force and the straight line between the
points marked is considerable.

8. If, on the other hand, too small a compass is used, the direction
of the needle is not as easy to note.

4 The experiment roust be done on a IstcI surface, and the com-
pass so tipped as to prevent the needle from striking the top
or bottom of its case.

Apparatus. Two sheets of paper, each one-half the size
of the sheets in Experiment 57, in Electrostatics; a bar

846 A MANUAL OF EXPERIMENTS IN PHYSICS

magnet ; a small pocket-compass ; two pins ; thumb-tacks ;
a thread about one metre long.

Hanipolation. (1) Resultant Field of Magnet and Earth.
— Choose a place away from masses of iron of any kind.
Fasten one sheet with the tacks to a smooth table or draw-
ing-board. (If a drawing-board is used it should be firmly
secured, so that its position will not change during the ex-
periment.) Remove the bar magnet to a distance and
draw with the help of the compass an east-and-west and
a north-and-south line through the centre of the paper.
Place the magnet on the centre of the sheet with its axis
east and west on the line already drawn. Trace the out-
line of the magnet on the paper, in case it should be
disturbed, marking which is its north and which its south
pole. (N. B. The north pole is that which seeks the north
and repels the north-pointing pole of the compass.) Be-
gin at any convenient point and mark ofl twenty points
on the outline of the magnet, each of which will be made
the starting-point for one of the lines of the field as drawn.
Place the points much closer together at the poles than
near the middle.

Place the compass close to the magnet so that it points
to one of the marked points. Mark the position of the
other end of the compass-needle as nearly as the case of
the compass will allow. Move the compass so that the end
of the needle nearest the magnet is as close as the case will
allow to the point just marked, and points towards it
Mark the opposite end of the needle as before. Continue
similarly until you reach the magnet again or the edge of
the paper. Mark from time to time the way the arrow-
head of the compass is turned. Finally, draw a smooth
curve through the points marked.

When one line is thus drawn, proceed similarly to locate
the one from the next marked point on the boundary of the
magnet, and continue until the whole field is drawn.

Two points will be found in diametrically opposite cor-
ners of the field, where the force due to the earth and that

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 347

dne to the magnet are exactly equal and opposite, and the
position of the needle is therefore indeterminate. Locate
these points as closely as the size of the compass permits,
drawing extra lines of force in that neighborhood.

(2) Held of Magnet Alone. — Remove the bar magnet to a
distance^ and set two straight pins vertically in the table
to mark a north-and-soath line. The pins must be far
enough apart to admit of the sheet of paper on which the
field is to be plotted being laid on the table and rotated
between them. The best way to lay off the line is to set
the compass about the middle of it, sight very carefully
along the needle, and stick the farther pin in position.
Sight again along the needle and set the nearer pin so that
it exactly hides the farther one. Then join the two pins
by a thread, running about one centimetre above the paper.
Place the magnet on the centre of the she^t and fasten
it with '* universal" or other soft wax. Draw its outline
again as a precaution. Lay ofE twenty points around it as
in Part 1, as starting-points in plotting the field. Place
the compass near one of the starting-points. Shift the
paper until the pivot of the compass lies exactly in the
north-and-south line marked by the thread. Now rotate
the paper until the needle lies under the thread. If, in
doing so^ the needle turns away from the point selected as
the beginning of the line, shift the compass on the paper
sideways until it points towards it, and move the whole
paper again without turning until the pivot is again under
the thread. Continue similarly until the desired starting-
point and the piv«t of the needle are exactly in the north-
and-Bouth line marked by the thread, and the needle also
lies exactly in this line. When this is secured mark the
position of the end of the needle away from the magnet as
in Part 1, and proceed in a precisely similar manner to find
another point on the same line of force. In doing so, the
point just marked, the pivot of the needle, and the direc-
tion of the needle must be brought undep the thread.
When all the points on a line are plotted, draw a curve

846 A MANUAL OF EXP£RIM£NT8 IN PHTSIGS

through them and indicate by an arrow the direction in
which a north pole would move along the line.

When lines have been drawn similarly from points all
around the magnet, take both sheets home and draw the
equipotential surfaces with red ink. Locate the equipo-
tentials around the neutral points in Part 1 very carefully.
In the rest of the field a comparative few will answer.

Date and sign the sheets, and fold them to fit in the re-
port books. Answer the questions in the report books as
usual.

Qaa«tioiui and Problema.

1. £zplaln tbe peculiarities in tlie equipotential surfaces around

the neutral poinia.

2. Knowing the strength of the earth's field at the place where

the experiment was 'performed, how would you calcuUte
the strength of either pole of the magnet from the direction
of the line of force at any point on your diagram of Part 1 ?
Assume the poles to be equal.
8. Why cannot two or more lines of force inteisect?

4. How could a field due to a single pole bf mapped ? Show by

a sketch what would be the direction of the lines of force.

5. Would there be any difference between a north and a sonth

pole?

EXPERIMENT 64

Ol^jeel To measare the magnetic inclination or dip.

Oeneral Theory. The magnetic inclination is the angle
which the line of magnetic force, dne to the earth, makes
with the horizontal at any point on the earth. To measure
this, it is necessary to so suspend a magnetic needle that
it is perfectly free to turn-about a yertical and also a hori-
zontal axis, and to determine the angle it makes with a
horizontal plane. Another method is to suspend a magnetic
needle so that it is free to turn about an axis which is per-
pendicular to the magnetic meridian, and to measure the
angle below the horizon made by the direction which it takes.

The difficulties in this experiment may be described as
follows :

1. The axis around which the needle turns may not pass
directly through the centre of the circle on whose circum-
ference the scale is diyided.

In this case the extremities of the pointers do not meas-
ure the angles correctly; but if
the scale is divided as shown, the
reading of one extremity of the
needle will be as much too great
as that of the other is too little.
This is apparent from the figure
in which the dotted line repre-
sents the true diameter. There-
fore the average of the readings
of the two extremities gives the
correct angle. wn. m

850

A MANUAL OF EXPERIMENTS IN PHTSICS

2. The centre of gravity of the needle may not coincide
with its axis of rotation, in which case the needle will be
influenced in its position when it is turning about an axis
perpendicular to the magnetic meridian.

If the centre of gravity comes at a point, P, in a line
perpendicular to the axis of fig>
nre of the needle, correction may
be made by reversing the needle —
». tf., in the figure — changing from
position 1 to position 2 ; for, in the
first case, the fact of the centre of
gravity being at P tends to make
the dip less by a certain angle,
while in the second position it
tends to increase it by the same
angle.

If the centre of gravity comes
at a point, Q, in the axis of figure
of the needle, correction must be
made by remagnetizing the needle,
so that the poles are reversed. This
changes the magnet from position
3 to position 4 ; and in these two
cases the influence of the position
of the centre of gravity is equal,
but opposite.
Therefore, in the general case, when the centre of grav-
ity is in any unknown position, it is necessary to make the
needle assume the three positions shown in 1, 2, and 4, and
to take the average angle of dip.

3. The axis of figure of the needle may not coincide with
the magnetic axis — i, e., the line joining the two poles of
the needle. In this case the extremities of the needle do
not record the true dip. As is shown in Figure 119, in one
position a is too high and b too low ; but if the needle is
reversed in its bearings, the magnetic axis does not change
its direction, but a comes as much too low as before it

Fio. 118

SXPERIMENTS IN ELECTRICITY AND MAGNETISM

801

was too high, and b vice versa. Therefore, taking the mean
position of the needle, direct and reyersed, corrects for
this error.

It follows that the
general method in-
volves these four steps :

1. Place the needle
horizontal.

2. Locate the mag-
netic meridian.

3. Place the needle
in this meridian, with
its axis of rotation per-
pendicular to it.

4. Make the readings
as just descnbed.

Sources of Error.
1. FrictioD in tlie bearings and slipping of the clamps are the

principal sources.
3. The above-mentioned difficulties in the use of the needle must

all be carefully overcome.
8. After remagnetization the magnetic axis may not coincide with

its former position.

Apparatus. A dip-circle and a small reading-lens. The
dip-circle consists essentially of a well-balanced magnetic
needle, pivoted with its axis approximately through the
centre of a divided circle and at right angles to the plane
of the same. This circle is arranged to move both about
a horizontal and about a vertical axis. These axes pass
through the centres of graduated circles at right angles to
their respective planes ; and these circles are divided into
degrees of arc by marks which are numbered from zero to
ninety. There are, besides, several screws on a dip-circle,
the uses of which will be explained in the appropriate places,
as they are needed in this experiment.

Manipulation. To simplify the explanation, let the three
circles be distinguished in the following manner : Let the

362

A MANUAL OF £XP£RI1I£KTS IN PHYSICS

circle which contains the needle be designated as ''the
movable circle ''; let the vertical circle be called simply
"the semicircle/' because it is graduated only over half of

its circumference ; and let the
fixed horizontal circle be known
as " the fixed circle." Make
the adjustments as follows :
Place the dip -circle on the
table and slip the screw -feet
into grooves prepared for this
purpose. Remove all magnetic
sul&stances from the neighbor-
hood of the table which aap-
ports the dip -circle. If the
needle does not move freely
in its bearings, or if it is too
loose, adjust the screw which
regulates the pivot, and clamp
it by means of the nut.

1. To place the needle hor-
izontal. Loosen the screw
which projects from the plane
of the semicircle below its arc,
and which, when tightened,
firmly clamps it to the vertical axis of the whole instru-
ment. Orasp the large milled head on the opposite side
of the semicircle, and turn it until the zero of the semi-
circle appears to coincide with its index. Clamp the semi-
circle tightly by means of the screw just noticed, and com-
plete this adjustment by turning the "tangent screw''
beneath the semicircle until its zero mark coincides exactly
with the index. (View the scales of the circles through
the lens in making all accurate adjustments or readings.)
Turn the three screws which form the feet of the dip-circle
until the plumb-line hangs in the middle of its i-ing. Turn
the upper part of the dip*circle around its vertical axis
through about ninety degrees, and adjust the screw -feet

Fio. 190

EXPERIMENTS IN ELEOTHICITY AND MAGNETISM 353

nntil the plumb-line again passes throngh the centre of the
ring. Continue to turn the dip-circle through the quad-
rants and to adjust its feet until the thread passes ap-
proximately through the middle of the hug in all positions
of the instrument — i. e., until the dip-circle is practically
leyel. The axis of the needle is now truly vertical.

2. To locate the magnetic meridian. Although the
grooves in the table are intended to prevent any move-
ment of the instrument; it is best to hold the fixed circle
firmly in position while turning the movable circle around
either its horizontal or its vertical axis. For convenience^
turn the movable circle about its vertical axis until the
needle points somewhere near either of its zero divisions.
(Pound on the table with your fist, thus jarring the needle
and giving it freedom of motion, so that it may assume its
proper position. ) Read both ends of the needle, estimating
to tenths of a degree, and take the arithmetical mean of
these positions. For consistency, call all readings around
the horizontal scales in one. direction from the zero marks
positive, and prefix a negative sign to all readings in the
opposite direction — e.g., clockwise 4- , anti -clockwise — .
Loosen the screw which holds the needle in its bearings,
remove it, turn it over and replace it, thus causing it to
be reversed relatively to the scale for the movable circle.
Again set the needle to vibrating, and when it comes to
rest note the positions of its ends and take the half-sum
of the readings thus obtained. Take the mean of these
two positions of the needle, direct and reversed ; let it be
a. Then a diameter of the movable circle which passes
throngh the scale at this angle, a, marks the magnetic north-
and-sonth line.

Tarn the instrument around its vertical axis through an
angle, a, as shown by the fixed circle at the base. This
places the horizontal axis of the semicircular scale directly
in the magnetic meridian.

3. To place the needle in the magnetic meridian and its
axis of rotation perpendicular to it. Keeping the index
ss

864 A MANUAL OF EXPERIMENTS IN PHYSICS

on the fixed circle unchanged, nnclamp the semicircle and
rotate the movable circle about its horizontal axis until
the index of the semicircle coincides exactly with either
one of its ninety-degree divisions — that is, tarn it through
a right angle, and clamp it. (Whenever the plane of the
movable circle is in a vertical position the clamp -screw
must be turned very hard to overcome the tendency of the
circle to slip into an oblique position due to the moment
acting upon it.) The needle is now approximately in the
magnetic meridian. Beat on the table and then record
the positions of both ends of the needle, as indicated by
the movable circle. Next turn the instrument around its
vertical axis exactly 180^ from its initial or zero position,
and note the readings of the ends of the needle. Turn
the movable circle around its horizontal axis through two
right angles, so that the index of the semicircle coincides
with its other ninety-degree division. Be very careful not
to move the index of the fixed circle in making this ad-
justment. Jar the instrument^and read the ends of the
needle. Again revolve the dip-circle around its vertical
axis through 180^ — i. e., back to its first position— and
record the positions of the ends of the needle.

Now, as before explained, reverse the needle relatively to
the movable circle, fiepeat all of the operations described
in the last paragraph, and record the four pairs of results
so obtained.

4. Ask an instructor to reverse the magnetization of the
needle ; and, after this reversal has been accomplished, re-
peat the entire set of adjustments and readings described
in the preceding paragraphs. Before levelling, turn the
instrument around its vertical axis through 180°, so that
the zero division of the fixed circle which was not used
before as the principal reference - mark shall now be so
used. This will vary the conditions of the experiment
slightly. Take the arithmetical mean of the thirty -two
readings of the ends of the needle and record it as the true
dip.

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 855

ILLUSTRATION

F9h. 11, 18M
Dip at Baltimork

Index zero = + 1.1*
Direct,

Naedte RmmUoss

^'^ J 70.^, 71.0° ^*«^^M71.6^71.8°

Needle Readings

£(/BWt9ML,

J ,^ 5 80.0^ 80.1° I -,, . < 79.2°, 79.4°

^''J71.0°.71.r I «^«^*hl.8°,71.1°

T^,STO.8°.72.4° I n\.ht^'^'^*'

^M 57.0°. 66.2° I ^'«'^M56.9°,1

B&magnstiged,

Index zero = — 8.2°

Direct.

; 72.2°. 72.2°

57.9°

Beoersed,

. -, S 71.1°. 71.6° I t>. w S 71-3°. 71.4°

.^" J 56.9°. 56.7° I ^^«^* 157.9°. 57.2°

Dip = ^ = 70.00°

Questioiis and Problems.

1. Give a practical method for diminishing the error due to the

sticking of the needle in its bearings when it is allowed to
come to rest.

2. What are meant by the '*magDet{c elements." and how do

they vary with the time and place of observation ?

EXPERIMENT 66

Olgeot. To compare the intensities of fields of magnetic
force. (See '' Physics/' Art. 269.)

Oeneral Theory, If a magnet be suspended free to oscil-
late about an axis perpendicular to a field of force, tlie
period of vibration is

V MR'
where A is the moment of inertia
of the magnet about its axis of
vibration, M is the magnetic mo-
ment, R is the intensity of the
magnetic field, M and A are con-
stants for a given magnet, if it is
not jarred or otherwise altered
magnetically.

Therefore, if this same magnet
is suspended so as to make oscil-
lations in another field of force
whose intensity is i?i, and if the
period of vibration is T,, then

Fig. 131

'•■=Vm-'

and hence

R^
~R

rp2

In this experiment, therefore, the same magnet is to be
made to perform oscillations in different fields of force;
and their intensities may be compared by measuring the
periods of vibration in the different fields.

EXPERIMENTS IN ELECTRICITY AND MAUNETLSM 857

fikraroes of Bnor.

1. If anything happens to the vibrating magnet, its magnetic mo-
ment will be changed. Therefore avoid all jars, changes in
temperature, contact with other magnets, etc.

3. The supporting fibre must be as free as possible from torsion.

Apparatus. A bar-magnet, about 3 centimetres long ; a
large glass jar ; a piece of silk fibre ; a sheet of paper or a
piece of pasteboard ; and a glass tube, the length of which
is somewhat greater than any diameter of the glass jar.

Manipulation. Make a stirrup of a short strip of paper,
and suspend it from the middle of the glass tube by means
of the fibre. Place the tube across any
diameter of the upper open end of the jar
so that the stirrup and fibre hang near the
middle of the jar. The fibre should be of
such a length as to support the stirrup
about five centimetres above the bottom of
the jar. When the torsional oscillations
of the fibre have practically ceased, place
the magnet in the stirrup. Take great
care not to drop or abuse the magnet in
any way, or else all of the results will be
vitiated. There must not be any magnetic substance in
the neighborhood of the jar other than the one magnet un-
der consideration. If the magnet under the action of the
earth's field alone turns around abruptly, reversing its posi-
tion, it must be taken out of the stirrup and replaced with
its ends interchanged relatively to the stirrup. This is
done to avoid producing undue torsion in the suspending
fibre. Then carefully balance the magnet in a horizontal
plane, and cover the jar with the sheet of paper to hin-
der draughts of air around the magnet. Cause the mag-
net to vibrate (not swing) in very small arcs about the
fibre as Vk fixed vertical axis, and record the number of sec-
ouds which elapse while the magnet makes one hundred
complete oscillations. Follow the method of Experiment 1.

Fig. 122

358 A MANUAL OF EXPERIMENTS IN PHYSICS

It is best to mark two vertical lines on opposite sides of
the jar^ and then make the fibre come in between them.
In this way the exact period may be measured. Repeat
this reading several times^ and deduce the mean period of
vibration.

Move the jar and magnet to different parts of the room,
or rooms, if there are several adjoining, and in a similar
manner measure the period of vibration. Do this in par-
ticular near the following places ;

1. A brick wall. 2. A window-sash, if there are window
weights. 3. A gas or steam pipe. Also at points, some
five feet apart, on a line leading through a doorway.

Assuming the intensity to be known at some standard
position, calculate the intensity at each of the other posi-
tions, and plot the results on a diagram of the rooms.

Questiona and Problema.

1. How does the presence of large masses of iron in the aeighbor-

hood of the oscillating magnet affect the results T

2. Explain why amall vibrations must be used.

8. Explain what would happen if an astatic system, the two mag-
nets of which are not quite parallel, were suspended and set
vibrating. What is the position of equilibrium with refer-
ence t<) tlie magnetic meridian?

4. A dipping-needle nisikcs 116 oscillations in a certain time when
vibrating in the mii^netic meridian, and 100 nscillatloDS in
an eqinil Inierviil of lime when its plane of viliration is per-
pendicular to the magnetic meridian. Calculnte Ihe dip.

EXPERIMENT 66

OTgeet. To measure the horizontal intensity (H) of the
earth's magnetic field. (See '' Physics/' Arts. 269, 270.)

General Theory. The horizontal intensity of the earth's
magnetic field is the horizontal component of the force due
to the earth which would act upon a unit north - pole if
placed at the given point on the surface of the earth.

It is shown in treatises on physics (see '* Physics," Art.
271) that it is possible to measure the magnetic intensity
of any field of force by two experiments.

1. Suspend a bar-magnet so that it is free to oscillate
about an axis perpendicular to the field of force ; its period
of vibration is

where A is the moment of inertia around the axis of oscil-
lation, M is the magnetic moment, R is the magnetic in-
tensity of the field.

2. Place the bar-magnet at rest, perpendicular to the

■ ; ^'

I p^

^..„.,..

~ -A

^ J -f

— ^ ^. _ — — — ».

Fio. 138

360 A MANUAL OF EXPERIMENTS IN PHYSIO?

field of force, and at a distance, r, from its centre in the line

of its axis suspend a small magnetic needle, so as to be free

to turn about an axis which is perpendicular to the field of

force and to the axis of the bar-magnet. Then the angle

of deflection, a, of the needle is such that

M r^ tan a ^i^ IW tan a

^ = T^ — , or, more exactly, ( r« - - j j^ ,

if /is the length of the bar-magnet.

From these two formulag R may be calculated.

The general method, then, in this experiment is, first,
to suspend the bar-magnet free to vibrate about a vertical
axis — then, if H is the horizontal intensity,

V HM
secondly, to suspend a small magnetic needle, free to turn
about a vertical axis, and to place the bar-magnet in a hori-
zontal plane magnetically due east or west of the needle.
If the deflection of the needle is S,

M r3 tan ^ .1 / . I'^V tan ^

~= —- — , or, more

,. / , ZA2 tun 5
exactly, (r=--j —

Sources of Error.

1. Tlie second formula above is derived on two as-cumptions—

that the distance, r, is immenaely great in compnrison with
the lengths of either magnet, and that the macnctism of the
bar -magnet is concentrated at its two poles. Neither of
these assumptions is true.

2. The distance, ?% is difBcult to measure exnctly.

Apparatus. "A magnetometer" ; a cylindrical bar-mag-
net; a reading- telescope; scale and adjustable stand; a
wooden metre- bar; a fishtail -burner with rubber tubing;
a piece of string. The magnetometer consists essentially
of a small, short magnetic needle fastened at right angles
to a small plane mirror, which is suspended vertically by
means of a delicate silk fibre. The needle hangs horizontally
inside a box with plane glass sides, and the fibre passes up
through a vertical glass tube to an adjustable metal head.

EXPERIMENTS IN ELECrTRIClTY AND MAGNETISM 861

MaBipnlation. 1. To measnre MHy

The moment of inertia (A) oi the bar-magnet should be
obtained from an assistant. Measure at least three times
the length {I) of the bar-magnet, by direct comparison with
a toooden metre-bar. Handle the magnet very carefully.
Hang the bar-magnet in a paper stirrup at the end of a
long fibre. (See preceding experiment.) The magnet
must be well balanced horizontally, and the thread must
be free from torsion. Surround the magnet with a large
glass jar and cover it with a sheet of paper or of pasteboard,
so as to prevent draughts of air around the magnet. Of
coarse, a slit must be cut in this cover to allow free mo-
tion of the suspending thread. Make sure there arc no
magnetic substances in the neighborhood of the apparatus.
Give the magnet a very small angular displacement from
its position of equilibrium, and determine by the method
of Experiment 1 the period of vibration of the magnet.
To do this properly, as explained in Experiment 1, it is
necessary to make a sharp vertical line on each side of the
glass jar, and so place it that the fibre which suspends the
magnet comes in between these lines ; then the exact in-
terval of one period may be easily determined. If the arc
of vibration is not small, allowance for the fact must be
made. (See Tables.)

Bepeat several times this process of counting the vibra-
tions, and deduce the mean value of the period T,

2. To measure -jj?

M r3 tan S x, / , ^\' tan S

-^ = — ^ — , or, more exactly, ^^' - ^ j -^^•

The tangent of the angle of deflection is best measured
by fastening a light plane mirror to the magnetic needle,
and measuring its deflection by means of a telescope and
scale, as is shown in the figure. It iR evident that the
*8cale-reading, p^ which is caused by a deflection, d, of the

862 A MANUAL OF EXPERIMENTS IN PHTSIGS

mirror, corresponds to an angle 2d between the lines drawn
to the mirror from the telescope and the scale division.
(See "Physics," Art. 307.) If ^ is the scale-reading, and
n the distance from the zero of the scale at the telescope
to the mirror,

tan2S=^ = -l^.
n 1 — tan* ^

But in these measurements tan ^ is small, and so its sqnare

may be neglected compared with 1, and tan ^ = - ^.

The magnetometer is fastened upon a long wooden frame
in such a position that the centre of its needle is equi-

I i-«*^\

I \

.._„_AJ

: ; \

|iiii|iiii[im|iiii|iiii[iiii|int;iiniTiii|iiM|riiqiiHpiii|iiii}ini|iiii|iiii}iiii[ini]^

no.iM

distant from the centres of two screws near the ends of
the frame. The groove in the upper surface of this frame
should be at right angles to the magnetic meridian. To
secure this position, place a large compass on a stand
whose top is on a level with the mirror and at a distance
from it of not less than one metre. Sight along the axis

EXPEKIMEKtS IN ELSGTRICITT AND MAGNETISM 863

of the compass-needle, and move the compass-box nntil the
centre of the mirror is in the line of vision. The centres
of the needle in the magnetometer and in the compass are
then in the magnetic meridian. Mark the position of one
end of the compass-needle by means of a pin or tack stnck
ap in the stand. Then measure, by means of a tightly
stretched string, the distances of the centres of the screws
at the ends of the magnetometer frame from the pin or
tack. Keeping the centre of the magnetometer altogether
unchanged in position, tnm its frame around in a horizon-
tal plane nntil the distances just mentioned are exactly
equal. Then the frame will be normal to the magnetic
meridian passing through the centre of the mirror. Set
a reading-telescope on the stand with its axis in line with
the pin or tack and the middle of the mirror. Clamp an
inyerted millimetre scale to the front of the telescope sup-
port, so that it projects about the same distance on both
sides of that instrument. Then ask an instructor to ad-
just the telescope, gas-light, etc. Finally, make the scale
parallel to the magnetometer frame. (Connect by threads
two points near the ends of the scale, which are equidis-
tant from its middle, to the centre of the mirror, and ad-
just the scale until these distances are equal, as before
explained for the frame.)

When these preliminary adjustments haye been made,
place the bar-magnet some considerable distance away (20
feet, say) and note the position on the scale of the vertical
cross-hair in the telescope. In general, the mirror con-
tinually swings a little, so that the reading must be found
by the method of vibrations, as explained in Experiment 11.
Now place the bar -magnet in the groove of the magne-
tometer frame, with one of its ends in close contact with
the near side of either one of the fixed screws before men-
tioned—c. g., the west one. Record the mean position of
the vertical cross-hair ; the difference between this read-
ing and the one just made gives the deflection, in centi-
metres, of the needle, caused by the presence of the bar-

S64 A MANUAL OF EXPERIMExVTS IN PH78I0B

magnet. Interchange the positions of the ends of the
bar -magnet, and note the resulting deflection. In like
manner record the deflections when the bar -magnet is
placed against the other (east) fixed screw and then re-
versed. Take the arithmetical mean of these four deflec-
tions and call it p. Next measure, by means of a piece of
string or a metre-bar, the exact horizontal distance between
the centre of the magnetometer- needle and that scale di-
vision which is nearest to it — i, e,, approximately the one
directly under the axis of the telescope tube, or line of
vision. Express this distance in centimetres, and denote
it by w. Finally, measure the mean distance between the
edges of the two screws by means of the metre-bar. Do
this several times ; and, from a knowledge of the mean
value and of l, the length of the bar-magnet, calculate r.

Substitute the experimental values of I, n, p, r, and T,
together with the known values of A and tt, in the formula,
and calculate JI.

ILLUSTRATION

MK48URBMKNT OF ff

100 vibrations iu 11^88 seconds

1291 •• J- .-. r= 12.86.

.. 1284 **

Mareh3,l»l

Magnet east, mean deflections \^'^ ^°^
** i 19.0

west ** *' i

p=: 18.9 cm.

19.0 cm.

17.8 cm.

18.7 cm.
r= 87.4 cm.
» = 140.2 cm.
/= 9.6 cm.
A = 278 C. G. S. unils. Hence, H= 0.197 C. G. 8. unite.

QueatlonB and Problema.

1. Calculate if, m, and A for the bar-magnet, using the data of

your experiment.

2. Why must a wooden metre-bar be used?

EXPERIMENT 67

Olgeot. To prove thai the resistance of a uniform wire
varies directly as its length. (See " Physics,'* Art. 254.)

Qeneral Theory. Ohm's Law states that if i be the cur-
rent flowing through a given conductor or several conduct-
ors joined so that the current passes through them suc-
cessively, and if E is the difference of potential between

any two points on this current, then -r-= a constant as long

as the conductor is not changed in the least by rise of
temperature, etc. This constant is called the *' resistance *'
of the circuit between those points, and, being denoted by
Ry we have

^=/2, or E=Ri.
t

Whence, for a constant current the difference of potential
between any two points is proportional to the resistance
between them. The difference of potential can be meas-
ured, as will be described below ; and the object of this
experiment is to prove that the resistance varies directly
as the length of the conductor at whose ends the poten-
tial difference is measured, if the wire is of uniform cross-
section.

To measure the difference in potential, the E. M. F., be-
tween any two points — c.jr., Jf and S — of a uniform wire
through which the current is passing, the following method
iB used : Connect the two points M and S by wires to a
galvanometer whose deflections, if small, are proportional
to the current through it. If the resistance of the galva-
nometer circuit is made extremely large, the current taken

866 A MANUAL OF EXPERIMENTS IN PHYSICS

away from that through the nniform wire will be too small
to produce any sensible effect on the fall of potential in the

Fia. 136

wire> and its variations will be minnte as iS^ is moved along
the wire ; but these variations may still be large enough to
be detected by the galvanometer if it is sensitive. This
current, however, through the galvanometer is caused by
the E. M. F. between Jf and S, the two points on the wire,
and is proportional to it by Ohm's Law, because the resist-
ance of the galvanometer circuit is practically constant.
Therefore, the deflection in the galvanometer measures the
E. M. F. between its terminals, which may be made to span
different lengths of the uniform wire through which the
constant current is passing.

The general method, then, is to keep one terminal of the
galvanometer permanently connected to the end of the uni-
form wire, M, and to read the deflections of the needle as
the other terminal, 8, is moved along the wire from M to
N, The deflections are proportional to the E. M. F. be-
tween Jf and S; and this is proportional to the resistance
R between M and S along the wire. The length, M S,
may be measured, and if the resistance varies directly as
the length, the deflection should be proportional to the
corresponding length. Consequently, if the lengths meas-
ured from M are plotted as abscisssB, and the correspond-
ing deflections as grdim^tes^ tbe series of points should

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 867

lie on a straight line passing through the origin of co-
ordinates.

The stndent must distinguish carefully between the
steady deflection and the first outward swing or "throw*'
of the galvanometer. The steady deflection is the differ-
ence between the position of eq[uilibrium when there is no
current flowing and the position of equilibrium with the
given current. It is not necessary to wait for the needle
to come entirely to rest in either case ; but the point of rest
is found by the method of vibrations^ as in Experiment 11,
as soon as the range of vibration has diminished to one or
two centimetres.

The battery circuit contains a resistance by which the
current through the uniform wire can be regulated to a
value at which it will remain fairly constant.

A key, K\ is placed in the battery circuit, and another,
K, in the galvanometer branch. These are usually com-
bined in one instrument, as shown in Fig. 126. Ay By and
C are three strips of brass, and Z> is a brass button on the
hard -rubber base a^p*

E. C and D are
connected by
wires, one to each
of the two posts
shown at 0 m the

figure. A and B are similarly connected to another pair
of posts not shown. A block of hard rubber separates A
and B at one end and keeps the other ends apart also, un-
less contact is purposely made by pressing down on the
knob. B and C are separated by blocks of rubber at both
ends, and can never be brought into contact. (7 and D are
separated at one end only, just as A and B are, but con-
tact cannot be made at the open end until after A and B
are pressed together ; consequently, if the strips A and B
are connected by means of their posts as a key in the bat-
tery circuit, and C and D similarly in the galvanometer
circuit, the battery circuit is always closed first and the

368 A MANUAL OF EXPERIMENTS IN PHYSICS

galvanometer circuit afterwards. Snch a key is called a
** Wheatstone's Bridge Key," and is most convenient in thia
experiment and essential in subsequent ones.

Souroes of Error.

1. If the battery is used contiDuously its elect romotive force

decreases, and, coDse4ueDtly, the current in the wire di-
minishes.

2. If the current passes contiDuously through the bridge wire,

it is heated, and its resistance increases.
8. Oare must be taken at each observation to make good elec-
trical connections by means of the sliding contact. Grease
or rust at the point of contact may increase the resistaDoe
of the galvanometer branch so that the deflection is greatly
diminisiied.

Apparatus. A high - resistance galvanometer; Wheat-
stone's wire-bridge and key; one or two battery-cells of
constant electromotive force (Daniel's cells will answer);
a resistance-box of as much as 100 ohms for the battery
circuit, and one of 1000 or more for the galvanometer cir-
cuit. (These resistances need not be accurately known.)

ManipulatioiL Connect the apparatus as is shown in
Fig. 127. If not familiar with the galvanometer, ask an
instructor to show you how it is put in working order.
Make t\ the resistance in the battery circuit, about 100
ohms.

Place the sliding contact so that the entire length of the
wire is included between the two galvanometer terminals.
In this position the deflection of the galvanometer will be
the greatest obtained during the experiment, and the re-
sistances r and r' should be so adjusted that when both
keys are cloeod the steady deflection is not more than one-
tenth the distance from the mirror to the scale and not
much less than this amount ; r, the resistance in the gal-
vanometer circuit, should not be less than 500 ohms.

Open both circuits. Place the contact about 10 centi-
metres from M, the fixed terminal of the galvanometer.
Note on the scale parallel to the wire the exact reading (^

£XP£Hltf£NTS lH £L£UTKIOiTY AND MAGNETISM 369

the poiut where contact is made^ and also observe and note
any difference between the terminal^ M, and the zero of the
aicale. Determine the zero of the galvanometer, taking
three swings one side and two the other^ if it is not at
rest. Close both keys, and, holding them down firmly, de-
termine tlie new point of rest, by vibrations as before if the
needle does not come to rest. As soon as it is observed

Fia 127

and noted, release the keys and open the circuits, so that
the current may not flow longer than necessary. Shift
the contact to a point .20 centimetres from M, and repeat.
Continue similarly to the other end of the wire and then
return, taking the same points exactly in reverse order.
The zero of the galvanometer should be redetermined
every two, or at most three, observations. Calculate the
steady deflections from the readings, and take the mean of
the two observed for the same point. Reverse the direc-
tion of the current by interchanging the wires joined to
the cell, and repeat the experiment. Plot the lengths of
wire between M and the various points as abscissse, and
the mean deflections as ordinates.

870

A MANUAL OF fiXPfiRIMENTS IN PHYSICS

ILLUSTRATION „ ^„,^

Mwcta 93, im

Resifltance in battery circuit, 100 ohms ; in galvanometer circuit,
1200 ohms.

Reading of point where wire is attached to terminal, Jf, + 0.2 ceoti-
metre.

RMding

of
SJIdlDg

CODUOt

Wtre
MS

Ortginal

BaturnlDg

Meu
DeAliOii

Zero

Coirent
on

Defl'iion

Zero

Current
on

Defl^Uon

10.2

28.68

24.61

0.98

• • • .

24.74

1.05

0.99

20.2

. . • •

25.64

1.97

28.69

25.68

1.99

l.W

80.2

28.66

26.72

8.06

....

26.72

8.02

8.04

40.2

• • • •

27.66

8.98

28.71

27.78

4.02

4.00

50.2

28.70

28.59

4.89

....

28.70

5.01

4.95

60.2

. . . •

29.71

6.08

28.68

29.85

5.67

6.85

70.2

28.65

80.72

7.07

. ■ • •

80.40

6.73

6.90

80.2

• • . ■

31.72

8.06

28.67

81.61

7.94

8.00

90.2

28.67

82.75

9.08

. . • •

82.66

9.00

9.04

100

99.8

....

88.68

9.96

28.66

88.60

9.94

9.95

Qtioationa and Problems.

1. How would a great change of temperature at the pdnt where

the Bliding contact touches the wire affect the readings of
the galvanometer?

2. What would be the effect of variations in the pressure with

which contact was made?

8. In what way is the error caused by the battery running down
diminished in doing the experiment as directed ?

4 Would you expect any difference in accuracy at different parti
of the bridge wire, and where would you expect this experi-
ment to show most accord with theoiy?

EXPERIMENT 68

(TWO OB8ERYBB8 ABB BBQUIBED)

Olgeot To determine roughly the effect upon resistance
of alterations in lengthy cross-section^ temperature, and ma-
terial of a conductor.

General Theory. Ohm's Law states that if a cnrrent, i, is
flowing through a conductor^ the difference of potential,
E, between any two points A and B of that conductor is

Ite. ISB

connected with i by a relation which may be expressed

-r = R,A constant for the given portion AB. Ris called the

resistance of the conductor between A and B, and it is eyi-
dent that it will vary for different conductors. It may be
proved that if the conductor is in the form of a cylinder of
length, ly and cross-section, a,

where p is a constant for a given material — {b, g., copper
at 10**) — but if the material is changed in any way, replaced
by another, hammered, heated, magnetized, etc., p the
" specific resistance'* — or "resistivity'' as it is called— will
change.

