LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA.
Accession th.M&/. clMS  V "'
\
PROBLEMS AND QUESTIONS IN PHYSICS
PROBLEMS AND QUESTIONS
IX
PHYSICS
BY
CHARLES P. MATTHEWS, M.E.
ASSOCIATE PROFESSOR OF ELECTRICAL ENGINEERING, PURDUE UNIVERSITY
FORMERLY INSTRUCTOR IN PHYSICS, CORNELL UNIVERSITY
AND
JOHN SHEARER, B.S.
INSTRUCTOR IN PHYSICS, CORNELL UNIVERSITY
goiit
THE MACMILLAN COMPANY
LONDON: MACMILLAN & CO., LTD.
1897
All rights reserved
JUbrary
COPYRIGHT, 1897,
Bv THE MACMILLAN COMPANY.
XortoootJ
J. S. Gushing & Co.  Berwick & Smith
Norwood Mass. U.S.A.
PREFACE
THERE is perhaps little that need be said prefatory to a work
of this character. The classroom experience of the authors
leads them to believe that any text in Physics needs to be sup
plemented by problem work in considerable variety. A nu
merical example in Physics serves a manifold purpose. It
takes the mathematical expression of a physical law out of
the realm of mere abstraction, by emphasizing the connection
between such a law and the phenomena of daily observation.
At the same time, it gives the student an idea of the relative
magnitude of physical quantities and of the units in which they
are measured. Lastly, it shows him the usefulness of his
previously acquired mathematical knowledge, while impressing
upon him the limitations which must be put upon this know
ledge when applied to physical relations. There would seem,
therefore, to be no lack of justification for the riot inconsiderable
labor of writing an extensive series of problems.
In the preparation of the following pages, the authors have
introduced a number of features which have seemed good to
them, and, it is hoped, may meet with general favor. The
problems are numbered consecutively throughout the book in
Arabic numerals. The paragraphs of the Introduction are num
bered in Roman numerals. This contributes to easy reference.
All tables of physical constants are placed in the Introduction.
To work the problems it will be necessary, not only to read the
Introduction, but to refer to it continually. The authors con
fess that in this arrangement they have aimed to abolish the
vi PREFACE
idea, prevalent in the student mind, that an "Introduction,"
like a "Preface," is something that no one ever reads. The
plan also shortens the statement of a problem, relieving it of
much reiterated information.
A few words should be said concerning the use of the cal
culus notation. As the tendency of writers of elementary
works in Physics seems to be towards a greater use of the
language of the calculus, it is only appropriate that a fair
number of problems should be inserted here which cannot be
satisfactorily worked by other than calculus methods. Their
number, however, is not large, and the usefulness of the book
to students not prepared for them will be in nowise dimin
ished. It is believed that the number of problems is sufficiently
large to enable the instructor to make an adequate selection for
any class.
Occasional questions not requiring numerical answers have
been asked. These are purposely few in number, and are put
in to indicate the general character of classroom and examina
tion questions, and with no thought of encroaching upon the
province of the instructor.
Here and there graphic methods have been suggested which
may be profitably extended by the student. On the other
hand, solutions and hints have been omitted in many cases
where the student might perhaps expect to find them. It is
felt that the methods preferred by the instructor in charge or
suggested by the text in use should be used rather than those
of the writers, since the general character of the course and the
degree of the student's advancement may be thus considered.
It is not expected that the student should work the problems
without suggestion, and inability to do so in particular cases
may indicate to both student and instructor just where some
law or definition is not clearly understood. There are undoubt
edly obscurities in the text and errors in answers, and the
PREFACE vii
authors would esteem it a favor if readers would call attention
to them.
Some criticism may be incurred because of the use of mixed
units. Many of the students who will use these problems are
pursuing engineering courses. In such case they must of
necessity use engineering units. The aim has been not so
much to train them in the use of these units, an abundance
of this training comes to them during their course, but to
bring out the relation of the socalled " practical" and gravita
tional units to the C.G.S. units of Physics.
Suggestions have been received from many sources, among
others the works of Jones, Jessop, and Everett. The authors'
thanks are due to Messrs C. D. Child, C. E. Timmerman, and
O. M. Stewart, Instructors in Physics at Cornell University, for
solutions of problems and many valued suggestions.
DECEMBER, 1896.
CONTENTS
MEASUREMENT AND UNITS ......... i
PHYSICAL TABLES 12
DIRECTED QUANTITIES . .21
GRAPHIC METHODS 26
AVERAGES . .31
APPROXIMATIONS  33
MECHANICS OF SOLIDS 37
LIQUIDS AND GASES 89
HEAT 100
ELECTRICITY AND MAGNETISM , . .121
SOUND AND LIGHT .191
MATHEMATICAL TABLES 225
ANSWERS 237
INDEX 245
PROBLEMS IN PHYSICS
I. INTRODUCTION
Measurement. Whenever, in the domain of physical science,
we step from the position of a mere observer of the phenomena
around about us to that of an investigator, we seek the aid of a
process known as measurement. Whether this process be sim
ple or complex, there is but one operation in it that is funda
mental, the determination of the value of one magnitude in
terms of another of the same kind. We may content ourselves
with the crudest approximation, as when we estimate moun
tain heights in terms of the highest peak of the range, or, we
may make a comparison with the utmost scientific accuracy,
using for such a purpose a quantity agreed upon among men as
a standard or unit. In either case the result sought is a ratio ;
namely, that existing between the magnitude and the chosen
unit of like kind. This ratio is the measure of the given magni
tude, and the process by which it is found is called measurement.
The accuracy with which measurements are made is governed
largely by practical needs. It should, however, be borne in
mind that the process is, at the best, an approximate one. Even
the most exact measurements of physics must be regarded as
attempts to determine numerical quantities whose true values
must ever/ remain unknown.
Units. It follows that the complete expression of a physical
quantity, so far as its magnitude is concerned, involves two fac
2 PROBLEMS IN PHYSICS
tors, one a concrete unit, the other a number or numeric. Thus
if L be a unit of length, the measure or numerical value of a
length /is n = , and the complete expression of the magni
tude of / is
The product of numeric and unit is constant. Whether a debt
be paid in dimes or in dollars, it is yet the same debt, but the
number representing it in the one case is ten times that repre
senting it in the other. The unit and numeric, in other words,
vary inversely.
Fundamental and Derived Units. Consider the case in which
the unit of length is taken as the foot, and the unit of area the
square yard. Then a rectangular area a feet long and b feet
wide is expressed as
A=\ab sq. yd.
And, in general, the area is given by
A = kab,
where k is a constant depending upon the units of length and
area involved. If, however, it is agreed that the unit of area
shall be the square foot, the value of k reduces to unity, and
A=ab sq. ft.
It thus appears that, in a system made up of arbitrarily
chosen units, transformations call into use a number of pro
portionality constants, many of which will involve endless deci
mals, introducing into computations much unnecessary labor
and liability of error. The earlier units were largely of this
character. They were chosen to meet the needs of practical
life at a time when simple and definite relations among them
were not deemed essential. Thus the origin of the foot is obvi
ous, as is also its variation in different countries.* Further,
* The Russian foot is 30.5 cm.; the Austrian foot, 31.6 cm. ; the Saxon foot,
28.32 cm.; etc.
INTRODUCTION 3
derived units based on powers of the fundamental are not
always convenient. The yard is a convenient length for the
measurement of cloth, but the cubic yard is too large a volume
for the grocer's needs. Yet the awkwardness of systems made
up of grains, scruples, drams, and ounces, of links, rods, and
chains, needs no comment. The metric system, now generally
used by physicists, obviates these difficulties by making all
change ratios multiples or submultiples of 10. All the complex
units of physics are thus bound together by ties that may be
easily manipulated.
The system in common use is based on three arbitrarily
chosen units. These are
the centimeter, the T ^g part of the length of a certain plati
num bar kept in the Archives of Paris ;
the gram, the YoVo P ai "t f a certain piece of platinum (the
kilogram des Archives) which is intended to have the same
mass as a cubic decimeter of water at the temperature of
maximum density (3.9 C.) ;
the second, the ^ art ^ ^ e mean
These units of length, mass, and time, respectively, are known
as the fundamental units of the C.G.S. system. Other units
based upon them are called derived units.
Another system, much less in use, is based on the same
physical quantities, but the units of length and mass are of
different value. They are
the /<?#/, as a unit of length ;
the pound, as a unit of mass ;
the second, as above denned.
These units are the basis of the footpoundsecond (F.P.S.)
system of units.
Referring again to the equation
A = kab,
4 PROBLEMS IN PHYSICS
we see that in the C.G.S. system in order to make k unity the
unit of area must be taken as the square centimeter. The
resulting equation is
A ab,
concerning the reading of which a word of caution is necessary.
When fully translated it affirms that the number of units of area
is equal to the number of units of length x the number of units
of breadth. In other words, it is the numerics that are actually
multiplied. So, force is measured by the acceleration produced
in mass. The equation
F= ma
is usually read force equals mass times acceleration. This is an
abbreviated statement of the fact that, in a consistent system
of units, the number of units of force equals the number of units
of mass x the number of units of acceleration produced.
Velocity is the rate of motion. The units of length and time
being the centimeter and the second, any other unit of velocity
than the centimeter per second \$> both awkward and unscientific.
Similarly the C.GlS. unit of acceleration must be an accelera
tion such that unit velocity is gained in one second. Accel
eration is measured, therefore, in centimeters per second per
second.
The more complex electrical and magnetic units are built up
in the same natural way. It is found that the force between
two magnetic poles varies as the product of their pole strengths
and inversely as the square of the distance between them.
That is, in air,
Whence unit magnet pole is a pole of such strength that it
repels an equal and like pole, placed I cm. away, with a force of
one dyne. This unit of force, itself a derived unit, has already
'been referred to.
Dimensions and Dimensional Equations. Suppose that for
INTRODUCTION
5
the unit of area in any system a square be taken one of whose
sides is the unit of length, and let an area a contain n such
units. That is,
a = nA.
Further suppose that it is desired to double the unit of length.
The new unit of area built upon the changed unit of length is
four times the old unit. In other words, the unit of area varies
as the square of the unit of length, or it is said to be of two
dimensions in length. To indicate this, the last equation may
be written
a = nL\
Let v be a concrete velocity such that a distance / is trav
ersed in time /. The numerical values of these quantities
are found by dividing each by the appropriate unit. Let V, L,
and T be these units. Then the numerical values are , , .
We have then two numerical values of this velocity, viz.,
v , L
v and T
but these values are to be equal, which gives
2=L L
V L ' t
Writing the equation so as to separate the units' part, we have,
Or, in words, the unit of velocity varies directly as the unit of
length and inversely as the unit of time. That is, the dimen
sions of unit velocity are LT~ l . In passing to dimensional
equations we may discard constant numerical factors, since the
units, and therefore the dimensions, are not affected thereby.