The object of this experiment is to verify these facts in
a somewhat rough manner. The method adopted will be
like that used in the preceding experiment, Various con*

872

A MANUAL OF EXPERIMENTS IN PHYSICS

ductors-— of different lengths, cross-sections, materials, tem-
peratures— will be joined in series, and, a current being
passed through them, the difference of potential at the ter-
minals of the various sections will be measured by a high-
resistance galvanometer. The measured values of E will
be proportional to the values of B.

Souroes of Xfaror.

1. The current must be kept constant.

2. The contacu must be good and constant.

3. There must be no accidental change in temperature.

Apparatus. A high - resistance mirror • galvanometer ;
wire connections ; two dial resistance-boxes ; a battery of
cells ; a key ; a board on which are fastened in series 6
copper wires of equal length, one being of less diameter
than the others and two joined abreast, 1 german-silver wire,
2 iron wires ; all the wires being of the same length, and
the german-silver and iron wires being of the same diam-
eter as one of the copper wires. One of the iron wires is

wound in a spiral and so
arranged as to dip into au
oil -bath, whose tempera-
ture may be altered at
will. Thermometer; Ban-
sen -burner; tripod; as-
bestos dish.

Manipulation. Arrange
the apparatus as shown.
B is the battery of cells;
Rf J?' are resistance-boxes;
J? is a contact key ; 0 is
the galvanometer ; W is
the board of wires; C'\i
the temperature celL

Place the oil -bath on
an asbestos dish and tri-
pod, and raise its tern*

EXFEWMBNTg IN IMOTBWITY AND JUaNETISM m

peratnre to abont 100°. Maintain the temperature as con-
stant ae poBsible during the entire escperiment.

Join the galTanometer terminals to the ends of one length
of the larger copper wire> and adjust the resistances B and
R' (keeping R as large as possible) until the deflection
prodnced whw the key ii9 closed 19 about 5 9cale diyisions.
Join the galvanometer terminals in turn to —

(a) One length of the larger copper wire ; (b) two lengths
of the larger copper wire in series ; (c) two lengths of the
larger copper wire in parallel ; (d) one length of the smaller
copper wire ; {e) one length of the german-silver wire ; (/)
one length of the iron wire at the temperature of the room ;
(g) one length of the iron wire at the temperature of the
bath.

In each case press the key and read the permanent de-
flection. Then reverse the order of experiment and thus
repeat the measurements. Note the temperatures of the
room and the bath.

Measure the diameter of the wires^ and verify the law
that the resistance varies inversely as the cross -section,

and that when joined in parallel 3-=-^ + "^ *

li Ri R2

Deduce the ratio of the specific resistances of copper,
iron, and german-silver.

Deduce the rate of increase of the resistance of iron with
the temperature.

ILLUSTRATION

Felk e, 1806
Comparison or Risistancbs

Copper wire, length I, diameter 0.04 cm., mean deflection 15.1 cm.
I, " 0.08 cm., '• " 4.2 cm.

*• I, *• 0.08 cm.,

2 in series, " " 8.4 cm.

2 in parallel, " *' 2.1cm.

Qerman -silver wire, length -^, diameter 0.08 cm., mean

deflectioD 82.6 cm.

SU A IfAKtTAL OF Eti^EfttifENTS IK PHTSlCd

Iron wire, lengih I, diameter 0.08 cm.

Temperature 18^, mean deflection 24.4 cm.

*'about98*, •* •• 85.8CIIL

Specific resistance of copper = p,

** " " german- silver = 15. 6p.

•* lron = 5.?p.
If a = " temperature coefficient " of iron, 87=24 (1 +80a).
/. a = 0.0082.

Qaeationa and Piobtems.

1. A current of intensity, 10, is divided and flows tbrougli two

tsonductors— one of resistance 10'*, tlie otiier of resistance
10" ; how much beat in calories is developed in each in one
hour ? If one branch is 5 times as long as the other/ both
being of the same material, what is the ratio of the rises in
temperature ?

2. Is it better to have the coils of a resistanoe-box long and thick

or short and thin ? Why ?
8. An incandescent lamp has a resistance of 20 ohms and requires
a current 0.0 ampere. Can it be worked by suitably groop-
ing 60 cells, each of which has an E. M. F. of 1 volt and an

internal resistance of 2 ohms ? (1 ampere = j -

EXPERIMENT 60

Ol(jeet. To measure a resistance by the Wheatstone
wire-bridge method. (See "Physics/' Art. 257.)

General Theory. A Wheatstone bridge is a combination of
conductors^ as shown —
viz, , a quadrilateral with
its opposite corners
joined. This bridge is
adapted for measuring a^
resistances in the fol-
lowing manner : a gal-
yanoscope (a simple
type of which is shown
in the illustration) is

FiaUl

placed in one diag-
onal branchy a cell
in the other. Then,
if rj, r^ rj, r^ are
the resistances of the
four side branches,
r,r4 = r2r3 if the re-
sistances are so ad-
justed that no cur-
rent flows through
the galvanoscope.
The bridge is then
said to be "bal-
anced.** This ad-
justment may be
made by altering the

S76

A MANUAL OF EXPERIMENTS IN PHTSICS

Fiaim

resistances of the arms. So if r„ r,, r^ are known resist-
ances, T] may be determined.

One method of ad>
JQstmenty known as the
''wire-bridge" meth-
od, is to make the
branches, ACdJiA CD,
continnons portions of
a uniform wire^ so that
rjr^ equals the ratio
of the two lengths, AC
and CD, yi^. Then,
in order to balance the
bridge with the known
resistance, r,, and the unknown one> r,, the method is to
more the terminal of the galvanoscope, C, along the wire
until there is no deflection of the instrument. Then

r,, of course, includes the resistance between the galvanom-
eter terminal, B, and the battery terminal, A ; r, includes
the resistance from C to A, etc. Consequently, the uni-
form wire {| + {4 must end in massive metal blocks, whose
resistance may be neglected ; and all the connecting wires
in the four arms should be short and of large cross-section.

Another mode of arranging the bridge is to use what is
called the "Post-oflSce Box'* method, in which three
known resistances are balanced against the unknown one.
This method will be described in full in the next experi-
ment.

The known resistance, r^, is generally a ''resistance-box,"
consisting of many coils of wire placed in a box, the ter-
minals of each coil being joined to large metal blocks.
These blocks are mounted on an insulating base, such as
ebonite or marble, and are sepftirated by an air gap which
may be closed by m^ns of brass plugs. These coils of

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 877

wire are wonnd doable, as shown, so as to have no self -induc-
tion (see *' Physics/' Art. 287), and tho ralne of the resist-
ance of each one is supposed to be accu-
rately known and marked on the box.
There are many precautions which
must be taken in using a resistance-
box, and the most important may be
thus stimmarized :

1. The temperature must be kept
constant, otherwise the resistance will

, Fio. 133

change, and the insulating base will also
expand and loosen the plugs.

2. The pings and openings must be carefully cleaned,
otherwise there is additional resistance introduced.

3. The surface of the box and metal blocks must be kept
dry and clean, otherwise there is leakage. (It is best to
cover a valuable box with a glass case like a balance case.)

4. Whenever a plug is put in or withdrawn, the top of the
box bends slightly, and the contact of all the other plugs
is altered. Therefore, each time one plug is changed, all
the others should be pushed into position again by a twist-
ing motion.

5. In time the plugs wear away, and unless they are so
shaped that their necks come below the top surface of the
metal Jt>locks '' shoulders'' will gradually form on them,
and then there is no longer good contact.

6. Each plug is ground so as to fit its own opening, and
therefore plugs should never be misplaced. While they are
out of their holes they should be put in.regular places and
kept clean.

7. The box must never be used in any circuit where
there is even the possibility of a current larger than a few
tenths of an ampere passing through it. Never use a good
box with a storage-battery.

Bonroes of Error.
1. Ohauges in temperature must be carefully guarded against;
Uiey may arise from the hand, the current, or external bodies.

878

A MANUAL OF BXPERIMBNTS IN PHYSICS

9. Good contact at all Janctiona la eaaential; and the bridge-win
key muat make both good and aharp contact.

8. Allowance muat be made, if neceaaary, for the wirea connect-
ing the boxea, etc., to the terminal binding -poeta of tbe
bridge to which the battery and galvanoacope are Joined.

4. The bridge wire may not be uniform; and, in any caae, care
muat be taken to make the error introduced at the ends aa
amall aa poaaible.

6. The bridge wire muat not be scraped by tbe contact key.

0. If mercury contacts are used, the metal polea which dip into
. the mercury-cupa must be pressed firmly against tbe bot-
tom plates.

Apparatus. A mirror-galyanometer; a WheatBtone wire
, bridge and sliding contact; a battery of constant cells; two
large resistance-boxes — one accurate, the other not neces-
sarily so; a Wheatstone bridge key; and the nnknown re-
sistance.

no. 184

Xanipnlation. Arrange the apparatus as shown. (? is the
galvanometer; r2, the unknown resistance; r,, the known re-
sistance; B, the cells; K' and K, keys (or combined in one);
r, the inaccurate resistance-box. Keep the sliding contact
at the middle of the bridge wire while finding the approxi-
mate value of the unknown resistance, as follows : Start
with the resistance in the battery circuit (r) so large that
the spot of light will not be deflected off the galvanometer
scale for all values of T] which may be used. Make r^ zero;
close the battery key, K', first, and then K; note the direc-
tion of the deflection of the spot of light. Next make r i

UNIVERSITX

£XPSRIMENT8 IN ELEGTRICITT AND MAGNETISM S79

yery large, and observe the deflection^ always closing K' be-
fore K. If the connections are properly made^ and the
apparatus is in good condition^ these deflections will be in
opposite directions. Now try two yalnes of Vi which are
nearer together and for which the deflections are opposite^
and continue in this manner until very narrow limits are
obtained — say, within 2 or 3 ohms. (Of course r, the re-
sistance in the battery circuit, must be diminished repeat-
edly, as the limiting case of ho deflections is approached.)
Keeping Vi fixed at the smaller of the two values thus de-
termined, slide the galvanometer terminal along the bridge
wire to that point at which no deflection. is noticeable. Do
not have the battery circuit closed for a greater length of
time than is necessary to allow the current to become steady
and to read the deflections. To find the point of no deflec-
tion accurately, increase any slight deflection in the follow-
ing manner : Make the resistance, r, in the battery circuit
zero ; close the battery circuit, and tap the .galvanometer
key as the band of light approaches the middle point of its
path. Repeat several times this '^ forcing of vibrations'' as
the image of the light on the scale approaches in the same
direction its point of rest, and note whether the deflections
are increased or diminished. In this way the direction of
the current through the galvanometer may be ascertained
even while the mirror is swinging. When at length it
seems impossible to force the vibrations in either direction,
open the circuits, and read very carefully the values of {3
and {4, the lengths of the two sections of the bridge wire,
estimating to tenths of a millimetre. If no deflection of
the galvanometer is produced when the sliding contact is
moved over a certain small length of the bridge wire, take
as the correct reading the middle point of this length.

But, now, r, = j^rj. The resistance r^ is read oft the re-

sistance-box as so many ohms ; l^ and l^ are measured in
the same units of length, consequently the unknown re-
sistance may be calculated in ohms.

880

A MANUAL OF £XP£RIMBNTS IN PHYSICS

Interchange r^ and fs, and repeat the foregoing open-
tion& so as to obtain another valne of rj.

Interchange the battery and galyanometcr^ and find ex-
perimentally two more values for r, corresponding to the
two possible positions of r^ and r,. In this second arrange-
ment K must be closed before JT', for the bridge key has
become that of the battery circnit, and mast, as usual, be
closed first.

Record the mean of these four results as the true Talue
of the unknown resistance.

ILLUSTRATION
UMABUKfMmn or RcsisrAjioR

Jan. 1»,1

Mo. of

Bxp.

.-.r,

21 ohms

68.87 cm.

46.18 cm.

17.06 ohms

28 •*

55.84 "

44.66 "

18.56 '•

20 •••

52.59 "

47.41 **

18.08 '•

22 "

56.19 •

44.81 "

17.86 ••

Meau. 18.1 ohms

Resistance of given coil == 18.1 ohms.

QuMttona 9nd Problems.

1. Reduce your result to electromagnetic units (C. G. 8.).

2. If the mean cross-section of 2, were greater than that of /«, ex-

plain the error which would be introduced in r,.
8. What electrical phenomena prevent the current from starting

at its normal value the very instant the battery circuit is

closed T
4. Is it strictly necessary to have a batteiy of earuiant oeili?

Why?
6. Give a reason for keeping the battery circuit open as much as

possible.

6. Deduce the formula for expressing the condition that there

shall be no current in the galvanometer, when the ^Ifaoom-
cter and cell are interchanged.

7. Would there be any advantage in introducing Itnown rsiist-

ances at the ends of the bridge wire, between the ends sod
the two battery terminals ?

EXPERIMENT 70

i« iw iw» A

JCZJCHCM

_^^_ 1 t t 5 If « » 80

MOO iOOO 1000 IMO 500 200 100 100

paaacDaaatu

Fig. 135

Olijeot. To measure the resistance of a mirror-galyanom-
eter by Thomson^s Method, using a "Post-oflSce Box."

Oeneral Theory. A '' Post-office Box'' is a plug resistance-
box with the coils arranged in a particular way, and hav-
\ng binding - posts ^

. 'J. A ^ ^ P 1000 100 10

at points A, B, C, tgJZ)CZI?CI3C
and D, as shown.
The two sets of
coils, 10, 100, 1000,
AB and BD, are
called the *' ratio
arms.'' The meth-
od ol use for meas-
uring an unknown resistance is to join A and D through
the battery, B and C through the galvanoscope, and A and
C through the unknown resistance (ra). Then AB \% r^,
BD IS r,, and DC is r^, as in the previous experiment. It
is at once evident how the unknown resistance r^ may be

determined by suitably al-
tering rj, ra, and r^. There
is, however, a definite mode
of procedure which is ad-
visable.

1. Make r, = r, = 1000;
and, beginning with r^zz
6000, alter it by steps of
1000, 100, 10, 1, until two
Tidues are obtained whioh

882

A maKual of experiments in physics

produce opposite deflections of the galvanoscope. Let
these two valaes be^ for illustration, 46 and 47. Then
46<rs<47.

2. Make r^ = 100^ r, = 1000, and find two values of r^

between 460 and 470, which will produce opposite deflec-

T r 1
tions. Let them be 463 and 464. Then, since - = -' = —-,

r^ r, 10

rg must lie between 46.3 and 46.4, or 46.3 <r^< 46.4.

3. Make r, = 10, rg = 1000, and find two values of r^,
between 4630 and 4640, which will produce opposite
deflections. Let them be 4634 and 4635. Then, since

- = -=rKK9 ^a mnst lie between 46.34 and 46.35, or

r3 = 46.34+.

The next figure may be estimated by a comparison of the
deflections produced by 4634 and 4635.

To measure the resistance of a galvanometer, two methods
al*e possible : one is to place it in the branch A C, and meas-
ure its resistance as just described, by means of a galvano-
scope ; another is to place it in the branch A C, but to re-
place the galvanoscope in the branch BC by a contact
key. The theory of this second method, called " Thom-
son's Method,'* is as follows : As the current flows around

from AioD, there will
be, of course, a deflec-
tion of the galva-
nometer in the branch
AC^ but if the bridge
is "balanced" by suit-,
ably altering r^, r^ and
r^, so that the points
B and C have the same
potential, no change in
the galvanometer de-
flection will be made
when the conductor joining S to C is made or broken, he-
cause no current will flow from 5 to C. Therefore, the

PiO. 13T

EXPERIMENTS IK ELECTRICITY AND MAGNETISM 888

method is so to alter rj, r,, and r^ that, when a key in the
branch BC ib made and broken, there is no change in the
galyanometer deflection. In that case.

Bat Ta is the galyanometer resistance, O.

.:G = '

Somoes of Enor*
These are the same as la the previous experiment. It should be
noted that, since there is always a current through the galva-
nometer, the needle is alwayd deflected, and so does not stand
in a position in relation to the coils in which it Is most easily
affected by a change in tlie current. Therefore, it is often
necessary to bring the needle heick towards its normal position
by means of a magnet which may be placed near the galva-
nometer.

Apparatus. A mirror-galvanometer; a post-office box; a
battery of constant cells; two contact -keys; a magnet;,
wire ; an ordinary dial resistance-box*

Hanipniation. Arrange the apparatus as shown, putting
a key, J£, in the branch BC, and a key, K*, and dial resist-
ance, r', in the battery
circuit, J2>. Adjust r*,
the resistance in the
battery circuit, so that
when ri and r, are 1000
each, and when the bat-
tery key, iT', alone is
closed, the band of
light is deflected
through about ten cen-
timetres.

In taking all the following readings, first close the bat-
tery key, £'; note the steady deflection of the galvanome-
ter mirror, and then observe whether this deflection is in-

884

A MANUAL OF EXPERIMENTS IN PHYSIOS

creased or diminished by closing the cro88*circuit key^ K.
Of coarse^ the vibrations of the needle may be forced by tap-
ping the key, K, at proper intervals ; and the sensitiveness
may be increased by altering r\ or bringing a magnet near,
so as to neutralize part of the action of the steady current
in case it carries the spot of light off the scale. If making
the circuit containing K causes no change in the deflection,,
then no current flows tlirough this branch, BC, and the
Wheatstone net is ** balanced.*'

However, when Vx and r^ are each 1000, the resistance, r^,
usually cannot furnish such a value as will satisfy the rela-
tion, 0 =3 -^— * exactly ; consequently, a small current will

flow through the branch BC, and will increase or diminish
the current through the galvanometer, according as r^ is
too small or too large. Keeping the ratio arms, Tx and rj,
1 000, find two values for r^ differing by one ohm such that
the changes in the steady deflections are in opposite direc-
tions. Now, make r^ = 100, rjs 1000 ohms, and proceed ex-
> actly as directed above. Then make rj = 10, keeping f,=
1000, and again balance the bridge.

1000
1000
1000

ILLUSTIIATION
RmsTANOK or Galtanomkter

lbrcbl3,UM

1000 46-47
100 468-469
10 4688-4684

46 -47
46.8 -46.9
46.88-46.84

Hence, O = 46.88 + olims.

Questions and Problems.

1. Is it better to use for B a siugle cell of small £. M. F., with f'

proportionally small, or to have a battery of oomparatively
high £. M. F. with a correspondingly large resistance?

2. What is the essential condition to be satisfied in either esse?
8. Why cannot the deflections of the galvanometer mirror be re

versed, as could be done In the preceding ezperimentf

EXPERIMENT 71

Olgeot. To measure the resistance of a cell by Mance's
Method. (See " Physics/' Arts. 242, 257. )

Oeneial Theory. The resistance of an electrolyte or a cell
{e.g.,€k Danieirs cell) does not remain constant as a cur-
rent flows through it, owing to changes in the liquids and
at the metal electrodes, and therefore any measurement of
it must be made quickly. There are several methods for
its measurement, two of which will be described— one in
this experiment, and the other in the following one.

Resistance of a Cell. Mance^s Method* — If the cell, B, is
placed in the branch, A C, of the bridge, and a key, K, in-
serted in place of the
battery in the branch,
ADj there will, of
course, always be a cur-
rent through the gal-
vanometer^and its nee-
dle will be deflected.*
But if, on making and
breaking the key, K,
there is no change in
this deflection, the
bridge must be ''balanced "—i, e., r, r^ = r^ r^; for in this
condition the current through the galvanometer is in-
dependent of the E. M. F. in the branch A KD, and so
will be the same when the key is opened and when it is
closed.

Fjo. ia9

886

A MANUAL OF EXPERIMENTS IK PHTfflGS

Hence, if B is written for r,,
5 =

r,r

The adjustments, then, are obvions ; and either a wire
bridge or a post-office box might be used. A third method,
however, will be described, simply for variation. This
method is to nse three separate plug-boxes for r,, r,, and r^,
ttnd adjust them until the bridge is balanced.

Bouroes of Error.

1. Tlie same remarks apply here as in the two previous experi-
ments.
9. The polarization of the cell must be avoided if possible.

Apparatus. A high-resistance galvanometer ; a constant
cell; three plug resistance-boxes; one dial resistance-box;
a key ; a magnet.

Manipulation. Arrange the apparatus as shown, putting
the dial-box in the galvanometer branch. Connect the cell,

By to the adjacent re-
sistance-boxes by very
short, thick wires.
Keep r^ constant at
1000 ohms through-
out the experiment.
Start with r, = 20
ohms and r^ = 10,000
ohms. Adjnstr',the
resistance in the gal-
vanometer branch, 80
that when the key ifl
open the image of the source of light is on the galvanome-
ter scale near either one of its ends. If necessary, bring
the spot of light back on the scale by means of the magnet,
or by putting a shunt around the galvanometer. Keeping
rg fixed, vary r^ until two values are found differing by one
ohm^ such that the corresponding changes in the pernuwent

Fio. 140

EXPERIMENTS IN ELECTRICITT AND MAGNETISM 887

deflection, which occur when E is closed, are in opposite
directions. Usually these values of r^ will be less than 20
ohms. In general, r' (or the resistance of the shunt) must
be yaried slightly with the other resistances, so as to main-
tain the steady deflection at the amount above mentioned.
Change ri to two ohms less than the smaller of the num-
bers just obtained, and keep it fixed at this value. Then
adjust r, until no change in the steady deflection of the
mirror is caused by closing the key. If a change in the de-
flection always occurs when the key is opened or closed,
determine two values of r,, one of which increases, the
other decreases the deflection, and choose the one which
gives the least change. Under this condition no current (or
a minimum one) flows through the branch containing the

key, Ky so that B = -^-^. Whence, calculate and record

the resistance of the given cell.

ILLUSTRATION

March 12, 1886
RffiiSTANCX OP Damikll's Ckll

First approzimatioD, witb r,= 10,000 and r4=1000 ohms, gave

11< ri < 12, or £r = 1. 1 + olims.
Finally, with r^ — 1000 and ri = 9. the least deflection occurred

when r, = 7706 ; hence. 5 = 1. 16 + ohms.

Qaestions and ProblemB.

1. Explain, tuing symbols, how an overwhelming error might be

introduced into the final result by putting a comparatively
large known resistance in the branch contiiining the cell?

2. Why cannot a Wheatstone wire bridge be used to advantage

in this experiment?
8. Upon what quantities does the internal resistance of a cell de-
pend, and how ?

EXPERIMENT 72

Olgect. To measure the Bpecific resistance of solutions
of copper sulphate by Kohlrausch's Method. (See "Phys-
ics/'Art. 244.)

General Theory. If ^ instead of using a direct steady car-
rent with a Wheatstone bridge, an alternating or varying
current {e. g., one from an induction-coil) is used, the bal-
ance of the bridge can no longer be tested by a galvan-
oscope, but a telephone may be used in its place, as it
responds to slightly varying differences in potential. The
general arrangement is the same as before. To produce a
varying current an induction-coil may be used or any kind
of an interrupter in a direct current, such as a commatator
or scraping contact. The advantage of an alternating cur-
rent with an electrolyte is that there is little if any elec-
trolysis or polarization.

The specific resistance of any conductor is defined as
being that of a cube of the substance 1 centimetre on each
edge. Consequently, if the substance is in the form of a
cylinder of cross - section <r and length I, and if p is its

specific resistance, the resistance of the cylinder is p —

Any form of the bridge may be used, but the method
employing three separate boxes will be described. Instead
01 using a single box, rj, in one arm of the bridge, it is
sometimes best to put in another very high resistance-box,
r,', parallel with it. For the combined resistance of the

two is iJ, where —=— + —,; and so, if r, is nearly equal

J{ fi Ti

BXPERIMEXTS L\ ELECTRICITY AND MAGNETISM 88a

to the desired resistance^ B can easily be made exactly
equal to it.

SoQxoes of Brror.

1. The greatest uDcertainty enters from diflSculty in detecting

the minima in the telephone.
3. The temperature must be kept extremely constant.
8. Polarization should be absolutely prevented.

Apparatus. A small induction -coil; a storage circuit;
two telephones ; a thermometer ; two exactly equal resist-
ances of about 100 ohms each ; two good plug resistance-
boxes^ the one containing low-resistance coils and the oth-
er high-resistance coils ; some crystals of copper sulphate ;
a glass funnel ; a piece of filter paper ; a large beaker-glass ;
and the electrolytic cell. This cell consists essentially of
a snitably mounted cylindrical glass tube, the ends of
which are closed by copper disks, called electrodes. The
distance between these electrodes may be varied so that
they form the ends of the column of liquid contained in
the glass tube.

Fio.141

Hampulation. Set up the apparatus as shown, fj is the
electrolytic cell ; r,, r/, the low and high resistance-boxes
respectively, which are joined in ** parallel" or ** multiple
arc"; r^, r^, the equal resistances; and T the telephone
circuit. The wires (W", W") of the secondary circuit must

800 A MANUAL OF EXPERIMENTS IN PHYSICS

be suspended in the air so far as possible^ to prevent short
circuiting. The induction-coil (/) must be placed on the
sill outside of a window as far from the observer as con-
venient^ in order to muffle the distracting sound of the
interrupter. The amount of copper sulphate required de-
pends upon the size of cell to be nsed^ and should be indi-
cated by an instructor. Pulverize the crystals and make
five and ten per cent, solutions (by weight), using tap water
as the solvent. After the copper sulphate is thoroughly
dissolved, filter the solution so as to remove impurities.
Pour a part of either one of the solutions into the electro-
lytic cell, and set the electrodes parallel to each other at
the ends of the column of liquid. Put an infinite resist-
ance in the plug-box (r/) which contains the high -resist-
ance coils — i. e., throw it out of the circuit by unbinding
an end of the wire (o), which joins the two boxes. Vary fj,
first by hundreds, then by tens, and finally by units, un-
til an approximate value of the electrolytic resistance (fa)
is found, as will be indicated by a minimum vibration in
the telephone. Using this result as a guide, regulate the
distance apart of the electrodes and the quantity of liqnid
in the cell, so that its resistance is B,few ohms less than that
of either of the equal arms (r,, r^) of the Wheatstone
bridge.

Now suspend the thermometer vertically alongside the
glass tube of the cell. Then adjust the electrodes very
carefully, so that their opposing surfaces are parallel, at
the same time that the electrolyte completely fills the
space between them, but does not extend beyond them.
After this, do not alter in the least the relative positions
of the parts of the electrolytic cell, and avoid, as mach as
possible, changes in the temperature of any part of the ap-
paratus. Find two values of r^, differing by one ohm, be-
tween which the unknown resistance lies. Join the other
plug resistance - box (r/), in parallel by the connecting
wires, and vary the resistances in the plug-boxes until the
bridge is balanced as nearly as possible — i. 0., until there is

EXPERIMENTS IN ELECTRICITT AND MAGNETISM 891

no sensible vibration in the telephone. Better results are
obtained by using two telephones — one for each ear.'

Finally, record the mean temperature of the solution as
indicated by the thermometer outside the cell, and meas-
ure the mean distance (/) between the electrodes. Empty
the cell, wash it with tap water, and measure the resist-
ance of the other solution in the manner just described.
When no current passes through the telephone branch,

' 7

r, = — J2 = ■■ ^ ' ^ ohms, since r3= r^. Also, r2= p - ; con-

seqnently, the specific resistance, p= ,. '^ ^ ,. x 10* C. 6. S.

units. The internal cross-section of the tube {9) is a con-
stant furnished with the cell.

ILLI78TRATION

April 2, 18M
RI8I8T1.HCI Oy GOPPKR SULPHATK

Mean temperature = 1S° O. ^ = 19. 60 sq. cm. r, = r4= 100 ohms.

6 % Solotion
Ti = 98 ohms

r/=4000 ♦' }■ .•.p,=6.48xl0'»
I =84.84 cm.

10 je Solution
r, =97 ohms ,

r,'=4200 " \ .•.p„=8.20xl0»»
/ =67.70 <jm.

Questions and Problems.

1. What is the adyantage of an alternating current over a direct

current in this ezperimenti?
8. Should the interrupter of the induction-coil emit a high or a

low note to cause the most ahrupt minima in the telephone?
8. Of what metal should the electrodes be made to give the best

results? Why?

4. What physical causes are there why the specific resistance of

an electrolyte should vary with the temperature?

5. If there are JV cells avaihible, each of E. M. F., E and resist-

ance r, how should they be Joined so as to give a maximum
current through a conductor whose resistance is i2? How
much energy is supplied in one second, and how is it spent?

EXPERIMENT 78

Oljeot. To compare electromotive forces by the high-
resistance method. (See ''Physics/' Art. 278.)

E
General Theory. Since by Ohm's Law, i = -b> &^d since

the deflections of any one galyanometer are the same for
the same current, the E. M. F.'s of two cells may be com-
pared by placing each in turn in circuit with the galya-
nometer and a resistance-box^ and varying the resistances
until the deflections are the same. Then^ if i2| and R^ are
the entire resistances in each circuit which correspond to

rr p

El and ^g, -i = ^, because the currents are equal.

A galvanometer - needle is never quite symmetrically
placed in the coils, and so, if a current is reversed, the di-
rect and reversed deflections may not agree exactly. To
multiply, then, the number of readings and eliminate as
many errors as possible, it is best to reverse the current
through the galvanometer and repeat the observations.

A good form of current reverser, or commutator, known
as Pohl's, is shown in the drawings. It consists of a board

with six metal cups containing
mercury, and four binding-posts,
the cups and posts being connect-
ed by conductors as^ shown. A
rocker is supported as shown so
as to make connection from Cj to
Ca and C, to C 2, or from C, to C3
FiGv 149 md C'a to C'3. The two metallic

ends of the rocker are separated by an ebonite handle.

EXP£RIM£NTS IN ELRCTRIOITV AND MAGNETISM 898

When connection is made from C, to Ca and C, to C^
Pj and P, are in metallic connection, as are also P'j and
PV lU however, the rocker is tipped over so jis to join
C, to Cs and C'a to C's, Pi
is joined to P', and P'| to
P,; and consequently a
current flowing from Pj
to P,, through the galva-
nometer, to P'^ to P'„ and J^ Ma«
through the cell to Pj, will have its direction through the
galvanometer reversed*

Sonioes of Error.

1. Two deflectioDB can never be made to agree exactly.

d. The comniutator must be clean, and the copper rocker must
be well amalgamated, so as to make good and constant con-
nection.

8. The resistance of the battery and the connecting wires is in
general neglected, hence it should l>e made small and kept
the same in the two experiments.

Apparatus. A sensitive mirror -galvanometer; two plug
resistance -boxes, containing low and high resistance-coils

Fia 143 °

respectively j a commutator (Pohrs) ; and several cells of
different kinds.

llanipulation. Arrange the apparatus as shown. G is
the galvanometer; B is the cell; C is the commutator;
the two resistances joined in series are shown at r.

Start with the circuit incomplete by removing the rocker
of the commutator from the mercury-cups. First use a
single Daniell's cell as the source of E. M. P. — i. e,, for
the battery B. Make the resistance r very large — say.

894 A MANUAL OF EXPERIMENTS IN PHTSIGS

10,000 ohms. Then complete the circuit by replacing the
commutator bridge, and note roughly the resolting de-
flection of the galvanometer mirror. Vary the amount of
the effective resistance in the boxes until the image on the
scale of the source of light is deflected to a division near,
but not beyond, either end of the scale. Keep the resist-
ance fixed and record its value. If this resistance is much
less than 5000 ohms, it must be increased at the cost of the
magnitude of the deflections. Remove the commutator
bridge, note the zero point of the galvanometer by the
method of vibration; replace the bridge and note the de-
flection produced. These deflections should be read by the
*' method of vibrations,'^ and the arithmetical mean of at
least four deflections thus obtained must be recorded.

Next reverse the current through the galvanometer by
the aid of the commutator. Again take the mean of the
same number of deflections, which will be, of course, in the
opposite direction to those just recorded. It may be neces-
sary in order to obtain deflections of the desired magnitude
to readjust r, in which case its new value must be re-
corded.

Substitute for the DanielFs cell one or more cells of a
different kind. Repeat the entire process just explained,
being careful to adjust the respective resistances so that
the deflections have the same values, both in magnitude and
direction, as the corresponding ones found for the Daniell's
cell. This equality should be attained to within the dif-
ference made by a change of one ohm in the resistance.

If 7?' is the resistance of the galvanometer, r, and r\
the mean resistances for the right and left deflections with
the Daniell's cell, and r^ and r'j the resistances for the
right and left deflections with the other cell, then

The value of R' can be learned from an instructor, or
may be directly determined ; and the resistance of the bat-
tery, commutator, and wires may be neglected.

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 3d6

Calculate the ratio of the electromotive forces of the cells
Qsed^ keeping the resistances corresponding to the left de-
flections separate from those for right deflections. Report
the mean of the two determinations of the ratio EyjE^^ thus
found, as the final result for each cell.

It is sometimes convenient to keep the resistance con-
stant and to allow the current to vary. Then, assuming
the deflections £?, and d^ to be proportional to the currents,
which is reasonably true if they are small, the relation be-
tween the E. M. F.'s is EJE2 = djd^. This process is often
called the "equal-resistance method** to distinguish it from
the ''equal-deflection method** explained above.

ILLUSTRATION

FebL8,1884

E. M. F. or Gills

R* = 130.7 ohms. For the Danieirs cell, r^ = 8000 ohms.
To obtain the same mean deflection with a Leclauche cell,
r, = 12080 and 11084 for left and right deflections respectively.

B 8120

E = M^ ~ ^'^^ '®^' deflections.

K 8120

^=j2jQg- =0.671 right deflections.

Mean, 0.670

Qnastioiift and Problenui.

1. Calculate the £. M. F. of the cells in volts, in C. O. 8. electro-

magnetic units, iind in 0. G. S. electrostatic units on the
assumption that the E. M F. of a Daniell's cell is 1.08 volts.

2. What is the theoretical advantage of the *' equal -deflection

method" over the " equal -resistance method"?

8. Discuss, in brief, the sources of error arising from the cells fur-
nishing a current.

4. Upon what does the E. M. F. of a cell depend ?

EXPERIMENT 74

Otyect. To compare electromotive forces by the *' con-
denser method/' (See " Physics/' Arts. 235-237.)

General Theory. If a condenser of capacity C is charged
to a difference of potential E, the quantity of charge on
either plate is

Q^EC.
As will be explained below, Q can be measured by dis-
charging the condenser through a ballistic galvanometer.
The method, then, is to charge the same condenser to the
potentials E^ and E2 of the cells, and to measure the cor-
responding quantities Q^ and Q^, Then,
E,IE^=QJQ^.

A ballistic galvanometer is designed to measure quanii-
ties, not currents — i, e., not t, but the product it, where
t is the number of seconds the current of intensity i flows.
The needle must have a long period of vibration, and the dis-
charge must take place quickly. (See Chapter "Galvanom-
eters.'*) The sine of half the angle through which the mirror
of a ballistic galvanometer is thrown by the sudden passage
of a quantity of electricity around the coil is a measure of
this quantity. Also, for small arcs the sines are propor-
tional to the deflections in scale divisions of the band of
light. In symbols, Q^zkd. The swings of the mirror are
"damped" by the resistance of the air, by induced cur-
rents, and by the viscosity of the fibre which suspends the
mirror. Therefore the deflection {d), which would have
occurred had there been no damping, is equal to the ob-
served initial **fling" (r/,) of the band of light, increased by
a correction term. The approximate value of this correction

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 897

may be obtained by the following considerations : Let Jj
denote the nnmber of scale divisions passed over by the
band of light in moving from its zero position to the first
turning-point in the opposite direction to that of the initial
fling, di. The difference between d^ and dz is caused by
the damping of the mirror while it is making two semi-
vibrations, — dj and +d^. But these vibrations are so nearly
equal that the retardation for either one of them may be
tiiken as half that of both — i. c, as Hd^ — dz)- Conse-
quently, of = rfj 4- i{d^ — f/2).

Similarly, before the mirror completes its oscillation in
the same direction as the initial swing, it has made four
nearly equal semi - vibrations ( — rfi, 4- </2» — ^ -f-da) ; so
that in this case d = di + ^{di — rfg) = dj + i^.

Of course there is less error involved in assuming an
equality of damping for two semi-vibrations than for four,
but since d^ can usually be read more accurately and easily
than ^3, it follows that the correction ih is the more ad-
vantageous in practice.

Therefore

^2 d\-hiy

In using a ballistic galvanometer it is sometimes a min-
ute or more before the needle comes to rest again after a
flingy or it may be kept constantly vibrating slightly, owing
to magnetic or mechanical disturbances. To obviate this
difficulty, always discharge the condenser at an instant
when the needle is at one of the turning-points of its small
vibrations, and note the deflection, not from the true zero,
but from the turning-point. The needle may be brought
approximately to rest by a " damping key.**

Various keys have been arranged to charge and discharge
a condenser. The requirements are that by one motion
the battery which is charging the condenser may be thrown
out of circuit, and the two plates joined through the gal-
vanometer. For this purpose the key must have three
binding-posts and two contacts, as shown in Figs. 144 and

896

A MANUAL OF EXPERIMENTS IN PHTSIGS

145. When the key is pressed down, the two plates of the
condenser are joined to the cell ; by releasing the spring
the key flies up and makes contact with the galvanometer
terminal, so that the condenser is discharged through it.

Fw. 144

Bouroea of Error.

1. The capacity of a condenser depends slightly upon the time

of charge, and the quantity discharged varies also with the
time of discharge.

2. There must be nu leakage through or over the key.

Apparatus. A ballistic galvanometer ; a condenser ; a dis-
charge-key ; a standard Clark cell ; a battery, the E. M. F.
of which is desired ; a damping-key, dry-cell, and coil.

-0"

UTv

■r*-K

FiO. 146

Manipulation. Arrange the apparatus as shown, first
using the standard Clark cell as the source of electro-
motive force. Arrange the coil of the damping circuit at
a convenient place near the galvanometer needle and ap-
proximately parallel to the coils of the instrument. Con-
nect it with the damping -key and dry -cell, as in Fig.
145a, placing the key where it can be most conveniently

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 899

reached while observing the instrument. Pressure on one
lever of the key will deflect the needle in one direction ;
and on the other^ in the opposite. If the needle is vi-
brating, it may be stopped by applying pressure alternately
a on one and on the other lever, so as

to counteract the swing each time.
If the damping is too powerful^ move
the coil farther from the needle.
When the lever of the discharge-

Fialtfa

key touches the lower stop^ the key is said to be in the
** charge*' position. When the lever touches tlie upper
stop, the key is said to be in the ''discharge'* position.
When the lever touches neither stop, the key is said to be
in the '' insulate'* position. Keep the discharge-key in the
insulate position while setting up the apparatus.

Record the initial temperature of the Clark cell as in-
dicated by the thermometer on its case. Push down the
lever of the key to the charge position by means of its in-
Bulating-knob, and keep it so for about one minute. Dur-
ing this time bring the needle as nearly to rest as possible.
The exact time of charging should be noted. At the ex-
piration of the chosen interval, when the needle is at a
turning-point of its vibration, instantly, without hesitating
at the insulate position, discharge the condenser through
the galvanometer by pressing the trigger. Becord the
fling and the first swing of the mirror back in the same
direction. After this, while the mirror is coming to rest,

400 A MANUAL OF EXPERIMENTS IN PHYSICS

thoronghly discharge the condenser by simnltaneously
earthing both plates through the fingers. !N^ow set the
key in the insulate position^ then charge and repeat the
process just explained so as to obtain another pair of read-
ings. Always charge the condenser for the same length of
time. In this manner take ten observations at least, and
denote the mean 1st and 2d deflections by d^ and d^ re-
spectively, and di — d^ by 5.

Once more note the temperature of the cell, and average
this value with the like quantity read at the start.

Replace the Clark cell by the battery of three cells, the
E. M. F. of which is unknown. Join the cells in series,
and find the corresponding deflections d\ and i\ both of
which must be the arithmetical means of the same num-
ber of readings as were involved in d^ and 3. The times
for charging must be the same in this as in the preceding
case. Substitute the values of «?,, d'j, 5, h', together with
the B. M. F., -^ of the standard cell, in the equation given
above, and calculate the E. M. F. of the given battery as a
whole. Next, join the three cells in parallel, and measure
the E. M. F. Also calculate the E. M. F. of each cell of
the battery, on the assumption that all the ceils are equal.
The absolute value of B, the E. M. F. of the Clark cell,
for the recorded mean temperature will be furnished by an
instructor.

Before putting away the apparatus, study the influence
of the time of charging the condenser upon the quantity
of electricity received by it. Let the chosen times be abont
1, 15, 30, 60, and 120 seconds. Record the corresponding
corrected deflections, each of which should be the average
of two or more readings. Finally, study the phenomenon
of "electric absorption '' by keeping the condenser iu cir-
cuit with the battery for five minutes and then discharging,
at once insulating and discharging again at the end of one
minute, insulating again and discharging again at the end
of one minute, and so on, until no further fling can be ob-
served. Record the fling for each discharge.

EXPERIMENTS IK ELECTRICITY AND MAGNETISM 401

ILLUSTRATION ^p«,x..l«

Mean temperature of Clark cell = 20.8° C. ; hence, i?= 1.428 volts.

E. M. F.

1.428 volts.
4.92+ volts.
1.65 volts.

4.1

0.8

14.2

0.8

4.9

0.2

Clark cell

8 QoDda cells in series

3 €k>Qda cells in parallel ....

Mica Condenser— Time of Charging

Iwo. 15 MC. 90MO.

Ck>rrected deflections 14.0 14.2 14.3

imto.
14.4

9mliL
14.4

Corrected deflectioDs.

Besidoal CbaiigeB at Inteirals of One Minute
.. 14.4 8.2 0.9 0.0

Qnestioiia and Protaleina.

1. Assuming C^\ microfarad, calculate the mean quantity of

electricity wliich was discharged through the galvanometer,
giving the answer in coulombs, in electromagnetic units
(C. O. S.), and in electrostatic units.

2. Mention the essential characteristics of a good ballistic gal-

vanometer.
8. Give an experimental method for comparing the capacities of
two condensers, also one for measuring 0 absolutely.

EXPERIMENT 76

Otgoct. To determine the "galvanometer constant" of
a tangent galvanometer. (See " Physics/' Arts. 246, 278.)

General Theory. It is proved in treatises on Physics that,
if a steady current of intensity i is passed around a tan-
gent galvanometer when its coils are in the magnetic me-
ridian, the needle will be deflected through an angle ^ such
that

* = -^tan^,

where H is the horizontal component of the earth's mag-
netic force at that point, and 6r is a constant for a given
instrument. 0 depends only upon the size and number of
turns of wire of the galvanometer. If the instrument has
n circular turns of radius r, ail of the same size, and placed

closely side by side, 0 = 2r njr,
(A simple form of a tangent
galvanometer is shown in the
illustration.)

If a current is passed throagh
any electrolyte, a quantity of
matter is liberated which is pro-
portional to the quantity of elec-
tricity carried over, and also to
the chemical equivalent of the
matter liberated. (Faraday's
Laws.) The quantity (number
of grams) liberated by a unit
current in one second is called
ita. 146 the electro-chemical equivalent

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 408

of that substance ; and its value is known for most sub-
stances (which can exist as electrolytes) — e.g.y the electro-
chemical equivalent of hydrogen is 0.00010352 grams, and
as the density of hydrogen under standard conditions is
0.0000895, the volume of hydrogen liberated by a unit cur-
rent flowing for one second is 1.156 cubic centimetres.

The values of the masses or vol-
umes for any other element may
be calculated at once by means of
Farada/s Laws.

Therefore, if the same current
is passed in series through a tan-
gent galvanometer and a voltam-
eter (an instrument devised to
measure quantities liberated by
the passage of a current through
electrolytes), the quantity of mat-
ter liberated may be measured ;
this gives the current which has
passed ; the angle of deflection of
the galvanometer-needle may be

measured, and therefore ^ may

be calculated. If H'\% known (see
Experiment 65), G may be at once
determined.

Two methods will be described,
in one of which a gas voltameter
is used, and in the other a copper
voltameter.

Gas voltameters are of two kinds.
In one, the two gases liberated at
the cathode and anode are kept
separated. Thus, in the instru-
ment shown, if the electrolyte is
a solution of sulphuric acid, hy-
drogen will collect in the tube fio. 147

404 A MANUAL OP EXPERIMENTS IN PHYSICS

over the cathode^ oxygen over the anode. (The oxygen h
not qnite pnre.)

In the other kind of gas Yoltameter the two gases formed
at the two poles are giv^n off in the same tube, and there-
fore mix as they rise throagh the liquid.

If the tabes in which the gases are collected are filled
with the electrolyte at the beginning of the experiment,
the Tolnmes of water displaced by the gases as the corrent
passes measure the yolumes of the gases ; and so the num-
ber of grams may be calculated if the pressure and tempera-
ture are known.

The use of the copper voltameter will be described in the
next experiment.

SouroeB of Bnor.

1. The pressure of the gases must be accurately read, takiog into

account the fact that the gases are wet

2. A steady current must be maintained.

8. The commutator which is used to reverse the current through
the galvanometer, and thus correct for some of its errors of
construction, must have its poles clean so as not to alter il»e
resistance when it is turned.

4. All wires carrying currents should be so twisted around each
other as to produce no magnetic effect — t. 0., always twist
together two wires which are carrying the same current in
opposite directions.

Apparatus. A tangent galvanometer ; a compass ; a water
voltameter; an iron coil resistance - box ; a commutator;
a storage circuit ; a centigrade thermometer, and a large
beaker-glass.

F10. 148

Manipulation. Set up the apparatus as shown. V is the
voltameter; G, the galvanometer; C, the commutator; R^
the iron wire resistance ; B, the storage cells.

Adjust the galvanometer in the magnetic meridisn bj

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 406

means of a compass (or asing the magnet of the galvanom-
eter itself as a guide, if its pointer is long enongh). Fill
the voltameter with dilute sulphnrio acid, made by adding
5 grams of sulphuric acid (H2SO4) to each 100 cubic cen-
timetres of water. Allow the current of electricity to flow
while regulating the galvanometer and the resistance R, so
that the deflections of the galvanometer are nearly equal to
30**. Usually, under these conditions, the gases are evolved
at a desirable rate, but if it should happen that the libera-
tion is too slow, the conductivity of the solution may be in-
creased by adding a little more sulphuric acid.

Break the circuit by means of the commutator, and open
the upper tap (or taps) to permit the enclosed gases to es-
cape. Next fill the open tube so full that the liquid rises
above the taps, and then jar the voltameter until every
visible bubble of gas is driven out. Close the upper taps;
note the zero position of the galvanometer, reading both
ends of the index. Close the circuit through the commu-
tator, and note the exact second at which the current is
made. Also note the resulting deflection of the galvanom-
eter, and quickly reverse the current through the galva-
nometer so as to obtain a deflection in the opposite direc-
tion. Continue to read pairs of opposite deflections, at
intervals not greater than flve minutes, until the tubes con-
tain a little less than their volume of gas.

The volume of gas in the voltameter is indicated by the
position of the top of the column of liquid in the burette
with reference to the scale divisions which are etched on
the outside of the glass tube. For convenience in meas-
uring the volume and pressure of the enclosed gases, it is
well to keep the level of the free surface of the solution in
the open tube a little below that of the menispns of the
liquid in the burette. This may be accomplished very
easily by opening the lower tap from time to time and al-
lowing the necessary quantity of liquid to flow out of the
voltameter into the beaker-glass, or by lowering the bulb
which is connected by means of rubber tubing. (If the

406 A IfAKUAL OF EXPERIMENTS IN PHYSICS

apparatus is different, the water may be removed by a si-
phon.) When the desired volume of gas is set free, break
the circuit by the aid of the commutator, and note the ex-
act time to the second at which this is done. The differ-
ence between this reading and the like one taken at the
start is the interval of time (7" seconds) during which the
current (i) decomposed the electrolyte into hydrogen and
oxygen. Tap on the burette near the electrodes with a
finger so as to cause the bubbles which stick to the inner
surfaces of the voltameter below the tap of the liquid to
rise and mix with the main volume of gas. Then hang the
thermometer vertically alongside the burette, and while it
is assuming the temperature of the enclosed gases vary the
quantity of water in the open tube until, when the burette
is truly vertical, the free surfaces of the liquid are at ex-
actly the same level in it and in the closed tube (in one of
the two tubes if it is a double-tube voltameter). Eecord
the volume of gas (F) in the burette as indicated by the
position on the scale of the under side of the meniscus.
Of course the usual precautions to avoid parallax must be
taken. In a similar way bring the free surface in the open
tube to the level of the water in the other tube in case the
voltameter is a double-tube one, and read the volume of the
enclosed gas. Note the temperature of the gases {i^ C.)
and immediately read the mercurial barometer '^correct-
ing " the observed height.
The volume of gas under standard conditions is given

by the expressions v, = ^ __i__; P = h'-p;

where /?' is the tension of aqueous vapor, at t^ C, in cen-
timetres of mercury, and h' is the '* corrected " height of
the barometer. A unit current in one second liberates
1.156 cubic centimetres of hydrogen, and therefore 0.578
cubic centimetres of oxygen under standard conditions.
Hence, if it is a single tube voltameter, the volume of the
combined gases liberated by a current in T seconds under
standard conditions must be

Bat

EXPEIUMENTS IN £LECTRIGITT AND MAGNETISM 407
*^^"" 76 1 +0.003665 r

76 X 1.734 X T{1 + 0.003665 t)

MoreoTer, « = j^ tan ^.
Consequently,

0 76x1.734x7(1 + 0.0036650 . a

If the voltameter is a double-tube one, the volume of the
hydrogen under standard conditions liberated by a current
t in T seconds must be

t;o= 1.156 tT.
If V is the observed volume of the hydrogen, it follows
at once that

g _ 76 X 1.156 X T{\ + 0.003665 f) ^ .

(Further, if V is the volume of the oxygen, Fshould equal

If the tangent galvanometer is a standard one, 0 may be

calculated, because it equals , where n is the number

of turns of wire and r is the radius of the coil, both meas-
urable quantities. In this case H may be determined. If
U is known, O may be calculated at once.

ILLUSTRATION May % 1897

Galtanomitkr Constant bt Gas Voltametkr

Observed cubic centimetres of hydrogen, 47.5, 18.2° C , 77.2 cm.

Corrected, 44.24.

Time, 7 m. 62 sec. = 472 sec.

44 24
/. t =- ^^^ — = 0.0810.
472x1.156

3, mean of ten observations, 86.4°.

E L-.= -^^^^_ -^^^15-0 110

C" tan a ~ tan 36'4° ~ 0.7373 "" *

408 A MANUAL OF EXPERIMENTS IN PHYSICS

Qaeationa and Problems.

1. What are Faraday's and Joule's laws concerning electrolyteif

2. Calculate the value of i used in your experiment, in amperes

and also in the electrostatic system.

3. Is 6^ a simple number, or what is it ?

4. Does the tension of the vapor above the liquid in the barelte

become greater or less us the concentration of the solutioD
is increased ?

5. From a knowledge of the electro - chemical equivalent of

hydrogen and the heat of combination of hydrogen and
oxygen calculate how much energy is required to carry a
unit quantity of electricity through the water voltameter.
What is the least E. M. F. which will enable a current to
pass through the voltameter ?

6. If the current in the above experiment had been farnished

by DanieU's cells, how much zinc would have been dis-
solved ? How much copper would have been deposited?

EXPERIMENT 76

Olgect. To determine G or Hhy the deposition of cop-
per. (See " Physics/' Arts. 246, 278.)

Copper Voltameter. A copper voltameter consists of a so-
lution of copper-salt in water, into which dip an anode and
cathode of copper. When a current is passed through,
copper is deposited on the cathode, and it increases in
weight while the anode decreases. (These two changes
should be equal and opposite.) It has been found by ex-
periment that the electro - chemical equivalent of copper
(from copper sulphate) is 0.003261; and, therefore, if the
increase in weight, m, of the cathode is known while a cur-
rent i has passed for t seconds,

m = 0. 003261 i/,
and so i may be determined.

BooroM of Brror.

1. Copper dissolves in copper sulphate unless special precautions

are taken.
8. There must be no leakage across the top of the instrument.
8. The commutator must be clean.
4. The current must remain steady.

Apparatus. Tangent galvanometer; commutator; iron
coil resistance - box ; sliding resistance to
keep the current constant, either wire or ^^^r"^,^^
liquid; storage cells; wires; copper vol- f Irn^n
tameter with an extra plate ; glass jar.

The voltameter consists, essentially, of
an ebonite top which bears three copper-
plates dipping in a glass jar containing the i.^o. 149

LIU

410 A MANUAL OF EZPERIMENTS IN PHTSIG8

electrolyte. The two outside plates are connected by a cop-
per strip, but the inner one is insulated from them and
can be screwed in or out of the cap. This inner plate is
always made the cathode ; the outer ones the anode.

Manipulation. Make a clean solution of the following
constituents in the proportion : 100 grams of water^ 15
grams of copper sulphate (CuSO^), 5 grams of snlphuric
acid, and 5 grams of alcohol. About 600 cubic centimetres
of water will make enough of the solution to fill the jar to
a convenient depth. (Alcohol is easily oxidized, so that it
annuls the action of certain secondary products which form
at the anode and tend to polarize the cell.) If the copper
sulphate crystals are at all large, crush them to powder, so
that the solvent may act more readily upon them. Next
clean the cathode with extreme care by rubbing it with
emery or fine sand-paper until every portion of the plate is
as bright as possible. Never touch the clean surface of the
plate with the bare fingers, but hold it by the screw at its
upper end, or grasp it with a piece of clean paper. Wash
the cathode by allowing water from a tap to flow over its
surface, and dry thoroughly by pouring alcohol over ii
Weigh the plate as accurately as possible on a convenient
balance; record its mass; and keep it wrapped in clean
paper until needed. In like manner the anode plates and
the trial cathode plate must be polished and washed (not
weighed, however), but for them such care as was bestowed
upon the weighed plate is unnecessary. Screw the trial
plate between the anode plates to their support, and fix it
roughly parallel to them. Ascertain the direction of flow
of the positive current by putting the ends of the battery
circuit wires in the copper sulphate solution, and note
which one becomes brighter by having metallic copper de-
posited upon it. This is the negative pole of the battery,
and must be joined (directly or indirectly) to the cathode
binding-post. Adjust the galvanometer in the magnetic
meridian, immerse the three plates in the solution, join
the apparatus, as shown in the diagram, and start the CQ^

EXPERIMENTS IN ELECTRICITT AND MAGNETISM 411

rent. Regulate the resietancej R, so that the deflections
of the galvanometer in either direction are equal to about
30^. It is best to have the E. M. F. of the storage battery
circuit such that, when the preceding conditions are fol-

«(/-.

'i^i^V^S^ — |lff

Fiaiso

filled^ the total metallic resistance of the circuit is, at least,
equal to ten ohms. Break the circuit by means of the com-
mutator, C, and substitute the weighed cathode for the
trial one, remembering never to touch it at any point which
will be beneath the surface of the liquid in the voltameter.
Make the circuit, noting both the exact second at which the
current starts to flow and the resulting deflection of the
galvanometer needle. Reverse the current through the
coil as rapidly as possible, and note the deflection. Repeat
the process of reading pairs of opposite deflections every
few minutes — e. g., five, always noting both ends of the in-
dex. Keep the current accurately constant by means of
the sliding resistance. If, at any time, it changes very
much, the experiment must be begun anew; otherwise
continue the readings for at least an hour and a half. Let
the deflection noted last be in the opposite direction to the
first, so as to have equal numbers of left and right deflec-
tions. Finally, break the circuit by removing the commu-
tator, noting the precise second at which this is done, say
t seconds later than the instant of starting the current.
Carefully remove the cathode plate from the voltameter;
wash it thoroughly by allowing water to flow gently over
its surface, and dry it by using alcohol as explained above.
Once more weigh this plate as accurately as possible, and
let its gain in mass be m grams.

A unit current (G. O. S. electromagnetic sjrstem) de-
posits 0.003261 grams of copper in one second. Hence,

418

A MANUAL OF KXPERIMEMTS IN PHYSICS

the mass of copper electrolyzed by a currant i in t secondB
is equal to 0.003261 it = m grams. Substitute in the equa-
tion the experimental values of m and t, and determine t.

For the tangent galvanometer t = 7^ tan S C. G. 8. elec-
tromagnetic units. Finally, therefore, substitute for tan ^
and i their values, and determine H/G.

1. To find 0, obtain the value of H from an instructor,
and substitute it in tlie value of H/O.

2. To find H, calculate 0 from the dimensions of the
galvanometer coil, if possible, and employ the above rela-
tion between O and H to obtain the latter.

ILLUSTRATION

Oalyanomitib Cokstant

By Copper VoKameier

Mass of cathode, 4b. 5m. 30b., 59.885 grama.

•• " " 4h. 56m. 80a., 60.985 "

51in. 1650 grama.

== 0. 165 CG.S. units.

UMji^vun

1.65

000826 X 61 X 60

ChUvanameUf Deflectiong

Time

Current Direct

Cttrrent BeTeiaed

N. PoJe

S.M»

N.Pole

apoie

a M. a

Degrees

Degraee

Degrn

4 5 80

51.7

51.8

51.6

51.7

4 11 80

51.6

51.7

51.6 1 51.7

4 16 80

51.8

51.9

51.7 51.8

4 21 30

51.9

52.0

51.8 1 51.9

4 26 30

51.7

51.8

51.6

51.7

4 31 30

51.6

51.7

51.5

51.6

4 86 80

51.5

51.6

51.4

51.6

4 41 80

51.7

51.6

4 46 80

51.8

51.9

51.7

51.8

4 51 80

51.6

51.7

51.5

51.6

4 56 80

61.7

51.8

51.6

51.7

....

51.7

51.8

61.6

61.8

Mmu, 5i.r>.

KXfKHlMENTS IN ELECTRICITY AND MAGNETISM 418

Hence. ?=-i- = -5i«- = M» =0.108. Butff=0.19«.
xiBiKx. Qt - ton » tan Ol.T" 1.2668

Questions and Problems.

1. What assumptioDB are made to deduce the formula t s= ^ tan ^?

2. What is the advantage in having ttoo anode plates and one caih>

ode plate ?
8. Explain how a compass-needle could be used to find the direc-
tion of flow of a positive current, If the latter were strong
enough.

4. Express the current in your experiment in amperes, and also

in the electroHtatic system.

5. From a knowledge of the heats of combination of zinc sul-

phate and copper sulphate calculate the E. M. F. of a
Danieirs cell.

EXPERIMENT 77

Olgect To determine the mechanical equivalent of heat
by means of the heating effect of an electric current.

General Theory. If ^ 0. O. S. units is the difference of
potential between the ends of a wire carrying a current t
C. G. S. units, the amount of energy necessary to maintain
the current in the wire for one second is Ei ergs. If the

resistance of the wire is R C. G. S. units, t =— or EzziS;

and hence the expenditure of energy per second is x^R
ergs. If the coil is stationary and enclosed in a calorim-
eter filled with water, this energy must all be expended in
heating the calorimeter and its contents, with the excep-
tion of the energy which escapes by radiation. If the
water-equivalent of the calorimeter, stirrer, coil, and ther-
mometer is a, if it contains m grams of water, and if it be
heated from t^ to t^ degrees while the current is flowing,
the energy received by it in heat is {m •{■ a){t^^ t^ c&'
lories.

By the mechanical equivalent of heat is meant the ratio
between the ordinary mechanical unit of energy, the erg,
and the heat unit of energy, the calorie. Calling this ratio
/, the definition may be stated thus : When one calorie
is converted into mechanical energy it becomes J ergs.
(See " Physics,'' Art. 180.)

If suitable precautions are taken, so that no energy
escapes from the calorimeter, and none is receiyed by
it except from the electric current in the coil, and if
the above change in temperature of the calorimeter and
its contents takes place in t seconds, {m + a)(t2—t\)

EXPERIMENTS IN ELECTRICITY AND MAGNETISM 416

caloriee = PRt ergs, or, reducing both sides to ergs,
{m+a){t^^t,)J=i'Rt. Whence J = .

The general method of the experiment is therefore as
follows : A current is passed through a suitable coil of fine
German-silver wire, enclosed in a calorimeter filled with
water, for a time t sec, which is carefully noted, as is also
the rise in temperature, t^^ — ti^, of the calorimeter and
its contents. The current is measured by including in the
circuit a copper voltameter, just as in the preceding ex-
periment. It must be noted, however, that it is even
more important in this case to keep the current constant,
for the heating effect is proportional to i^, whereas the
copper deposited is proportional to t, and it is a mathe-
matical fact that the mean of the squares of the successive
values of a quantity which varies is not equal to the square
of the mean of these values. For this reason some obvious
indicator, such as a galvanometer, must also be included
in the circuit. The constant of the galvanometer need,
however, be known only approximately, as it is used mere-
ly to show whether the current remains constant and for
one other purpose, to be indicated further on. The actual
measure of the current is obtained from the copper vol-
tameter. In order to aid in keeping the current exactly
constant, a sliding resistance is also placed in the circuit.

The water-equivalent of the calorimeter and the weight
of water it contains are determined as is usual in heat
experiments. (See Experiment 49.) The resistance of
the coil is determined by a Wheatstone bridge. In order
to insure that any possible leakage of the current through
the water may be taken into account, the resistance should
be measured with the coil in water up to the level used in
the experiment. Error from this source is further guarded
against by making use of the fact that water cannot be
electrolyzed unless there is a difference of potential of over
1.6 volts between the terminals which dip into it. Hence,
by using a coil of very low resistance, but of fine enough

416 A MANUAL OF EXPERIMENTS IN PHYSICS

wire to be heated by a low current, the experiment can be
80 arranged that there need not be 1,6 volts' difference of
potential anywhere inside the calorimeter. A coil whose
resistance is about 0.9 ohms, and a current of 1.5 amperes
— not more— will give very good results. For thia reason,
also, the constant of the galvanometer has to be roughly
known in order that the current may not be allowed to ex-
ceed this amount.

Sources of Brror.
1. The very low resistance is hard to determine accurately.
3. The usual errors iu using a copper Yoltameter and in all heat
experiments have to be guarded against.

Apparatus. A small and large calorimeter as described
in Experiment 49, with a stirrer for the smaller one. The
coil of wire is a loosely wound spiral of 82 centimetres of
No. 22 Qerman - silver wire, soldered to two stout copper
terminals which pass through the cover of the small calo-
rimeter, and which end in binding -posts. A thermome-
ter ; copper voltameter ; an iron wire resistance ; a sliding
resistance ; key suitable for a heavy current ; the tangent
galvanometer of the previous experiment (any other in-
strument which will indicate the approximate value of
the current will answer) ; a storage-battery circuit giving
an E. M. F. of 10 or 12 volts ; distilled water ; a watch.
During the early part of the experiment a Wheatstone
bridge (either slide wire or post-office box), key, battery,
mirror -galvanometer, and a ^'wire connector" are also
needed.

Manipulation. Place in a beaker on ice enough distilled
water to more than fill the small calorimeter. Garefnlly
dry the small calorimeter and stirrer, and weigh them with-
out the top. Fill the calorimeter with distilled water to
such a depth that, when the cover is put on, the coil will
be entirely under water. Insert the coil, press the cover
down, attach it in a Wheatstone bridge by short, very stout
wires, and determine the resistance to within 0.01 ohm.

EXPERIMENTS IN ELECTRICITY AND MAGNETISM

417

Fio. 161

Disconnect the lead-wires from the coil, connect them to
each other tightly by means of the wire connector, and thus
determine the resistance of the lead-wires themselves, and
subtract it from the resistance just found so as to get the
resistance of the coil alone. Clean and weigh the cathode
plate of the voltameter. Set up the apparatus as shown
in the diagram, with
the trial plate of the
voltameter in place.
B is the storage-bat-
tery; R is an iron
wire resistance; S is
the sliding resist-
ance; G is the tan-
gent galvanometer;
C is the calorimeter
and coil; K is the
key ; and V is the voltameter. Close the circuit and regulate
the resistance, R, so that the galvanometer shows approxi-
mately 1.5 amperes, and not over. (The constant of the
instrument should be obtained from the instructor unless
already determined in a previous experiment.) Break the
eircait, remove the trial plate, and note whether as set up
it was the cathode ; and, if not, change the connections so
that the middle plate will be the cathode.

Note the temperature of the room near the place of ex-
periment. Empty the water in the calorimeter and fill
with distilled water at a temperature about 12 degrees lower
than that of the air, up to the same mark as before. The
correct temperature can be obtained by mixing distilled
water from the regular laboratory supply with that cooled
for the purpose. Weigh carefully. Dry and replace the coil
and cover, being careful to see that its spirals do not touch
one another. Connect the coil in the circuit as before,
and put in the weighed plate of the voltameter. Stir the
water and read the temperature of the thermometer placed
inside the calorimeter; close the key and note the exact

418 A MANUAL OF EXPERIMENTS IN FHYSIG8

time. j^^tV continuously, and watch the galyanometer care-
fully, keeping the needle absolutely in the same position
throughout the experiment by regulating the current with
the aid of the sliding resistance. Bead the thermometer
from time to time ; and, as it approaches a temperature as
much above that of the room as the initial temperature was
below, prepare to break the current. When the exact tem-
perature is reached, break the current, and read the time
and the temperature exactly simultaneously. Wash, dry,
and weigh the cathode plate of the voltameter. From the
data calculate J, remembering that 1 ohm = 10* C. 6. S.
units. With a coil whose resistance is 0.9 ohm and a cur-
rent of 1.5 amperes, the temperature of 100 cubic centi-
metres of water will be raised about 20^ C. in a little over
an hour, which gives an idea of the time the current mast
be on.

ILLUSTRATION iityia,l»6

Mechanical Equitalknt or Hkat
Mass of calorimeter and stirrer =09.68 grama

*• ** " full of water and stirrer = 189.12 "

Mass of water = m = 09.44 grama
Water-equivalent of calorimeter :

a = 69.68x 0.095= 6.flS
.*. m + a = 76.06

Resistance of coil and leads = 0.984 ohms.
•• leads alone =0.084 "

*• coil. =0.900 ohDia.

.-. iJ = 0.9xlO*C. G. 8. units.
Temperature of room, 19. 4^
Current was made at 8b. 25m. Os. Temperature of water, 9.V

" broken at 5h. Im. Os. ** " " 29T

Time current was on, Ih. 86m. Os. = 57608. locreaBe, S0.5'

Weight of vollameter plate after, 61.284 granas.
" " •• " before, 59.180 "

Copper deposited = 2.104 grams.

A The current i = -—^^-—^OAl^ C. G. 8. units.
0.00826x5760

. ,_(0.1125)«x.9xl0*x5760_,^,^,^, 0
•• •^- 76.06x20.5 -^.-Sixiu. .

EXPERIMENTS IN ELEOTRICITT AND MAGNETISM 419

Questioiia and ProblemB.

1. What is the mean activity of the current in the coil during

the experiment ?

2. The resistance of an incandescent lamp filament is 56 ohms,

and the £. M. F. between its ends is 110 volts (both when
the lamp is lit); how many such lamps can be lit by a
dynamo giving 10 horse-power, assuming 90^ of the energy
to be expended in the lamps ?

EXPERIMENT 78

(TWO OBBBBYBRS ABB BBQUIBBD)

Olject. To determine, by means of an ''earth indnc-
tor/'the inclination of the magnetic force of the earth to
the horizontal, usually called the "Magnetic Dip." '(See
''Physics/' Art. 286.)

General Theory. An "earth inductor ''consists essential-
ly of a coil of wire capable of rotation through 180° aboat
an axis in its own plane. The axis may be' made either
vertical or horizontal at pleasure, since the bearings in

which the axle turns are
set in a frame which may
be attached to a fixed
stand in either position.

Consider the coil to be
placed first- in a vertical
plane perpendicular to the
magnetic meridian, and
with the axis of rotation
truly vertical. The result-
ant magnetic force of the
earth may be represented
by a number of straight
lines threading the coil at
an angle of ^° to the hori-
zontal, ^ being the magnetic dip. If F is the magnitude
of this resultant, there are said to be F " lines of magnetic
force '' passing through each square centimetre of a plane
at right angles to the direction of the force; and, there-

EXPERIMENTS IN ELECTRICITY AND MAGNETISM

421

fore, F cos d pass through each square centimetre of a
vertical plane at right angles to the magnetic meridian.
Hence, if A is the average area of a turn of
the coil, the number of lines threading it is
iV= J/'cosS.

Now rotate the coil through 180°. There are
A' lines of magnetic force passing through it
as before, but in an opposite direction rela-
tively to the coil. Hence, each turn of the
coil has cut through J^iV lines of force ; and,
if there are n turns in the coil, the circuit of
which it is a part has had the number of lines of force
threading it in the original direction changed by
2iVw = 2;i^/'cos d.

Experiments show that while the number of lines of
force passing through a closed circuit is changing a cur-
rent flows in the wire such that, if Q be the total quantity
of electricity which flows around the circuit while the cur-
rent lasts, and JV,— Nq the change in the number of lines.

Fia. 153

e =

y where R is the resistance of the entire circuit.

Hence, if the coil of the earth inductor be made part of
a closed circuit, the quantity of electricity which will pass
around it when turned through ISO'', as described above,
will be,

^ 2nAFoo%d

^>= R

If, now, the frame be turned so that the coil lies in a hori-
zontal plane with the axis in a horizontal line in the mag-
netic meridian, AF^xn^ lines will thread it. Hence, when
it is rotated 180^ the quantity of electricity passing around
the circuit will be,

^^" R

/. tand = |?-

422 A MANUAL OF EXPERIMENTS IN PHYSICS

Hence, to determine d, we hare to measure the ratio of

f^y which is done, as in Experiment 74, by inciading a
Vi

ballistic galvanometer in series with the coil.

The conditions for measuring a quantity of electricity
by a ballistic galvanometer require that the entire quan-
tity shall pass before the needle moves from rest appreci-
ably. This is secured by making the revolution as rapid
as possible.

Sources of Brror.

Accuracy in this experiment requires:

1. That the coil when in a yertical position be rotated through
180°, from a positioo in a plane exactly |>erpendicuhir to
the magnetic meridian into the reverse position in the same
plane.

3. That tlie coil when in a horizontal position be rotated through

180°, from one position in a plane exactly horizontal into the
reverse position in the same plane.
8. That in each case the revolution be completed before the gal-
vanometer needle starts appreciably from rest.

4. That the observations in the two positions follow each other

before any change can take place in the sensitiveness of the
galvanometer or strength of its magnetic field.

5. That the resistance of the circuit remain the same.

Apparatus. An earth inductor; ballistic galvanometer;
damping-circuit, with its proper key and battery ; spirit-
level and L-square^ compass; clamp -stand and piece of
card-board ; plane mirror.