6 PROBLEMS IN PHYSICS
So, the dimensions of the unit of acceleration are readily seen
to be L T~ 2 ; of the unit of force, ML T~ 2 ; of the unit of work,
ML*T 2 ; and so on.
It becomes apparent at once that dimensional formulas show
the powers of the fundamental units that enter into derived
units. Hence dimensional equations are of much use in facili
tating change of units.
EXAMPLE. The numerical value of the acceleration due to
gravity, when the centimetre and second are used as units of
length and time, is 980. Find the value in terms of the foot
and minute.
The dimensions of acceleration, it has been seen, are L 7^~ 2 .
We have
= 980 x .033 x 3600 ^
_min.
= II6424 ^
Lmin. I J
That is to say, the acceleration due to gravity is 116424 ft.
per minute per minute.
Whenever problems involving change of units occur in the
following collection, the student is strongly advised to work
them in this way, until the processes become so familiar as not
to need formal statement.
The two members of every equation must reduce to the same
dimensions, otherwise the equation is absurd. Or, what amounts
to the same thing, every term of an equation is homogeneous
with respect to each fundamental unit involved. The equation
of the motion of a particle having uniform acceleration in the
direction of motion is
INTRODUCTION ;
wherein / and a are of the dimensions L,
b, a velocity, is of the dimensions LT~ l ,
and c, an acceleration, is of the dimensions LT~ Z .
Thus each term of the expression for / is of the dimensions L
of / itself.
This gives a very convenient check upon our work in deriving
such an equation.
Mass and Weight. These words stand for two distinct phys
ical concepts. Thus, mass is quantity of matter, while weight
is force. Physically, then, they are no more alike than length
and time. Not infrequently the beginner fails to apprehend
this fact. Confusion arises partly because masses are compared
by comparing their weights, and partly because the same word
is often used ambiguously to name both a unit of mass and a
unit of force.
If a point move over equal spaces in equal times, any con
stant distance corresponds to a constant time. Or, in other
words, distance traversed and time vary in direct proportion.
For example, when, in railroad parlance, two stations are said
to be "four hours" apart, every one understands roughly what
distance is meant. Now it is precisely this relation that exists
between mass and weight, and it is largely because of their
proportionality in any one locality that some license is admissi
ble in naming their units.
Masses attract each other according to the fundamental law
of gravitation. To the attraction between the earth and the
bodies upon its surface the special name weight is given. The
weight of a body, therefore, is the force with which it is drawn
towards the earth, or with which the earth is drawn towards it.
When two bodies are placed in opposite pans of a beam balance
and do not destroy its equilibrium, they are said to be of equal
weight. That is, the forces acting at the ends of the beam are
equal. Further, by the law of proportionality, the bodies are of
8 PROBLEMS IN PHYSICS
equal mass, since we have for each force (or so much of it as
may be due to the added mass),
F= MS,
wherein g is the acceleration with which the mass M would fall
if released. The balance thus serves to determine equal masses,
and it is evident that if the system were carried to any other
locality the equilibrium would remain perfect, the masses re
maining unaltered and the weights varying with g. It is in this
way that masses are compared through the agency of their
weights.
As to units of mass, there are two in common use :
the pound,
the gram,*
each of which is the quantity of .matter in a certain carefully
preserved piece of platinum. To obtain the weights of these
masses we must multiply by the value of g appropriate to the
system of which the unit is a fundamental, and to the locality
at which the weight is desired. Thus the weight of a pound
where g = 32.2, is
W P = mg= i x 32.2
= 32.2 units of force in the F.P.S. system
= 32.2 poundals.
The weight of a gram where g = 980 is
W g mg = i x 980
= 980 units of force in the C.G.S. system
= 980 dynes.
All this is clear enough. But unfortunately, perhaps, the terms
pound and kilogram are used in such expressions as, "a body
weighs 16 pounds" or "a weight of 12 kilograms." The
* The original standard is the kilogram 1000 grams.
INTRODUCTION 9
pound and kilogram being units of mass, such usage, taken
literally, is absurd. The expressions, however, are elliptical,
their full meaning being "a body weighs the same as 16 pounds
weigh," or "a weight equal to the local weight of 12 kilo
grams." Or, we may say, with equal correctness and greater
brevity, " 16 pounds' weight" or " 12 kilograms' weight." So,
a grocer is said to weigh out tea; but he does not sell weight
he has no force for sale but mass.
A still greater source of confusion arises from the fact that the
engineer finds the poundal (3^2 P oun d's weight)* and the dyne
(QFO x ToVo kilogram's weight) too small for practical needs as
units of force. The engineering unit of force among English
speaking people is the weight of a pound (called simply a
pottnd), and among people using the metric system the weight
of a kilogram (called simply a kilogram}. Since these units
depend on the value of g, they are slightly variable, but the
variation is so small as to be usually negligible for engineer
ing purposes.
As illustrating this last usage, suppose that the piece of plati
num which the English people have agreed to call a pound were
hitched to a spring balance and the whole arrangement carried
to different points on the surface of the earth. The registry of
the balance would evidently vary to a slight extent. The engi
neer says we will neglect this variation as being of negligible
importance, and say that any agent which stretches the balance
spring ten times as much as does the freely suspended pound
mass is a force of 10 Ib. Let us suppose, then, that in this
way a body is found to weigh 10 Ib., and let us inquire what
the mass of this body is. By Newton's second law this force
is measured by the mass of the body times the acceleration
which it would possess if allowed to fall freely. Taking g=$ 2. 2,
we write
* The accepted value of g at Ithaca is 980, which corresponds to 32.15 in foot
second units. 32.2 is commonly used, however. See Church's " Mechanics of
Engineering."
OP 1 THK
10 PROBLEMS IN PHYSICS
10 = m x 32.2,
whence, mass = =
g 32.2
This makes the mass of the body invariable, as it must be.
To the unit mass in this system no name has been given, but it
is readily seen to be the mass of a body weighing 32.2, or more
generally g, pounds. With this understanding it is quite cor
rect to say that a body weighs G pounds, to speak of a pull or
thrust of G pounds, a pressure of G pounds per square inch, etc.
The pound and the kilogram are sometimes called gravita
tional units of force. Likewise the footpound and the kilo
grammeter are gravitational units of work, and the horse
power is a gravitational unit of power.
As illustrating this system we may consider the following
problems :
A body weighing 12 Ib. is moving with a velocity of 193.2 ft.
per second. What constant force must be applied to bring it
to rest in 3 sec. ?
The acceleration is
"*' = 64.4 ft. per second per second.
The mass of the body must be found. Since the weight 12 Ib.
would produce an acceleration of 32.2 ft. per second per second,
if the body were allowed to fall, we have
12 = m 32.2,
32.2
Finally F= ma
12
~ 322
x 64.4 = 24 Ib.
INTRODUCTION 1 1
A force of 12 kg. is overcome through a distance of 20 m.
Find the work done. We have
W= Fl
12 X 20
= 240 kilogrammeters.
This result is dependent on the value of g at the place at which
the work is done.
The physicist solves this problem as follows :
A force equal to 12 kg. weight where g = 980 is
F= 12 x io 3 x 980 dynes,
and the work done is
W= Fl 12 x 2 x 980 x io 6 ergs.
a. What two elements are necessary for the complete expres
sion of the magnitude of a physical quantity? Explain fully
in what the process of measurement consists.
b. What is the logical objection to a system of units in
which the inch is taken as the unit of length, the square rod as
the unit of area and the cubic metre as the unit of volume ?
c. A certain surface is a units long and b units wide; the
general expression for the area is
A = kab.
Under what conditions will the area be expressed as ab simply?
d. If in the last example a and b are given in feet, what will
be the value of k if the unit of area be taken as I square mile ?
e. Explain what is meant by fundamental and derived units.
f. Imagine the unit in which a definite magnitude is meas
ured to vary continuously. Plot values of the unit as abscissae
and corresponding values of the numeric (or measure) as ordi
nates. Discuss the locus.
12 PROBLEMS IN PHYSICS
NOTE. Many examples involving change of units, use of dimensional
equations and like matters are to be found further on in this book. It has
seemed better to place such examples, with the exception of the few general
ones above, where they may be used after the student is in some degree
familiar with the ideas involved.
UNITS OF LENGTH.
NOTE. The student is advised to study the approximate values. They
are of assistance in mental calculations, and are frequently sufficiently exact
for problem work.
Roughly
approximate
values.
I in. = 2.54 cm 2*.
i ft. = 30.48 cm 30^.
i mi. = 160933 cm.
= 1.6 km.
i cm. = .394 in .
i cm. = .0328 ft T fo.
i m. = 39.37 in 40.
i km. = .6214 mi f .
UNITS OF AREA.
i sq. in. = 6.45 sq. cm
i sq. ft = 929.01 sq. cm.
i sq. mi. = 25899 x io' 2 sq. m.
I sq. cm. = .155 sq. in
= .001076 sq. ft.
i sq. m. = 3.861 x io~ 7 sq. mi.
UNITS OF VOLUME.
i cu. in. = 16.387 cu. cm. . . . i6\.
i cu. ft. = 28316. cu. cm.
i gal. = 4541. cu. cm.
= 4. 54 litres 4$.
i cu. cm. = .061 cu. in ^.
= 3.532 x lo 6 cu. ft.
INTRODUCTION
UNITS OF MASS.
i ib. = 453.59 g
i oz. (av.) = 28.35 g
1 g = I 543g r 
.0353 oz.
= .0022 Ib.
UNITS OF FORCE.
[g = 980 in all gravitational units.]
i poundal = 13825 dynes,
i gram's weight = 980 dynes,
i pound's weight = 444518 dynes,
i kilogram's weight 9.8 x io 5 dynes.
 2249 x io~ 9 pound's weight.
UNITS OF WORK.
i footpound = 1.35485 x io 7 ergs
= 13825 gramcentimeters
= .138 kilogrammeters.
i kilogrammeter =7.233 footpounds.
i joule = io 7 ergs.
i watthour = 36 x io 9 ergs.
i horsepowerhour = 26856 x io 2 joules.
UNITS OF POWER.
i horsepower = 746 watts
= 746 x io 7 ergs per second
= 33000 footpounds per minute.
i watt io 7 ergs per second.
UNITS OF STRESS.
i Ib. per square foot = .48826 grams per square centimeter
= 478.5 dynes per square centimeter.
i Ib. per square inch = 70.31 grams per square centimeter
= 68904 dynes per square centimeter.
i in., mercury at o = 34.534 grams per square centimeter.
i cm., mercury at o =13.596 grams per square centimeter.
PROBLEMS IN PHYSICS
THE MECHANICAL EQUIVALENT OF HEAT.
g. through iC. = 4.2 x io 7 ergs
= .4281 kilogrammeters.