Manipulation. Set up the earth inductor on a firm table,
with the frame vertical and screwed tight to the fixed
stand, so that the coil, when it rests against one of the
metal stops which limit its rotation, is, as closely as can
be told by the eye, perpendicular to a horizontal north-
and -south line. Make the plane of the coil accurately
vertical by means of the levelling-screws, testing it with
the spirit-level and L-square. Fasten a plane mirror flat
against the face of the brass bobbin on which the wire is
wound, if there is not already one on the instrument

EXPERIMENTS IN ELEGTRICITT AND MAGNETISM 428

Cut a narrow slit in the card-board, and by the aid of the
compass place the slit vertical, due north in a horizontal
line from the piece of mirror and eight or ten feet away.
Place a light behind the slit either at its upper or lower
half, and look through the other half of the slit at the
mirror in a line directly parallel to the needle of the com-
pass, which should be placed between. (The line of the
needle may be magnified by two large pins placed upright
due north and south of it.) Now adjust the plane of the
coil exactly so that when it rests against its stops the bright
slit is reflected back to itself, and hence to the eye placed
behind it. Now place the mirror flat against the opposite
face of the coil, and rotate the latter on its axis until it
stops against the other screw which limits its motion.
View the mirror again through the slit along the compass-
needle, and turn the limiting-screw in or out until the coil
is stopped with its plane exactly perpendicular to the north-
and-south horizontal line.

The coil is now adjusted for the first set of observations.
Connect it in series with the ballistic galvanometer, twist-
ing the lead-wires around one another so as to include as
little area between them as possible outside of the coil.

One observer now prepares to read the galvanometer,
which he does exactly as in Experiment 74, giving a sig-
nal to the other observer each time when he has noted the
zero and when he is ready to note the throw. At the sig-
nal, the other observer rotates the handle as quickly as he
can without danger of jarring the instrument out of ad-
justment, and stops it exactly in contact with the screw
adjusted for that purpose, lie holds it firmly there while
the first observer notes the throw, and also the second
swing in the same direction, as in Experiment 74. The
swing of the needle should now be damped ; and when the
first observer is again ready another signal is given and
the coil turned back in the reverse direction, through 180**
exactly. Take twenty readings in this way.

Now unfasten the frame from the stand and revolve it

424

A MANUAL OF EXPERIMENTS IX PHYSICS

nntil it is in a horizontal position. With the coil tight
against one of the stops place a leyel on its upper face^ and
level it by the screws on the stand without touching either
stop, as you have already carefully adjusted them so as to
allow a revolution of exactly 180^. Reverse the coil and
test the other face, which should also be perfectly leTel
unless the stops have been jarred out of place. When the
instrument is adjusted, take twenty observations in this
position just as in th6 other.

ILLUSTRATION
Maonkhc Dip

April 10, 1»7

Axis vertical

Axis horitarUdl

Deflecttons

DeflectioDg

Left
3.62

Right
2.70

Left
7.85

Right
780

2.70

2.67

7.80

2.70

2.65

7.40

2.60

7.85

2.65

2.66

Mean. 7.85

7.85

Mean. 2.66

2.64

Mean, 7.85

Mean, 2.65

.-.^ = 70^0'.

EXPERIMENTS IN LIGHT

INTRODUCTION TO LIGHT

UnitB and Definitions. As is explained in treatises on
Physics^ light is a sensation due to waves in the ether ;
and the only quantities which can be measured in abso-
lute units are those which involye length and time. All
lengths should be measured in centimetres ; but, since the
wave-lengths of those ether waves which produce vision
are extremely minute, they are generally expressed in a
certain fraction of a centimetre which is called an Ang-
strom unit. This unit is 0.00000001 centimetre, or lO-^
centimetres ; thus, the wave-length of a certain line in the
spectrum of sodium, called />], is 5896.357 Angstr5m units.
A table of standard wave-lengths will be found in the Tables.

Otgect of Bxperiments. The experiments in this section
have two objects : 1. To verify certain of the laws of ether
waves — e. g., laws of reflection and refraction. 2. To meas-
ure certain quantities, such as the indices of refraction of
certain solids, the focal lengths of lenses, the wave-length
of light

EXPERIMENT 79

(THIB EZPBBIICBNT SHOULD BU UADB IN A DABKBNJBD BOOM)

•

Object To compare the intensities of illamin'ation of
two lights by means of a Joly photometer. (See '^Phys-
ic8,"Art. 297.)

General Theory. A Joly photometer consists of two small
rectangular blocks of paraflSne, about 4 centimetres square
and 1 centimetre thick, placed side by side, so as to form a
block 2 centimetres thick. If the intensity of the illumina-
tion on one of the faces is more than that on the other, the
fact is manifest by a difference in brightness of the paraffine
on the two sides of the separating surface; whereas, if the
intensity is the same, there is no difference in brightness.

The method, then, is to place the two sources of light,
one on each side of the paraffine block, and, keeping one
in a fixed positron, alter that of the other until there is no
difference in brightness of the two halves. If the distances
of the two sources from the photometer are r| and r^ and
if the illuminating powers are /| and /,, then

For, since the intensity of light varies inversely as the square
of the distance, and since in this case the intensity at the
photometer due to each source of light is the same, the il-
luminating power of the sources must vary directly as the
square of the distance.

Soarces of Brror.

1. The two halves of the paraffine photometer may not be alike
in quality or thickness.

EXPERIMENTS IN LIGHT

429

2. The illuminating power of the two sources must be kept
constant.

8. If the lights are of different colors, there is difficulty in com-
paring their intensities.

4 All extraneous light must be kept from reaching the parafflne
blocks.

Apparatus. A Joly photometer; a fish-tail barner; a
candle ; two tin screens, with equal rectangular openings ;
a metre-rod.

ManipnlatioiL Mount the photometer, the candle, and the
gas-flame so that they are in the same horizontal line, the
photometer being between the two lights, with the di-

FICWIM

viding surface perpendicular to the line joining them.
Place the two lights about 80 centimetres apart, and hold,
by means of suitable clamps, a tin screen immediately in
front of each light, on the side towards the photometer, at
such a height that the opening exposes equal areas of the
brightest part of each flame to the photometer.

View the photometer from the end, so as to see both
the halves equally well. Move it nearer one of the lights,
and, observing the effect, place it finally in such a position
that the illuminations of the two halves are apparently
equal. Measure the distances from the openings in the
tin screens to the plane of separation of the two halves.

Displace the photometer, and repeat the observation and
measurements. Turn the photometer 180° around a verti-
cal axis, and repeat the experiment, making three observa-
tions in each position. Gall the means r, and r^*

Place the two lights at a different distance apart, and re-
peat the experiment as above. Gall the means E^ and J^.

480

A MANUAL OF EXPERIMENTS IN PHYSICS

The illuminating powers, /j and /,, should then satisfy
the two equations,

E,t'

Calculate 1^1 1% from the two equations.

ILLUSTRATION ^„.

Comparison of Edison Lamp and 6as-Flamb

Ri B,

88.0

43.0

22.0 28.0

88.6

41.5

22.6 27.6

88.6

41.6

22.6 27.6

Mean. 88.8

41.7

Mean, 22.8 27.7

= 1.66.

Mean

1.66.

QaeBtionB and Problema.

1. Describe a pbotometer wbich depends upon the equality of

two penumbras. •

2. Why are the measurements taken to the opening in the scxeeu,

and not to the flames themselves ?

3. What diflSculties are introduced by the flames baving different

colors ?

4. Discuss the conditions for a standard of illumination.

EXPERIMENT 80

Otgect To verify the laws of reflection from a plane
mirror. (Bee " Physics," Arts. 306, 308.)

General Theory. There are two cases to be considered —
(1) reflection of plane waves, (2) reflection of spherical
waves.

1. Plane Waves. — The laws, as deduced from the wave-
theory, are that the reflected waves are plane, and that
their normal lies in the same plane as the normals to the
Borface and to the incident waves ; also, the angles made
with the normal to the surface by the normals to the inci-
dent and reflected waves are equal.

2. Spherical Waves. — The law to be verifled is that the
reflected waves are spherical, seeming to come from a
centre on the opposite side of the surface from the source
of waves, the centres of the incident and reflected waves
being in a straight line perpendicular to the surface, and
at equal distances from it.

(Read " Physics,'' Art. 302.)

As waves of any kind, plane or curved, advance in any
direction in an isotropic medium, such as air or water, the
disturbance at any point, P, of the wave front, as it exists
at a definite instant, produces disturbances at later times
at points Pj, P2, ©tc,
where P, Pi, P^ etc., ""

all lie in a line normal
to the wave front. This
is called the rectilinear
propagation of light,
and can be verifled by

-r

I
I

■^

FUklM

432 A MANUAL OF EXPERIMENTS IN PHYSICS

any shadow experiment. To trace a particular line of
disturbances (called a "ray'*), the following method may
be adopted: Make P a centre of disturbances {e, g.y a point
of light, a pin brightly illuminated) ; place a small ob-
stacle, such as a pin, at a random point, P^, then there

will be no disturbance at a
point Pj, in the line PP^
because the line of dis-
turbance— the ray PPx—
has been stopped at Py
Consequently, if an ob-
server moves his eye in
Fro^iM such a direction as to

make the pin at P^ hide the bright point at P, he will be
sighting along the line PP^P^ — i. «., he will be tracing the
ray PP,P,.

The fundamental property of a plane wave is that all its
** rays'* are parallel ; therefore, to trace the progress of a
series of plane waves, it is sufficient to follow the path of
any one ray.

In studying the reflection of plane waves, then, the gen-
eral plan is to follow the path of the waves by tracing the
progress of a ray, both be-
fore and after reflection.
This is done by placing
two pins so that the line
joining them falls oblique-
ly upon the mirror ; then,
looking at the reflected

images, moving the eye so that the pin farthest away is
hidden by the other, and fixing this direction of sight, by
two other pins. The two pairs of pins give the directions
of the normals to the incident and reflected plane wares
by tracing the path of one particular ray.

In the case of spherical waves (see Pig. 168) proceeding
from a source Q (e. ^., a pin brightly illuminated), the centre
of the reflected v^aves, 0', is called a "virtual*' image; and

KXFISKIMKNTS IK UGHT

488

its position may be aconrately located by using a transparent

mirror^ snch as a pieoe of glass^ or by scratching off a small

horizontal slit from the silvering of an ordinary mirror, for

a pin may be moved

around back of the

mirror until it is

exaotly in the spot

where the virtual

image seems to be.

This. agreement

may be tested by

the pin and the im-
age-^'. 0. ^ that^when
the eye is moved
sidewise, the two
remain coincident. fio.i58

Sonroea of Brror.

1. The mirror may not be plane.

%. There may be more than one reflecdng surfaoe, and so con-
fusion may arise.

8. The illuminated bodies must be sharp and distinct.

Apparatus. A small rectangular block of wood ; plane
silvered mirror, with a horizontal slit removed ; a drawing-
board ; paper ; pins. (A plane glass mirror will do in
place of the silvered one.)

KaniptllatioiL Place the drawing-board on a table, and a
piece of paper on the board ; draw a straight line across the

middle of the paper, and place the
mirror so that the reflecting sur-
face coincides with this line and
is perpendicular to the board.
This may be done by fastening
the mirror to a rectangular block
of wood by means of rubber bands,
no. ifef and then placing the block suita*

484

A MANUAL OF EXPERIMENTS IN PHYSICS

Fl0.159a

bly. Place two pins vertical in the board at such dig-
tances that the line joining them falls obliquely upon the
mirror.

1. By means of two other pins locate a line such that,

when looking along it at the
reflection of the first pair of
pins, the image of one hides
that of the other. Motc the
mirror one side, and draw
two straight lines, one pass-
ing through each pair of pins.
These should intersect at a
point of the base-line which

\p marks the reflecting surface.
Prom this point draw a line
perpendicular to the base-line,
and compare the angles made
with this line by the two lines
through the pins. This may be done by drawing around
the point of intersection a circle of radius 10 or 15 centim-
etres, and comparing the lengths of the intersected arcs.
The angle of incidence should equal the angle of reflec-
tion.

2. Replace the mirror in its position along the base-line,
taking care to place the mirror so that a transparent portion
projects beyond the block and is met by normals dropped
from the two pins. Locate by means of two pins the vir-
tual images of the first two pins mentioned in part 1. Do
this very carefully, taking pains to avoid all parallax be-
tween a pin and a virtual image. It is best to make the
points of the pins coincide, because the smaller the ob-
jects are, the better can their coincidence be determined.

In order to determine the position of the virtual image,
it is not absolutely necessary to have a transparent por-
tion of the mirror; for if a pin is placed parallel to the
length of the mirror, with its point in a normal to the
mirror at its edge, its image will just reach to the edge of

FiCk10O

EXPERIMENTS IK LIGHT 486

the minror, and a Becond pin may be so placed (behind the
mirror) as to seem to form an unbroken line with the image,
when yiewed from a . ^

point in the normal to
the mirror at its edge.

Having fixed the po- ,^^^^y^^
sitions of the two im-
ages> draw a straight
line joining them. The
two images shonld be
at the same distances
back of the mirror as the original two pins are in front of
it ; and the line joining the two images should be the pro-
longation, backward, of the reflected ray.

Repeat both experiments, using different angles of inci-
dence.

(Methods will be given later for testing the mirror, to
see if it is plane. See p. 441.)

Qaestlona and Problems.

1. What would be the effect in the two experiments if the mirror

had been concave or convex ? Bhow by drawings.

2. Eh>w may the tbiciiness of a piece of glass be determined by

means of images formed by reflection from its two surfaces?

8. Give the drawings for the images formed by two plane mir-
rors inclined to each other at an angle of 46^

4 What is the smallest plane mirror in which a man may see
his entire figure?

S, Bhow by graphical construction the reflection from a plane
mirror of flpherical waves winch are converging apparently
to a point at a distance h behind the mirror.

EXPERIMENT 81
(this rxpbrtmknt should bb made in a dabksnkd boom)

Olgeot. To verify the laws of reflection from a spherical
mirror, (See '' Physics/* Arts. 311-317.)

(General Theory. For any spherical mirror, concaye or
convex, and for any train of waves, converging, diverg-
ing, or plane, the same formula applies with a proper
understanding as to the signs. It is this: d-|-C=
20, if Ci is the curvature of the incident waves, C, the
curvature of the reflected waves, C the curvature of the
mirror; the quantities, Ci and 6', have a + or — sign ac-
cording as the centres of the waves are on the same
side of the mirror as its centre of curvature or on the
opposite.

Expressed in terms of distances from the mirror to
the centres of the spherical surfaces this formula be-
comes

1 1_2

where u is the distance from the mirror to the centre of
the incident waves, v is the distance from the mirror to
the centre of the reflected waves, r is the distance from
the mirror to its centre of curvature, with the same
understanding as to the signs of u and v as before for
Ci and Cv.

This formula leads to a graphical construction for im-
ages, which is here illustrated by two •

KXPmrMKNTS IN UGHT

487

Fia 161

rick im

where S is the centre of the imrror, VS^r,
OP is the object, ^ = w

O'P' is the image, ^ssr.

It will be noticed from the formnla,

— + — — — >
u V r

1. If w =2co» ». €.y if plane waves are lit^ident, v=r/2.
This point, F, shown above, is called tne principal focus,

2. If snrface is concave,

t; is + > i' e.y there is a real image, if u >?,
V is — , i. 6., there is a virtual image, if u<x»

4d8 A liANUAL OF £XP£EIli£NTS IK PHYSICS

3. It surface is convex,

V is always + , i. e., there is always a yirtaal image, bo
long as the waves are diverging from a real object — i. e.,
so long as u is negative.

If w is+, t. e., if the waves are converging, v may be—,
and so the image may be real.

To verify these laws of reflection, a brightly illnminated
object — e. g., a pin — is placed in front of the mirror, so that
a line passing through it and the centre of curvature of the
spherical surface will intersect the mirror near its middle
point ; the position of the image, virtual or real, is found
by moving a second pin until there is no parallax between
it and the image of the first — i. e., until they do not shift
relatively to one another when the eye is moved sidewise
while looking at them. They then evidently occupy ex-
actly the same position, and neither is behind nor in front
of the other. In finding the position of a virtual image, it
is necessary, of course, to have a strip of the silvering re-
moved from the mirror so as to have a transparent portion.*

A real image may also be located by making the re-
flected waves fall upon a screen, and moving the screen
until the image is as sharp as possible. By having an ob-
ject of a known size the magnification may be thus deter-
mined ; for the size of the image may also be measured*

BoQxoeB of Bnor.

1. UDless the incideDce is normal — i. e,, unless the line joiniog
the centre of curvature to the source of the light meets the
central portion of the mirror— the above laws do not hold.

8. The object must have a sharp outline, so as to admit of aoca-
rate focusing.

Apparatus. A concave and a convex mirror; a gas-
burner ; pins or needles, with suitable stands ; metre-rod.

Manipulation. Adjust the mirror so that the line joining
its middle point to its centre of curvature is horizontal and
so that the transparent slit is horizontal.

* This plan was saggested by Mr. Wilberforce, of the Gaveudisb Laboca*
torji Oambrldge^ England.

EXPERIMENTS IN LIGHT 489

1. Beat Image. Concave Mirror. — Get an inverted re-
flection of the gas-flame in the concave mirror by moving
the flame backward or forward, and locate the image ap-
proximately by means of a piece of paper used as a screen.

This flxes the fact that the flame is beyond the principal
focns {u > r/2). Place the pin in its stand near where the
flamd is, and move the flame one side so as to illuminate it.
Adjnst the pin quite accurately, so that its extremity lies
in the horizontal line joining the centre of curvature of
the mirror to its middle point. Place the other pin (or
needle) in a stand, at the same level as the flrst, and move
it until its extremity coincides with the image of the first
pin. This position has been approximately determined by
the image of the flame on the paper screen.

Another method is to illuminate a piece of wire gauze
by a flame, and receive its image on a suitable screen. The
dimensions of the wire gauze and its image may be meas-
ured by a caliper, and the magnification calculated.

Measure the distances from the middle point of the mir-
ror to the object and image. Keeping the illuminated
pin (or gauze) stationary, move the pin (or screen) which
locates the image, and redetermine its position. Do this
three times. Gall the means of the three sets of readings
u^ and Vj.

In a similar manner, place the first pin at a different dis-
tance, u^, and determine the corresponding v,. Do this for
three distances.

Finally, move the illuminated pin until it coincides with
its own image, and measure its distance from the mirror.
In this case t^ = v. Heuce, « = v = r.

Therefore the following equations should be verified :

2. Virtual Image. Concave or Convex Mirror. — By
means of the gas-flame find a position for which there is
no real image. Place the pin or needle close to the flame,
and alter its position until the image of its point is seen

440 A MANUAL OF £XPERIM£NTS IN PHYSICS

diatinctly, apparently through the slit whioh has been re-
moved from the silvered surface. Locate the exact position
of the image by means of a second pin or needle, carefullj
avoiding parallax. (To make the second needle more visi-
ble, it is best to hold baok of it a white screen^ snoh as a
piece of paper.)

Measure the distances from the mirror to the object and
image, and repeat twice, without moving the object. If the
mean distances are «, and y^u^ — jPi, vzstfi for a convex
mirror ; and w = jTi, v :^ — y, for a concave mirror. Hence,
in both cases.

^ •

In a similar manner, measure the distanoee for two other
positions of the illuminated pin, and call them X2, y^, and
X2 and yy (r cannot be measured by a simple direct ex-
periment for a convex mirror.)

Verify the fact that

1.1 1.1 1,1

«i y\ ^ y^ ^ y%

and calculate r from the three separate measurements.
Take the mean.

ILLUOTRATIOK

OONOAVI MlRBOB

1. BmI Image
« V r

96 46 82.4 cm.

80 86.5 89.6 "

81 84.8 83.8 "
88.4 82.4 82.4 "

Hean, 82.40 cm.

0. ViiiuaHmage.
In a similar mimner

OoMVtx MxRioa
In a simila^^amier.

EXPERIMENTS IN LIGHT 441

Qnastioiia and Problems.

1. What connection is there between the focal length of a con-
cave mirror and the size of the image formed of an object
at a distance?

3. If a telescope is focused on a distant object, and is then turned
so as to see the image of this object as formed in a slightly
concave or convex mirror, will it be in focus? How will ii
be if it is a plane mirror ? Illustrate by diagrams.

3. What is spherical aberration ? How can it be guarded against

in mirrors ?

4. Under what conditions will a convex mirror produce a real

image ? Give a diagram illuatrating the answer.

EXPERIMENT 82

0t(J60t To verify the laws of refraction at a plane sur-
face. (See '' Physics," Arts. 319-323. )

OenenJ Theory. As before in the case of reflection, there
are two cases to be considered : 1. Refraction of plane wayes.
2. Refraction of spherical waves.

Since the incident and refracted waves are in different
media — e, g,j ait and glass — it is, in general, impossible to
trace the rays and mark the images directly, as was done
in Experiment 80. The obvions method is to locate the
direction of the waves before they enter and after they
leave the refracting medium, and to mark the points where
any definite ray enters and leaves ; this will give the direc-
tion of the ray and waves inside the refracting medium. If
this is done for more than one ray corresponding to any
point, the image of this point may be determined.

The laws of refraction at a plane surface are as follows:

1. Plane Waves. — The refracted waves are themselyee
plane ; and if the normal to the surface makes an angle

^1 with the normal to the in-
cident waves, and ^, with the
normal to the refracted waves,

the ratio, — — ^, is a constant for
sm ^,

the two media and for the train

of waves used. (If the waye-

length is changed, this ratio

changes ; and, consequently,

for white light the ratio is not

exactly definite.)

FM.1M

EXPERIMENTS IN UGHT

448

This ratio is independent of the angle of incidence, and
is called the '^ index of refraction of the second medinm
with reference to the first/'

2. Spherical Waves. — If spherical waves proceed from a
point at a distance h above a plane surface, the refracted
waves will also be spherical, with
a virtaal centre at a distance h'
above the snrface, and so situated
that it and the source are in the
same straight line normal to the
surface, the length, A', being such
that

A' = /iA _________„

where /a is the index of refraction ^^^^^^^^^^
of the refracting medium. tab U4

Sontooa of Bnor.

1. The surface may not be plane.

d. Since the above laws for spherical waves hold for normal in-
cideDce only, care must be taken to make use only of that
portion of the refracting surface around the foot of the
normal let fall from the source of light.

Apparatus. A plate of glass with plane parallel sides (or
a glass prism); a drawing-board; paper; pins. (The pins
should be so long that they project above the plate or
prism.)

Haaipulation. Place the board on a table, the paper on
the board, and the piece of glass with its refracting sur-
faces perpendicular to the board. Mark the position of
the two refracting surfaces by pencil lines. Place two
pins so that the line joining them falls obliquely on the
refracting surface; but do not let this line differ much
from the normal.

1. By means of two other pins locate the direction of
the ray as it emerges from the second surface. Draw the
incident and emerging rays, and join by a straight line the
two points where the two rays meet the lines marking the

444 A MANUAL OF fiXPfiRIMENTS IN PHYSICS

refraoting anrtaoe&u ThiB line is the path of the refracted
ray, and, conaeqaently, fixes the direction of plane waves
when refracted.

The sines of the angles di and ^2 conld be measured ;
bat the accuracy attainable is not great enough to warrant
the labor.

If the piece of glass need has parallel faces, the emerging
and incident rays should be parallel ; but one is not the
continuation of the other unless the incidence is normaL

Fio.165

If the piece of glass used is a prism, the emerging ray is
deviated through a certain angle, which should be noted
in the drawing. If the angle of the prism is great enough,
show that there may be total reflection inside the prism,
and draw the rays from theoretical considerations, if there
is not time for the actual experiment.

2. In a similar manner locate the incident, emerging, and
refracted rays for a different angle of incidence, keeping
one of the first pair of pins fixed and moving the second
slightly.

EXPERIMENTS IN LIGHT 446

Prolong backward the refracted rays found in this and
the preceding experiment; they should meet in a point
which is the virtual image of the pin which has remained
fixed. The line joining this point of intersection with the
fixed pin should be normal to the surface, and V should
equal /i//.

Prolong backward also the two emerging rays in the two
experiments; these should meet iu a pointy which is the
one from which the waves sent out by the fixed pin seem
to come as they emerge. If the piece of glass has parallel
sides, this point will be in the same normal as the pin and
its refracted image ; if it is a prism, it and the refracted
image will lie in a normal to the second face of the prism.
This image of the emerging waves may be located experi-
mentally by means of a pin which is so placed on the same
side as the fixed pin as to coincide in position with the im-
age. To do this, it is necessary for the piece of glass to
have a sharp, fiat upper face, and for the fixed pin to pro-
ject above the piece of glass. The point as determined
experimentally should agree with that found graphically.

Questions and Problems.

1. Explain the actioo of spberical aberration in the case of flat

plales and prisms.

2. Devise some experiment by which the critical angle may be

measured.
8. What properties of a train of waves change as it passes from

one medium into another ?
4. Construct graphically the image of a train of converging waves

whose centre lies inside the refracting surface.

EXPERIMENT 88

Olgeet. To measure the index of refraction of a solid
which is made in a plate with plane parallel faces. (See
*' Physics," Art. 324.)

General Theory. It was shown in the preceding experi-
ment that, if spherical waves are emitted by a source 0
at a distance h from a plane surface separating two trans-
parent media, the waves refracted into the second mediom
will seem to come from a centre, 0', where the line Off
is perpendicular to the plane surface, and (7 is at such
a distance, h', from it that h' = /Ah, /i being the index
of refraction of the second medium with reference to

the first. Therefore, if
a point, 0, on one face of
a plate of glass is sending
out waves, those which
emerge into the air from
the other face will seem
to come from a point 0',
where 00' is a line per-
pendicular to this face,
and the distances from
0 and 0' to the surface,
called h and h' respec-
tively, are such that A' = /ijA, fi^ being the index of refrac-
tion of the air with reference to the glass. But if fi is
the index of refraction of the glass with reference to air^
/I = llfii, hence,

^=h/h'= — L=.

h-OO'

Fro. 166

EXPERIMENTS IN UGHT 447

The distance h can be measured by a vernier or microm-
eter caliper, and 00' can be easily measured in the follow-
ing manner : If a microscope whose axis is perpendicular
to the plane surface is focused so as to see the source 0, it
is in such adjustment that, if the plate were removed, it
would be in focus for a source at ff ; therefore, if the plate
is removed, the microscope mu8( be lowered a distance
00', in order to again see the source 0.

8ouro«s of Biror.

1. The axis of the microBcope may not be exactly perpendicular

to the plane surface.
3. The surface may not be exactly plane.
8. The source of light must be on the face of the glass, not near it

Apparatus. A plate of glass with plane parallel faces;
a low-power microscope which is movable in a sleeve, such
as a microscope of a comparator or dividing-engine ; a
millimetre scale ruled on paper ; vernier caliper.

Manipulation. In choosing a suitable microscope, select
one which does not require the objective to be placed closer
than 1 or 2 centimetres to the object. Place the ruled
paper on a platform under the microscope, and hold it
fixed by means of some *' universal wax.^' Be sure that the
portion viewed is perfectly flat, and perpendicular to the
axis of the microscope ; focus carefully on a line. Place
the glass plate over the scale and press it closely against
the scale, being sure that the plate is perpendicular to the
axis of the microscope, and that it touches the mark which
has just been focused in the microscope. Raise the micro-
scope in its sleeve until the same line as seen bi^fore is
again in focus. The distance it is raised should be meas-
ured with the greatest care, and for this purpose a scale
should, if possible, be engraved on the side of the micro-
scope. Another method is to gum to the microscope tube,
when it is focused directly on the scale, a piece of paper
with a straight edge placed along the edge of the sleeve.

441 A MANUAL OF EXPERIMEKTS IN PHTSIGS

and to measare by a short paper millimetre scale the dis-
tance from this mark to the edge of the sleeve after the
microscope has been raised so as to focas through the glass.
This adjustment should be repeated many times^ removing
the glass, lowering the microscope so as to focus directly
on the scale, then introducing the glass, raising the mi-
croscope, etc., making measurements each time.

Having thus obtained the distance Offy measare by
means of the vernier caliper A, the thickness of the glass
at the point through which the scale was viewed. This
reading should be made twice or three times, but not more
accurately than 0(y has been measured.

From a knowledge of 00' and A, calculate /i.

ILLUSTRATION

IIV«,1IW

Imdix or RirRACTioN or Glass

* o&

0.8d5 0.81 g05

0.895 0.82 '* = ^-^-^"'-

0.695 OJll

Mean, 0.895 0.818

QuestionB and Problems.

1. What would be the difficulty if the microscope had a very

sbori focus ?
3. Is there any neoessitj of the plate being flat on both aides?

EXPERIMENT 84

(THU ■ZPBBnOINT SHOULD BB 1C4DB W A DABXXNB]) BOOV)

Object. To verify the laws of refraction through a spheri-
cal lens. (See '' Physics/' Arts. 328-339.)

Oeneral Theory. For any spherical lens — i. e., a lens whose
snrfaQes are portions of spherical surf ace8*-and for any train
of wares (of definite period)^ the aame formula applies^ if
the lens is thin, with a proper understanding as to the
signs. It is this,

Ci is the curvature of the incident waves;

C, is the curvature of the emerging waves ;

C/ is a constant for any one lens and any defiuite train of

waves with constant period.
C/ is always to be considered positive ;
Ci is positive if the centre of the incident waves and the

centre of curvature of the first surface they meet are

on opposite sides of this surface ;
C^ is positive if the centre of the emerging waves and

the centre of curvature of the second surface are on

opposite sides of this surface.
Expressed in terms of distances from the lens, this for-
mula beoomea

u^v'-f
where u is the distance from the lens to the centre of the
incident waves,
V is the distance from the lens to the centre of the

emerging waves,
/ has the obvious meaning of being the value of v

460

A MAxXUAL OF EXPERIMENTS IN PHYSICS

when u is infitiite, or the valne of u which will make v in-
finite. In other words^ if plane waves fall npon a lens, the
centre of the emerging waves is at the distance / from the
lens, and is called the principal f ocns ; or, if the waves as
they emerge are plane, the centre of the incident waves is
at a distance / from the lens. (Proper regard mnst be had
for the meaning of the fact that/ is always positive.)

The graphical construction of images which follows from
this formula is illustrated by two cases :

Fio.167

Fio. les
In case I., a double convex lens, OC=zu, and is +;
0^=t;, andi8+; OF=f.

In case II., a double concave lens, OC=zu, and is — ;
D^=t;,andi8 + ; CF=f.

With a double convex lens, it is evident that
u is always + if the waves come from a real object; hence,
t; is + if w >/; i. e., the image is real, because the emerg-
ing waves converge.
V is — if u<f] i.e., the image is virtual, because the
emerging waves diverge.

EXPERIMENTS IN LIGHT 451

With a double concave lens, it is evident that u is al-
ways — , if the waves come from a real object ; hence,
V is always positive; f. e., the image is virtaal, because the
waves diverge.

It is evident from geometry that the 'linear magnifica-
tion ^ is the ratio of v to u.

To verify these laws of lenses^ the same general plan as
in Experiment 81 (spherical mirror) is followed. If the
image is real^ it can be located by means of a screen. If,
however, the image is virtual, a somewhat different plan
mast be adopted. It is this, cut the lens in halves by
a plane passing through the axis of the lens; then the
image may be seen through the half -lens, and a pin may
be so placed as to coincide with it, the pin being seen
past the edge of the lens, not through it.*

It would be well to have the lenses and objects so mount-
ed as to be at the same height, and to be movable along a
fixed beam parallel to the axis of the lens.

Sources of Error.

1. The portious of a lens used mast be those near its axis.

2. If possible, light of one wave-lenglh should be used, as white

light does not give sharp foci.
8. One half a lens is rarely exactly like the other.

Apparatus, A concave lens ; a convex lens ; a half-con-
cave lens ; a half-convex lens ; pins or knitting-needles in
suitable stands; gas-fiame; screen; wire -gauze; metre-
rod.

ManipuIatioiL The lenses, as used, should always be
mounted so that their axes are horizontal ; and the different
objects used should all be placed along a straight line.

1. Beal Image. — Set up the convex lens, and place the
gas-fiame in such a position that a real image is formed on
a screen suitably placed. Put the wire-gauze in front of
the flame, and move the screen until the image of the gauze

* This plan wu suggested by Mr. WUberforoe, of the Gavendish Labora-
tofj, Cambridge, England.

462 A MANUAL OF EXPERIMENTS IN PHYSICS

is as sharp as poguible. Measure the distances between the
lens and the gauze and the lens and the image. Mots the
screen^ redetermine the image, measure again. Do this
three times in all. Let the mean distances be u^ and r,.
Measure also the size of the image by means of a caliper.

Place the gauze at a different distance and determine its
image. Gall the corresponding mean distances Ug and tv
Do this for a third distance.

Then the following relation should hold :

tti v, w, Va tts ^s
Calculate / from the three measurements and take the
mean.

Measure the size of the gauze (one linear dimension will
do), and calculate the 'Minear magnification.''

(In one experiment keep the gauze and screen in the po-
sitions which give a sharp image, and move the lens until n
and V are interchanged, and a second sharp image is formed.
Measure the magnification of this second image, and also
the distance through which the lens has been moved.)

2. Vtrttcal Image. — Mount the half-lens, either concave
or convex, with its edge vertical, and by means of the flame
find a position for which there is no real image. Place
near that position a pin or needle, held horizontally paral-
lel to the lens, perpendicular to its edge, so that the point
of the pin comes exactly opposite the middle of the lens;
that is, place the pin so that its point
lies on the axis and the pin itself is
perpendicular to the axis and the edge.
Illuminate the pin, and view it through
the lens; the virtual image is on the
same side of the lens as is the pin, and
its position may be determined by mov-
ing a second pin, suitably held, until
*^*'^* its point seems to coincide with the

point of the image. The eye must look directly along the
axis of the lens, and the second pin most be moved until

EXPSRlKfiNTS IN LIGHT 468

it Beemi to be oontinnons with the image of the flnt pin^
their points apparently touching. Measure the distances
from the lens to the pins, and make the setting of the sec-
ond pin twice more. Let the mean distances be x^ and jfii
then for a double convex lens

and for a double ooncare lens

In both cases, then,

= a constant.

Move through a short distance along the axis the first
pin which serves as an object, and make two more sets of
observations for two new positions. Let the distances be
0^2 and ^3, x^ and ^3. Then verify the relation

Calculate/, the focal length of the lenses used.

Draw diagrams in each case showing the formation of
the image.

May 10. 1884

ILLUSTRATION

CuMrSX LiMS

44.1
87.1
81.7

1. BdcU Image

88.5
46.8
59.7

Mean,

2. Virtual Image
In Bimilar manner.

CoNCATR Lkns
Id siinilur manner.

/
30.7
20.6
20.7

20.67

Qnestioiui and Problems.

1. Wbat is chromatic aberration ? How does it affect the foci of
lenses ? Would colored glass be of adrantage ?

464 A MANUAL OF EXPERIMENTS IN PHYSICS

2. Explain the use of diaphragms in photogi^hic lenses, micro-

soopes, etc.
8. Draw a " ray " at random on one side of a lens, and coostnict

its continuation on the other side.
4. Deduce the focal length of (1) two double-concave lenses, (3)

one double-concave and one double-convex lens, placed clrve

together. Do this graphically.
6. What advantage has a long-focus lens over one of short focus

in the magnification of a very distant object ?

6. Describe a method by which the focal length of (1) a cnnvoz

lens, (2) a concave lens, may be measured by means of n tele-
scope and metre-rod.

7. If the distancSe between the object and its real Image is kept

constant, but the lens moved through a distance d in onl*'r
to interchange u and «, and thus produce a second inrngeof
the object, oahriilate the connection between d and the focal
kngth of the leoA.

EXPERIMENT 85

Object. To coDBtrnct an astronomical telescope. (See
** Physics," Art. 341.)

General Theory. An astronomical telescope consists es-
sentially of two double-convex lenses — one of long foe as,
the other of short. The two are placed with their axes
continnous ; the lens of long focus is turned towards the
object to be viewed, and is therefore called the " object-
glass^'; and the observer, looking through the other lens,
called the " eye-piece," sees an image of the image formed
by the first lens. The lenses are so placed that the prin-
cipal focus of the object-glass coincides with the principal

/ F

Fio. 170

Fro. 170a

focns of the eye-piece, or comes a short distance inside the
focus towards the eye-piece ; and therefore the eye sees a
virtual inverted image of the object.

The "magnifying power" is the ratio of the angle sub-
tended at the eye of the observer by the final image formed
by the eye - piece to that subtended by the object itself 3

456 A MANUAL OF EXPEKIMENTS IN PHYSICS

and this ratio can be proyed to be eqnal to the ratio of the
focal length of the object-glass to that of the eye-piece.

Booroes of Brror.

1. An object far distant should be rlewed.

2. Spherical and cliromatic aberration may cause difficulties.

Apparatus. Two dou ble-oon vex lenses^one of focal length
15 or 20 centimetres, the other of less than 5 centimetres;
suitable stands ; a pin or a piece of gauze in a stand.

Manipulation. Place the lenses some distance apart, with
their axes horizontal and continuous ; turn the combina-
tion so that the lens of longer focus is facing some distant
object (if the object is viewed through a window, open it).
Locate the position of the image formed by the object-
glass ; this may be done by placing the pin or gauze so as
to coincide with it. Move up the second (short-focus) lens
until, looking through it, there is seen an inverted image of
the distant object (and at the same time a direct image of
the pin or gauze).

It should be noted that the angle subtended is much
magnified. If other lenses are available, another telescope
should be formed and its magnifying power compared with
that of the first, the focal lengths being compared at the
same time.

Draw accurate diagrams illustrating (1) the formation of
the images ; (2) the paths of the extreme rays which enter
the object-glass from any point of the object, so as to form
an idea as to how great a portion of the eye-piece is abso-
lutely essential.

Questions and Problems.

1. How can such a teleecope as this be made by the Introduction

of a lens or lenses to produce a direct instead of an inreited
image ?

2. Draw a diagram for a telescope made up of a concave mirror

and eye-piece.
8. Discuss relative adrantages of a reflecting and a refrsctifig
telescope.

EXPERIMENT 86

Otgeot. To constrnct a componnd miorosoope. (See
'' Phy dies/' Art. 340.)

General Theory. A componnd microsoope consistB of two
short^focus double-convex Ien8e8> placed with their axeft co-

PiAin

inciding. The lens which is turned towards the object to
be viewed is called the '^objective''; the other^ the eye-
piece ; and they are bo placed with reference to each other
thati when an object is just ontside the principal focus of
the objective^ its image is formed immediately inside the
principal foctis of the eye-piece. Therefore, a virtual in-
verted image of the object is seen by the observer.

The magnifying power is the ratio of the linear dimen-
sions of the final image to those of the object ; and it may
be proved to vary inversely as the product of the focid
length of the eye-piece and that of the objective.

Soiaroaa of Brror.

Chromatic and spherical aberration produce error.

Apparatna. Two. short -focus double -convex lenses on
suitable stands; a millimetre scale ruled on white paper;
a pin or wire gauze on a suitable stand.

4«8 A MANUAL OF EXPERIMENTS IN PHYSICS

IbfldpulatioiL Place the two lenses some distance apart,
with their axes horizontal and coincident ; place the paper
scale on a level with this axis, and just outside the focas of
one of the lenses. This can be done by first placing the
lens some distance-— 6.^., 20 centimetres — away, and not-
ing the image by the eye or by a screen ; then bringing
the lens up nearer to the scale until the image is 30 or 40
centimetres from the lens. Determine by means of the pin
or gauze the position of the real image formed by the ob-
jective, and bring up the eye-piece until an inverted image
of the scale is seen through it.

Notice the magnification, and observe how this changes
as the scale itself is moved relatively to the objective, and
also when the distance apart of the two lenses is changed.

Draw diagrams illustrating:

1. The formation of the image.

2. The paths of the extreme rays entering the objective

from any point of the object.

3. The variation in the magnifying power as the distance

apart of the lenses is altered, and also as the object is
made to recede from the objective.

Qaeations and FroblemB.

1. What effect would be produced if the space between the object
and the objective were filled with water and were illumi-
nated by light coming from beyond the object T

d. Is there any limit to the magnification of an object?— «'. «., cio
any two points of an object, no matter how near together,
be finally seen disthictly separated?

EXPERIMENT 87

(this SXFBRIICBNT SHOULD BB MADE IN A DARKENED BOOM)

Otgeot. To measure the angle between two plane faces
of a solid — 6. g.y a crystal or a prism. The adjustment of
a spectrometer.

General Theory of a Spectrometer. A spectrometer is an
instrument primarily designed to measure the angle be-
tween the directions of two beams of light. In order to
accomplish this end, it must be most carefully adjusted,
and the following method is recommended :

FiO. 172

As appears from the illustration, a spectrometer consists
primarily of a vertical axis around which are movable in
horizontal planes two circular platforms and two metal
tubes. One of these tubes is a telescope made up of two
converging lenses, the object-glass and eye-piece ; the other
tube is called the " collimator,*' and contains at the outer
end a metal slit with straight parallel edges, and at the
inner end a conyerging lens so placed that the slit is at its
principal focus. All these various parts are movable in

460 A MANUAL OF EXPERIMENTS IN PHYSICS

soYeral ways : the level of at least one of the circular tables
can be altered by means of three levelling - screws ; the
axes of the telescope and collimator may be raised or low-
ered; the eye-piece tube of the telescope contains cross-
hairs which may be adjusted, as may, also, the eye-piece it-
self ; the slit of the collimator can be revolved around the
axis of the collimator, and it can also be pushed in or out
of the tube. The position of the tables, the telescope, and
the collimator, at any instant, may be read off and recorded
from scales and indices attached to the various parts.

The spectrometer (or goniometer) may be used to meas-
ure accurately the angle between two surfaces, or betweeo
lines perpendicular to two surfaces. To do this, the instru-
ment must be so adjusted that the axes of the telescope
and collimator and the lines which are perpendicular to
the two surfaces are all four at right angles to the vertical
axis of the instrument. The focus of the collimator and
telescope must furthermore be adjusted so that the spherical
waves diverging from the slit of the collimator are transform-
ed into plane waves in passing out through its lens, and so
that plane waves falling on the object-glass of the telescope
are brought to a focus at the cross-hairs. These adjustments
should be carried out in the following order and manner :

1. To focus the telescope, — Remove it from its clamps;
remove the eye-piece tube, and shift the position of the
cross-hairs until they are distinctly in focus for the eye of
the observer ; replace the eye-piece tube ; direct the tele-
scope towards some extremely distant object— e. g,, a star,
or a spire some miles away — and then push the eye-piece
tube in or out until the image of the object falls exactly
upon the cross-hairs — t. e.y until there is no parallax be-
tween them. This focuses the telescope for plane waves.

2. To focus the collimator. — Replace the telescope in
its clamps ; turn and elevate the telescope and collimator
until their axes are approximately in the same straight
line ; turn the slit horizontal (or place a fine hair across
its middle point) ; adjust the slit by pushing it out or in,

EXPERIMENTS IN LIGHT 461

and raising or lowering the collimator until, when the slit
is illuminated by a flame or by being pointed towards the
daylight, it is sharply in focus on the cross-hairs. It may
be well to revolve the entire collimator (or the slit-tube)
aronnd its own axis through 180^ to see if the slit remains on
the cross-hairs, as it should if the instrument is well made.
This adjustment places the slit in the focus of the colli-
mator lens, so that, when the slit is illuminated, plane
waves emerge from the collimator. It also places the axes
of the telescope and collimator in the same straight line.

3. To adjust either the telescope or the collimator so that
its axis is perpendicular to the axis of the instrument. —
To do this, some polished plane surface — e.g,, the face of a
prism — must be placed vertically on the central table of the
instrument at such a height as to be at the level of the
telescope or collimator, and it must first be adjusted so that
its normal (t. 0., a line perpendicular to its plane) is itself
perpendicular to the axis of the instrument. This first
step is performed in the following manner :

{a)' Turn either telescope or collimator around the axis
of the instrument, being careful not to change the levelling-
screws of the tubes until the normal of the plane surface near-
ly bisects the angle between the telescope and collimator.

{b) Alter the position of the plane surface by the levelHng-
screws of the table on which it rests until, when the slit
(still horizontal) is illuminated, its reflection from the plane
surface falls exactly upon the cross-hairs of tlie telescope.

The normal to the plane surface is now perpendicular to
the axis of the instrument.

Now, leaving the plane surface absolutely untouched, it
is possible to adjust the axis of either the telescope or the
collimator perpendicular to it.

If the telescope has a Gaussian eye-piece — that is, if the
side of the eye-piece is cut away so as to allow the inser-
tion of a thin piece of glass between the eye-piece and the
cross-hairs — it is best to adjust the telescope perpendicular
to the plane surface. The method is as follows :

462 A MANUAL OF EXPERIMENTS IN PHYSICS

Insert the piece of glass at an angle of 45^ with the axis
of the telescope, as is shown in the figure-; place a lamp or

gas - flame at the side
' of the opening, so that

the light falls upon the

glass-plate ; this iUa-
,^ ,_ minates the crosshairs

hnghtly. Now turn
the telescope nntil it is approximately perpendicular to the
plane surface on the central table ; and raise or lower the
telescope by the levelling-screw, until there is seen through
the eye-pieco a reflected image of the bright cross-hairs.
This image must be made by means of the levelling-screw
of the telescope to coincide with the cross -hairs them-
selves ; and it is obvious that the telescope is then perpen-
dicular to the plane surface, and therefore perpendicular
to the axis of the instrument.

If the telescope has no Gaussian eye-piece, it is best to
adjust the collimator perpendicular to the plane surface;
and the method is the same in principle as that used with
the Oaussian eye-piece. The collimator must be turned
until it is approximately perpendicular to the plane sur-
face; turn the slit vertical and open it rather wide by the
side-screw ; illuminate it by means of a gas-flame, but cut
off half the slit by interposing a piece of tin or card-board;
then adjust the collimator by its levelling-screw until the
reflected image of the illuminated half of the slit appears
on the tin or card-board. By carefully adjusting the screw,
the central point of the slit may be made to coincide with
the image of itself ; and then the axis of the collimator is
perpendicular to the plane surface, and therefore perpen-
dicular to the axis of the instrument.

If the telescope is perpendicular to the axis of the in-
strument, it can be turned so as to be pointed obliquely
towards the plane surface, and the collimator can be turned
so as to make approximately the same angle with the plane
surface on t;he opposite side of its normal. Illuminate the

EXPERIMENTS IN LIGHT 468

slit (still horizontal), and raise or lower the collimator by the
leyelling-screw nntii the reflected image of the slit coincides
with the cross-hairs of the telescope. The axis of the colli-
mator is now perpendicular to the axis of the instrument.

If the collimator is adjusted perpendicularly to the plane
surface^ it is the telescope which must be adjusted until the
reflected image of the slit coincides with the cross-hairs, and
then it is also perpendicular to the axis of the instrument.

The spectrometer itself is now in adjustment.

If the instrument is to be used to study the dispersion
of a grating or prism, or if it is to be used to measure the
angles of a prism, the surfaces of the prism or grating must
be themselves adjusted so that their normals are perpen-
dicalar to the axis of the instrument. This can be done
according to the general plan described in 3a and Sb, by
placing the grating or prism on the central table and al-
tering its level by the three levelling-screws until the re-
flected image of the illuminated slit coincides with the
cross-hairs of the telescope. This makes the normal of
the reflecting surface perpendicular to the axis of the in-
strument. To make the normal to the second face of the
prism also perpendicular to the axis of the instrument, the
slit must be reflected in it and its image made to coincide
with the cross-hairs. But in the adjustment of this second
face by the levelling-screws of the platform, the first face is
thrown out of adjustment unless the prism is placed on the
platform as is shown in the figure —
i. «., unless the three faces are per-
pendicular to the lines joining the
three screws. In this position, the
position of any one face may be al-
tered without changing the plane of
the face already adjusted. Thus, if
the face A has been adjusted, screw "VioriTi^

1 can be turned without affecting the plane of A. After
the second face has been adjusted^ however, it may be nec-
ceasary to readjust the first face.

4i4

A MANUAL Of SXPKmM£NTS IK PHYSICS

Considerable thought is required as to exactly the best
poaitioa on the central platform for the grating or priim,
according to the U9e to which it is to be put ; but by carefol
consideration the most suitable position can be determined.

In certain spectrometers the telescope and collimator are
fixed BO that they cannot be turned around the axis of the
instrument ; and, although in this case the adjustment of
the instrument is more difficult, the general principles are
those which have been made use of in the discussion of the
ordinary form of instrument.

To HeasuFO the Angle of a Priam.
A solid which is bounded in part by two plane faces which .
are not parallel forms a prism. The line of intersection of
these faces, considered produced if necessary, is called the
''edge^' of the prism. The prism should be placed upon
tlie table of a spectrometer, with the edge parallel to the
axis of the instrument following the directions just gifen.
There are then three methods of proceeding :

1. If the telescope has a Gaussian eye-piece, the crow*
hairs may be illuminated, and the telescope turned until it
is in succession perpendicular to the two plane faces, The
angle through which the telescope has been turned equals
ISC'-*- A, where A is the angle of the prism.

2. Place the colli-
mator so that when
the slit is illumi-
nated by a flame or
lamp the light as it
emerges from the
lens falls upon Mk
the plane faces, be-
ing divided into
two sections, as it
were, by the edge.
JSs^h of these sec-
^^'i^^ tions is reieotsd

EXPERIM£NTS IN LIGHT 4d6

from the corresponding face; and the direction of eacli
may be determined by the telescope. The angle through
which the telescope must be turned^ in order that the im-
age of the slit may fall on the cross-hairs^ first when reflected
from one face and then from the other, may be proved
to be 2A, where A is the angle of the prism. For it is
apparent from Fig. 175 that ^i + '^a equals the angle be-
iwoeu the normals to the two faces, and, therefore, equals
180°— -4; but, as the telescope is turned from one re-
flected beam to t}ie other, it moTes through an angle
3G0°-2 (^1+ ^2), 01-2-4.

3. If the platform on which the prism rests has a scale
on its edge, by means of which its position may be read, the
following method may be used : Place the collimator so

that light is reflected from ^.^^ *-«%.

one face only of the prism, y^ >' ^ ^v
and focus the telescope up- / k \ \

on the'reflected beam; then, / | A ^_^X^^3^*

keeping the collimator and j ^^«/ ^h-^ZT^I^

telescope fixed, turn the \ y^^ ^^ ^^^fel

table which holds the prism \ y""''^^

until the second face of the N. /

prism reflects lightf rom the ^y.^^ ^^^^y^^
collimator down the tele- fm. m

scope. The angle through which the prism has been turned
is 180° ±-4, depending upon the direction in which it has
been turned, as is evident from the diagram.

Sonroes of Brror.
1. The faces of the prism may not be plane.
% The two halves of the collimator lens may not be alike, which
' would introduce an error in Method 2.

8. The divided scale may not be uniform.
' i. The centre of the circular scale may not coincide with the axia
of the instrument. This must be corrected for, by making
readings at both ends of the index which extends across the
scale. (See Experiment 64, Dip Circle.)
5. The clamping - screws may not hold, and so every reading
should be repeated.

4«6

A MANUAL OP EXPERIMENTS IN PHYSICS

Fio 177

Apparatus. Spectrometer ; prism ; fish-tail gas-bnmer.
Manipulation. Place the prism on the spectrometer table^

as shown, and adjust the in-
strument carefully, nsing the
method described in the in-
troduction. In making any
final adjustment of the tele-
scope, it is best to clamp it
with the screw, and complete
the adjustment by means of a
small screw and spring attach-
ed to the clamping -screw.
This screw and spring are
called the '* tangent-screw."
Method 1. Gaussian Eye -piece. — Adjust the glass
mirror in the telescope tube, and place the telescope so
that its axis is approximately perpendicular to one face of
the prism. Holding a gas-flame near the opening into the
side of the telescope, turn the latter until the reflected
cross-hairs coincide with the actual cross-hairs. (Do not,
of course, alter the levelling-screws.) In making this final
adjustment, clamp the telescope arm and use the tangent-
screw. Bead the position of the telescope by means of its
index and scale. Some instruments have verniers and oth-
ers have microscopes and micrometer screws. In any case,
make three settings and readings, and take the mean. If
there are two verniers or microscopes, read both each time.
Keeping the prism table fixed, turn the telescope arm
until it is approximately perpendicular to the other face
of the prism. By means of the flame illuminate the cross-
hairs and adjust the telescope exactly perpendicular to this
face. Make, as before, three settings and readings, and
take the mean. Then make three more readings on the
first face, and average with those made before.

The difference between the mean readings is 180®— i,
where A is the angle of the prism.
Tarn the table which holds the prism through about

EXPERIMENTS IN LIGHT 467

90% and repeat the measurements. Again turn the table
90% and repeat. This is done because it is possible that
the scale may not be uniformly divided on all sides.

Take the mean of the measurements.

Method 2. Prisfn and Collimator Stationary, — Turn
the collimator arm until its axis apparently bisects the
angle between the two faces. (If the collimator arm can
not be turned, turn the prism table.) Turn the slit until
it is vertical, if it is not so already, and place a gas-flame
in front of it. Determine by means of the eye the approx-
imate directions of the beams reflected from the two faces.
Place the telescope so as to receive these reflected beams
in turn, and by means of the tangent-screw adjust until the
image of the slit falls exactly on the cross -hairs. Make
three settings and readings in each position, using both
verniers or microscopes, and take the means. After set-
ting on the image formed by reflection from the second
face, turn the telescope arm back, and make three more
readings on the image reflected from the first face. Aver-
age the mean of these readings with the first set. The
difference of the means equals 2A.

Turn the prism table in succession through 90** and 180%
and make similar readings. Take the mean of the three
sets of measurements.

Method 3. Telescope and Collimator Stationary. — Turn
the collimator arm until it makes an angle of about 30^
with the normal to one face of the prism. (If the colli-
mator arm cannot be turned, turn the prism table.) Illumi-
nate the slit by a flame, and turn the telescope arm until
the reflected image of the slit falls upon the cross-hairs.
Make the final adjustment by means of the tangent-screw.
Clamp the telescope, and keep it and the collimator fixed
during the rest of the experiment. Read the position of
the prism table, using both verniers ; unclamp the table, re-
set, and make another reading; do this three times. Make
these final adjustmeuts with the tangent-screw attached to
the table, not the one attached to the telescope. Turn the

4M A MJkKUAL OF KXPERIMENTS IN PHYSICS

table nntU ttn image of the slit formed by reflection from
the other face is seen on the croas-hairs. Make a reading
for the position of the table ; anclamp, reset, and make an-
other reading. Do this three times, aud take the mean.
Unclamp the table, turn it back into its previous position,
and make three more readings of the reflection from the
first face. Average the meto of these with the mean of
the first three readings. The difference between these
mean readings is 180^ duA. Be careful not to set at any
time on a refracted image.

Place the collimator arm at approximately an angle of
45^, with the normal to the surface ; and repeat the read-
ings, as described above. Do this also for some other
angle.

Take the mean of the determinations for A, using as
many methods as possible.

ILLUSTRATION

Anglk op a Prism ibf 8,1897
l8t Face ad Fue Dfflferanoe

Mbthod 2. —Vernier A. .886^ 2' 00" W 60' 00" 119° 48* 00'

B,AfXP%'W 275^ WOO" 119* 48' 00'

Mean, 119*48' 00"
.% Angle of prism = K119* 48) = 59° 54' 00'.

iBfc Fue ad Face Dtfl

Method 8.— Vernier A.... 889° 68' 60" ^19° 55' 00" 119° 68' 50"

*• i9.... 159° 58' 50" 89° 55' 10" 119°58'40''

Mean, 119° 58' 46'
,\ Angle of prism = 180° - 119° 68' 45" = 60° 1' 16".

2(1 trial, different part of scale, 59° 46' 20"

8d ** " " •* " 69° 66' 10"

Mean, 69° 64' 15"
Hence, angle of prism is 69° 64'.

Queationa and Problema.

' 1. Give reasons why one of the above methods is more accurate
than the others.
2. Give reasons why readings taken on different parts of the scale
should difTer so widely. Describe possible errors which may
ariao when a scale is rolod.

EXPERIMENT 88
(THIS bzpbrucbnt should bb mabb in a DARKBNBD BOOIC)

Olgeot. To study the deviation produced by a prism.
To measure the angle of minimum deviation. (See " Phys-
ics/'Arts. 322,344.)

General Theory. The deviation is defined as being the
angle by which the direction of the emerging waves differs
from that of the incident ones, when plane waves fall upon

Fig. 178

a prism. It is evident from the figure that the deviation
2 = i + t'— ^, and since i' varies with ^, the index of re-
fraction, i varies with the angle of incidence, the angle of
the prism, the material of the prism, and the wave-length
of the light. In this experiment the first and the last of
these two facts will be verified. It is also proved by the-
ory that for a definite prism and for a definite train of
waves there is a certain angle of incidence for which the
deviation is a minimum. This is called the '^ angle of

470 A MANUAL OF EXPERIMENTS IN PHYSICS

minimnm deviation " for the given prism and light. This
fact will also be proved in this experiment^ and the angle
will be measured.

The method is to place the prism on the table of the
spectrometer^ and so arrange the collimator that light from
it falls upon one face of the prism, and is refracted into it
and out of the other face ; then the direction of the emerg-
ing light may be studied by the telescope. The angle of
incidence may be varied by turning the prism table ; and,
if light of several wave-lengths is to be studied, the slit
may be illuminated with white light ; while, if the angle
of minimum deviation is to be measured for any particular
wave-length, a sodium-fiame may be used, for it gives light
which is approximately homogeneous.

BonrceB of Error.

1. The most common source of error is the confusion arising

from reflected images, if the prism is small.

2. The minimum is always difficult to observe because the change

is so slow ; for a considerable change in angle of incidence
will produce only a slight change in deviation.

Apparatus. Spectrometer; prism; gas -flame or incan-
descent electric light; Bunsen- burner; a piece of fused
salt (NaCl) supported on a suitable stand.

Muiipalation. Place the prism on the table of the spec-
trometer and adjust the instrument.

1. To Study Deviation of Waves of Different Wave-length. —
Turn the prism table so that, when the slit of the collima-
tor is illuminated,
light will enter

I — "^-p^ /\^^ *^^ P*®® through

the prism. Make
the angle between
the axis of the
collimator and
the normal to the
Fm. 179 N/^\ face of the prism

EXPERIMENTS IN LIGHT 471

about . 30^. (The collimator should^ of course^ be on the
Bide of the normal towards the base of the prism, so that
the entering light is refracted towards the base.) Illami-
nate the slit, placed yerticallj, by the gas-flame, the elec-
tric lamp, or by light coming from the sky, if this is
possible. By means of the nnaided eye locate the re-
fracted light. This can best be done by turning the prism
table slightly and noticing the change, carefully guarding
against any possibility of obserring a reflected image in-
stead, by covering different surfaces of the prism with a
small piece of paper. Note the order of colors in the re-
fracted light, and record them in the order of their devia-
tion. Now turn the telescope arm until the refracted light
enters the tube. Again note the order of the colors, and
account for it.

2. To Study the Effect of Changing the Angle of Inci'
dence. — Darken the room and replace the source of light
before the slit by a sodium -flame. This consists of a
small piece of fused salt, supported just in the edge of
the middle part of a Bunsen-flame by a support and wire.
Care should be taken to see that the yellowest portion of
the flame illuminates the slit. A yellow image of the slit
should now be seen in the telescope. Make the angle of
incidence ks great as possible by turning the prism table,
and receive the refracted light in the telescope. Note,
roughly, the angles of deviation and incidence (to within
5°). Decrease the angle of incidence by steps of about
10° or less, and note the corresponding angles of deviation.
Continue as far as possible. Plat the results.

It will be noticed that there is a certain angle of
incidence such that any further decrease in the angle
will make the angle of deviation reverse the direction of
its change. This angle of incidence should be located quite
carefully, by making slight changes and noting their effect.

3. To Measure the Angle of Minimum Deviation. — Bring
the telescope to the angle which corresponds to minimum
deviation as determined approximately in Part 2 of the

47a A MANUAL OF EXPERIMENTS IN PHYSICS

experiment. Then^ by minute motions of the prism tabic
and corresponding ones of the telescope, make the angle
of incidence such that the refracted image is on the cross-
hairs of the telescope when the deviation is exactly a mini-
mum. This final adjustment should be made by clamping
the telescope and using the tangent screw. Sead the posi-
tion of the telescope, using both verniers or microscopes.
Displace the prism table and telescope slightly, and repeat
the measurement. Do this three times in all, and take the
mean of the readings.

There are now two ways of proceeding. One is to re-
move the prism ; turn the telescope arm until the telescope
and collimator are in line, as is shown by the image of the
slit being on the cross - hairs ; read the position of the
telescope, and take the difference between this reading and
the one made at minimum deviation, for this is obvionslj
the angle of deviation. The other, and the better, is to
turn the prism table around through approximately 180^, so
that the edge points in an opposite direction to that which
it did before ; then, to find, by means of the telescope, the
position of minimum deviation of the prism turned this way.
make the reading, as above, three times, and take the dif-
ference between this reading and that made when the devi-
ation was a minimum on the other side. This difference
is evidently twice the angle of minimum deviation, if the
readings mark tlie angle through which the telescope has been
turned. In certain instruments the verniers are attached
to the prism table ; so that, when the latter is turned, the
verniers measure the angle of rotation. The telescope is
rigidly fastened in such instruments to the circular scale,
over which the verniers move. The difference between the
vernier readings in the two positions of minimum devia-
tion would then be 180°— i>, where D is the angle of mini-
mum deviation.

If it is possible, repeat the entire experiment, using a
different portion of the scale. Oall the mean of the two
results for the angle of minimum deviation D,

EXPERIMENTS IN LIGHT 473

In a similar manner observe the minimnm deyiation for a
lithinm-flame, and compare it with the valae just found for
a sodium-flame. Is it different ? If so^ is it greater or less?

N. B, — It should be noted that Parts 1 and 2 of this experiment do not
require a spectrometer, but simply a prism and a slit — the latter can be made
bj cutting an opening in a blackened metal screen.

ILLUSTRATION ^.^^„^^

Anolk of IIinivitm Deyiation or Sopt-Glasb Paism

Sodium Light

let Position
Vernier ^...186^ 67' 00"
•• ** ...186° 65' 00"
" " ...186° 60' 00"

2d Position
48° 26' 00"
48° 22' 00 "
48° 24' 00"

Dlflference

Mean, 186° 57' 20"

48° 24' 00"

188° 38' 20"

Vernier 5... 6*5700"
" "... 6° 55' 00"
*' "... 6° 60' 00"

228° 26' 00"
228° 22' 00"
228° 24' 00"

....

Mean. 6° 57' 20"

228° 24' 00 "

188° 38' 20"

Mean, 188° 33' 20"
Minimum deviation, 2> = 180°- 188° 88' 20 "=41° 26' 40".
(In this instrument the TernieTB move with the prism table.)

Qnestioiu and Problems.

1. What is meant by chromatic aberration? How may it be cor-

rected approximately ?

2. What is meant by saying that the spectrum produced by n

prism is *' irrational"?
8. Describe an experiment to determine the absorption spectrum
of any liquid. Also describe an experiment to investigate
** anomalous dispersion."

4. What is the exact process by means of which a prism spectro-

scope identifies or records the spectrum of a gas ? Does it
give wave-lengths?

5. Why is it almost essential to make two readings for the angle

of minimum deviation, one on each side of the collimator
axis, especially in the case of a small prism ?

6. What would be the application of DOppler's principle to light ?

How could it be tested by observations made on the light
reaching the earth from the sun ?

EXPERIMENT 80

Object To measare the index of refraction of a trans-
parent solid made in the form of a prism. (See " Phys-
ics," Art. 322.)

General Theory, It is proved in treatises on Physics that,
if A is the angle of a prism, and D the angle of minimum
deviation for the same prism for waves of a certain wave-
length, then the index of refraction, /«, for this light is,

. AJtD

sin — ' —
2

. A

sm-

Therefore, if A and D are known, /* may be calculated.

SouroeB of Brror.

Those of the preceding two experiments.

Apparatus. Same as for the last experiment.

Manipulation. Measure A and i>, as described in the two
preceding experiments, taking particular precaution to use
the same angle of the prism in the two measurements. It
is well to mark it in some way — e. (jr., by a pencil mark on
the top, not on the faces. Do this at least three times.
Take the means, and substitute in the above formula.

ILLUSTRATION ^^^^

Imdix or RxriucTioM or Sorr-GLASs Prism for Sudium-Liobt

Angle of prism i4 = 69'46'201

<* *' miaimum deviation, D-^^AX^ 3T 00" for sodiuralight.

"°— 2~"

.-. ik^ 2-" = ^-^^-

sin^

EXPERIMENTS IN UQHT 476

Qnestlons and Problema.

1. Explain in detail why the following parts are need in a spec-
troBoope : (a) the slit ; (b) the ool lima ting lens ; (e) the prism ;
(<0 the object -glass of the telescope; (e) the eye -piece.
Which could be di8pensed witli ?

2l In studying the spectrum of a star, what apparatus is required?

EXPERIMENT 00

Otgect To study color - sensation. (See ''Physics,^
Art. 361.)

Oeneral Theory. There are several theories of color-sen-
sation— the Young-Helmholtz, the Hering, the Franklin—
but none can be regarded as entirely satisfactory. Apart
from all theory, however, it is possible to prove that by the
combination of certain color-sensations an entirely differ-
ent sensation may be produced. Three colors are, in gen-
eral, selected, which, when combined upon the retina of the
eye (not combined like a mixture of paints), will produce a
sensation of white or gray ; and the effect of combining
these color -sensations in different proportions and with
different amounts of white and black is studied. The one
requirement is that all the sensations which are to be com-
bined should be produced simultaneously on the retina of
the eye. This may best be secured by placing colored disks
in the form of sectors of circles upon a top which has a flat
upper surface, and spinning the top rapidly. Then, if the
top is brightly illuminated by sunlight, and if the observer
looks intently at the colored sectors, his eye will receive
different color - sensations which are practically simul-
taneous. The one serious source of error is in not having
the colored disks illuminated strongly enough.

With each set of apparatus, as furnished by the instru-
ment maker, come explicit directions, which need not be
repeated here.

EXPERIMENT 91

(THT8 BXPBBTMRNT SHOULD BE MADB IN A DABKBNBD BOOM)
(TWO OBSBBVBBB ABB BEQUIBBD)

Otgeet. To measnre the wave-length of light by means
of a grating. (See " Physics/' Art. 365.)

General Theory. The simplest form of grating is made
by ruling parallel straight lines by means of a diamond-
point at equal distances apart on a piece of plate -glass.
In order to get good results there should be several thou-
sand lines per inch, and the grating should be two or
three inches* long at least. If plane waves of a single
wave-length fall normally upon such a grating, there will
be several streams of emerging light determined by the
condition that

a sin ^ = n\,
where a is the "grating- space" — t. 0.,the length of the
raled surface of the grating divided by the entire number
of lines ruled in this length ; X is the wave-length of the
incident light ; n is any whole number, 0, 1, 2, 3, etc. ; and
^ is the angle made with the normal to the grating by the
direction of one of the diffracted emerging beams of light.
Consequently, there will be light leaving the grating at
the angles ^o' ^v ^%y ®^^*' where
a sin ^0 = ^
a sin S, = X
a sin ^2 = 2X^ etc.
The quantity, w, is said to mark the " order of the spec-
trum."

To measure X for any train of waves, it is necessary to

478 ▲ MANUAL OF EXPERIMENTS IK PHTSIGS

know a, n, and the corresponding value of ^. The most ac-
earate method to obtain ^ is obyioasly to place the grating
on the table of a spectrometer, perpendicular to the axis
of the collimator; to illuminate the slit with the light
whose wave-length is known, thus causing plane waves to
fall normally upon the grating ; and to locate and measure
^ by means of the telescope and its verniers. Another
method, not so accurate, but much simpler, is this : Cause
plane waves to fall normally upon the grating by means
of a slit and convex lens; place a- metre -bar parallel to
the grating, but some distance back of it ; and determine
some line on this bar which, with a point at the centre of
the grating, fixes the direction in which the eye must look
in order to receive one of the diffracted emerging beams.
This evidently gives a means of measuring ^ for a definite
value of n. The grating-space, a, is, in general, known from
the pitch of the dividing-engine which ruled the gratiug.
If, however, it is not known, two methods to measure it
are possible : one is to measure it directly by comparison
with a standard centimetre rule (a most diflficult task);
the other is to assume as definitely known the value of X
for some light — e. g., any one of the Fraunhofer lines — and

to measure the value of d for it, thus giving a | = - - V

\ sm^/

SouroeB of Error.

1. The slit must be exactly in the focus of the coUimating lens.

2. The face of the grating must be perpendicular to the axis of

the lens.
8. The lines of the grating should be parallel to the slit.

4. In mensuriiig d, great care must be taken to keep one poiotof

the grating constantly in the line of vision.

5. There is trouble generally from the spherical aberration of the

lens.

Apparatus. Slit; short-focus convex lens; grating; so-
dium-fiame ; 2 metre-bars ; telescope ; clamp-stands.

Kanipolation. Make the slit quite narrow by means of
its screw^ and cover it with strips of opaque paper except

EXPERIMENTS IN LIGHT

479

for abont 1 centimetre at its centre. Then proceed as fol-
lows : To place the slit in the focus of the iens^ the best
plan is to focns the telescope on a distant object; then,
placing the lens in front of the illuminated slit> to view
the slit through the lens by means of the telescope, and
to move the lens until the slit is seen clearly on the tele-
scope cross-hairs. (This is the ordinary spectrometer ad-
justment.) Turn the slit vertical, if it is not already so.
Place the grating £
at the same height
as the lens^ a few
centimetres away
and as closely per-
pendicular to the
axis of the lens as
the eye can judge;
turn it in its own
plane so th^t the
ruled lines are par-
allel to the slit.
Place back of the
slit a metre -bar,
and make it as
closely as possible
parallel to the grat-
ing and at nearly
the same level. Now illuminate the slit with the sodium-
flame; and, on looking through the grating towards the
lens, a yellow image of the slit will be seen for certain
positions of the eye. By first looking normally through
the grating, and then more and more obliquely, there will
be seen the various '^orders" of the spectrum correspond-
ing to » = 0, 1, 2, 3, etc., which will be, of course, sym-
metrical on the two sides of the normal. To determine
the angles at which these images are observed, proceed as
follows: Make some faint mark at the middle point of the
top edge of the grating, snch as a fine pointer of paper ;

Fro. ISO

480 A MANUAL OF EXPERIMENTS IN PHYSICS

and; keeping this in the line of vision^ look through the
grating obliquely in such a direction as to see one of the
images. Let a second observer move a vertical edge of
paper along the front face of the metre-bar until the edge
comes exactly in the line of vision. Record the reading,
and make two more determinations^ being careful to pay
attention only to the central portion of the image, not to
the extremes, because they may be distorted by spherical
aberration. Call the mean reading di.

Now look through the grating obliquely from the oppo-
site side of the normal, and determine the direction of the
diffracted image of the same order as before. Call the
mean reading t/,. Take the difference, d^ — d^.

Do this for as many orders as possible.

Remove the slit, flame, and lens, and measure as accu-
rately as possible the perpendicular distance from the ruled
surface of the gfating to the metre-bar. Call it A. Then,

for any order, tan ^ = ^ ^"7 ' ; and so sin d may be calcu-
lated.

In this way calculate sin ^ for as many orders as possible,
at least for n =zl and 7i = 2. An instructor will give the
value of the grating-space ; and so X may be deduced from
each set of measurements.

ILLUSTRATION „ ,^ ,^-

Maj 10, VSn

MXASUREMKNT OF WaTE-LeNOTH 01' SODIUM LlOBT

Grating-space, a = 0.0001759 centimetres.