Ib. through i F. = 1.058 x io 15 ergs
= 780.8 footpounds.
TABLE I
DENSITIES
SOLIDS
Aluminum 2.6
Antimony 6.7
Bismuth 9.8
Brass 8.4
Copper 8.9
Gold 19.3
Iron 7.8
Lead 11.3
Nickel 8.9
Platinum 21.5
Silver 10.5
Sodium 98
Tin ' . 73
Zinc 7.1
Asbestos . ,
Chalk . . ,
Coal . . . .
Cork . . . .
Glass, common
Glass, flint . .
Ice . . . .
Iceland Spar .
Ivory
Marble . . .
Paraffine
Quartz . . .
Oak . . . .
Pine .
2.4
2.332
1.41.8
I43
2.52.7
335
.917
2.75
1.9
2.7
.87. 9 i
2.65
71
5
Alcohol . . .
Ether ....
Carbon Bisulphide
Glvcerine
LIQUIDS, oC.
.806
736
1.29
1.27
Mercury . 13596 Oil of Turpentine
Sea Water . .
Sulphuric Acid .,
Nitric Acid . .
Hydrochloric Acid
1.026
!.8 5
1.56
1.27
.87
TABLE II
SPECIFIC HEATS OF SOLIDS
Aluminum .2122
Bismuth 0298
Brass 0940
Calcium
Carbon, diamond
Carbon, graphite
,1804
,1128
,1604
INTRODUCTION
Carbon, charcoal .
Copper
1935
OQ'2 7
Gold
O3l6
Glass
l8?7
Ice
rn/lO
Iron
Lead
.1124.
0315
Magnesium
2450
Nickel ......... 1092
Platinum ........ 0323
Silver ......... 0559
Tin .......... 0559
Zinc . . ........ 0935
TABLE III
SPECIFIC HEATS OF LIQUIDS
Alcohol 55
Carbon Bisulphide . 24
Ether 53
SPECIFIC HEATS OF GASES AND VAPORS
( Constant Pressure}
Air 237
Oxygen 217
Hydrogen 3.4
Nitrogen 244
Steam .
Marsh Gas
Alcohol
.48
593
453
TABLE IV
MELTINGPOINTS AND HEATS OF LIQUEFACTION
Aluminum
Copper
Glass . .
Gold . ,
Ice . . ,
Iron . .
Lead
Melting
point.
o
Heat of
Liquefaction.
Calories.
600
1054
1 100
Mercury . . .
Nickel . . . .
Platinum . . .
1045
o
80
Silver
Tin . . . .
1600
Zinc . . . .
326
54
Melting
point.
o
Heat of
Liquefaction.
Calories.
40
2.82
1450
4.64
1775
27.2
954
24.7
230
14.6
412
28.1
i6
PROBLEMS IN PHYSICS
TABLE V
BOILINGPOINTS AND HEATS OF VAPORIZATION
Boiling Heat of
point. Vaporization.
Calories.
Alcohol 77.9 202.4
Bromine 58 45 6
Ether 34.9 90.4
Mercury 350 62
Water 100 536
TABLE VI
UNITS OF HEAT
Ergs.
i calorie (gramdegree C.) = 4.2 x io 7
i major calorie (kilogramdegree C.) = 4200 x io 7
i pounddegree Centigrade = 1905 x io 7
i pounddegree Fahrenheit = 1058 x io 7
TABLE VII
COEFFICIENTS OF LINEAR EXPANSION
Brass 180 }
Copper 170
Glass . 085
Gold 150
Iron 120
Lead 280
Platinum 085
Silver 190
Tin 200
Zinc 290
x io
COEFFICIENTS OF VOLUME EXPANSION
Alcohol (mean o 78) 00104
Mercury (mean o 100 C.) 000182
Water (mean o 100) 000062
INTRODUCTION
TABLE VIII
THERMAL CONDUCTIVITIES
Relative
Conductivity.
Silver 100
Copper 74
Iron 12
Lead 8.5
Bismuth 1.8
Ice 0.2
White Marble .... o.i
Glass 0.05
C.G.S.
13
0.99
0.16
o.n
O.O2
0.003
0.001
0.0007
TABLE IX
COLLECTED DATA FOR DRY AIR *
Expansion from o to 100 at constant pressure as 273 : 373
Specific Heat at constant pressure 2375
Specific Heat at constant volume 1691
Standard barometric height 76 cm.
Density at o and 76 cm 001293
Volume i g. at o and 76 cm 7733 c.c.
* Everett.
TABLE X
RESISTANCE
Substance.
Specific Resistance.
Temperature Coeffi
cient (0100).
Aluminum (annealed) ....
Copper (annealed)
Gold
289 io~ 8 ohms
160 io~ 8 ohms
208 io~ 8 ohms
388 io 5
^6c io~ 5
Iron (pure)
964 io~ 8 ohms
Iron (telegraph wire) . ...
1500 io~ 8 ohms
Lead . . .
1963 io~ 8 ohms
187 io~ 5
Mercury ....
9434 io~ 8 ohms
72 IO~ 5
Platinum .
898 io~ 8 ohms
Silver . . .
149 io~ 8 ohms
377
German Silver . ...
2100 io~ 8 ohms
4,4. to 6; IO~ 5
Platinoid .... . . .
3200 io 8 ohms
21 IO~ 5
Mano^anin
4700 io~ 8 ohms
122 IO~ 5
i8
PROBLEMS IN PHYSICS
TABLE XI
UNITS OF RESISTANCE
i true ohm unit of resistance,
i legal ohm = .9972 true ohms.
i B. A. unit = .9867 true ohms,
i Siemen's unit = .9407 true ohms.
TABLE XII
SPECIFIC INDUCTIVE CAPACITIES
Air = i
Solids.
K.
Liquids.
K.
Glass
A to 7
Acetone
21 8
Gypsum
r 6
Alcohol
2 r
Ice
3***
2 8C
Aldehyde
"O
18 6
Iceland Spar ....
***3
74.
Benzine ....
7 2
Marble
64
Carbon Disulphide
2
Mica.
6 to 8
Ether
A 27
Paraffine
2 2
Glvcerine
c6 2
4.1:4
Oils
3^'^
2.2
Rosin
Rubber, soft ....
vulcanite
2.55
2.4
2.7
Petroleum ....
Turpentine ....
Water
2.06
2.23
7c r
Salt
5.8
Sandstone ....
Shellac
6.2
9
Gases.
Hydrogen
O QQO8
Sulphur
2 6q
Vacuum
w.yyyo
o oo8c
Wood
,.wy
2.QC
Vapors
w.yyo^j
i.ooi to i.oi
TABLE XIII
PRACTICAL UNITS EXPRESSED IN C.G.S UNITS
Let Kbe the velocity of light, about 3io 10 cm. per sec.
Electromagnetic
Electrostatic.
C.G.S.
Practical.
C.G.S.
Quantity ....
i coulomb
I/IO
I 7 / 10, i.e. 3io 9
Current . . . .
i ampere
I/IO
V/io 3io 9
Potential . .
i volt
I0 8
io 8 /y i/(3io 2 )
Resistance ....
i ohm
I0 9
io 9 /^ 2 i/(9io u )
Capacity ....
i farad
I/IO 9
F/io 9 9io n
Selfinduction . . .
i henry
I0 9
INTRODUCTION
TABLE XIV
SOUND
VELOCITY OF SOUND IN METERS PER SECOND
Solids (20 C).
Liquids (20 C.).
Gases (o).
Brass
2/180
Alcohol
1 1 60
Air 1^2
Copper . . .
3560
Water . . .
1440
Illuminating Gas, 490
Iron ....
5'3o
Petroleum . .
1395
Hydrogen. . . 1280
Steel, cast . .
5000
Oxygen ... 317
TABLE XV
LIGHT
Velocity of light  2 99 86 kilometers per sec. Nearly 3  10*.
( 186323 miles per sec.
TABLE XVI
WAVELENGTHS OF THE PRINCIPAL FRAUNHOFER LINES IN
TENTHMETERS *
Line.
Wavelength.
Line.
Wavelength.
A
7594.059
M
$ 3727763
B
6867.461
\ 3727. 20
C
6563.054
N
3581.344
A
5896.154
344LI35
D,
5890.182
P
3361.30
(5270.533
Q
3286.87
E
^ 5270.448
jg
^ 3l8l.40
(5269.722
'317945
F
4861.496
$i
3100.779
G
( 4308.071
s.
3100.064
\ 4307904
T
53021.19!
H
3968.620
\ 3020.759
K
3933809
U
2947993
L
3820.567
* i tenthmeter = io~ 8 of a centimeter.
20
PROBLEMS IN PHYSICS
TABLE XVII
INDICES OF REFRACTION [Z? LINE] *
Density.
Index.
Glass (hard crown) .......... 2.486 .517
Glass (soft crown) .......... 2.55 .5 145
Glass (light flint) .......... 3.206 .574
Glass (dense flint) .......... 3658 .622
Glass (extra dense flint) ........ 3889 .65
Glass (double extra dense flint) ...... 4429 .71
Rock Salt ................ .544
* Everett, C.G.S. Units and Constants.
LIQUIDS
Alcohol ....... 1.363 Ether ........ 1.36
Canada Balsam ..... 1.54 Olive Oil ....... 1.47
Carbon Bisulphide .... 1.63 Turpentine ...... 1.48
Chloroform ...... 1.446 Water ........ 1334
UNIAXIAL CRYSTALS
Ordinary Extraordinary
Index. Index.
Iceland Spar ...... 1.6584 1.4864
Tourmaline ...... 1.6366 1.6193
Quartz ........ 15432 i55 12
TABLE XVIII
NUMERICAL CONSTANTS
LOGARITHMS
= 2.7183 ........ Log 10 e = .434294
Log 10 W = Log e AT .434294
Log e lV =Log lQ AT 2.3025 85
i radian .... 57.2958
i ....... 01745 radians
Log 10 Log 10
^=3.14159 497H9 ^ " 98696 ^994299
TT approx. 22 : 7 I : ?r 2 ..... 10132 1.005700
I:TT ..... 3183 ^502850 2 * . . . . 6.283 .798179
\/TT .... 1.772 .248575 i:27r ... 1592 1.201820
I : VTT . . . .5642 1.751425
I: V2
1.4142
.7071
_.I5<>5I5
1.849485
I : V3
1.7321
.5773
1.761439
II. DIRECTED QUANTITIES, VECTORS
Many of the quantities considered in physics involve the idea
of direction, and require the statement of two things before we
can form any clear idea of them. First, we must state how
large they are as compared with a thing of like kind taken as
a unit ; second, in what direction they must be taken. The
familiar idea of motion from one point to another may be con
sidered as typical of this class of quantities. Suppose one asks
the way from one point in a city to another. The answer might
be to go a certain distance north, then a certain distance west,
etc. Or, if circumstances permit, he may be told to go a certain
distance in a specified direction without turns.