Order of spectrum, » = 2.

€k = 71.1; d.=24.5; A = a9.5.

77 1 — 24 5
tan ^ = 1^3—^ = 0.8911+.
2x29.5

A sin ^ = 0.665+.

a sin d

/. X = • = 0.0000686 + centimetres.

= 5860 IjigBtrOm units.

The correct value is 5898.

EXPERIMENTS IN LIGHT 481

Questions and Problems.

1. What is the advantage of having a large number of lines on

the grating? What of having a large number per eenti-
metre f

2. What differences are there between spectra produced by grat-

ings and those produced by prisms ?
£. Describe a method of using a reflecting grating — that is, one
whose lines are ruled on a polished metallic surface.

APPENDIX I

LABORATORY EQUIPMENT

There are several usefal pieces of apparatus and nsefnl
arrangements with which every laboratory should be pro-
vided^ and which should form, as it were, part of the per-
manent equipment of the laboratory. A few of these will
be mentioned.

ABpirator Pump. Such a pump, to be driven by water
from a tap under pressure, should be joined to various taps
in the laboratory. A sketch is given
of a most efficient one, which, with
water under a pressure of 30 pounds,
will give a vacuum of less than 2 cen-
timetres.

This pump may also be used as a com-
pression pump, if the end from which
the water is escaping is fitted into a
closed space from which the air cannot
escape ; for the water may be allowed
to escape through a trap; and the
pressure of the confined air, which
thus increases constantly, may be used
for various purposes.

PI itform Air-pump. A hand air-
pump provided with a plane brass plat-
form and a glass bell-jar with ground
edge should be constantly ready for
use. One of its main purposes is to enable a student to
exhaust air from water, which is to be used for density
determinations.

Fio. 181

484

A MANUAL OF EXPERIMENTS IN PHYSICS

Dr]ring Tubes. In order to facilitate the drying of glass
tubes and bulbs, permanent drying tubes should be fast-
ened to a shelf neal* an aspirator pump. They need con-
sist only of a sulphuric-acid bottle, into which the inlet

STOMHIUTM

Fig. l«2

tube dips, and from which a tube passes to a calcium-
chloride bottle, out of the top of which a tube leads
to a three-way cock. The other two branches of this cock
are joined to the aspirator pump and to the bulb to be
dried, as is shown. The three-way cock consists of a T-tube
with a ground -glass stopper at the junction, into which
are bored two holes at right angles to each other, as
shown. When the cock is turned so that the opening
2 faces the aspirator, and 3 the bulb, air will be exhausted
from the bulb ; if now the valve is turned so that 2 faces
the drying tubes and 1 the bulb, dry air will enter the bulb.
This process may be continued indefinitely. To stop dust,
it is prudent to place a loose plug of cotton-wool in the
tube by which the object to be dried is connected.

Distilled Water. Every laboratory should be provided
with an apparatus for distilling water. One may be easily
constructed, but it is as well perhaps to buy one from an
instrument maker. It should be fastened to a wall, and
kept running almost constantly. The apparatus, of course,
needs cleaning from time to time.

Olook Circuit Every physical laboratory should be pro-
vided with a good clock and a number of electric clocks

APPENDIX I 485

driven by this central one. The central clock shonld have
a compensating pendulum with a heavy bob^ and it should
be of such a length as to beat seconds or half -seconds — t. e»,
its period should be two seconds or one second. The nut
by which the position of the bob is regulated should have
a fairly large^ divided head, and to the bob should be fast-
ened a pointer resting on this head, so that the change in
the length of the pendulum can be measured. This clock
should be rigidly fastened to the wall of the building, and
should be regulated and rated by astronomical observations.

If, by a suitable arrangement, the pendulum closes an
electric circuit for a very short time once during each
complete vibration, or once during each half-vibration, an
electromagnet included in the circuit may be used for mov-
ing the hands of a clock. This is the principle of the elec-
tric clock. As many of these as desired can be run from
the one central clock, and, since they are all driven by the
same pendulum, they will all have the same accuracy — tn>.,
that of the central clock. The main trouble is with the
arrangement for closing the circuit by means of the pendu-
lum. Various forms of contacts have been devised, but the
simplest and the one most generally used is the *' mercury
contact.*'

In this arrangement the circuit is completed through the
pendulum rod. To its lower extremity is fastened a nar-
row piece of platinum, whose plane is in the plane of vibra-
tion. Below this, when in its position of equilibrium, is
the open end of a glass horn containing mercury, which
is so mounted that it can be rotated about a pivot in the
centre of the arc of the pendulum. The end of this Jiorn
is quite narrow in the direction of vibration, and is placed
as near the centre of the arc as possible, and in such a
position that the globule of mercury formed there, when
the other end of the horn is raised, shall at each half-
vibration of the pendulum be cut by the platinum edge.
If one terminal of an electric circuit, including an electric
clock, be connected with the bearings on which the pendu-

4^6

A Manual of ^xprrimknts in physics

lum swings, and the other extremity be connected with the
mercury in the horn, then the circuit will be closed at each
swing of the pendulum, and the electric clock will record
the half -vibrations of the central clock. The horn must
be placed so that the pendulum cannot touch the glass,
no matter at what angle it may be inclined. The circuit
should be open when it is not in use, and an arrangement
should be made for opening and closing the circuit without
opening the case of the clock. This may be done by means
of a key included in the circuit, or by means of a wheel
and ratchet which, by lowering the outer end of the horn,
allows the mercury to run back and thus break the circuit.
If intervals of time that are exactly equal are needed,
as in rating a tuning-fork, this contact is not satisfactory.
The mercury globule has a tendency to vibrate, and it is
very difficult to get it exactly in the centre of the pendu-
lum arc. lience, on the whole, the contact now to be de-
scribed is to be preferred. It is very easily made and
adjusted, and one used for over a year for determining the
frequency of forks by Michelson's method has given per-
fect satisfaction. It is easily understood from the figure.

Fio. 183

H is a weak steel watch-spring, soldered to the collar C,
which can be clamped to the clock-case in any desired
position by means of a screw. To H is soldered a plati-
num point (P), perpendicular to the flat surface of the
spring ; and perpendicular to both H and P is soldered
the point (N) of a fine steel sewing-needle. M is a mer-
cury-cup ; D is a vane of mica that dips into oil in the
cup 0, and is intended to damp the vibrations of the

APPENDIX I 487

spring. The oil-cup 0, the mercnry-cup M, and the col-
lar C are all fastened to a piece of brass that is screwed
to the back of the clock-case near the top of the penda-
Inm-rod, and in sach a position that the spring is hori-
zontal, and the needle^ N, projects horizontally and comes
near the centre of the arc of the pendulum- rod. The
mercury -cup is insulated from this piece of brass. E is
a light brass collar that can be fastened to the pendu-
lum-rod by means of the brass screw S. To the back of
this collar is soldered a short piece of flat steel, T, that
projects towards the back of the clock. The plane of T
is perpendicular to the plane of yibration of the pendulum,
and is inclined to the horizon at an angle of about 20*^ or
30^. The horizontal needle and T are well polished on
both sides, and are of such lengths that, when the collar is
in place on the rod and the pendulum is at rest, T laps
over the needle by an amount just sufficient to insure the
lifting or depressing (as the case may be) of the needle at
each swing of the pendulum. The collar is placed on the
rod at such a height that the upper edge of T is a little
higher than the needle when the pendulum is at rest.
One terminal of the electric-clock circuit is connected to
the mercury -cup M, and the other to the brass plate K,
which is in metallic connection with the point P. Then,
as the pendulum swings in one direction, it depresses P
into the mercury and closes the circuit; as soon as it passes,
the circuit is opened by the force of the spring ; as the
pendulum swings in the opposite direction it raises P
slightly, which does not affect the circuit, as it is already
open. Hence, when this contact is used, the electric clocks
record the complete vibrations of the pendulum; and,
therefore, N need not be centred accurately. If the yane
in the " dash-pot,'* D, works properly, the vibrations of the
spring will be '* dead beat." When this form of contact is
used, a key should be placed in the circuit so that it may
be opened when the electric connection is not desired.
In every case, the current through the clock should be

488 A MANUAL OF EXPERIICENTS IN PHYSICS

small, as otherwise the meroary rapidly oxidizes. If a
large current is needed, its circnit should be closed by a
relay, worked by the feeble clock circuit.

Sets of Ohemieals. Certain chemicals should be kept
on shelves ready for use at any moment; and there are
others which should be kept in a stock -room and dis-
pensed to the students in small quantities. Lists are
given of each :

LaboraioTy Sioek-room

Sulphuric acid. Mercury.

Hydrochloric acid. Benzene.

Chromic acid. Ether.

Bichromate of potassium, 1.2 kilos. Copper sulphate, C. P.

Sulphuric acid. 8.6 " Zinc sulphate, C. P.

Water, 8 " Calcium chloride.

Nitric acid (iu small quantities). Caustic potash, C. P.

Alcohol. Caustic soda (com.)

Salt. Kerosene.

Copper Sulphate. Fused salt.

Supplies. Certain supplies should be kept constantly on
hand. It is impossible to give a complete list ; but a few
of the most important should be mentioned :
Files — triangular and round. Clamp-stands.

Sand-paper. Rubber tubi ng— common and pure.

Emery -paper. Wooden blocks, assorted sizes.

Drying-paper. Qlass tubing, *'

Wire — copper, brass, and iron. Iron weights, ** "

Cork borers. Thread— linen and silk.

Corks— wooden and rubber. Sealing- wax.

String— good linen and also cotton.

Books of Beferenoe. There are a few books of refer-
ence which should be at the disposal of students. These
may be conveniently kept on a shelf near an assistant's
table :

Stewart and Cke, *' Experimental Physics."
Glazebrook and Shaw, '* Practical Physics.'*
Nichols, " Laboratory Manual of Physics."
Eohlrausch. "Physical Measurements.*'
'* Smithsonian Tables of Constants."
7-place Logarithm Tables.

APPENDIX I 489

Glass -blower's Table. A tabic with a metal top, fitted
with bellows and glass-blowing burner, is a great con-
venience; and all glass-blowing should be done upon it,
if possible.

Laboratory Tables. Suitable tables for physical labora-
tories may be made by any carpenter. All that is necessary
is a steady wooden table, about 6 feet by 3 feet, with a
frame of 2 inches by 4 inches carried over the table about
3 feet above it from end to end. This frame should be
supplied with nails, pegs, and holes. It is sometimes ad-
visable to have a shallow trough cut around the top of the
table near its edge, so that any mercury which is spilled
may be caught.

Balances. Platform - balances, sensitive to 0.1 gram,
should be available for use in every laboratory-room ; and
sets of nickel-plated weights, 1 kilo to 1 gram, should be
placed beside them.

Other balances, more accurate than these, should also be
provided. If there are many students in the laboratory, it
is unwise to furnish fine weights with the balances. It is
a better plan to have each student, or pair of students if
they work in pairs, rent from the stock-room a good box of
weights, 100 grams to 0.01 gram with riders; for otherwise
the injury to the weights and the number of those lost are
of considerable importance,

Oalvanometers. There should be mirror-galvanometers of
various types, according to the purposes for which they
are needed, provided and attached to the walls in suitable
places. The question of construction and selection of gal-
vanometers is so important that a separate chapter is de-
voted to it.

Storage-batteries. There is no part of the equipment of
a modem laboratory which is more useful than storage-
cells. These may be procured of the agents, and full di-
rections for their installation come with them. Every
room in the laboratory should have at least one line of
wires leading to and from the battery-room.

490 A MANUAL OF EXPERIMENTS IN PHTSICS

In using storage - cells, care must be taken not to
short -circait them; and open iron or german - silver re-
sistance coils should be used when current is taken from
them. An ordinary plug resistance -box is likely to be
burned out if the storage-cell current is passed through it

APPENDIX n

LABORATORY RECEIPTS AND METHODS

Cleaning Glass. The best method to clean glass — e. g,, the
interior of a bulb — ^is to wash it in turn with chromic acid,
distilled water, alcohol. Sometimes a mixture of alcohol
and ether is used in place of the alcohol alone. Caustic
potash or soda will clean certain things ; but they them-
selves adhere to glass, and must be removed by the most
thorough rinsing with water.

In some cases it is necessary to use hydrochloric acid, or
even nitric acid (or a mixture of the two), but this rarely
happens ; and, if nitric acid is used, the operation must be
carried on under a hood, so as to remove the noxious fumes.

Chromic acid consists of
8 parts water;

1.2 parts bichromate of potassium ;
3.6 parts sulphuric acid.
It may be used again and again for cleansing purposes.

C9eaning Mercury. The methods necessary to clean mer-
cury depend upon the nature and amount of the impurities.

If the mercury is pure — /. e.y has no amalgams on it, but
is dusty or wet — it may be cleaned by first drying it by
drying-paper and then filtering it through a cone formed
of glazed paper, pin-holes being made in the bottom. Care
roust be taken not to allow the last portion of the mercury
to pass through.

If the mercury is impure, there are two methods of clean-
ing : 1, purely chemical ; 2, by distillation in a vacuum.

402 A MANUAL OF KXP£RI]I£NTS IN PHYSICS

The chemical process is as follows:

The mercury is first shaken violently with dilate sulphu-
ric acid, to which, from time to time, drops of a solution of
potassium bichromate are added. It is then thoroughly
rinsed in water under a tap, partly dried by drying-paper,
and allowed to pass in fine drops through a column of dilate
nitric acid (6 to 10 per cent.), about 80 centimetres high.
This is best done by making a trap at the lower end of a
wide, long glass tube, setting it vertical, pouring in some
clean mercury, filling the rest of the tube with the acid,
and pouring in the mercury at the top through a funnel
which has a stop-cock, or which is drawn out into a fine
tube. The mercury now falls in minute drops, which col-
lect at the bottom and gradually pass oat through the trap
into a vessel placed to receive it.

To distill the mercury in a vacuum, a suitable still must
be placed in some permanent situation. One which has
proved useful is shown in Figure 184. (The design is due
to Professor Smith, of Oxford.) It consists o*f a large
mushroom -shaped glass bulb, in which there is a little
trough around the edge, a long glass tube being joined to
the trough at one point, and a larger glass tube being join-
ed to the bottom of the bulb. The first of these tubes is
about 100 centimetres long, and has a trap near its lower
end ; while the larger tube is about 70 centimetres long,
and is joined at its lower end by a flexible stout rubber
tube to a large open reservoir, which may be raised or
lowered. Wire gauze is wrapped around the lower part of
the bulb, and it is heated by a ring gas-burner, with small
openings on its top side. The whole apparatus is firmly
fastened by clamps and supports to some solid wall.

The method of use is as follows : The open reservoir is
filled with mercury, which is fairly clean; and the ex-
tremity of the trap, in which the other tube ends, is at*
tached to a good air-pump — e,g,, an aspirator. As the
pump is worked, mercury rises up into the bulb, and more
should be poured into the reservoir. When the pressure

APPENDIX 11

493

in the bnlb, as indicated by the mercury colnmn, is about
2 centimetres, light the ring burner, and let it heat the
mercury in the bulb gently. The mercury reservoir should
be 80 adjusted that the top surface in the bulb comes more
than 8 centimetres below the edge of the rim. As the air-
pump continues to work, minute drops of mercury may
soon be noticed condensing on the top of the bulb and

Fio 184

collecting in the shallow trough. Enough will soon col-
lect to flow over into the long tube connected with it.
This mercury wili collect in the trap at the bottom, and
soon back up a short distance ; and at this moment the
connection with the pump must be broken, otherwise the
mercury might overflow into the pump. When the trap
is opened to the air the mercury in it will rush back up

494 A MANUAL OF EXPERIMENTS IN PHTSIGS

the tnbe, and stand at the barometric height aboTe the
free surface in the trap. The mercarj will now continue
to vaporize, condense, and flow out of the long tabe. A
suitable clean bottle, into which no dust or dirt can enter,
should be prepared to receive it as it escapes.

The still is now in operation, and the mercury to be
cleaned can be poured into the open reservoir from time
to time, care being taken to dry the mercury before it is
put in and to keep the levels properly adjusted.

A mercury-still should be cleaned carefully at least once
a year; and it is best, if possible, to have two stills working
side by side, the second one being supplied with the mer-
cury distilled by the first.

A tray should, of course, be prepared below the still, so
as to catch the mercury in case the glass breaks or the re-
ceiving-bottle overflows.

To Fill a Barometer Tube with Mercuiy. In many experi-
ments it is necessary to have a glass tube which is closed
at one end completely filled with mercury, so that when
inverted and dipped, open end down, into a basin of mer-
cury there shall be no air in the tube. There are two
processes by which this may be done.

The tube should be carefully cleaned and supported,
closed end down. A long capillary tube may be drawn
having a small reservoir at one end, into which mercury
may be poured. This capillary tube should be placed in
the larger tube so as to reach to its bottom, and mercury
should be slowly admitted through it, care being taken to
exclude air bubbles. This process of exclusion may be
greatly helped by putting a small ring of glass around the
capillary tube, which will rise on top the mercury, and keep
the tube from touching the wall. In this way the mercury
will slowly rise in the tube and push out the air ahead
of it.

A better method is to place the tube, closed end down,
in a piece of cast-iron tubing of suitable length, which has
a Bcrew*cap at its lower end. The glass tube is separated

APPENDIX II 496

from the iron by a packing of dry sand, and two or three
Bunsen-bnrners are directed at the bottom and sides of the
iron pipe. Mercury is poured in slowly and boiled gently ;
and in an hour or more the tube may be filled. When this
is done it may be removed and inverted, care being taken
to allow no air to enter during the process. The best plan
is to cover the forefinger with a piece of black rubber —
e. g., Q, piece of dental rubber — press this tightly against
the open tube, squeezing out a drop of mercury, and then
to invert.

Fumes of mercury are injurious to the health ; and so,
when possible, a trap should be made at the open end of
the tube by bending it over and dipping it under the sur-
face of mercury in a shallow basin. This trap will stop
the mercury vapor, and yet will allow air to bubble through
it. By stopping the heating, mercury may be driven back
into the tube, and the process completed. When the tube
is filled the basin may be removed and the tube safely in-
verted.

Amalgamating Zinc. All zinc rods and plates used in
cells must be amalgamated with mercury, so as to prevent
local action. The process is as follows: Clean the zinc
carefully with dilute sulphuric acid by dipping the zinc
repeatedly in a battery -jar containing the acid, using a
piece of cloth tied to the end of a stick as a mop, if neces-
sary ; then pour a few drops of mercury upon the zinc,
while holding it over a glass tray, and spread the mercury
as uniformly as possible over the zinc by means of a cloth
and stick, repeating this process until the zinc has a clear,
bright surface at all points. Keep the mercury which runs
off the zinc in the glass tray, and use it for amalgamating
other zincs.

Amalgamating Copper. Electric connections are often
made by dipping copper wires into cups of mercury; and,
in order to insure good connection, the copper terminals
must be amalgamated with mercury. This is done as fol-
lows : Pour nitric acid into a bottle which has a glass

496 A MANUAL OF £XP£RI)CENTS IN PHYSICS

stopper and add a few drops of mercary, thus forming
mercuric nitrate (there should be an excess of mercury
in the acid) ; clean the copper wire and dip it for a mo-
ment into the liquid^ or by means of a splinter of wood vet
the wire with the liquid ; the wire becomes blackened, but
if wiped off by a cloth will appear 'brightly amalgamated.

A test of perfect amalgamation is that the extremity of
the wire be able to raise a small drop of mercury off the
table.

"Universal Wax." A most useful soft wax is made by
thoroughly mixing and working together 1 part by weight
of Venetian turpentine and 4 of beeswax. The wax should
then be colored red by mixing best English red vermiliou
with it. This wax can be used to hold almost any two sub-
stances together ; but it is soft and yields to any consider-
able stress.

CementBy etc. Sealing-wax is often used to fasten various
things to glass — 6. g., an iron tube to a glass one — and the
only precaution necessary is to heat the glass thoroughly
so as to destroy some of its glaze and then to rub the rod
of sealing-wax over it, thus forming a thin layer of wax on
the glass before trying to make the glass stick to the iron
or other substance. After this preliminary layer is ob-
tained, others may be added, and they will make an air-
tight joint. (The metal must also be heated.)

Sealing-wax makes a water-tight joint, but dissolves in
contact with kerosene.

Damping Keys and Magnets. It is often inconvenient to
wait for the vibrations of a galvanometer needle to die
down, so that the instrument may be used again ; and to
hasten this process several methods have been devised, two
of which will be described here.

One is to place close behind the coils of the galvanom-
eter a few turns of wire parallel to the coils, and to join
these turns through a *^ damping key " to a cell of some
kind. This key is a simple form of commutator, and con-
sists (as shown) of two inclined wire springs which may be

APF£KPXX U

407

pressed down so as to move between two horizontal wires.
The connections are made as shown, the terminals of the
coil of wire being in the two springs, and the battery being
joined to the two
horizontal wires. If
now one spring is
pressed down so as
to make contact with
the lower wire, the
other spring being in
contact with the up-
per, a current is sent ^^' ^^

through the coils of wire in a particular direction, which
is reversed if the relative position of the two springs is re-
versed. Therefore by tapping in turn, first on one spring,
then on the other, a series of impulses may be given the
.galvanometer needle, and, if these, are properly timed, the
needle may- be brought to rest very quickly. The strength
of the damping may be varied by altering the distance of
the coil from the galvanometer needle.

The other method depends upon the fact that when a
coil of wire is moved along a magnet currents are induced
in it depending upon the direction and rate of motion. A
magnet is accordingly made in the shape of a long narrow
U, and it is clamped by one of its arms to a wooden frame

^ which may be fastened

] to the wall or to a table ;

over the other arm

slips a brass collar on

M I which are wound fifty
or a hundred turns
of fine wire. The ex-
tremities of this wire
are joined directly to
the coil of wire placed parallel to the galvanometer coils. By
sliding the brass collar first in one direction and then in the
other the needle may be brought to rest almost instantly.

Kio. 186

500 A MANUAL OF EXPERIMENTS IN PHYSICS

around the tube two pieces of damp paper with straight
edges facing each other, but a slight distance apart, so as
to include the scratch in the gap between ; then, by means
of a finely pointed flame, start a crack at the scratch and
carry it around the tube.

2. To Bend Glass Tubing, — Hold the tube horizontal in
a flame from a fish-tail burner — not a Bunsen-flame — and
turn continually and rapidly around its axis until it
begins to be soft ; then let the tube bend slowly under
its own weight, by letting go one end ; or, at least, if
force is used, use very little, and take care to make a
smooth bend.

3. To Draw a Capillary Tube, — Take a piece of tubing
about 7 or 8 millimetres in diameter and 20 centimetres
long, and heat it in a Bunsen - flame, keeping it turning
continually. When the central portion has become red
and quite soft, withdraw it sidewise from the flame; and,
after it is out of the flame, rapidly extend the ends, thus
drawing the tube into any capillary size desired.

4. I'o Make a Small Opening in the Side of a Tube.^
Cork up all openings of the tube, and by means of a finely
pointed flame — e,g., from a blow-pipe — carefully heat one
point on the wall of the tube. It will soon become soft,
and the air inside expanding will blow the soft wall out,
thus making an opening whose size depends largely upon
the area which was heated by the flame.

5. To Join Two Tubes of the Same Size Together. — It is
necessary that the two tubes should be of the same kind of
glass, otherwise, although they may stay joined for a few
hours, they will surely crack apart. Close the end of one
tube by means of a cork ; heat the other end and the end
of the second tube in the hot blue flame of a blast-lamp,
turning one by each hand and holding the two ends almost
touching. When both are quite red, withdraw them from
the flame, place the two ends squarely against each other ;
blow slightly down the open end of the tube so as to force
the hot walls at the junction slightly outward; place the

APPENDIX II 601

junction again in the flame^ and by repeated heating,
blowings and extension make a smooth joint.

6. To Join One Tube to the Side of Another. -^Goxk both
ends of the tube to whose side the other tube is to be fast-
ened ; make a hole in its side as described above, taking
care to make the opening nearly as large as, but no larger
than, the cross-section of the tube which is to be joined.
Break off the ragged edges of the opening, and join the
tube exactly as described in the last section.

Standard Cells. The best standard cells are those made
according to the specifications of the International Elec-
trical Congress, 1893. These are published in the Proceed-
ings of the National Academy of Sciences, 1895. The cells
are called Clark cells, and have an £. M. F. of 1.434 volts
at 16^ C.

Another standard cell is the Danlell. It consists of a
glass jar which holds a porous cup ; the porous cup con-
tains a solution of zinc sulphate into which dips a rod of
zinc, and is surrounded by a solution of copper sulphate
into which dips a copper rod. In setting up the cell the
following precautions are necessary :

The porous cup should be cleaned by being boiled in
water and then allowed to soak ^n cold water.

The zinc rod should be well amalgamated.

The copper rod should be cleaned and polished by sand-
paper and tap water.

The zinc -sulphate solution consists of 44.7 grams of
crystals of C. P. zinc sulphate (or 25.08 grams of the
anhydrous salt) dissolved in 100 cubic centimetres of dis-
tilled water.

The copper-sulphate solution consists of 39.4 grams of
C. P. copper sulphate dissolved in 100 cubic centimetres
of distilled water.

The zinc rod should be put in the porous cup; then the
zinc-sulphate solution poured around it to a depth of one
or two inches ; the cup should now be placed in the glass
jar, the copper rod inserted, and the copper-sulphate solu-

602 A MANUAL OF EXPERIMENTS IN PHTSIGS

tioQ carefnily poured in to a depth slightly less than that
of the zinc snlphate in the porons cnp. (No copper snl-
phate must splash into the porons cnp.)

Short - circnit the cell for fifteen minutes> then let it
stand on open circnit for fiye minutes. It is now ready
for use and will give an E. M. F. of 1.105 volts within .2
of one per cent.

The cell should not remain set np for more than two or
three hours. In taking apart^ remove the porous cup^ rinse
the outside under a tap, and pour the zinc solntion back
into the stock-bottle, unless the zinc rod is blackened, in
which case throw it away ; then pour the copper-sulphate
solution back into the stock-bottle, and dry and clean the
zinc and copper rods.

APPENDIX m

GALVANOMBTBRS

Oalvanombtebs are of two types : in one the coils of
wire are fixed and the magnet movable ; in the other the
magnets are fixed and the coil of wire movable. The first
type is ordinarily called the *' Thomson reflecting galva-
nometer "; the second, the " D'Arsonval galvanometer," al-
though its principle was also first made use of by Sir Will-
iam Thomson (now Lord Kelvin) in the siphon recorder.
Sections of each of these types are given in the figures.

FMuUn

Fia.188

004 A MANUAL OF EXPEEIMENTS IN PHYSIOS

In this chapter a description will be given of galyanome-
ters designed for special use — e, g., tangent, ballistiC| dif-
ferential instruments; details of the constrnction of the
various parts, coils, needle - systems, fibres, etc. ; and in-
struction as to testing and use.

Tangent Galvanometers. This type of instrument con-
sists of one or more turns of wire wound in a circle whose
radius is large compared with the length of the magnetic
needle placed at the centre. If the plane of the coil is in
the magnetic meridian, we have for equilibrium, if there

are n coils of radius r, i = ^ tan d, where (? = ; it is

called the galvanometer constant. If it is used as a mirror

galvanometer, i s -^ d, very nearly, since the deflection is

small.

In the deduction of this formula it has been assumed
that the magnetic force of the current on the needle is the
same whatever be the angle d. This cannot be assumed
unless the length of the needle be small compared with the
diameter of the coil.

One of the great advantages of a tangent galvanometer is
that it enables one to measure currents in absolute units,
and it should, therefore, be constructed in such a manner
that its constant can be accurately calculated. The rings
on which tangent galvanometers are wound, are nsuaUy
made of brass, and turned up in a lathe with rectangular
grooves for the winding. (These brass forms frequently
contain sufficient iron to cause large disturbances on ac-
count of the induced magnetization.)

The Differential Galvanometer. This type of reflecting
galvanometer is composed of two coils which act in op-
posite directions upon a magnetic needle. The two coils
are usually made of equal resistance, and so placed witb
respect to the magnetic needle that, if the same current
pass through each coil, the deflection will be zero. In
some forms of these galvanometers these conditions are

APPENDIX III 606

realised by winding the coil with a strand of two equal
wires. A better method is that found in some forms of
Thomson galvanometers, in whioh a portion of one ooil
is wound as a small auxiliary coil whioh can be displaced
towards or away from the needle> and the action of the
two coils thus made equal.

The Ballistic Galvanometer. This form of reflecting gal*
yanometer is used to measure quantities of electricity^ and
hence is employed in the study of the distribution of mag-
netism, the flow of magnetic induction through any circuit,
the discharge of condensers, etc.

It can be shown that the total quantity of electricity that
passes through the galvanometer is proportional to the sine
of half the angle of deflection, provided that the moment
of inertia of the suspended system is so great that it has
not moved appreciably from its position of equilibrium
before the current has died down to zero.

In order to correct the throw of the ballistic galvanome-
ter for damping we must multiply sin i$ by 1 + ^X, if the
damping is small, where X is a quantity depending on the
construction of the instrument. If the needle be set in
vibration and Op o^ . . . a« be the lengths of successive

swings, X = rlog,— , and hence is called the "logarith-
mic decrement

From the above consideration we see that the damping
in a ballistic galvanometer should be made small, especially
that due to the resistance of the air, the exact effect of
which is very uncertain.

FArsonval Oalvanometen. The essential parts of this
type of galvanometer are a coil suspended in a magnetic
fleld by means of a very fine wire or strip, which serves at
the same time to convey the current to the coil and to fur-
nish the couple which opposes the rotation. The current
is usually led away from the bottom of the coil by means
of a loose spiral or loop of fine wire, and sometimes by an
accurately centred wire dipping into a mercury oup« In

606 A KULNUAL OF EXPERIMENTS IN PHTSIGS

order to obtain great Bensibility we mnst have (1) small
torsion in the saspeusion wires or strips^ (2) a strong mag-
netic fields and (3) a coil giving the maximum taming
moment with the least moment of inertia.

Since the torsion yaries as r*, it diminishes rapidly with
decrease in the size of the wire ; this must not be carried
too far^ however, for ultimately the resistance of the sus-
pension becomes too large a part of the total resistance.
The bifilar suspension has been used in instruments of
this type with great success. Thin phosphor-bronze strips
have been used by many makers. Temperature changes,
however, produce a change in the zero, owing to the varia-
tion of the coefficient of torsion with temperature.

The small traces of iron found in the wire and insula-
tion of the suspended c<9il (even when special precautions
have been taken in drawing the wire and insulating it)
exert a '^ magnetic control '' which has prevented the use
of strong magnetic fields, inasmuch as it increases as the
square of the field strength. Hence, high sensibilities
have been sought by diminishing the diameter of the sus-
pending wire and the use of comparatively weak fields. In
some instruments the moving coil is surrounded by a very
thin silver tube to increase the damping. It Yum been
shown that the best form of coil is one whose horizontal
cross-section is two circles tangent at the axis of suspen-
sion.

Proportionality of Deflection with Current. In all accurate '
work, where a reflecting galvanometer is used to measure
currents, the law connecting the deflection and current
must be found experimentally, and the results expressed
in the form of a curve called the "calibration curve" of
that instrument. For practical work, however, it is desir-
able to have an instrument in which the deflections are
very nearly proportional to the current. Special precau-
tions must therefore be taken in the design of the instru-
ment, or else a scale with divisions of different lengths
suited to the peculiarities of the instrument may be used.

APPENDIX ni 607

Ghoioe of a Galvanometer. In localities subject to mag-
netic disturbances the D'Arsonval type of instrument, which
is now made of extremely high sensibility, possesses many
advantages. Where, however, the very greatest sensibility is
required, as in bolometer and platinum thermometer meas-
urements, the Thomson galvanometer must be used. The
proper choice of galvanometer resistance depends on the
work for which it is intended. If the galvanometer is
to be used for the measurement of resistance in a Wheat-
stone bridge circuit, the best resistance for the galvanome-
ter in order to attain the highest sensibility depends on the
resistance in the other circuits. Speaking generally, a low-
resistance galvanometer is best when low resistances are to
be compared, and a high-resistance galvanometer for the
comparison of potentials and high resistances. If the gal-
vanometer resistance is five times greater or less than the
best galvanometer resistance, the sensibility is only reduced
about 25 fi. It is well to remember that a galvanometer
may be too sensitive for the purpose at hand.

For use in measuring electromotive forces by the ''high-
resistance " method, a galvanometer with a high resistance
should be chosen; while for most ballistio work a low-re-
sistance instrument is better.

For use in thermo-electric work and with bolometers
low-resistance galvanometers must be selected.

OontroUing Magnet The action of the directing magnet
may be best shown by means of a diagram. Let OH rep-
resent in direction and magnitude the horizontal intensity

no. 189

608 A MANUAL OF EXPERIMENTS IN PHYSICS

of the earth^s fields and OA the direction in whioh the qrs-
tem flhonld stand.

If, then, OC represents in direction and magnitude the
action of the directing magnet on the suspension system,
the resultant will be represented by 01. In order to
lengthen the period of vibration, the directing magnet
will have to be placed so that this resultant is smalL

Since Hi is equal and parallel to OC, the action of the
directing magnet can be represented by HI. In order to
diDiinish the resultant controlling moment, 01, the action
of the directing magnet, HI, must be diminished — i. «:,
the magnet moved farther from the suspension system ; but
to keep the resultant along the line OA, the magnet must
be turned from HI towards H2. In this way the resultant
becomes in succession 01, 02, 03, as the directing magnet
is moved farther away and takes the direction Hi, H2, H3.
When the position HB is reached, the directing magnet
must be turned in the same direction as before, but now,
in order to make the resultant 04, its action on the system
must be slowly increased from HB to H4, etc. (i. e., the
magnet must now be brought nearer to the suspension sys-
tem). By moving the magnet very slowly when this po-
sition is reached, the control OB, 04, etc., may be made as
small as desired In passing through OH, the direction in
which the system stands will be reversed.

The Suspended System. Among the first questions that
have to be considered in the construction of the magnet
system is that of astaticism, whether it be necessary, and
what are its advantages and disadvantages.

Consider a suspended system made of two sets of magnets.
Let M^ = magnetic moment of upper set of magnets.
Ml = magnetic moment of lower set of magnets.
11^ = strength of controlling field at upper set of magnetfiu
Ht = strength of controlling field at lower set of magnets.
0« = strength of field due to current in upper coiL
&i = strength of field due to current in lower coiL
^ =: resulting permanent deflection.

APPENDIX m 609

We then ha?e^ if the two fields are perpendicalar to each
other.

In order that the sensibility may be great, tan d must
be as great as possible for a given current through the gal-
vanometer— i, e., the numerator of the above expression
must be large and the denominator small. The sensibility
may therefore be increased almost indefinitely by weaken-
ing the controlling field.

In an astatic system the upper and lower magnets are
set in opposition — i. e., J3",= -^Hi^R, say— and the coils
are so joined up that they both tend to produce a defleo*
tion in the same direction — $. e., G^=:Oi^ 0. We there-
fore have for an astatic system,

R{M.-M,)
Hence, in this case, to secure great sensibility use strong
magnets — t. e,, make M^ + Mt great — ^and make them as
nearly equal as possible — i. e., make M^—Mi small.

Non-astatic systems are more easily constructed, and by
means of a controlling magnet equally great sensibilities
may be attained, but they cannot be used where there are
local magnetic disturbances ; for, to attain the high sensi-
bility required, the strength of the controlling field must
be so far reduced that the zero becomes unsteady. If a
system were perfectly astatic, it would be in equilibrium in
any position in a uniform magnetic field, and would be un-
infiuenced by a uniformly varying field ; hence in localities
subject to magnetic disturbances the only system that can
be satisfactorily employed is an astatic one.