The answer is one based on the experience that we may go
from one point to another either by a series of connected "steps"
or courses such that they begin at the startingpoint and end at
the final one, or by a single step, the straight line joining the
points. Or, since the result is the same so far as change of
position is concerned whether we take the crooked path or the
straight, we may call the latter the resultant of the former.
In considering the geometry of the problem, it may be noted
that if we are given the steps I and 2 we may (Fig. i) find
their resultant in either of
two ways : from A we may
lay off i in its proper direc
tion, and from the end of
i lay off 2 in like manner.
The line joining the ends
of i and 2 is then the resultant required. Or we may form
a parallelogram with one corner at A, and whose sides are i
21
22
PROBLEMS IN PHYSICS
and 2. The diagonal drawn from A is the equivalent step or
resultant.
The student should remember that the problem of finding the
resultant of a given system of steps is perfectly definite, and
only one solution can be found ; but the converse is not true, as
a given step may be made up of any one of an endless number
of step systems.
The process of finding the resultant of a given system is
often spoken of as the composition of steps; while that of replac
ing a single step by a system, usually two, is called the resolution
of steps.
The simplest, and, at the same time, the most useful case of
resolution is when the step is resolved into two at right angles
Fig. 2.
to each other. Or the line is said to be projected on two rectan
gular axes X and Y.
Then X component = AB' = AB cos 0,
Y component = BB 1 = AB sin 0.
The name vector (i.e. carrier) is usually applied to this class
of quantities, and the resultant of a system of vectors is spoken
of as the vector sum of the components. A thorough under
standing of the geometrical ideas involved in adding and resolv
ing vectors is of the greatest importance to the student in
physics, and must be acquired before any real progress in the
DIRECTED QUANTITIES, VECTORS 23
subject is made. The following simple problems are added to
assist the student toward this end.
1. Which of the following quantities are vectors? Force;
mass ; acceleration ; momentum ; energy ; volume ; velocity ;
current ; weight ; time ; interest.
2. Show by diagram the vector sum (i.e. the equivalent
straight path) of the following set of paths : E. 4 mi. ; N. 2 mi. ;
N.W. 3 mi. ; S.W. 5 mi.
3. Draw the same set of paths in the reverse order; i.e.
S.W. 5 mi. ; N.W. 3 mi. ; etc.
4. When the vectors are not in the same plane, show how
the vector sum is found.
5. What is the vector sum of the length, breadth, and
height of a room ?
6. Two vectors at right angles to each other, of lengths 4
and 3 respectively, have what vector sum or resultant ? If at
60 ? 180? o?
7. Six vectors equal in length are placed end to end so
that the angle between each pair is 120. What is the vector
sum ?
8. Show that the order in which "steps" are taken in no
way modifies the sum.
A vector may be given in either of two ways, by its components or by
its length and direction, or the angle it makes with a given line. In the fol
lowing examples the line of reference is the horizontal line drawn to the right
(.raxis) .
9. Find the resultant of the following vectors : 3, 25 ;
4, 100 ; 2, 200 ; 5, 300. The work may be conveniently
arranged as follows :
ATcomp. Fcomp.
Length Dir. cos sin /cos /sin
3 ... 25^
4 ... 100
2 ... 200
5 .. 300
2 4
PROBLEMS IN PHYSICS
10. Draw the following vectors : 3, 90 ; 4, 180; 5, 190.
11. Draw the vectors whose components at right angles to
each other are 2 and 3, 4 and 6, 2 and 3.
12. A vector 10 units in length makes an angle of 30 with
one of two perpendicular lines. Find the component along
each line.
13. A given vector is to be resolved into two at right angles,
such that one component is double the other. Find the angle
which the longer must make with the given vector.
14. Could the vector AB be considered as the vector sum
of the set of short vectors parallel to the axes ? Those parallel
to X are called what in calculus ? Those parallel to F?
Y
Fig. 3.
15. Two vectors, a and b, are given, making angles l and 2
with the reference line. Find the sum of their X components.
Find the sum of their Y components. From these find the
Y
Fig. 4.
' resultant of a and b. Reduce to the formula given in trigo
nometry for the cosine of an angle in terms of the sides.
DIRECTED QUANTITIES, VECTORS 25
16. Show that the resultant of two vectors may be found
from the theorem in geometry : The square on any side of a
triangle is equal to the sum of the squares on the other two
sides twice the product, etc.
17. n coplanar vectors are drawn from a common point.
A polygon is formed by joining their extremities. Prove that
the resultant is given in magnitude and direction by n times
the vector joining the origin and the center* of gravity of the
polygon.
18. Test the above statement for two, three, and four
vectors.
19. If the vectors were so numerous that their ends formed
a continuous curve, what method could be used to find the
resultant ?
* See 197.
III. GRAPHIC METHODS
It is frequently impossible to keep in mind the complete
time history of variable phenomena, or to readily compare the
values of quantities which alter with time or position. A
clearer conception in such cases may often be obtained by some
geometrical method of representing the relative values of two
quantities at different times or places. Take, for example, the
motion of a ball struck by a bat ; we may wish to compare any
two of the various quantities which are involved in its motion.
The height above the earth may be compared with the hori
zontal distance from the startingpoint, or with the time since
it was struck, or with its vertical velocity, etc.
In the first case, we might draw an actual picture of its path
to reduced scale, as (Fig. 5). If we wished to compare height
Fig. 5.
at any instant and time since the ball was struck, we might
measure a series of lengths to suitable scale, along a straight
line, to represent heights, and label each with the time required
to reach that height. This would, however, be confusing, since
the ball is at the same height, in general, twice. Suppose we
26
GRAPHIC METHODS 27
displace each height h as many arbitrary units to the right
as units of time have elapsed since starting, as / , t lt / 2 , etc.,
and the corresponding heights // lf // 2 , // 3 , etc. (Fig. 6). We know,
however, that the ball took, in succession, every height between
those indicated ; hence if we were to erect a perpendicular at
every point between ^ and / 2 , and measure along each the
corresponding height of the ball, the ends of these perpendic
ulars would form a continuous curve. This process is known
as " plotting " the curve, and is of fundamental importance in
Fig 6.
the study of physics. The two lines of reference from which
distances are measured are called the axes of coordinates, and
are usually chosen at right angles to each other. One is often
called the axis of x, and the other the axis of y, and the lengths
measured along the .raxis are called abscissas or x's. Those
measured along or parallel to y are called ordinates or j/s, and
any x with its corresponding y are called the coordinates of the
point which they determine.
" Selfregistering " instruments usually draw a curve by some
mechanical device. An example is the selfregistering ther
mometer, where a pen is made to rise and fall with the temper
ature, while the paper is drawn at a uniform rate in a line
perpendicular to the motion of the pen. A curve such as the
following is the result (Fig. 7). Both time and temperature are
continuous, and the curve is a fairly true picture of the time
temperature relation.
28
PROBLEMS IN PHYSICS
In case we had observed the temperature at 2, 2.30, 3, 3.30,
4, etc., and had no knowledge of intermediate temperatures,
we would draw a continuous curve through the observed points,
which would be more and more reliable as the time intervals
were made smaller. In general, the more irregular the changes
in the observed quantity, the shorter these intervals must be
made to ensure that no sudden variation escapes notice.
Fig. 7.
We may expect in each case certain peculiarities in the
curve, depending on the physical relations which determine
it, and, conversely, any peculiarity, as a maximum or minimum,
change of curvature, asymptote, etc., will usually have a physical
meaning. For example, every change of temperature requires
a certain time interval, so that no portion of the timetemper
ature curve can be vertical. Time never decreases, and tem
perature has only one value at a given instant, so there are no
"loops " or multiple points in such a curve.
When we consider the quantity of heat supplied to a gram
of ice, for example, and the resulting change of temperature,
we find a curve with certain abrupt changes (see Fig. 8).
Starting at o, 80 heat units are used with no increase of /.
The line AB shows the quantitytemperature relation after
melting (approximately straight). At 100 we have an abrupt
rise to C then, another straight line whose slope is depend
ent on conditions of pressure, etc. The amount of heat
GRAPHIC METHODS
2 9
required per gram for any temperature change may be read
from the curve.
EMPERATURE
Fig. 8.
Curves are used in physics for various purposes ; as,
(a) To represent graphically general laws.
Ex. Path of a projectile. Laws of falling bodies.
(b) Asa record of results of observation of two related varying
quantities.
(c) For use in computation. As a sort of numerical map of
simultaneous values.
The student should not rest content with simply drawing the
curve, but should endeavor to associate the changes or peculi
arities in form with the underlying physical conditions. If
familiar with the methods of analytic geometry and calculus, he
may apply these methods to their study.
In particular, if the curve is a graphic representation of a
general law, he should note whether all portions of the curve
have an actual physical interpretation, whether the physical
conditions indicated by certain portions of the curve could be
realized ; if it cuts the axes, what the intercepts mean ; whether
the direction of the tangent line at any point has a physical inter
30 PROBLEMS IN PHYSICS
pretation ; does the area of a given portion represent some physi
cal quantity ; etc. When it is drawn from observed values, the
relation between the coordinates may often be expressed as an
algebraic equation, either from its general appearance or from
a knowledge of the physical law involved.
20. Draw a curve showing the relation between the side of
a square and its area. Interpret its " slope." Should it pass
through the origin ?
21. Draw a curve showing the relation between simple inter
est, principal, and time. What is the slope ? How would the
curve of amount and time differ from this ? Interpret the inter
cepts in this case.
22. Given the curve of displacement and time, how could
you find the velocitytime curve ? the accelerationtime curve ?
IV. AVERAGES
When we have to deal with a series of values of the same
quantity at different times or places, it is convenient to substitute
for the series a single quantity, so chosen that the result will not
be changed. Such a quantity is known as an "average" or a
mean value. For example, we may wish to consider the temper
ature of the air at a certain point during a certain period of time,
as an hour. Some of this time the temperature may have been
rising and some of the time falling, and these changes may have
been more or less rapid and irregular. To find the temperature
which may fairly be taken to represent the temperature at that
point during the hour, we would be obliged to add together a
great number of observed temperatures and divide the result by
this number. The greater the number added, the more nearly
correct the average. We might also have required the average
temperature at a given instant along a given line, over a given
area or throughout a given volume. In all these cases we should
take the sum of an indefinitely great number of separate values
and divide by the time, length, area, or volume considered. We
actually only approximate this by taking a smaller number. The
actual addition of these quantities can in certain cases be avoided.
As when the values to be averaged increase or decrease at a
constant rate, the terms then form an arithmetic series, and the
mean value is onehalf the sum of the first and last. Examples
of this will be found in problems on velocity, force, etc. Again,
when a curve is drawn showing the relation between the two
variables, if by means of calculus or otherwise we are able to find
the area ABB'A', we may divide this area by AB and get the
average ordinate.
31
For,
PROBLEMS IN PHYSICS
Area = f ydx
= AB average height.
(Fig 9)
The student should be very careful in averaging quantities to
first find the actual values to be averaged. For example, the
V
dx
Fig. 9.
average of a series of fractions is not the average of the numer
ators divided by the average of the denominators. The average
of a series of quantities each the product of two factors will not
be the product of the average value of each factor.
V. APPROXIMATIONS
The computation of results from physical data is often labori
ous, on account of the number of decimal places involved in the
constants required. In many cases, however, we may diminish
the work by the use of suitable methods and approximate
formulae. Not only is the labor of computation increased by
the retention of too many decimal places, but the results so
obtained are actually misleading, in that they give an appearance
of accuracy not warranted by the data. For example, any re
sult obtained by data accurate to one part in one hundred will
not be accurate to any higher degree.
Suppose that two sides of a rectangle have been measured by
a metre bar divided to hundredths, and that the tenths of a divi
sion have been estimated, giving 4.258 and 6.543 . The last
figure in each case is only approximate, and if the area is com
puted the result contains six decimal places, only three of which
should in any case be retained. The labor of writing these
superfluous figures may be easily avoided by using only those
partial products giving the orders we wish to retain. We see
that 4 units x .003 gives a product which we 4 . 2 5 g
require, while .2 x .003 is of secondary impor
tance. The lowest partial products required are
readily seen from the diagram, in which we 6 . 5 4
"step down" one in the multiplicand as we Fi ? 10 
"step up" one in the multiplier (Fig. 10), the arrows connecting
the factors of the products required.
The simplest arrangement of work is that given in textbooks
of advanced arithmetic, and may be stated as a rule thus :
Write the multiplier in reverse order, placing the units' figure
under the figure of the multiplicand of the same order as that to
D 33
34 PROBLEMS IN PHYSICS
be retained in the product. Multiply cacJi figure of the multi
plier into the figure of the multiplicand next to the right above,
and "carry" the nearest 10 ; then proceed as in ordinary multi
plication, only writing the initial figure of each partial product
in the same column, which is of the lowest order in the product.
EXAMPLE. 4258
345^
25548 [Multiply by 6 as usual.
2 1 2 Q [Multiply 8 by 5 and carry 4, then proceed
" as usual, placing 9 under 8.
I 70 [4 X 5, carry 2. o under 9.
13 [3X2, carry i, etc.
27.860
Multiply 85.39738 by 1.00295, retaining four decimal places.
85.39738
59200. i
853974 t 8 x x cari r x . etc 
1708
768
43
85.6493 Ans.
Many examples of this nature occur in connection with ap
proximate formulae, expansion coefficients, etc. The student
should perform several multiplications by each method, and
observe carefully the details of the shorter process.
Expressions of the form (i ), where is a small quantity,
are of frequent occurrence in physics. Whenever any power of
such an expression is used as a multiplier or divisor, an approxi
mate multiplier can be found by means of the "binomial
theorem." <
Since [i ] n = i + n ( ) + n ( H ~ ^ ... for all values of
n, whether positive, negative, integral, or fractional, and, when
is small in comparison with unity, we may neglect 2 and all
APPROXIMATIONS 35
higher powers of , the approximate multiplier consists of
i na.
EXAMPLE. The edge of a wroughtiron cube is 20 cm. at
o C. What will be its volume at 15 C, the coefficient of linear
expansion being .0000122?
The length of each edge at 15 is
L 15 = 20 [i + 15 .0000122]
= 20 [ I + .000182].
Whence volume at 15 = 2O 3 [i + .000 182] 3
= 20 3 [i + 3 .000182
+ Higher powers of small quantities.]
= 20 3 [1.000546]
= F [1.000546].
Had the volume at 24 C. been given and the volume at o
been required, we have, in like manner,
= V^ [i 3 24 .0000 122]
= F 24 [i .0008784]
= FiJ9991216].
When V^ is given, the approximate method of multiplication
gives the result easily. It is to be observed that when the
original length or volume is large, i.e. when the multiplicand is
large, more decimal places in the multiplier are of importance.
As another example, consider the area of a rectangle of sides
a and b when each side is slightly increased.
36 PROBLEMS IN PHYSICS
If a is increased by , and b is increased by /:?,
the new area = (a 4 a) ( 4 /3)
= # + tf/3 + ba + a/8 (Fig. 1 1)
= ab + afi + a,
when a/3 can be neglected ; z>. when the corner rectangle is
very small in comparison with those on the sides.
The student will be able to form approximate formulae similar
to those given in many cases,
and these, in connection with
the various tables, will greatly
reduce tiresome numerical
computations which in them
j3 selves give no insight into
physical laws and phenomena.
In addition to these, a few points in connection with arrange
ment of work and notation may be useful.
It is customary and convenient in expressing very large or
very small numbers to write only the few figures actually
observed or derived, and to indicate their position by a power of
10 used as a multiplier ; as,
45630000000 = 456.3 io 8 ,
.0000122 = 122 io~ 7 , etc.
In every case where numerical work is required, spend a little
time and thought in a general survey of the problem.
Note in what order it is best to perform the various parts,
whether factors can be cancelled or approximate values used.
It is often best to write out the entire expression before any
numerical work is done. Bear in mind that the understanding
of the method and the facts involved is of primary importance,
and numerical results are often only secondary.
MECHANICS
VELOCITY, ACCELERATION, AND FORCE
23. Express a velocity of 22 mi. per hour in (a) feet per
minute, (b) kilometers per hour, (c) centimeters per second.
24. An express train leaves Albany at 10. 13 A.M., and arrives
in Buffalo at 4.45 P.M. The distance is 297 miles. Find the
average velocity of the train over this distance.
25. Using velocities as ordinates and times as abscissas,
draw a curve which might show the changes in velocity between
any chosen time limits in a train's run. 'What is represented
by the area included between the curve and the ^raxis ? What
by the steepness (pitch) of the curve at any point ?
26. Which is the greater velocity, 40 mi. per hour or 12 m.
per second ?
27. A railway train reaches a speed of a mile a minute.
What is the value of this speed in kilometers per hour?
28. Speaking of the time required for light from the sun to
reach the earth, Lodge says : * "If the information came by
express train it would be three hundred years behind date, and
the sun might have gone out in the reign of Queen Anne
without our being as yet any the wiser." Verify this and com
pute the time which is actually required for light to reach us
from the sun. (Mean distance to sun 928 io 5 miles.)
* Pioneers of Science.
37
38 PROBLEMS IN PHYSICS
29. The side of a cube increases at the uniform rate of
10 cm. per second. After 2 sec. at what rate is the area
of one side increasing ? the volume ?
30. A gun is fired on board a ship at sea ; an echo is heard
from a cliff after a lapse of 7 sec. Find the distance of the ship
from the cliff. (Velocity of sound = 332 m. per sec.)
31. A man of height h walks along a level street away from
an electric light of height b. If the man's velocity is v miles
per hour, find the velocity of the end of his shadow.
32. What is acceleration ? What are the dimensions of
acceleration ? What is the C.G.S. unit of acceleration ?
A particle has unit acceleration when it gains (or loses) unit velocity in
unit time. The C.G.S. unit of velocity is a velocity of one centimeter per
second. The corresponding unit of acceleration may therefore be called one
centimeter per second per second. This is a somewhat cumbersome name,
but it is conducive to clearness.
33. Show that the general expression for acceleration is
Take a as constant, integrate twice, and discuss the resulting
equations.
34. A body acquires in 4 sec. a velocity of 300 cm. per
second. What is the value of its acceleration ?
3^ = 75 cm. per second per second.
35. A train having a speed of 64 km. per hour is brought
to rest under the action of brakes in a. distance of 510 m.
What is the acceleration, if assumed to be constant ?
36. What is the final speed of a body which, moving with a
uniformly accelerated motion, covers 72 m. in 2 min., if
(a) the initial speed = o,
(b) the initial speed = 15 cm. per second.
VELOCITY AND ACCELERATION 39
37. Plot a curve showing the relation between distance
passed over and time in the case of a body having a constant
acceleration. What is shown by the pitch of such a curve at
any given point ?
38. Find the distance passed over in the /th second by
a body having a uniformly accelerated motion.
We have
space described in t seconds = \ at 2 ,
space described in / i seconds = % a(t i) 2 ;
whence space described in the /th second
= I at 1  \a(t i) 2
(*/*>
If the body has an initial velocity z> , we have, obviously,
space passed over in the /th second
39. What are the ratios of the spaces passed over in succes
sive seconds by a body moving with a constant acceleration ?
40. If a body starting from rest has an acceleration of 36
cm. per second per second, over what distance will it pass in
the seventh second ?
41. A body has a uniform acceleration of 36 cm. per second
per second. Initial velocity = o.
(a) How far does it travel in 8 sec. ?
(b) How far does it travel during the eighth second ?
42. With an initial velocity of 14 cm. per second, how
answer the preceding problem ?
43. A train acquires 8 min. after starting a velocity of 64
km. per hour. Assuming constant acceleration, what is the
distance passed over in the fifth minute ?
44. A body starting from rest with a constant acceleration
passes over 18 km. the fourth hour. Find the acceleration.
40 PROBLEMS IN PHYSICS
a (2 x 4  i)
/ 4 th =18 ^~
1 = 18,
p er h our p er hour.
45. A body starts from rest with a uniformly accelerated
motion. In what second does it describe five times the distance
described in the second second ?
46. A and B are initially at the same point. If A move to
the right with a uniform velocity of 6 km. per hour, and B
move to the left with a uniform acceleration of 3 km. per hour
per hour, what is the distance between them at the end of
4 hr. ?
47. Suppose in the preceding problem that at the expiration
of the 4 hr. A turns and follows B with a uniform acceleration
of 4 km. per hour per hour, how long before A overtakes B ?
48. A body moving with uniform acceleration passes over
distances of 13 and 23 m. in the seventh and twelfth minutes
respectively. Find its initial velocity and its acceleration.
49. A body starting from rest passes over 1.2 m. in the first
second. The acceleration being uniform and the initial velocity
zero, how long has it been in motion when it has acquired a
velocity such that 6 m. are described in the last second of its
motion ?
50. A body m has an acceleration of 40 cm. per second per
second ; a body n has an acceleration of 56 cm. per second.
Provided both bodies start from the same origin at the same
instant and travel (a) in the same direction, (b} in opposite direc
tions, how long before they will be 6 m. apart ?
51. What definition of force is implied in Newton's first
law ? What quantitative definition of force is embodied in New
ton's second law ?