Several types of astatic suspension systems are shown in
the following diagrams :

Fig. (1) shows a multiple magnet system built up of ten
short magnets made from small sewing-needles or tempered
watch-springs. These magnets are first fastened by means
of shellac to a thin piece of mica, which is afterwards at-

510

A MANUAL OF EXPERIMENTS IN PHTSIGS

4' nI

8 N

Nc8 Nc8

(1)

(8)

8 N

(3> (4)

Fni. 190

8 N
(5)

N S

(8)

tached to a thin, sti-aight staff, preferably glass tnbing suit-
ably drawn out. In building up the system hard shellac
should be used, as wet shellac often distorts the system in
drying, and thus destroys the astatioism. A light mirror,
M, is then fastened to the staff.

Fig. (2) is similar in construction to (1) with a mica vane
back of the mirror to increase the damping.

Fig. (3) shows a system easily constructed and which
damps rapidly. The magnets are fastened upon a long
lamina of mica.

Fig. (4) is a system in which the magnets are of the form
of a split cylindrical bell, or horseshoe shape. These sys-
tems have a large moment of inertia, and are frequently
used in ballistic galvanometers. Systems of this kind are
damped by surrounding the magnets with copper in which
currents are induced.

Figs. (5) and (6) show the vertical magnet systems used
first by Weiss and Broca.

Magnets. The astaticism of the system cannot be ex-
pected to remain very long unless the magnets are very
permanent. The permanency of a magnet depends not
alone upon the quality of the steel, but also upon the tem-
per, which should be different for different kinds of steel
The best temper for any particular specimen of steel can
only be decided by experiment. After the magnets have
been ground to the required size and tempered, they should
be strongly magnetized, then boiled in water fpr seyeral

APPENDIX m 011

hours; if this process of magnetizing and boiling be re-
peated several times, the magnetization approaches a max-
imum and is very permanent. On account of the difficul-
ties encountered in securing a high degree of astaticism,
the time and trouble taken to prepare the magnets will be
well spent. Watch-springs, properly tempered, make ex-
cellent magnets. Sewing-needles will also be found satis-
factory, but the very best magnets are those made of tung-
sten steel. By the use of tungsten - steel magnets the
sensibility may be yery nearly doubled.

Small magnets, like those required for galvanometer
systems, can be tempered by laying them in a groove in
a piece of charcoal and heating with a blow-pipe until a
cherry red is reached, when they should be quickly dropped
into water or mercury. The magnets for the Weiss and
Broca systems can be made of needles or tempered piano-
wire. In order to secure straight pieces the wire must be
heated uniformly and tempered under tension ; this can
best be accomplished by means of an electric current. It
is best to prepare these magnets, also, by successively mag-
netizing and boiling.

The Staff The staff upon which the magnets are mounted
can be prepared by heating a glass tube in a Bunsen-flame
and drawing it out very fine. Care must be taken to select
a straight piece. The hook should be made of very fine
wire and attached to the glass staff by means of shellac.

ICrrors. Good mirrors for galvanometer magnet systems
may be made by silvering thin microscope cover glass, from
which pieces of the desired size may be cut by means of a
diamond point. If these small mirrors are to be used with
a telescope and scale a number of them should be cut out
and tested before they are mounted on the staff, for, unless
they are perfectly plane, the definition will be bad. These
yery thin mirrors should be mounted on the staff with
some soft wax, such as *' universal," in order to prevent
distortion. If a spot of light on a ground-glass scale is to
be used, then a lens of the proper focal length must be

0X9 A MANUAL OF KXPSItlMKNTS IN PHYSICS

placed iu front of the mirror, or the lens may be dispensed
with aad a concave mirror used.

Snspenaion Fibres. Good fibres can be obtained from
Japanese floss-silk, which should be well washed to remove
the gam. A single fibre of silk (one-half of an ordinary
cocoon fibre) will easily support several grams. The diam-
eter of these fibres varies from about 0.0008 centimetre
to 0.0015 centimetre. They will be found satisfactory in
all cases except for galvanometers in which the highest
attainable sensibility is sought, in which case the torsion
of the fibre becomes a serious factor. In this case it must
be unduly lengthened, or, what is better, one may resort
to quartz fibres. Quartz has a much higher coefficient of
rigidity than silk, but as the torsion varies as the fourth
power of the diameter, and quartz fibres can be obtained
so fine as to be beyond the power of the microscope, their
torsion may be made negligible. These fibres are made by
lieating quartz and then shooting it out with a bow and
arrow. In all cases, on account of steadiness, compact*
ness, etc., it will be found more eatisfactory to use short,
fine fibres than to diminish the torsion by lengthening out
the fibre.

Astatioifm. The two essential requisites for astaticism
are that the magnets shall be of equal magnetic moment
and shall be parallel.

1. Horizontal Systems, — After having completed the sys-
tem it should be suspended in a glass tube and astatioized
before being placed in the galvanometer. On first suspend-
ing the system it will be found that one of the sets of
magnets controls ; this set should be slightly weakened by
successive approaches of a magnet, until the system stands
east and west. If the period is then not as great as de*
sired, one of the sets of magnets must be slightly twisted
around the staff in such a direction that the same set
as before again controls. This set is again slightly weak-
ened until the system once more stands east and west.
This process must be continued until the period of the

APPENDIX III 618

system is sufficiently great. Each time the controlling
magnet is weakened the magnetic moments of the two sets
of magnets are made more nearly eqnal^ and each time
they are twisted they are brought more nearly into the
same plane. The twisting of one set of magnets around
the glass staff upon which they are mounted can best be
accomplished by laying the system on a plane surface,
placing a small wedge under the end of one set of magnets
and heating it until the shellac becomes viscous.

The length of the period which must be obtained de-
pends upon the sensibility required and the location of the
instrument. When very great sensibilities are required,
the system must be astaticized to a long period, for by
doubling the period the sensibility is increased four times.
On the other hand, if the location of the galvanometer is
in the neighborhood of electric railways, transformers,
machine-shops, etc., where large masses of iron are moved,
it will often be necessary to astaticize to a long period, not
for the purpose of attaining high sensibility, but to re-
duce to a minimum the effect of outside magnetic dis-
turbances. The length of period also depends upon the
moment of inertia of the system. With systems weighing
from 20 milligrams to 50 milligrams, a period of 15 to 20
seconds is about as great as can be maintained for any
length of time.

2. Vefiical Systems. — Systems of type (4) were first suc-
cessfully used by Weiss, who attained great sensibilities.
They consist of two or more long magnets fastened to a
thin lamina of mica, and suspended so that the magnetic
axis of the magnets shall be vertical. Each magnet, if its
magnetic axis is vertical, will be in neutral equilibrium with
respect to a horizontal field. The astaticism of these sys-
tems does not depend, as in the horizontal magnet systems,
upon the equality of the magnetic moments of the two mag-
nets, and they therefore have the advantage that a slight
weakening of one of the magnets does not destroy the as-
taticism, provided the magnetic axes remain parallel. These

514 A MANUAL OF EXPERIMENTS IN PHYSICS

systems^ when constructed^ should be snspended in an astat-
icizing tube, when it will be seen which set of poles controls.
The system should then be placed upon a plane surface, and
the controlling poles pressed nearer together ; the period
is then again taken. By a series of steps of this kind,
which is often long and tedious, the system may be astati-
cized until it becomes aperiodic. These systems will be
foHnd very satisfactory in places subject to local magnetic
disturbances.

On account of the difficulty in getting the magnets of
the Weiss system perfectly parallel, Broca has proposed his
consequent pole yertical system, in which the parallelism is
not of so much importance. Before mounting these mag-
nets on the system, they are suspended from their centre
in a horizontal position, and the consequent pole displaced
towards the centre by stroking with the same magnet used
to magnetize it. Obviously, if this pole is exactly at the
centre, the magnet will be in neutral equilibrium in any
position in a uniform magnetic field. Two, or four, such
needles are then mounted on a thin lamina of mica, and
the astaticism completed as for the Weiss systems. Such
systems may be used with one, two, or three pairs of bob-
bins. If used with three pairs of bobbins, the diameter of
the central bobbin should be equal to -/^ x diameter of the
outer bobbins. For galvanometers having equal resistance,
that with one pair of bobbins is 1.4 times as sensitive as
that with two ; with three pairs of bobbins, it is about twice
as sensitive as with two pairs of bobbins. Hence, on ac-
count of its greater simplicity, that with one pair is to be
preferred.

Sensibility. Thereare severalfactors thatenterintothesen-
sibility of a galvanometer, among which may be mentioned :

The magnetic constant of the coils, depending on the
form, winding, etc. ; the magnet system ; the method of
observing the deflections.

The sensibility is defined as the current required to pro-
duce 1 millimetre deflection on a scale 1 metre distant,

APPENDIX III

615

when the period is 10 seconds. If this current is observed
for any other period, T, it is reduced to a lO-seconds period

by multiplying by — rp:. (The magnetic moment of a mag-
lUU

net varies inversely as the square of its period.)

It is obviously unfair, however, to the heavy systems to
compare a light system and a heavy system at the same
period of 10 seconds, as a heavy system can generally be
used at a loi^ger period.

The resistance of the galvanometer is another factor en-
tering into the sensibility. In order to compare galvanome-
ters with coils of the same form and volume, but wound with
wire of different sizes, the sensibility (as defined above) may
be reduced to that of a galvanometer of the same type, whose
resistance is 1 ohm, by multiplying by ^^Ti; for, assuming
that the thickness of the insulation bears a fixed ratio to
the diameter of the wire,

the sensibility oc number of turns.,
the resistance oc (number of turns)' ;
i. e., the sensibility oc ^721

The sensibility may be obtained as follows :

E is a standard ^ ^ E

cell, or one whose
E.M.F. is approx-
imately known;

Fia. 191

Rj and r, two re-
sistances in series
(Rj generally 10,-
000, and r the 1-
ohm coil of an or-
dinary resistance-box ; if G is a high-resistance galvanome-
ter, then Rj may have to be 100,000 ohms).

j?2 is & resistance connected in series with the galvanome-
ter.

i20=: resistance of galvanometer.

i = deflection observed on scale 1 metre distant,

Q 3= ourr^ut to pro^UQO I QiiUimetre d^flectiosu

51« A MANUAL OF EXPERIMENTS IN PHYSICS

B,^'-^^

( Rq \1 1

R,)\R^+ Ro) R^l*

r+Jtt+Rt,
Br 1 I, . *, X

ILLUSTRATION

JTs 1.4 volte.

Bis= 10,000 ohms ; r a 1 ohm. %

B^Tz 3000 (^msi ^■BSohma.

Period sr 10 seoonda
Scale distance s 76 centimetrea

<^-l» X 8x S'y X W X lOO''^-^ ^ 10-»ampen.

This sysUm was a Weiss system, made of two No. 13 sewing-needlei,
27 millimetres long, 1.5 millimetres apart. Mirror was about 4 milli-
metres X 2 millimetres, and weighed 8 milligrams.

Galvanometer contained 4 bobbins, each 18 millimetres external
diameter, and 4 millimetres internal diameter, containing 600 tnms of
wire. Three sizes, 86, 88, 80, of copper wire were used in winding the
bobbins. Resistance of each bobbin was 12 ohms

TABLES

Mensuratiom
Circle: radius, r; circumference, 2itr\ area, arr-.
Ellipse: axes, 2a and 2b; area, vab.

Sphere: radius, r; surface, 4rr*; volume, -irr".

Ellipsoid: axes, 2a, 2b, 2c ; volume, -robe.

8
Spherical segment: radius, r; height, a; area, 2irra.
Cylinder: i-adius, r; height, a; surface, 2ir?-a + 2irr« ; volume, irr»a.
Circular cone: radius of base, r ; height, a; surface, »rVr*+a*+irr»;

volume, lirr*a.
8

Mbchanical Units
Lengt^i
1 inch s 2.540 centimetres.

1 centimetre = 0.3987 hich.

1 mile = 160981 centimetres = 1.61 kilometres.

1 kilometre = 0.6214 mile.

Area
1 square inch = 6.451 square centimetres.

1 square centimetre = 0.1550 square inch.

Volume
1 cubic inch = 16.886 cubic centimetres.

1 gallon = 4548 cubic centimetres = 277.46 cubic inches

I cubic centimetre = 0.0610 cubic inch.
1 litre =1.7608 pints.

Mass
1 pound =458.69 grama. •

1 ounce ss 28.85 grams.

1 gram = 0.08527 ounce — 0.002205 pound.

518

A MANUAL OF SXPERIMEKTS IN PHYSIG8

Force
1 pouDdal = 13825 dynes.

1 gram's weight =980 *'
1 pound's weight = 444518 "

Work and Energy
1 foot-pound = 1.883 x 10* ergs = 1.883 joules.

= 0.1388 kilogram -metres.
1 kilogram-metre =7.283 foot pounds.

P&uier or Actitiiy
1 horse-power = 746 waits.

= 38000 foot-pounds per minute.
1 watt = 0.0013406 horse-power.

Ill
Elastic Gonstamts or Solum

Bulk- mod aloe

Coefflcient of Rigidity

Toang^Modnlot

Brass

10 X 10"

4 xlO"

14.6 X 10"

18,4 X 10"

8.7 X 10"
2.4 X 10"
7.7 X 10"
8.2 X 10"

10.4x10"
6 xlO"

Glass

Iron (wrought). . .
Steel

12.6 X 10"
22 xlO^'

DlMSITIlS

MidM

Aluminium 2.58

Brass (about) 8.5

Brick 2.1

Copper 8.92

Cork 0;24

Diamond 8.52

GliLSs, common 2.6

•* heavy flint 8.7

Gold 19.8

IceatO°C 0.91

Iron, cast 7.4

Iron, wrouglit 7.86

Lead 11.8

Nickel 8.9

Oak 0.8

Pine.. 0.5

Platinum 21.50

Quartz 2.65

Silver 10.53

Sugar 1.6

Tin 7.»

Zinc 7.15

Mean density of earth is 5.5270.

TABLES
Liquids

619

Alcohol at aO^ C 0.789

Carbon bisulphide. 1.39

Ethyl ether at 0° C 0. 785

Glycerine 1.26

Mercury

Sulphuric acid

Water at 4®C

Sea water at 0"" C .

18.596
1.85
1
1.026

Water at other temperatures, see below.

Gases eUO^ G. and 76 eenHmetres of Mercury Pressure

Air, dry 0.001293

Ammonia 0.000770

Carbon dioxide 0.001974

Chlorine 0.008188

Hydrogeu 0.0000895

Nitrogen 0.001257

Oxygen 0.001430

Water at Different Temperatures

Degrees

OC 0.999878

1 0.999988

2 0.999972

8 0.999998

4 l.OOQOOO

5 0.999992

6 0 999969

7 0.999938

8 0.999882

9 0.999819

10 0 999739

11 0.999650

12 0.999544

18 0.999430

14 0.999297

15 0999154

Degrees

16 C.

17 .

18 .

19 .

20 .

21 .

24
25
26

27
28
29
80
81

0.999004

0.998889

0.998668

0.998475

0.998272

0.998065

0.997849

0.097628

0.997886

0.997140

0.99686

0.99659

0.99682

0.99600

0.99577

0.99547

52U

A MANUAL OF EXPERUfENTS IN PHYSICS

Surface Tension
Liqvidi with Air

Liqaid

Alcohol, Ethyl

Benzene

Glycerine

Mercury

Olive-oil

Petroleum

Water

Temperature

Degreos
20 C.
15
17
20
20
20
0
20

7*. in Dynefl per Cm.

21.7
28.8
63.14
450
31.7
28.9
76.6
74

VI
Acceleration Due to GRAvrrr

Latitude

Equator

0°0'
20° 52'
85° 41'
87° 20'
88° 58'
40° 28'
41° 49'
45° 81'
48° 50'
61° 28'
52° 80'

978.07

Sandwich Islands

Tokio

Lick Observatory

Washington. D. C

Allegheny, Pa

ChicftfiTO

978.85

979.94

979.92 (reduced to sea-level)

980.10

980.16 (reduced to sea-level)

980 87 *• *• •• •*

Montreal

980.75

Paris

980.97

Kew

981 20

Berlin

981.27

TABLES

OonaonoM ior Laroi Abcb or Vibration
If observed period of vibration is 2* for arc of swing a, the period

for an arc infinitely small is (r-iTr). where /r=|8in» j + ^sln*y

Degrees

0.00190 •

0.00012

0.00261

0.00080

0.00822

0 00068

0.00400

0.00008

0.00487

0.00188

0.00683

DtameCer

Capillart DitPRnsiON or Mrrcurt ih Glass

Hrighi cf Menisetis in MiUimetres
I 0.4 I 0.6 I 0.8 I 1 I 1.2 I 1.4 I 1.6 | 1.8

Gorrectians to be Added

mm.

0.88

1.22

1.64

1.98

2.87

....

0.47

0.66

0.86

1.19

1.46

1.80

....

0.27

0.41

0.66

0.78

0.98

1.21

1.43

• • . •

0.18

0.28

0.40

0.68

0.67

0.82

0.97

1.18

• • • •

0.20

0.29

0.88

0.46

0.66

0.77

• « * •

0.16

0.21

0.28

0.88

0.40

0.46

0.62

....

0.16

0.20

0.26

0.29

0.88

0.87

....

0.10

0.14

0.18

0.21

0.24

0.27

. • • •

....

0.07

0.10

0.18

0.16

0.18

0.19

....

0.04

0.07

0.10

0.12

0.18

0.14

6S2

A MANUAL OF EXPERUIENTS IN PHTSIGS

Baboiotrio Corueotiohs

1. Correction for Temperctture

Mercury— Brass scale correct at 0° C.

TempenitQre

Degrees

15 C.

0.178

0.181

0.188

0.186

0.188

0.191

0,198

0.190

0.198

0 196

0.198

0.201

0.208

0.206

0.202

0.206

0.208

0.210

0.218

0.216

0.218

0.214

0.217

0.220

0.228

0.226

0.229

0.231

0.226

0 229

0.282

0.285

0.288

0.241

0.244

0.288

0.241

0.244

0.247

0.251

0.254

0.257

0.250

0.268

0.256

0.260

0.268

0.267

0.270

0.261

0.265

0.269

0.272

0.276

0.279

0.283

0.278

0.277

0.281

0.284

0.288

0.292

0.296

24.

0.286

0.289

0.298

0.297

0.801

0.805

0.309

Corrections are to be subtracted from observed readings — ««., if
reading at 19'' is 76 centimetres, the *' corrected " reading is 76 - 0.235
= 75.765 centimetres.

. Correction for

TdrioHon in g.

Latitude

85° or 55°

40° or 50°
. 45°

0.065
0.082
0

0.066
0.088
0

0.067
0.084
0

0.068
0.085
6

0.069
0.035
0

0.070
0.085
0

TABLES

Frxqubmcus or Middu Ootatb

628

Name

Interval

SUndard
Freqaency

K6nig*8
Frequency

Tttnpered Intervals

Do, Ut,

261

266

2^ = 1.06946

1.126

293.6

288

2* = 1.12246
2^ = 1.18921

1.26

326.26

820

2^ = 1.25992

1.883

848

841|

2^ = 1.88484
2* = 1.41421

Sol

1.6

391.6

884

2^ = 1.49881
2^ = 1.68740

1.667

436

427|

2^ = 1.68179
2** = 1.78180

Si
Do

1.876
2

489.4
622

480
612

2^ = 1.80776
2

Velocity of Souhd

Air

Hydrogen 0

IllumiDatiDg gas . . 0

Oxygen 0

Alcohol (absolute). 8.4

Petroleum 7.4

Degrees Cm. per Sea

.. OC. 83.260

128,600

49,040

81,720

126,400

139.600

Degrees Cm. per Sea

Water 4C. 140.000

Brass 350,000

Copper 20 366,000

Glass 606,000

Iron 609,300

ParafBne 16 130,400

XII

AtIRAOI COKPtlCIBllTB OT LlMSAB EXPIHSION BkTWXEN 0® AND 100^ C.

Aluminium. 0.000023

Brass 0.000018

Copper 0.000017

Glass 0.000009

Gold 0.000014

Iron (soft). 0.000012

Iron (cast). 0.0000106

Lead 0.000029

Platinum... 0.000009

Silver 0.000019

Steel 0.000011

Tin 0.000022

Zinc 0.000029

5S4

A 1£ANUAL OF EXPERIMENTS IN PHYSICS

xni

Atsraoi CoimciKimi or Cubical Ezpinsion or Liquids

Alcohol 0'-80*C. 0.00105 [Mercury.. OMOO^C. 0.0001818

Bthyl ether . . 0*^88* 0.00210 | TurpeoUDe 9"*-!^ 0.00106

XIV

Atkriok SpBcinc Hkatr

Alcohol....
Aluminium

Brass

Copper

GermaD-silrer

Glass

Gold.

Ice

IroD

OMOO^
0^-100^

0*-100«

0.616

0.3186

0.09

0.0038

0.0046

0.20

0.0816

0.504

0.1180

Lead ....
Mercury.
Parafflne.
Platinum
Silver....

Tin.

Turpentine
Zinc

0^-100**
20"- 60**

OMOO*
0^-100*
0^-100*

00815

0.0883

0.688

0.0828

0.0568

0.0659

0.467

0.0085

XV
SpBCiriG HiATS or Gists

XVI
FunoN Constants

Aluminium

Copper

Gold

Ice

Iron

Lead.

Mercury . . .
Platinum. . .

Silver

Sulphur. . . .
Zinc

Fasfon Polot

600° C.
1050°
1045°
0°
1400«^160a>

825°
-89°
1775°

954°

1160

416°

Ut«nt RMt of

28<83
5.86
2.82

27.2

24.7
9.87

TABLEii

5%5

xvn

TiroEitATioN CoHCTAirra

Alcohol (ethyl)
Carbon dioxide
Chloroform . . .

Cyanogen

Ether (ethyl). .

Hydrogen

Mercury

Oxygen

Sulphur

Water

XVlli
VAFOK-PiutssuBi or Watsk

Tetnpemtiire

PraMore in Centime-
trw of Mercury

Temperature

Preesore in Centime-
tres of Merrury

DegreM
-5C.

0.810

Degrees
98 C.

70.718

0.467

78.816

0.651

99.2

78.846

0.914

99.4

74.880

1.786

99.6

74.917

8.151

99.8

76 467

6.486

100

76.000

9.108

100.2

76 547

14.888

1004

77.096

28.881

100 6

77.660

»>487

100.8

78.207

52.647

101

78 767

68 886

log

81.609

68.188

110

107.54

fi26 A MANUAL OF EXPERIHENTS IN PHYSICS

TTT

YATcm-TwuBUBM OF Hbcdbt
In mUifnetret^EegnauU and HerU (a); Banuay and Taung {b)

D«irMi

Degreet

OC.

0.0002

• • • •

200 C.

18.26

17.02

0.0006

, , , ,

210

26.12

•• • .

0.0018

• • • •

220

•84.9

81.96

0.0029

. . • •

280

46.4

. • • •

0.007

0.008

240

68.8

• > • •

0.014

0.016

260

75.8

. . > •

0.028

0.029

260

96.7

. . ■ •

0.061

0.062

270

128.0

128.9

0.098

280

166.2

167.4

0.168

0.160

290

194.6

198.0

100

0.286

0.270

800

. 242.2

246.8

110

0.470

• • . •

810

299.7

804.8

0.779

0.719

820

868.7

878.7

180

1.24

. . * •

880

460.9*

454.4

140

1.98

1.768

840

548.4

548.6

160

2.98

....

860

668.2

65ao

leo

4.88

4.018

860

797.7

....

170

6.41

. . . •

870

964.7

... a

180

9.28

8.686

880

1189.7

....

190

13.07

....

890

1846.7

....

HkAT or GOMBXHATION

1 gram cfmbstancd eombinea with equivalent O or BO«

TABLES

527

XXI
Tbibmal Gonductititiis

Silver 1.8

Copper 96

Iron. 20

Stone 006

Ice 008

Water 003

Glass 0006

Wool 00012

Paper OOOOW

Air 000049

XXII

Dui.BGTRio Constants (Electrostatic Ststrm)

Snbelance

SubBtaoce

Glass (about)

Mica

2.6

2.7

Water

Alcohol

Parafflne. ... . . . r r - . .

Turpentine

2.4

Rubber

Petroleum

2.1

Shellac

Hydrogen

0.9998

Wood

lUuminutitig gas

Carbon dioxide

(Vacuum)

1.0004

1.0004
0.9994

XXIII
Elbctroltsis Constants

Atoraio Weight

Valency

Chemioal Eqaivalent

Chlorine

Copper (cupric).

Hydrogen

Iron (ferric)

Lead

Oxygen

Potassium

Silver

Sodium

Zinc

85.87

68.18
1

66.88
206.89

16.96

89.08
107.66

64.88

1
8
2
2

1
1
1
2

86.87

81.69
1

18.68

108.20

7.98

89.08
107.66

82.44

Elkctro-Cbkmical Equivalents

Chlorine 0.008676

Copper (cupric). .. . 0.008261
Hydrogen 0.00010862

Iron (ferric) 0.001962

Oxygen 0.000828

Silver 0.011180

Zinc, 0.00888.

026 A MANUAL OF EXPERIMENTS IN PflYSICS

XXIV

Sf AHDABO BiBUTAHOn

s 0.9408 international ohma.

Siemens unit
B. A. unit
Legal ohm (1884)

= 0.9868
= 0.9972
International ohm (1898) = 1

= 106.8 centimetres of mercury, croBS-section
1 square millimetre, at (P G.

XXV

Bpioino Ck>in>uoTiynT, Referred to Mbrourt

Aluminium (soft) 82.85

Chopper (pure) 69

Iron 9.75

Meicuty 1

Nickel (soft) 8.14

Platinum 14.4

SiWer(80ft) 62.6

Tin 7

BicsiSTAiioi, ni Ohms at 0* C. of Wire 100 cm. Long, 1 mm. Diameter

Raie of Change in ReaisUnoe
per Degree CenUgnule

0.00888

0.00044

0.00065

0.00072

Aluminium 0.(

CJopper 0.02062

German-silver 0.2660

Iron 0.1284

Mercury 1.198

Platinum 0.1150

fiilver 0.02019

0.00877

XXVI

E. M. F. OF Ck)MMox Cells
Name K.]f.F.

Voltaic (zinc, acid, copper) 0.98 volta

Daniell (zinc, acid, copper-sulphate, copper) 1.09 "

Grove (zinc, acid, nitric acid, platinum) 1.70 "

Bunsen (zinc, acid, nitric acid, carbon) 1.86 **

Chromate (zinc, acid, chromic acid, carbon) 2 *'

Leclanch6 1.46 "

Edison-Lalande 0.70 "

Dry cell 1.8 "

Chloride of Silver 1.08 "

TABLES
XXVII

IHDICIS of REFEACnON

$%9

Substance

Wavelength

Index

Air, pressure 76 cm

•> << <t tf

•. «< (< *«

IMium, " *•

llv^irogen, " '*

Niirogen, ** **

Oxyiren, " •»

Alcohol

Centimetres
0.0000589
0.0000485
0.0000484
0.0000589
0.0000589
0.0000589
0.0000589

0.0000589
0.0000589
0.0000589
0.0000485
0.0000589
0.00004^5
0.0000434

0.0000589
0.0000485
0.0000484
0.0000589
0.0000485
0.0000484
0.0000589
0.0000485
0.0000484

1.0002922

1.0002948

1.0002962

1.000048

1.000140

1.000297

1.000272

1.868
1.449
1.624
1.648
1.384
1.888
1.341

1.5441

1,5881

1.5607

1.651

1.665

1.677

1.517

1.524

1.529

Degrees
OC.
0
0

0
0
0 .

Clilnroform

Carbon bisulphide

Water •. V

25
25
16
16

Rock salt

t( it

i( ti

Flint glass

ti t<

a «i

Crown glass

44 «(

xxvm

WlTS-LSNRTHS IN CkNTIMXTRIS

0.00003938825

/^.

0.00005269723

0.00003968625

^^i

0.00005270500

0.00004226904

0 00005890186

000004808000

0 00005896357

0.00004861527

0.00006568045

0.00005183791

0.00006870188

580

A MANUAL OF EXPERIHRNTS IK PMYSIG8

XXIX

NUMBRICAL CONSTlllTS

» = 3.14159 ; lo-,oir = 0.497149.
Vir =1.772; l/Vn =0 5642.
ir« = 9.8696 ; l/n- - 0,10132.

The base of the natural system of logarithms.
€ = 2.7183; K)ff,o€= 0.434294;

log«a- = -^''"? = 2.302585 log,o«.
i«gio«

- = 0.368.

XXX

Numerical Tablu

n«

yfn

i/n

n«

y/n

V»

1.414

50000

784

5.291

35714

1.732

833iW

841

5.385

34483

2.000

25000

900

5.477

33333

2.236

20000

961

5.568

322.%

2.449

16667

1024

5.657

31250

2.646

14286

1089

5.745

80303

2.828

12500

1156

5831

29412

3.000

Hill

1225

5.916

28571

100

3.162

10000

1296

6.000

2;;v8

121

3.317

90909

1369

6083

27027

144

3.464

83333

1444

6.164

26316

109

3.606

76923

1521

6.245

25641

196

8.742

71429

1600

6.325

25000

225

3.873

66667

1681

6.403

24390

256

4.000

62500

1764

6.481

23810

289

4.123

58824

1849

6.557

23256

324

4.243

55556

1936

6.633

22727

361

4.359

52632

2025

6.708

22222

400

4.472

50000

2116

6.782

21739

441

4.583

47619

2209

6.a56

21277

484

4.690

45455

2304

6.928

20833

529

4.796

43478

2401

7.000

2O408

676

4.899

41867

2500

7.071

2(X>00

625

5.000

40000

2601

7.141

19608

676

5.099

38163

2704

7.211

19231

729

5.196

37037

2809

7.280

18868

TABLES

531

XdifRRiCAL Tablks — {Continued)

y/n

l/n

2916

7.348

JN->li)

8025

7.416

18182

8136

7.483

17857

8249

7.550

17544

8864

7.616

17241

8481

7.681

16949

8600

7.746

16667

8721

7.810

16893

8844

7.874

16129

8969

7.987

15878

4096

8.000

15625

4225

8.062

15385

4856

8.124

15152

4489

8.185

14925

4624

8.246

14706

4761

8.307

U4U3

4900

8.367

14286

5041

8.426

14084

5184

8.4a5

13889

5329

8.544

13699

5476

8.603

ia514

5625

8.660

13333

5776

8.718

13158

1 »

n»

V«

l/n

! 77

5929

8.775

12987

6084

8.832

12821

6241

8. 888

12658

6400

8 944

12500

6561

9000

12346

6724

9.055

12195

6889

9.110

12048

7056

9.165

11905

7225

9.220

11765

7896

9.274

11628

7569

9.827

11494

7744

9.881

11364

7921

9.484

11286

8100

9.487

11111

8281

9.539

10989

846-1

9.592

10870

8649

9.644

10753

8830

9.095

10688

9025

9.747

10526

' 96

9216

9.798

10417

9409

9.849

10809

9604

9.899

10204

9801

9.950

10101

LOGARITHMS lOOO TO 1 lOO.

100

101
102
103

104
106
106

107
108
109

ooooo

130

4
173

604
030

452

870
284
694

100
902

217

647
072
494

912

325
735

141
543
94 r

389

043

087

260

689
115
536

303

732
157

578

995
407
816

346

432

86o

oi 2S4

475
903
326

518

945
368

56 i

(r8
410

823
243
^'53

4''3
h02

775
199
620

036
449

857

262

^^
060

817
242
662

078
490

898

302
703
irx>

703

02 119

531

745
160
572

979

3H3
7S2

787
202
612

019

423

S22

953
366

776

181

583
981

938
03342

743

222
623
021

682

LOGARITHMS 100 TO 1000

12
13!
141
15i
16!

oooocx)43

0414 0453
0792 0828
11391173

17
18
19

461
1 761

2041 2068

2304 2330

25532577
27882810

3010 3032

1492
1790

0086

0492
0864
1206

0531

0899 0934
1239

1523
1818
2095

2355

2833

3054

01280170

1553
1847
21

238<

2601 "2625

2856

0 2405
2648
2878

3075

0569

0934
1271

1584

1875

148

3096

0607
0969
1303

1614
1903
2175

2430
2672
2900

3118

0253 0294

0645
1004
1335
1644
193 1

2201 2227

24552480
26952718
2923 2945

3139 3160

0682
1038
1367

1673
1959

0334

0719
1072
1399

1703
1987
2253

2504
2742
2967

3181

123 4 6 6

0374

0755
1 106
1430

1732
2014
2279

2529
2765
2989

3201

789

Use preceding Tabic

37 10
3 6 10

369
368
35 8

15 19 23|26 30 34

14 17 2IJ24 38 31

13 16 iQ 23 a6 39

12 15 18121 ?4 77

II 14 17 20 23 25

13 16

2 5 7|To 12 15
257 9 12 14
9 II 13

247

246

18 21 34

17 30 22
16 19 21

16 18 20

»» 13 '5 >7 '^J

21
22
23

3222 3243

3424b444
361

24
25
_26

27
28
29

73636

3S02 3820

3979 39Q7
41504166

3263
3464
3655
3838
4014
4183

3284 3304
3483 3502
3674 3692

3324
3522

37"

3345
3541
3729

3365
3560
3747

3385
3579
3766

3404
3598

3784

246
346
346

10 13
10 13
9 "

403

420042

4314 4330
4472 4487
4624 4639

4346
4502

4f>54

43624378
45184533
4669 4683

4771

47864800

48144829

31
32
33
34
36
36
37
38
39

491
5051

5185
5315
5441
5563
5682

44928

14942
5065 5079
519815211
5328 5340

5453
5575

5694

5798 5809

5911

5922

j4q

41
42
43
44
45
46
47
48
49
50
61
62
53

6021

6031
61286138
6232 6243
6335r34_5

)444
65326542
6628 6637

67'30
6812I6821

690269

6990 6998

076 7084
7160 7168
72437251

7324 7332

5465

5587

5705

5821

5933

38563874

4048
16

3892
4065
4232

3909

3927

4082 4099
4249 4265

3945
4116
4281

3962

4133
4298

2 4 5

2 3 5
2 3 5

9 "
9 10
8 10

4393
4548
4698

44094425

4564 4579
47134728,

4440
4594
4742

4456
4609
4757

235
235

I 3 4

4843

48574871

4886

4900

« 3 4

8 9

7 9

7 9 10

i6 ii<
«5 »7
15 17
14 16
14
n »5

13 M
12 14

12 11

49554969
50925105

52245237

4983
5119
5250

4997 501 1
51-
5276

5132 5145
5263

5024

5159
5289

5038
5172
5302

> 3 4
« 3 4
134

53535366
5478 5490

5599 561 1

5378
5502
5623

57175729
5944

58325843

6042

6149
6253
6355

6454
6551
6646

5951
6053 6064

61606

6263 6274
6365

170

274
6375

6464 6474
6561 6571
6656 6665

6739I6749I6758

6830 6831

6920 6928J6937

7007 70167024
71 10
7193
7275

7356

7093 7101

71777185
72597267

73407348

5740
5855
5966

6075
6180
6284
6385
64~84
6580
6675

5391
5514
5635

5403

5527
5647

5416

5539

5658

5428

5551
5670

« 3 4
1 2 4
I 2 4

57525763
5866 5877
5977 5988

5775
5888

5999

5786
5899
601

60856096

6107

6117

6767
6857
6946

7033
71 1 8
7202

7284

7364

6191 6201
6294 6304
6395 6405

6212
6314
6415

6222

6325
6425

6493 6503
6590 6599
6684 6693

6776678
6866 6875
6955

7042

6964
7050

6513 6522

66096618

6702 671

6794.6803

688416893

697 2 1698

70591706

123
» 2 3
1 2 3

» -^ 3

1 2 3

» 2 3

I 2 1

I 2 3I
I 2 3|
» 2 3

7126 7135
72I072I8I
7292 73001

7143

7226

7308

737273807388

7152
7235
7316

I 2 3

1 2 3

13 3

12 3

7396

5 6

S
5

45
4 5
4 5
4 5
4 5

II I
10 12
10 II
to II
10 II

9 lO
9 la

9 lO

7
7

7 S

7
6 7

LOGARITHMS 100 TO 1000

688

123

4 6 6

7 8 9

7404 7412

7419

7427

7435

7443

7451 7459

7466

7474

345

s 6 7

66
67
68

7482 7490
7559 7566
7634 7642

7497
7574
7649

7505 7513
7582 7589
7657 7664

7520

7597
7672

7528 7536
7604 7612
7679 7686

7543
7619
7694

7551
7627
7701

4 5
4 5
4 4

69
60
61

62
63
64
66

7709 7716
7782 7789
7853 7860

7924 7931
79938000
80628069

7723
7796

7868

7731 7738
78037810
78757882

7745
7818
7889

7752 7760
7825 7832
7896 7903

7767
7839
7910

7774
7846

7917

4 4
4 4
4 4

7938
8007

8075

7945 7952
8014 8021
8082 8089

7959
8028
8096

7966 7973
8035 8041
8102 8109

7980
8048
8116

7987
8055
8122

8129 8136

8142

8149815618162

8169 8176

8182

8189

' I

3 41
3 4

6
6

66
67
68
69
70
71
72
73
74
75

8i<
826]

95 8202

I8267

8325833

8209
8274
8338

8228
8280I8287I8293
8344835118357

821582228

8235

8241

8299 8306
8363 8370

8248
8312
8376

8254
8319
8382

3 4
3 4
3 4

6
6

8388 8395

76
77
7Q
79
80
81

82
83
84

9138 9143
9191 9196
9243 9248

JB5
86
87
88

89
90
91

92
93
94

96
97
98

8451

851

8457

85738579
8633 8639

8692
S75'i

88088814
8865 8871
8921 8927
8976 8982
9031 9036
90859090

9294

38519

8401
8463

8525

840784148420
847084768482
8531 8537I8543

8426 8432
8488 8494
85498555

8439
8500
8561

8445
8506

8567

3 4
3 4
3 4

8698

8585,8591
864518651
8704871087161

8597*8603

8657,8663

I8722

8609 8615

866g 8675
8727 8733

8621
8681
8739

8627
8686
8745

3 4
3 4
3 4

8756

8762I 8768 8774! 8779

8785 8791

8797

8802

8820:8825883118837
888288878893
8938 8943 8949

8876^
8932'^

8842 8848
8899 8904
8954 8960

8854
8910
8965

8859

8971

8987
9042
9096

9149
9201
9253

9299

9304

9345 9350
9395 9400
9445 9450

9355
9405

9455

9494 9499
9542 9547
959c 9595

9504
9552
9600

9638 9643
9685 9689
9731 9736

96529657,1

9647

9694,969919703:

9741

9777 9782

9786 9791

9823 9827
9868 9872
9912 9917

99569961

89938998

9004

9058

9101 9106 9112

9047 9053

91549159I9165
92069212I9217
9258 926319269

90099015
90639069
91179122
9170 9r75
9222 9227
9274 9279

9020

9074
9128

9025
9079
9133

9180
9232
9284

9186
9238
9289

3 3.
3 3
3 3

930^93159320

9325 9330

9335

9340

936o|9365;9370
.20

946o[9465,'9469
9518
9566957
9614

9375 9380
9425 9430
9474 9479

9385
9435
9484

9390
9440
9489

3 3'

3 3

« 3,

2 3

95099513

9557 9562

9605

9609

9523 9528
9571 9576
96199624

9832
9877
9921

9661

9708

974519750 9754

9795 9800

9533 9538

9581 9586

9628:9633

119675 19680

97i3|97i7;9722 9727

9759I9763 9768(9773

6
6

_5
5
5
5

5
5
5
5

9666 967

983698419845
988 1I9886 9890

3
3

3 3

3 '3 '

3 3

3 $

9805 98o9'98i4 9818

9850 9854'9859 9S63
,9894 9899 9903 9908
9926 9930;9934 9939 9943 9948 995

« 3
2 3
a 3

9965 9969 9974 9978

9983 9987.9991 99',6

684

3 U tJ
NATUKAL SINES

11 looS I

LI
12
13

14
16
16
17
18
19

21
22
23
24
25
26

27
28
29

30
31
32
33^
34
36
36
37
38
39
JO
41
42
43

00000017

175019

0349^366

0523

92; 0209
' 0384

o54i|o558
06980715 0732
0889I0906
1063! 1080
1236

0872

1219
1392

1564I1582

754
1925
207912096
2250 2267
24192436

2756

2605
2773

2924 2()40

3090 3107
3256 3272

3420 3437

4384

4695
4848

544^
5592
573^'
58781589'
60 iS

^Ji57
6293

6428
6561
6691

6947

0035

0052 0070,0087

1409

3584 3600
3746I3762
3923
4083

39^
4067
4226 4242

4399

45404555

4710
4863
5000 5015

5150 5165
5299 5314

54^i

5G06

5750

603

6170

6307

. J0262<

041910436

0610

"75010767,0785 oSo2|o8

094io<;5S;o976|oc;93
1097 1115 1132

0227|0244|(

0401
o576fr>593

1253
1426

1599

^171
1942
2113
2284

2453
2622
2790

2957
3123
3289

1271
1444
1616

1788

1959 1977
2130 2147
23002317

14702487
2639 2656

345J
3616
3778
3939
4099
4258
44J5

4571
4726
4879

5030
5180
5329
547(>

5621

5764
5906

f)046
6184
6320

^U55

6441

^)574'^'587|

670416717!

682068336845!

2823
2990
3140 3156
3322

2807
2974

3305
3469
3633
3795
3955
4115

4741
4894

50^5
5195
5344
5490
5633
5779
5920

6198
6334
6468

6959

68^8

24 30

0105 0122

0279J0297 0314 0332
0454 0471 1048810506
)62S 0645 cy)63 0680

1288

[461

1633

805

1305
1478
1650

149 1167

1323

1495
if>68

1340
1513

68q

1 1 84 1 201

I357II374
I530JI547

I702II719

1822II840 1857 1874 1891
2028 204512062

2I98l22l5!2232
23681238512402

2538:2554,2571

34S6 3502
364936^
3811,3827
3971 '3987
4131I4147
4274 4289 4305
4446I4462

4617

199412011
21642181

2334j235'

250412521 _ ^^ _ „
2672126892706 272312740 3
2840 2857 2874I2890I2907
30073024
3i73!3»9o|32o6
3338I3355
3518

45864602

4756
4(P9

4772
4924

5o6()|5075
5210

5358
5505

368

3843
4003

4163
4321
4478
4633

50905

5650
5793
5934

5225
5373
5519
5664
5807
5948
6088
6225
6361
648 1 1 6494 65
6600 66 1 3 1 6626 1 663
6730674316756'

60606074

62 1 1
6347

6871 6884' 6S96 6909

6972 I6984I6997 7009 7022

54'

01400157

Si9,o837|0854
loii 1028

30403057,3074

337

I 3223
13387

3239
3404

35351355^3567 3
3697137143730 3
3859 3875 3891 3
401914035405^1 3
4179J41954210 3
4337 4352 4368 3
4493I45094524
4648!4664|4679 3
47874802I4818I4833 3
493049554970,4985

52405255

5388

5534

5678

582

5962

6101

10515 120' 5 135
5270I5284
5402j54i7|5432
5548|5563'5577
5693|5707|572i
58355850,5864
59761599016004

6115,61296143 257

6239I6252 6266 62S0 257

')388|64oil64i4 247

650816521165346547 247

)6652 6665 6678 247

769I6782 6794 6807 246

6921:^)934 246

7034

704617059 246

12 3

369

6 9
6 9
6 9

6 9

6 9
6_9

6 9

6 8

6 8
6 8
5 8

5._8
5 8
5 8
5 8
5 8
5 8
5 8

12 15

5 8
5 8
5 8

5 8

5 7

8 10

NATURAT. SINES

586

12'

36 42'

12 3

4 5

46°

7071

7083

7096

7108

7120

7133

7145

7157

7169

7181

246

46
47
48
49
60
61^
^2^
63
64
55

7193
7314
7431

7206
7325
7443

7218

7337
7455

7230 7242
7349 7361
7478

74(>(i

7254
7373
7490

75477558
7660 7672

7782
7880 7891
7986I7997
100

8202

66
67
68^

60
61

62
63

^4^
66
66
67

J8
69
70
71
72
73
74
76
76
77
78
79

81
82
83
84
85
86
87
88

80908
8192
8290
8387

757o|73^'^'
7683 7^)9^

7793 7^«>4
791
80T8
8121

7902
8007
8111

8211I8221

7593
7705

7815

7923

8028

1311

8231

7C04
7716
7826

7278
7396
7513
7627
7738
7848

7934
8039
8141

8241

8290830018310

9272

933^'

9397

9455

951

9563

8396
8480 8490

8572 8581
86f)0 8669
87468755

882<
8910
8988
(/)63

.^8838
8918
8996!
;>070

8406
8499

8590
8678

87_63
8846
8926
9003

8320 8329

8415
8508

91359143
9205 9212
i;278

1)342
W03
i)4(n

9516
19568
9613J9617
9659 9^)64

9703 9707
9744 9748

978
9816
9848
:)877
9'>o3
)925
;945
9()62

9976 9977
)98(
)90^

9785
9820

)85i
)88o

W28
)047
)963

9078

9150
9219
9285
9348
9409
9466

9521
9573
9622

9668

9711

9751
9789

8425
8517

85998607
8686 8695
8771 8780
8854 8862
8934 8942
9011 9018
()o85 9092
9T57 9164
9225 9232
9291 9298

8339
8434
8526

7266
7385
7501
7615
7727
7_837

7944 7955
8049 8059
8151 8161

8251 8"26i
8348 83"58
8443 8453

7290
7408

7524
7638
7749
7859

79^>5
8070

8j_7i
8^1

7302
7420
7536
7649
7760
7869

246
246
246

246
246
2 4 5

7976
8080
8181
8281

245
235
235

2 3 5

8536

8545

8616
8704

8788

8870
8949
r)026

8625 8634
8712 8721
8796 8805

887S

8368
8462
8554
8643
8729
8813

8957

8886
8965

9354 9361
94159421
9472 9478

9HV)
9171
9239
93«4

9367
9426

9483

90339041
9107 91 14
9r78 9i84
9245 9252

8894
8973
994^
9121

931

9317

9191
9259
9323

9373 9379
9432 9438
9489 9494

9385
9444
9500

95279532
95789583

9627
9673

2^3
iK)77

9537
9588
9636
9687

9542 9548

9593I9598
964]

9646
9686 9690

9553
9603
9650
9694

8377
8471
8163
8652
8738
8821

235
235

2 3 5

I 3 4
I 3 4
I 3 4

8902
8980

9128

9265
9330

I 3 4
I 3 4
I 3 4

2 4

I 2 3
123
123

9391
9449
9505

123
I 2 3
I 2 3

9558
9608
9655

123
12 2
12 2

97159720

9755 9759
9792 9796

)987
)995

9823,98

9854:9857
9882^9885

9<P7|
9930
9949 iW
9965 ()9f:

J9978 997(

9988

99959996

6 98 29

9724
9763
9799

9833

9863

9890

9914

9936

9954

6 996S](/)69

[) 9980' 998 1

9989|r)9()o|9990

99969997

9732

r986o

)8S8

99iol()9i2

9932 9934

9952

9728

9767 9770
)8o3 9806

<)836 9839
9866 9869
9893 9895

99179919

9938

9940

W569957

9736
9774
9810

984^
9871
9898

9921
9942
9959

9699
9740
9778
9813

9845
9874
9900

9923
9943
9960

W9'

)999, 9999,9999 9999 '•«»,

' ' nearly

997
99S2
9991
9997

1.000

nearly

I I
I I
I I

I I
O I
O I

O I
O I
O I

)972:9973 9974
)9S3 9984,9985

)<792

)997

o o

II"^

! 1 9993 1 9993
''9998 99(j8

o o

000

i.ooo 1. 000 1. 000

iiearlv nearly near!)

8 10

8 10
8 10
8 10

8 9
7 9
7 9

7 9
7 9

7 8

6 7
6 7
6 7

5 7
5 6
5 6

2 3

2 2

J2 2

2 2
2

6S6

NATURAL COSINES

9994 9993
9986 9985

4
5
6
7
8
9^
10

9976 9974

11
12
13

16^

19^

30^

36^

39^

9903

9877 9874

9998

9962
9945

9960
9943

9925 9923

9900

9848 9845
98169813

9781

9778

9744 9740

9703

9699

96599655

9613

9608

12' 1 18

i.oc»

nearly

1.000, 1. <
nearly nearly

9998

9993
9984

9973
9959
9942

9921
9898
9871

9997 9997
9992 9991
9983 9982

99729971
9957 9956
99409938

99199917
9895 9893
9869 9866

984219839 9836
98 10; 9806 9803
9774 9770 9767
9736I9732 9728

9694.9690 9686
9650 9646 9641
9603959819593

24 30

[.000

learly

9999

9999 9999

9996 9996 9995 19995
9990 99891998819987
9980997999789977

9997

99901

99811

9969]

99549952J9951

9936*9934 9932

)!9968

299

9914
98901
986319860^857

H99I
)9888

983398299826

9799,97969792
9763I97599755
972497209715

968 1 1
9636

9588

9677
9632

9583

967396689664

9627

9578

48 54

9999,9999

9966

99659963
994919947
993019928

9885

99079905
98829880
98549851
98239820

9789,9785
97519748
97 1 1 19707

9622*9617
9573I9568

12 3

000

000
000
o o

o o

O I
O I

O I

0 I

1 I

112
112
112

112
12 2
12 2

2 2
2 2
2 2

95639558
9505
9-W9
9391

95
9455

955319548954'

95009494

94441943^

939:

9336 9330
9272 9265

9205

9135
9063

29537
94899483
9432,9426

938519379 9373'9367

95219516
94669461
9409I9403

9361I9354 9348:9342

9532
9478
9421

9527
9472
9415

123
I 2 3
123

9198

9128

(>056
8980

8910 8902
88298821
8746 8738

9323931

9259
9191191

7 93

92529245

849

9298 9291

11I9304

,9239
I78!9i7il9i649i57

9232 9225

9121I911
9048, 904T
897318965

8660 8^52

8572 8^63
8480 8471

838:

8377

4910719100

' 9033 9026,

^95718949

887818870,
8796,8788
S712I8704

8901

8290 828

8192
8090
798C
788c
7771

S181
3o8o

7060

79761

7869I
7760,
7649

754
743

73M

7536
7420
7302

8894,8886
8813*8805
872918721
86438634

8554I8545

84628453

8368I8358

8271 1826

8i7i|8

8070^805918049,

79657955
785978487837
7749|7738
7638,7627

75247513

7408

729017278

3i6i8

S625'86i6]
85368^2^6
8443 8434
83488339
8251 82411
151 8141I
8039

7944' 7934

7826

77277716

9285I9278

92199212

9' 50,91 43

9085 9078I9070

11I90038996

8934.89261 891 8

8854J8846I8838

878087711876318755
86861 8678I 8669

85998590,8511

: 2 3
I 2 3
I 2 3

9092
901"
8942
8862

I 2 4
I 3 4
I 3 4

8695
8607

I 3 4
I 3 4
I 3 4

85178508
84258415
8329 8320

^|_

I 3 4

84998490
8406 8396
83io'83oo

235
235
235

8231
8131
8028

7923
7815
7705

822l|82Il|8202
2l'8lIll8lOO

8018 8007 1 7997

7615,7604
7501 7490

38517373
72667254

79 J 2
7804
7694

7i93
7478
736
7242

235
2 3 5
235

7902
7793
7683!

7891
7782
7672

245

245
246

7581 7570

7466 7455

734917337
7230I7218!

7559

7443
7325
7206

246

246
246
246

4 5

4 6

5 6

5 6
5 6
5 6

5 7

6 7
6 7

8 10
8 10
8 10

193

7181I71697157

7145I7133 7120 7108I7096 7083

246

8 10

A^.^.— Numbers in difference columns to be subtracted, not added.

NATURAL COSINES

587

45=

46
47
48
49'
60
61
'52
63
54
55"

56
67
58
69
60
61

62
63
64

65
66
67

6947 6934
6820 6807
66gi 6678

7071

6561

6428 6414
6293 6280

6157
60

58781

143

004

5864

86004

6508

64946481

654765346521

6401 16388 6374I6361 6347I6334
6266 6252 6239

61296115 6101
5990^5976 596;

5850:5835

5736|571»!_5707 5693

55'63'5548

55925577
5446 5432
5299 52845270:5255

5150 5135 5120 5105

5000 4985 4970 4955 493914924

4848 4833 4818 4802 4787I4772

4695 4679 4664 4648L633I4617
4540 4524 45094493 4478 4462
4384I4368 4352 4337 4321 4305

4226 4210:4195 4179 4163

,3907

68 ,3746

69 3584

' 2250 2233

982

723005

24i9|2402|2385'2368 2351,23341231

n _ gj 21642147J21302113I2096
I 1994' 1977
840 182211805

75 "2588
76

77 '

78 2079I2062

79 1908

80 1736
8a^ji5^4
82
83
84

'85
'86

1374 I357|i340
1201 1184 1167
1028,1011 099
0872108540837^08 1 9[o8o2 0785I0767

1392
1219
1045

7059

7046

7034 7022

692

679416782
6665 '6652 6639

6909 6896
6769

594S 5934

5807,5793

5678:5664 5650

2
5821

5534I5519,

5388'5373|5358

5240^5225

50905075

40674051 4035401
38913875

37303714

3567|355i

3697

3387
3223

30571
2890!
2723,

3535 35

32063

70 .342013404 3387 337 J

71 I 3256 3239

■72 I 3090 3074

73 12924 29071

74 ''2756 27401 2723^2706126891

2028 201

2215 21
2045'

189111874,1857
1719 1702 1685
1547 1530 1513

0175 0157

24'

30'

7009 6997 6984 6972 6959

6858 6845 6833

6884I6871
675616743
6626I6613 6600 65876574

6756 6743 6730,671716704

aaaaIa/;./! A/Lrx^ a»q.'a..»^

"1^

6225 6211
6088 6074 6060

6468 6455 6441

6320:6307

6198 6184 6170

60466032
592059065892
5779 5764 5750
56355621 5606

5461
53M
5165

r.1^

4147

94003

38593843

3681
18

3355

3987
3827
3665

5505154905476

53445329

52io|5i95 5180

50305015
4863
4710I

5060 5045
4909 4894 4879
4756 4741 14726

46o2 4586|457i|4555
44464431I4415I4399
4289 4274 4258 4242

4131

411540994083

3502:34863469
33383322
90317331

1563

3040 3024

13007,29902974

2874I285712840 2823 2807

2672I26562639

257i|2554|2538|252i|2504;2487|2470 2453 2436

2284J2267

1 668
1495
1323
1 149

1650 1633
1478 1461

1305I1288
I 1132 1115

I I0993J0976 0958 094 1

0698I0680 0663 0645I0628 o6io|o593Jo576

)6 0488 047
034910332' 03 j 4 029710279 0262 '0244)0227 '02090 1 92

01400x22 0105 0087 0070 0052 0035 0017

42'

I
48 ; 64

3971

381 1 3795:

3955,3939

,3778

3649 3633 3616

3453

3305 3289

140.3123

3923
3762
3600

343

3272

3107

2957j294<
27902773
2622 2605

19591942:1925

1788,1771 1754
1616 1599 1582
1444 1426 1409

1271 1253 1236
1097 1080,1063
0924 0906,0889
O75o'o732|o7i5

,055810541

12 3

246

2 4

2 4
2 4
1_5

2 5
2 5
LJ_
2 5
2 5

2 5

3 5.
3 5
3 5
3._5.
3 5
3 5
3_5..
3_5._
3 5
3 5
3 5

4 5

8 10

6
6

-7_,.

7';"

7 ,
_7I

71
.11.

S|
8

81
'S\

81
J_l

8 I

9 I
J? J

9 »

9 I

9 12
9 12
9 12

36
3 6
3 6
36

3 6
3 6
3 6

3 6

3 6_
3^6

3 6

369

o 12

O 12

o 13
o 13
o_i3

o 13
o 13

_I3.
14
14

_.^±
14
14

_M_
14
14

_lt

14
14

2 14
2 14
2 14

2 14
2 14

2_I4

2 15
2 15
2 15

12 15

N.B. — Numbers in difference oolomnt to be subtracted, not added.

688

NATURAL TANGENTS

12' 18

24'

30'

42'

48'

54'

12 3

4 5

00000017

0035

0052 0070

1
2
_3
4
5
6
7
8
9

11
12
13

0175 0192

49
0524

0349 0367
0542

0209
0384
0559

0227 0244
04020419
0577 0594

0087

0105 [)I22

0140

0157

369

0262

0437
0612

0279 0297

0454 0472
0629 0647

0314
0489
0664

0332
0507
0682

J 6 9

369
369

069c) 0717
0875 0892
1051 1069

0734
0910
1086

0752 0769
09280945
1 104 II22

0787
0963
II39

0981
II

571

0998
175

I22S
1405
1584
17^3

1246
1423

r6o2

1781

1263

1441
1620

1281

459
1638 1655

1299
1459 1477

1317
1495
1673

0805I0822 0840
1016
1192
1370
1548
1727

369
369
369

1334
1512
169

1352
1530
1709

1799

1944 1962
2126 2144
23092327

1980
2162
2345

1817 1835
199^ 2016
2i8g 2199
2364 2382

1853

I87I

890

2035

2217
2401

2053 2071
2235 2254
2419 2438

1908
2089
2272

2456

0857
1033

I2I0

1388
1566

1745

1926 369

369
369
369

23^

29^

3b
31
32
33
34
36
36^
37
38
39^
40
41
^42
43

24932512
2679 2698
2867 2S86

3057
3249
3443
3640

3839
404(

1^5

4452
4663

4877

2530
2717
2905

254912568 2586
2773

2943

3076 3096
3269 32881
14f)3l3482!

2736 2754
2924

3115
3307

3134
3327

35023522

3659! 3679 3699
3899
4101

3719

3859
4061
4265

3879389913919
408 1 1

4286I4307I4327

^45I5

453^>

44734494

468447061472711748

4899492149424964

2962

2623

3000

3153
3346
3541

2605

2792281

2981

3365
3561

2642
2830
3019

2107
2290
2475
2661
2849
3038

369
369
369

369

369
369

723

191
3385
3581

3211
J404
3'

3739

3759

3779

3799

3230
3424
3620

3819

3 6
3 6
3 6

3 7 >o

3939
4142

4348

4557
4770
4986

3959

3978

4163 4183
4369 4390

4000
4204
4411

4020
4224
4431

4578 45^
4791 4813
500S 5029

4621

4834
5051

4642
4856
5073

3 7

4 7
4 7
4 7

5095
531

5543

5117:5139
3340I5362I
5566I5589;

5774
6oo<;
6249 ^^
6494 6

6745
7002
7265

753^^
7813
8098

8391
S693

5161

538

562

5844

8415206
53841540715430

5635 1 5658

8<?7 5890

6128

52285250

54525475
5681

5704

5272
5498
5727

579J5820'
603216056^)080 6104

fj2 73 1 6297 63 2 2 6346

fJ529|6544!6562J6594
6771 6796,

7028
7292

7563
7841
S127

^!6;
i6822l6

1 1 7080 7

-)847
107

7054

73J9 7346I7373

7590

7869178987926
8156" -

)l 76 1 817646

; 98

>8i85

3214

5914
61

5938 5961

6371
6619

68^

7133
7400

7673
7954
8243

8421^845118481

851]

^^72418754
9004 903619067

932593589391

9(^s:

)(n)\

9725

8541
8847
9163
9424194579490

8785
9099913

S8i6

526176
6395 6420
6644 6669
6899 6924
7159
7427
7701

7983

8273

8571
887
9195
9523

7186
7454

7729
8012
S302
S601
S910
9228
9556

975997939827

5295
5520

5750

5985

7 "

8 II

8 12

6200

6445
6694

6224
6469
6720

8 12

8 12
8 13

6950
7212
7481

6976

7239
7508

4 9 >3

4 9 '3

5 9 '4

7757 7785
8040,806c)
83328361
8632 '8662
894118972
926019293
9590I9623

5 »o 15

5 10 16

5 II 16

611 17

9861 9896,993019965

6 II 17

"3
»3

»3
»3
>3

»3
»4
>4

"14 18
,4 18
15 18

IS 18
15 19
15 '9

16
16
17

»7
18
18

18
'9

Natural tangents

5H9

I.OOOG

46
47
48
49
50
61
62
63

^-^
55

56
57
68
59
60
61
62
63
64|!

65
66
67
68
69
70
71
721
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88

I.0355
1.0724 0761
I.I 106

1.1504
1.1918

13349

2.24()0
23559

2-4751
2.6051

2.7475
2.9042

0035

0392
0761
"45

1544
i960

2393

2846
3319
3814

1.2799
1.327C1
1.3764
1.4281
148'26 4882

I-5399I
IJ

4335

5458

6003 6066

7391
8115

1.6643 6709

1.7321

1.804G

778807
1.9626
2X>503
2.1445

8887
9711
0594

1543
2566

3673
4876

3.0777] 0961
2914
5105

32709
3-4874
37321

4.01 oS 0408

4331
4.7046

5.1446
5.6713
6.313S

7.1154
8.1443

951449

n.43
14.30
19.08
28.64

6187
7625
9208

12'

0070

0428
0799
1 184

1585
2002

2437

2892
3367

3865
4388
4938

5517
6128

6775
7461
8190

8967
9797
0686

7583

0105

0464

0837
2224

1626
2045
2482

2938
3416
3916

4442

4994

5577
6191

6842 6909!

7532
8265

9047

9883
0778

3662
7453
1929
7297
3859
2066
2636
677
1.66
14.67
19.74
30.14

57.29 63.66

1642

2673

3789
5002

6325
7776
9375
1146
3122
5339
_78_48

0713
4015
_7?67
2422
7894
459^'
3002
3863
9845

11.91
15.06
2a45
31.82

71.62

£742

278
3906

5I2CJ

7929
9544

1334
3332

5576
811

1022

8288

2924

85.

135«

3962

5126

IO.(

I2.lf^J

24'

0141

0501

0875
1263

1667
2088
2527

2985

3465
3968

4496
5051
5637
6255

30'

0176

7603I
8341;
9128
9970
08721

0538
0913
_I303
1708
2131
2572

3032
3514
4019

4550
510S

5697
6319

6977

767

8418

1842
2889
4023

5257

6464 6605 1

8083
9714

36'

0212

1524
3544
5816

8391

1335

437414737

8716

3435
9124
6122

4947
6427
10.20

69 35

1243
15.89'

!.02
.80'

81.85 95.49

9210

0057
0965

1943
2998
4142
5386
6746
8239
9887
1716

3759
6059

8667

1653
5107

395"5
9758
6912

"5958
7769
10.39

12.71

T6;^3'5
22.90

38.19
114.6

0575
0951
234?

175^
2174
2617

3079
3564
4071

4605

5l6(;

5757
6383

7045
7747
A495
9292

0145
lo6ci

2045
3109

5517
688(j
8397

ao6i
1 910
3977
6305

42'

0247

0612
0990
1383

48'

0283

0649
1028
1423

1792
2218
2662

3 J 27
3613
4124

1833
2261
2708

3175
3663
4176
405()lj7i5
5224 5282

5818
6447

5880
6512

7113; 7182
7820I 7S93
8572'^65o

93751 9458
6233 0323

II55!_I25
2T48J 2251
3220J 3332
4262(4383 4504

5649! 5782

54'

031CJ

7034
8556

7179
8716

0237 0415
2106

4197
6554

2305
4420
6806

89471 92321 9520

26351 2972

1976 2303
5483 5864
9594 0045
448O 5026

12 3 4 5

6 13 18

6 12 18

6 13 19

7 13 90

7 M a»

7 M aa

8 15 23
Xi6' 23

8 16 25

9 17 26
9 18 27

10 19 29

10 ao 30

11 21 32

" 23 34

12 24 36
'3 a6 38

14 27 41

15 29 44

16 31 47

»7 34 5»

»8 37 55,

20 40 (mi

22 43 ^"^'

0686
1067
1463
1875
2305

^22

3713
4229

477<'
534<^
5941
1>577

7251
796(.

_8_7_28

9542
0413
134S

2355

3445
4627

59if>

7326 24 47 7»|

24 30

39
4

8878

i>595
2506
4646
7062
9812

143.2

6252
0504

5578
1742

9395

0405 T066
7920 8548
6996 80621 9158
9152 0579! 2052
10.58 10.7H 10.99
1300 1300 1362I13 95
16.83 17.34I 17.89' 18.46
23.86 24.90I26.03 27.27
40.92 44.07 47.74 52.0S

6646
0970
614c
2432
0264

6285
3572

II.2<J

I9I.O

286.5^73.0

26 52 78
29^ 58 87
32 64 </>
36 72 108

41 82 122

46 94 KV)
53 107 itx)
62 124 186
73 146 21 u
87 175 262

25 3'

25 3»

26 33

~28 34

29 3t>
30
3»
33

34 43

36 45

"38 48

40 50

43 53

45 5^

48 60

5» ^4

~55 68

58 73

63 78

68 85

74 92

79 99

87 n>P

95 118

104 130

«»5 M4

129 161

144 180

162 203

"186 232

214 267

248 310

292 365

350 437

Difference - col-
umns cease to be
useful, owing to
the rapidity with
which the value
of the tangent
changes.

INDEX

Aberration. Chromatic. 458, 478

Al)errati<)n. Sphericnl. 441. 445

Absorption, £lectric, 400

AcceleratioD, Angular, Measure-
ment of, 79

Acceleration, Linear, Measurement
of, 70

Advantage, Mechanical. 1 11

Air-pump, Platform, 483

Amalgamating Zinc and Coppor,
495

Ampere. 318

Angle,Mt»nsurementof,VJ 1.149,464

AngstrOm Unit, 427

Anode. 403

Archimedes' Principle. IHO

Arcs, Corrections for Laii:c. 521

Area, Measurement of. 2o

Aspirator, 488

Astaticism. 513

Atwoml's Machine, 98

*' Back-lash," 50, 52

Balance, Theory and Use of, 17,

61,15l0<A!9.,4d9
Ballislic Galvanometer, 396
Barometer-corrections, Tables, 522
Barometer, To Fill a, 494
Barometer, To Read a, 157
Boiling-point, 262, 800
Boiling-point, Tables of. 525
Books of Reference, 488
Boyle's Law, 209
Broca. System of Magnets, 514
Bunsen-burner, 257

Caliper, Micrometer, 89

Caliper, Vernier, 34

( 'alorie, 258

Calorimeter, 284

Capacity, Electrical, 817, 384

Capillary Correction, 18i3, 521

Capillary Tubes, 204

Catiietomeier, 58

Cathode, 403

Cells, E. M. P., Tables of, 528

Cells, Standard, 501

Cements, 496

Centigrade Scale, 255

( 'en I i metre, 2

Centrifugal Motion, 105

Clie?nical8, Useful, 488

(lark Cell, 318. 501

Ceaning Glass, 23, 491

Clock-circuit and Contact, 484

Coincidences, Method of, 168

Collimator, 459

Color-sensation, 476

Combination, Heat of, 526

Commutator, Pohl's, 892

Comparator, 58

Condenser, 384, 896

Conductivity, Heat, Tables of, 627

Controlling Magnet, 507

Cooling Curve, 811

Coulomb, 818

Current- sheet, 840

Curvature, Measurement of, 44

Damping-key, 897, 399, 496

D' Arson val Galvanometer, 503,505

Density of Gas, 206

Density of Liquid, \SSet seq.

Density of Solid, 189 et seg., 193,

197, 201
Density, Tables of, 618
Deviation, 469

Deviation, Minimum, Angle of, 471
Dielectric Constant, 334, 336
Dielectric Constant, Tables of, 527
Dip-circle, 349
Dip, Magnetic, 849, 420
Discharge-key, 898
Dispersion, 470. 478

544

INDEX

Resistance of an Electrolyte,

Mance's Method. 885
ResistaiLce of Uniform Wire, 865
Resistance, Specific, 871. 388
Resistance. Specific. Tables of. 528
Resistance, Variation of, with

Clianges of Temperature, 871
Resistances in Parallel, Multiple

Arc. 388
Resistances, Sliding, 499
Resistances, Standard, Tables, 528
Restitution, CoeflScient of, 90
Rider of Balance, 156
Rigidity, CoetQcieut of, 168

Satxtbated Vapor, Law of, 806
Scale. Musical, 282, 528
Second. 2

Sensibility of Qalvnnometer, 514
Sensitized Paper, 76, 498
Sliunt, 320, 888
Siiverinp Mirrors, 498
Siren. 246
Smoked Glass, 74
Sonometer, 227

Sound. Velocity of, in Air, 236
Sound, Velocity of, in Rods, 239
Sound, Velocity of, in Wires, 243
Specific Heat of a Solid, 282
Specific Heat of Turpentine, 288
S[)ecific Heat. Tables of, 524
Spectrometer, Adjustments of, 459
Spectroscope. 475
Spectrum, 471, 477
Spherometer, 48

Spring-babince, the Use of a, 125
Standard Coniiitions of a Gas, 406
Stationary Vibrations, 218
Still, Mprcury.492
Still, Water, 484
Supplies, Useful, 488
Surface-tension, Measurement of,

204
Surface-tension, Tables, 520

Table, Laboratory, 489
Telescope, Construction of, 455
Temperature, Definition of, 255
Thermometer, Air, 279
Thermometer, Mercury, Correc-
tions of, 254
Thermometer, Mercury, Errors of,
255

Thermometer, Weight, 275
Tliomson's Method—Gal vanometer

Resistunce, 88d
Time, Measurement of, 18. 70, 168
Tuning-fork, 70, 229

Unitb, Electrical, 817, 318, 819

Units, Mechanical, 67

Units, Mechanical, Tables of, 517.

518
"Universal" Wax, 496

Vacuum, Reduction of Weights to,

156
Vapor-pressure, 306
Vapor - pressure, Mercury, Water,

Tables of. 525, 526
Velocity, Angular, Measurement

of, 79
Velr>city, Linear, Measurement of,

70.91
Velocity of Sound in Air. 286
Velocity of Sound in Brass, 289
Velocity of Sound in Iron, 243
Velocity of Sound, Tables of, 523
Vernier, Theory and Use of a, 30

efseq,, 34
Vibrations, Method of. 61, 394
Vibrations of Air in Organ -pipe,

233.246
VibraiioFis of Cord, Transyerse,

218, 226, 243
Vibralionsof Rod. 239
Vibrations of Spiral Spring, 111
Vibrations of Water, Surface,

222
Virtual Image, 432. 439, 452
Volt. 818

Voltafneter, Copper, 409
Voliameter, Gas. 403
Volume, Measurement of, 22, 24

Water-equivalent, 282

Watt, 68. 819

Wave-length of Light. 477. 529

Wave-length of Sound, 218

Weighing, 17, 160

Weiss, System of Magnets, 518

Wheatstone Bridge, 375. 881, 886

Wheatstone Bridge Key, 367

Whirling-table, 106

Young's Modulus, 163

THE BND

N^'

UNIVERSnY OF CALIFORNIA LIBRABY
This book is DUE on the last date stamped below.

OCj 61947
OCT 17 1947

LD 21-100n»-12,'46(A2012il6)4120

re V[2I8

>/ 158371

, UNIVERSITY OF CAUFOKNU UBRAHV

• i

I #

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