52. Discuss Newton's third law, giving one or more familiar
examples.
VELOCITY, ACCELERATION, AND FORCE 41
53. Define the C.G.S. unit of force, the dyne.
54. Define the dyne in terms of momentum and time.
55. What is the character of the motion produced by a con
stant force acting on a given mass ?
56. What constant force will give to a mass of 40 g. a
velocity of 4.8 m. per sec. in 12 sec. ?
57. A force of 30 dynes acts on a mass of 2 g. Find the
velocity acquired in I sec. :
30 = 2 a,
a= 15.
Find the velocity acquired in 6 sec. :
v at = 6 x 15= 90 cm. per sec.
58. Explain fully the difference between mass and weight.
59. A body of 6 g. mass is moving with a velocity of
3.6 km. per hour. Find the force in dynes that will bring it
to rest in 5 sec.
The application of a constant force to the body will produce a constant
(negative) acceleration. Since the body is to lose all of its velocity in 5 sec.,
the rate of change of velocity, i.e. the acceleration is
a= 3.6 x io 5
36 x io' 2 x 5
= 20.
And the force necessary to produce this acceleration is
f = ma = 6 x 20
= 120 dynes.
60. A mass of 500 g. moving at the rate of io m. per
second is opposed by a force of 1000 dynes. How long must
this force act in order to bring the body to rest ?
61. A mass of 4 kg. falls freely. What is the value of the
force acting upon it ?
The acceleration due to gravity is 980 cm. per second per
second. We have
F = Ma
= 4000 x 980
= 392 x io 4 dynes.
42 PROBLEMS IN PHYSICS
62. Show that the dyne is, roughly speaking, the weight of
i mg., and that the unit of force in the F.P.S. system (called
the poundal) is the weight of  oz. approximately.
63. Engineers use the weight of a pound 2& the unit of force.
Taking g as 32.2, what is the value of the unit of mass in this
system ?
64. Reduce a force of 2 kg. weight to dynes.
65. Find the weight in dynes of a man who gives his weight
as 140 Ib.
66. What is the value of "the acceleration due to gravity"
in terms of (a) the centimeter and second, (b) the foot and
second, (c) the meter and minute ?
67. Would any change occur in the weight of a ball if it
were, carried to the center of the earth ? Imagine the ball to
be in motion at the center of the earth ; is the same force
required to stop it in a given time as would be required under
the same conditions at the surface of the earth ?
68. Aside from any possible difference in value, would there
be any advantage in buying silver in Philadelphia and selling it
in Berlin, provided weighings at both places were made with the
same spring balance ? Explain your answer fully.
69. A force equal to the weight of 2 kg. acts on a mass
of 40 kg. for half a minute. Find the velocity acquired, and
the space passed over in this time.
70. A force equal to the weight of a kilogram acts on a
body continuously for 10 sec., causing it to describe in that
time a distance of 10 m. Find the mass of the body.
71. The weight of a pound being taken as the unit of
force (the engineer's unit, called by him simply a pound},
find the constant horizontal pull necessary to draw a block
of 12 Ib. weight over a frictionless horizontal table, with an
acceleration of 8.05 ft. per second per second.
VELOCITY, ACCELERATION, AND FORCE
In the fundamental relation
43
.we have
whence
The force required is
F= 12 and a = 32.2;
M = units of mass.
32.2
12
F=Ma' = 8.05 = 3 Ibs. weight.
72. How far will a body fall from rest in five sec. ? What
is its final velocity ? What is its mean velocity during this
time ?
The acceleration due to gravity is sensibly constant in any one locality.
Problems in falling bodies, therefore, come under the head of uniformly
accelerated motion, and the same formulas apply.
73. The Washington monument is 169 m. high. In what
time will a stone fall from top to bottom ?
74. What velocity does a body acquire in falling through a
distance of 100 m. ?
75. From what height must a body fall to acquire a velocity
equal to that of an express train making 96 km. per hour ?
76. A stone dropped from the top of a building strikes the
ground in 3 sec. What is the height of the building ?
77. A pebble thrown vertically downward from the top of a
tower with a velocity of 3 m. per second, strikes the earth in 4
sec. What is the height of the tower ?
78. Show that if two bodies A
and B be let fall a time interval
of 6 apart, As velocity relative to
B is constant.
After a time /, A has acquired the velocity
But B has now been falling a time / and
has acquired the velocity
Fig. 12.
44 PROBLEMS IN PHYSICS
Their relative velocity is therefore
V A  V B = gO,
that is, simply the velocity acquired by A before B was allowed to fall.
Graphically A's velocity is represented by the line OA drawn at a pitch g Ws
velocity is represented by BC drawn at the same pitch but having an inter
cept on the jraxis of + 0. The constant intercept MN represents their rela
tive velocity.
79. Extend the foregoing problem to the case in wbich
both A and B have initial velocities, and discuss the conditions
under which their relative velocity may be +, o, or .
80. A body is thrown vertically upward with a velocity V Q .
Find an expression for its velocity at any time /.
The student should here remember that the conditions differ from those of
a body thrown downward with an initial velocity only in the direction of this
velocity. In time / the body acquires the velocity gt irrespective of its initial
velocity. If we count velocity upward as positive, we must have then
v VQ gt.
81. A body is projected upward with a velocity of 30 m.
per second. Find its velocity after 2 sec. ; after 4 sec.
82. A body is projected upward with a velocity V Q . When
will it reach a given height // ?
The equation of this motion is
* = /***
Its solution gives two roots which, if real, are both positive. The smaller root
is the time required to reach a height h during the ascent. The greater one is
the time required to reach the same height during the descent. If the roots
are imaginary, V Q is not great enough to carry the body to the height h. The
student will readily interpret the case in which the roots are equal.
83. A body is projected vertically upward with a velocity of
24 m. per second. When will it reach a height of 10 m. ?
84. Show that when a body is thrown upward it has, at a
height h, numerically the same velocity, whether the body be
rising or falling.
85. A body is projected upward with a velocity of 20 m. per
second. How high will it rise before beginning to descend ?
^\\ BRA/;
" OF THB
fVERS***
VELOCITY, ACCELERATION, AND FORCE 45
86. A ball is thrown upward with a velocity of 20 m. per
second. How long before it will cease to rise ? How long
before it returns to the hand ?
87. The velocity of a body varies as the square of the time.
If in 2 seconds after starting it has acquired a velocity of 40
cm. per second, how far will it go in 5 sec. ?
88. The velocity of a particle varies as its distance from
the startingpoint. Find the distance traversed in time t.
Velocity at startingpoint given as ?; .
NOTE. In the following problems on the inclined plane friction is not
considered ; that is, the plane is assumed to be perfectly smooth.
89. Explain how the acceleration due to gravity may be
studied by means of a body sliding down an inclined plane.
Show that the body's acceleration along the surface of the
plane varies as the vertical height of the plane. Discuss the
limiting cases of this relation.
DEFINITIONS. The pitch of an inclined plane is the ratio of its height to
its base, i.e. pitch =  Or, again, the pitch of a plane is the tangent of its
b
inclination to the horizontal, i.e. pitch = tan <f>.
In connection with roads the word grade is com
monly used by engineers to denote the relation of
the height of an incline to its length, i.e. grade
=  A " 3 per cent grade," for example, means
that
7 = 03 b
Fig. 13.
Obviously, grade sin <.
90. The pitch of a plane is .75. With what acceleration
would a body slide down its surface ?
a =sinec = 980 f = 588.
91. Which is the steeper, a 6 per cent grade or a 6 per
cent pitch f
92. A body sliding down an inclined plane describes in the
third second of its motion a distance of 122.5 cm  Find the
grade. .
46 PROBLEMS IN PHYSICS
,
2
# = ?^ii = 49 cm. per second per second
^ 40 i
Grade   = ~ = = 5 per cent.
g 980 20
93. A body slides down the plane
OA. Show that the velocity acquired
on reaching A is the same as that
which would be acquired in a free fall
through the distance OH.
94. A heavy particle slides from rest H
down an inclined plane whose length is
4 m. and whose height is 1.2 m. What is the velocity of the
particle on reaching the ground ? What is the time of fall ?
95. A man can just lift 150 Ib. What mass can he drag at
a uniform rate up a frictionless grade of 7.5 per cent ?
100
x = 2000 Ib.
96. A body slides down a plane 2.1 m. long in 3 sec.; to
slide down another plane of the same height requires 5 sec.
What is the length of the latter plane?
97. A body slides freely down an inclined plane. The dis
tances passed over in successive seconds are in what ratio ?
(Compare with 40.)
98. A board is 4.95 m. long. To what angle must it be
tipped in order that a body shall slide the full length in 3 sec. ?
99. The height of an inclined plane is 426 cm. and its grade
is 30 per cent. With what initial velocity must a particle be
projected upward along the plane in order to come to rest just
at the summit ?
VELOCITY, ACCELERATION, AND FORCE 47
100. A number of planes have lengths
and inclinations equal to the chords OA,
OB, etc. Show that if a number of parti
cles are allowed to slide down these planes,
all starting from O at the same instant and
without initial velocity, they will all reach B N
the circumference in the same time.
101. A point and a line lie in a vertical
plane. Find the line of quickest descent from the point to
the line.
102. A freight train is moving at the rate of 8 mi. per hour ;
a train man runs over the cars towards the rear of the train, a
distance of 220 ft., in 30 sec. What is his speed relative to the
surface of the earth ?
103. Two trains of the same length are running with the
same velocity on parallel tracks, but in opposite directions.
Their combined length is 800 ft., and they pass each other in
6 sec. What is the velocity of the trains relative to the track ?
104. A and B are at one corner of a square. They desire to
reach the diagonally opposite corner at the same instant. A
chooses the diagonal path, while B follows around two sides.
(a) What ratio must exist between the magnitudes of their
velocities ? (It is assumed that these magnitudes are constant.)
105. The component of a ship's velocity in an easterly direc
tion is 7.2 mi. per hour ; the component in a southerly direction
is 4.6 mi. per hour. What is the total velocity of the ship ?
What is its direction of motion ?
106. When a ship is sailing northeast at the rate of 10 mi.
per hour, with what speed is it approaching a north and south
coast lying to the east ?
107. A steamer is moving due north with a velocity of
25.6 km. per hour. The smoke from the funnel lies 35 south
of east. If the wind is due west, find its velocity.
48 PROBLEMS IN PHYSICS
108. A body is moving upward along a path inclined 30 to
the horizontal with a velocity of 60 m. per minute, (a) What is
its velocity in a horizontal direction, (b) in a vertical direction,
(c) at right angles to the direction of motion ?
109. A street car is moving at the uniform rate of 6 mi. per
hour up a 5 per cent grade. Find the velocity in feet per minute
with which the car is rising vertically.
no. Find the resultant of the velocities 8 and 10 m. per sec
ond when the angle between them is 30, 45, 150, and 180.
in. Given four velocities a, b, c, and d of magnitudes 6, 8,
12, and 20 units respectively. The angle between a and b is
30, that between b and c is 15, and that between c and d\s 80.
Find by resolving these velocities along any two rectangular
axes their resultant in direction and magnitude. (See Intro
duction.)
112. A man starts to row across a stream at a velocity of 4.4
mi. per hour. If the velocity of the current at all points be
3 mi. per hour, at what angle to either bank must he make his
course in order to land at a point directly opposite that from
which he started ? If there were no current, at what speed
should he row directly across in order to make the trip in the
same time as under the foregoing conditions ?
113. A point is moving along a straight line with an accelera
tion of 22 cm. per second per second. Find the acceleration of
the point in directions 30, 90, and 180 from this line.
114. A particle is projected upward at an angle of 45 to the
horizontal with a velocity of 120 m. per second. In what time
will it reach its greatest height ?
SUGGESTION. When the body reaches its greatest height, the vertical
component of its velocity must be zero. Hence find the vertical component
of the initial velocity, and divide by the loss of velocity per second; that is,
find the time required for the body to lose all of its initial velocity in a vertical
direction.
VELOCITY, ACCELERATION, AND FORCE 49
115. A particle is projected upward at an angle of 30 to the
horizontal with a velocity of 70 m. per second. Find the time
of flight, i.e. the time elapsing before the particle again reaches
the horizontal.
116. A body is projected with a velocity Fat an angle a.
Find the horizontal distance (the range] described.
Without considering the nature of the path, the range is readily obtained by
multiplying the horizontal velocity, which is constant, by the time of flight.
117. For a given initial velocity, show that the range is a
maximum when the body is projected at an angle of 45.
118. A body is projected at a given angle a to the horizontal.
If the initial velocity be doubled, how does the range vary ?
119. Show that any two complementary angles of projection
give the same range.
120. Find the greatest height to which a body will rise and
its range, if it is projected with horizontal and vertical velocities
of 40 and 80 m. per second.
121. A body is thrown horizontally from the top of a tower
100 ft. high with a velocity of 200 ft. per second. Find
(a) the time of flight,
(b) the range,
(c) the velocity with which the body strikes the ground,
(d) the angle at which it strikes the horizontal.
122. Find the equation of the path of a projectile, and show
that the trajectory is a parabola.
123. Find an expression for the angle at which a particle
must be projected with a velocity of given magnitude in order
that it shall pass through a given point in the plane of the
motion. What indicates that the given point is out of range ?
124. (a) Define angular velocity, (b) Find the angular ve
locity of a wheel making 1000 revolutions per minute.
In engineering practice it is common to express rate of rotation in revolu
tions per minute. In these units the angular velocity would be simply 1000.
But in physics the velocity would be taken in radians per second.
50 PROBLEMS IN PHYSICS
125. Compare the angular and linear velocities of two points
distant I and 2 m. respectively from the center of a wheel mak
ing 40 revolutions per minute.
126. What are the dimensions of angular velocity ?
127. A wheel makes i revolution in .5 sec. What is its
angular velocity ?
128. Express the angular velocity of the rotation of the earth
on its axis in radians per second.
radians per second.
24 x 3600
129. What is the linear velocity of a point on the surface
of earth at 60 north latitude ? (Rotation alone considered.
Mean radius of earth 6366.8 km.)
130. A pinion having 16 teeth is geared to another having
66 teeth. Compare the angular velocities.
131. The driving wheel of a locomotive is 1.5 m. in diame
ter. If the wheel makes 250 revolutions per minute, what is
the mean linear velocity of a point on the periphery? What is
the velocity of the point when it is vertically above the axis of
rotation ? When it is vertically below ?
132. A freely falling body acquires a momentum of 12,054
C.G.S. units in 3 sec. What is its mass ?
133. The velocities of two bodies are as 6:4, and their
momenta are as 9 : 2. What is the ratio of their masses ?
6 m _ 9 .
4 m' ~ 2 '
=
m' 12
134. The mass of a gun is 4 tons. If a shot of mass 20 Ib.
be fired with an initial velocity of 1000 ft. per second, what is
the initial velocity of the recoil ?
VELOCITY, ACCELERATION, AND FORCE
135 What pressure will a man weighing 150 Ib. exert on
the floor of an elevator descending with an acceleration of
4 ft. per sec. per sec. ? Explain the sensation of being lifted
which one has in an elevator suddenly arrested in its descent.
136. A balloon rises with a uniform acceleration of 4 m. per
second per second, carrying with it a spring balance upon the
hook of which is hung a ball of 7.35 kg. weight, (a) What is
the reading of the balance in kilograms' weight ? (b) What
reading would the balance show if the balloon were descending
with the acceleration named ?
137. Two masses M and m are connected by an inextensible
string passing over a smooth peg. Neglect
ing the mass of the string, find : (a) the
acceleration of M and m, and (b) the ten
sion of the string.
M<
Fig. 16.
Since the string is without mass, and since it does
not stretch, it has the same tension T at every point in
its length. Further, the downward velocity of M must
equal the upward velocity of /, and their accelera
tions must be numerically equal. Let a be this com
mon value.
Consider the forces acting on M. These are: (i)
the weight of M downwards, and (2) the tension T
upwards. And there are no others. Hence we write
Mg  T = Ma.
Again, considering the forces acting on ;//, we arrive at a similar relation,
and, from the two equations thus found, the values of a and T are readily
deduced.
138. Show that the value of a found above is independent of
the unit in which M and m are measured. Can the .same be
proved of 7\ ?
139. If the masses M and m are equal, what kind of motion
is possible ? What is the value of the tension 7\ ?
140. Two masses are connected by a weightless cord hanging
over a smooth peg ; the sum of the masses is twice their differ
ence. Find the common acceleration.
52 PROBLEMS IN PHYSICS
141. Show that, in order to derive the expression for the
acceleration in 137, it is not necessary to consider the tension
in the cord.
142. A cord passing over a frictionless pulley has fastened to
its ends masses of 5 and 10 kg. respectively. Find the pull on
the hook sustaining the pulley when the masses are in motion.
(Neglect weight of pulley itself.)
143. Explain how the value of g may be determined by
Atwood's machine.
144. One has weights aggregating 10 kg. ; it is required to
divide the total into two parts such that when connected by a
string passing over a pulley, the whole will have an acceleration
\ that due to a free fall.
145. A mass m is drawn horizontally along a smooth table by
a cord passing over a small fric
tionless pulley and attached to
a mass M. Find expressions for
the acceleration of both masses ^
and the tension in the cord.
146. In the last problem what
must be the ratio of M to m in Fi s 17 
order to produce an acceleration equal to f that of a freely
falling body ?
147. A mass of 20 g. hanging over the edge of a table draws
a mass of 84 g. along the horizontal surface. Assuming no
friction, find the tension in the cord. In what time will the
second mass traverse the length of the table if this latter is
3 m. long ?
148. Two masses m l and m% are connected by a string. m 1
hangs freely while m 2 rests on a plane inclined at an angle a to
the horizontal. If the string passes over a small frictionless
pulley at the summit of the plane, find the resulting acceleration.
VELOCITY, ACCELERATION, AND FORCE 53
Consider the forces acting on m r These are : (i) its weight m^g and (2)
the cord tension T. If f be the common acceleration, we must have
So, the forces acting on m z are the resolved part of its weight acting along the
Fig. 18.
plane and the cord tension. This gives another and similar equation in
which f and T are unknown. By eliminating these quantities are readily
found.
149. Show that when a = 90, the results are identical with
those obtained in 142 ; also that when a = o, the results are
identical with those in 145.
150. In order to pull a mass of 1000 kg. up an incline of 30,
a rope and pulley are used as in 148. Neglecting all friction,
compute the tension in the rope when a mass is used sufficient
to cause an acceleration of 0.4 m. per second per second.
151. Find the resultant of two forces of 6 and 9 kg. weight :
(1) Acting in the same straight line and in the same direction.
(2) Acting in the same straight line but in opposite directions.
(3) Acting at angles of 30, 45, 90, 120, and 150.
152. A force is inclined 36 to the horizontal. What is the
ratio of its vertical to its horizontal component ?
153 Three concurrent forces of 8, 30, and 12 kg. weight are
inclined to the horizontal by angles of 32, 60, and 143 respec
tively. Find the horizontal and vertical components of their
resultant.
154. Two forces acting at an angle of 60 have a resultant
equal to 2V3 dynes. If one of the forces be 2 dynes, find the
other force.
54 PROBLEMS IN PHYSICS
155. Two equal forces act on a particle. If the square of their
resultant is equal to three times their product, what is the angle
between the forces ?
156. At what angle must two forces act so that their resultant
may equal each of them ?
157. Find the angle which shall make the resultant of two
forces of constant magnitude a maximum.
158. Let the angle between two forces of constant magni
tude increase continuously from o to TT. Discuss the variation
of the angle between the resultant and one of the forces.
159. Show that when three forces in the same plane are in
equilibrium their lines of action meet in a point.
160. Show that when three forces are in equilibrium each
force is proportional to the sine of the angle between the other
two (Lami's theorem).
161. Find by graphic construction the resultant of four
forces of 3, 7, 5, and 12 Ib. weight acting on a particle, and
represented in direction by the successive sides of a square.
162. Two forces of 3 and 4 units respectively are balanced
by a third force of "N/37 units. Find the angle between the
first two forces.
163. A mass of 4 kg. is suspended at the middle of a cord
whose two halves make an angle of 30 with the horizontal.
What is the tension in the cord ? (Mass of cord neglected.)
The mass remaining the same, how may the tension in the cord
be varied ? Discuss the law of variation.
164. A weight of 14 kg. hangs at the end of a string ; a
force is applied horizontally deflecting the string 30 from the
vertical. What is the value of this force and what the tension
in the string ?
VELOCITY, ACCELERATION, AND FORCE 55
165. A string connecting two equal
masses hangs over three smooth, equi
distant pegs. Neglecting the weight
of the string, find the resultant pressure
on each peg.
166. Why is a long line desirable in
towing a canal boat ? To pull a canal
boat at a uniform rate requires a force
Fig. 19.
in the direction of motion of P Ib.
weight. If the rope make an angle of 10 with the line of
motion, and if the weight of the rope be neglected, what pull
must the horses exert ?
167. A body of weight 30 kg. is suspended by two strings
of lengths 5 and 12 m., attached to two points in the same hor
izontal line whose distance apart is 13 m. Find the tensions in
the strings.
168. A mass of 40 g. rests on a plane inclined at 30. Find
in grams' weight the force parallel to the plane : (i) neces
sary to hold it there, (2) necessary to draw it uniformly up the
plane, (3) necessary to cause an acceleration of 30 cm. per sec
ond per second up the plane.
169. A block having a mass of 100 g. is prevented from
sliding down an inclined plane by means of a cleat. Find the
inclination of the plane which will make the pressure on the
plane equal that on the cleat, and give the numerical value of
their sum.
170. A block is held from sliding down an inclined plane by
a cleat. Plot two curves showing the variations of the pressure
exerted by the block (i) on the plane and (2) on the cleat, with
variations of the angle of the plane.
171. Determine analytically the angle for which the sum of
the cleat pressure and plane pressure is a maximum.
56 PROBLEMS IN PHYSICS
172. A ball is held at rest on an
inclined plane of given angle a by means
of a cord. Find the cord tension when
the angle between the cord and plane
is 6. For what value of 6 is this tension
a minimum ?
173. The upper end of a ladder rests
against a smooth vertical wall ; the lower
end on a smooth horizontal floor, slip
ping being prevented by means of a p . 2Q
cleat. The ladder is of uniform cross
section, weighs 100 lb., and is inclined at 60 to the hori
zontal. Find the reactions of the different surfaces against
which the ladder rests.
174. When a person sits in a hammock the tension on either
sustaining hook is greater than the person's weight. Explain.
Does the tension increase or decrease as the hammock is made
more nearly horizontal ?
175. A string hanging over a pulley has at one end a mass of
10 kg. and at the other masses of 8 kg. and 4 kg. When the
system has been in motion for 5 sec., the 4 kg. mass is re
moved. Find how much farther the weights go before coming
to rest.
176. The ram of a pile driver weighs 500 lb. It is allowed
to fall 20 ft. driving a pile 6 in. Find the value of the resist
ance, assuming it to be uniform.
[Consider the acceleration needed to bring the body to rest in the given
distance.]
177. Show graphically how to find the resultant of two
parallel forces, (a) when the forces are like, and (b) when the
forces are unlike.
178. Apply the graphical construction to the case of two
equal, unlike forces and interpret the result.
VELOCITY, ACCELERATION, AND FORCE 57
179. A man carries a bundle at the end of a stick placed over
his shoulder. If he varies the distance between his hand and
his shoulder, how does the pressure on his shoulder change ?
180. The resultant of two like parallel forces is 16 kg. weight
and its point of application is 6 cm. from that of the larger
force, which is 10 kg. weight. Find the distance of the smaller
force from the resultant.
181. Equal weights hang from the corners of a triangle which
is itself without weight. Find the point at which the triangle
must be supported in order to lie horizontally.
SUGGESTION. The forces at the corners are all equal and parallel. The
resultant of any two must act at the midpoint of the side connecting them.
Combine this partial resultant with the force at the third corner.
182. A teamster considers one horse of his pair as 25 per
cent stronger than the other. At what point should the bolt be
placed in the "evener" in order that each horse may draw in
proportion to his strength ?
183. A bridge girder rests on two piers distant a feet apart.
The girder is of uniform crosssection, / Ib. weight per linear
foot. At a distance  a from one end a load of P Ib. weight
is placed. Find the reactions of the piers.
184. What is a couple and what is the moment of a couple ?
185. Show that the algebraic sum of the moments of the two
forces forming a couple about any point in their plane is
constant.
1 86. One of the forces of a couple is 60 dynes ; the distance
between the forces is 0.3 m. Find the moment of the couple.
187. A straight bar is acted upon at its ends by two equal
and parallel but opposite forces of 12 kg. weight each. The
bar makes an angle of 45 with the direction of the forces and
is 3 m. long. Find the moment of the resulting couple.
CENTER OF INERTIA (OR MASS) (OR GRAVITY;
1 88. Two equal weights are connected by a light, stiff rod.
Find the center of inertia.
189. How would the center of inertia be moved if one of the
weights were doubled? if both were multiplied by three ?
190. Three weights,, 4, 5, and 7, are joined by stiff weightless
rods. Find the center of mass of the system.
191. What is the center of gravity of a triangle ? a square ? a
parallelogram ? a trapezoid ? Test your answers with pieces of
cardboard.
192. The diagonals of a square are drawn, and one of the tri
angles resulting is removed. Find the center of gravity of
the remaining figure.
193. Two lines are found on a surface such that the surface
will " balance " about each. What point is determined by their
intersection ?
194. Four masses are supposed concentrated at the points A,
B, C, D\ masses 9, 5, 6, 10, respectively. The lengths OA, AB,
BC, CD are 5, 8, 4, 10, respectively. Find the distance of the
center of mass of the system from the point O.
A o D
O
^i in __?!__
Fig. 21.
We have 59+ 135 + 176 + 2710 = sum .of massdistance products,
9+5+6+10 = sum of masses.
.. distance required is Vo 2  = 16+.
58
CENTER OF INERTIA 59
The distance from O to the center of gravity may be found from an equa
tion expressing the fact that about that point the sum of the moments of the
couples due to gravity is zero.
Let x : = distance required.
Then lever arm for gravity action on A is ~x 5.
Whence moment of couple due to A is gQc 5)9,
couple due to B is g(x 13)5,
couple due to Cis^(^ 17)6,
couple due to D is g (x 27)10.
Sum equals o. .. 30 .r = 482, ~x = 16+, as before.
195. A body is suspended by a flexible cord. What position
will the center of gravity assume ? Explain.
196. Explain the connection between the center of gravity
of a body and its stability.
197. Express the fact of no resultant couple about the center
of gravity in the notation of the calculus.
xx
35 dx x z
Fig. 22.
When the body is linear or is symmetrical about a line.
Let x= the distance of C.G. from O,
x = the distance of any mass element from O,
dx= the length of element.
Then pdx mass element,
x ~x lever arm.
.. mom. of couple = pdx(x ~x)g>
Sum of mom. = I 2 pdx(x ~x) = o. [By def. of C.G.
Jx r
(a) Find ~x for a uniform rod of length /.
(b) Find x for a rod where p increases from x^ to ;r 2 , i.e. where
= kx + p Q .
(c) Find x for an isosceles triangle.
6o
PROBLEMS IN PHYSICS
198. Show directly from the definition of C.G. that its coordi
l pxdv
nates are given by three equations of the form x = *
pdi>
199. Explain the distinction in meaning and use between the
f  ^mx
above expression for x and x =
200. Find C.G. of a cone of revolution.
Fig. 23
Take dv as a slice  to base. Then
rjp**dx
201. Find C.G. of a sector of a circle.
202. Find C.G. of a segment of a circle.
203. Find C.G. of an arc of a circle.
204. Apply the general formula for the coordinates of the
C.G. to the square, the circle, the rectangle, the triangle.
205. Two bodies, attracting each other with a force measured
by m i* t move toward each other. Where will they meet ?
206. Show that the momentum of any system of bodies, each
of which has motion of translation only, is the same as the
momentum of the sum of the masses moving with the velocity
of the center of gravity of the system.
CENTER OF INERTIA 6l
207. Two masses are joined by a rigid rod ; the system is
thrown in the air so that it whirls. What will be its center of
rotation ?
208. Two spheres glide freely on a light, rigid rod, and are
joined by a spiral spring sliding freely on the rod ; the system
is thrown so that the rod has an initial angular velocity <w .
Discuss the relative position of the two spheres with reference
to the center of gravity of the system.
WORK AND ENERGY
209. A constant force of 20 dynes moves a body 100 cm.
What work is done ?
210. A force of 9000 dynes is exerted constantly on a body,
and moves it 4 m. per second. How much work is done in
1 min. ?
211. How much work is required to lift I kg. from the sea
level to a height of I m. where g = 980 ? 3 kg. ? 8 kg. ?
212. How much work is required to raise i kg. 2m.?
2 kg. 5 m. ?
213. What work is required to raise 80 kg. 3 m. against
gravity ? 10 m. ?
214. Raising 80 kg. 8 m. is equivalent to raising 40 kg. how
many meters ? To lifting what mass 5 m. ?
215. 98 io 10 ergs are expended in raising 100 kg. How high
were they raised ?
216. A force of 40 dynes is applied at an angle of 60 to the
path along which the point of application moves. What work
will be done when the point is moved 1000 cm. ?
217. 8 io 8 ergs of work are required to move a body 400 m.
in a straight line. What force is required if applied at an angle
of 10 with the path ? of 20 ? of 30 ? of 80 ?
218. 4 io 8 ergs of work are required to move a body 8 io 4
cm. What was the average force required ?
219. 6 io 10 ergs of work have been expended in moving a
body against a resisting force of 3 io 5 dynes. How far was it
moved ?
62
WORK AND ENERGY 63
220. A stone of volume io 3 c.c., sp. gr. 2.6, is raised from
the bottom of a lake to the surface, a distance of 20 m. Find
the work done. See Ex. 422.
221. Find the work done in forcing a block of wood, volume
8 io 4 c.c., sp. gr. .7, to the bottom of a tank of water 4 m. deep.
What if tank were filled with mercury ?
222. Show that if gravity be the only resisting force, the
work done on a given mass in raising it a given height is in
dependent of the path. Or that the force required always
decreases in the same ratio as the path increases.
223. Show why it is easier to draw a load up an inclined
plane than lift it vertically, neglecting friction. What element
is decreased ? What increased ?
224. A vertical tank having its base in a horizontal plane is
to be filled with water from a source in that plane. The area
of the crosssection is 4 sq. m., the height is 6 m. Find the
work required to fill it.
225. Show that the work required to raise a system of bodies
each to a certain height is the same as the work required to
raise the entire mass to a height equal to that through which
the center of gravity of the system is raised.
226. A body is raised 80 m. against a force which constantly
increases. The initial value of the force is 40 dynes, its final
value 460 dynes. If the force increased uniformly with the
distance moved, how much work was done ?
227. In an ordinary swing is the force required to displace
the swing constant ? If not, how could the work be computed ?
228. A uniform rod io m. long, and mass per centimeter
length 5 kg., is drawn vertically upward a height of io m.
How much work is done? How much work would be'required
to raise the rod from a horizontal to a vertical position ?
NOTE. Consider the average height of elements of mass.
64
PROBLEMS IN PHYSICS
229. A plank 4 m. long is hinged at one end. The plank
is raised so as to make an angle of 45 with the horizontal.
What work is done ? (Mass of I cm. of plank 9 kg.)
230. Express work in terms of mass, acceleration, and dis
tance.
231. If the unit of time were taken as 2 sec., how would the
unit of work be altered ?
232. Show that power = force x velocity. What does the
statement mean when the velocity is changing ? In what units
must force and velocity be measured so that power may be
expressed in ergs per second ?
233. In what two general ways is the energy of a railway
locomotive expended while the train is acquiring velocity ?
234. The force required to overcome the friction of a wagon
on a certain road is 2  io 10 dynes. How much work is done in
drawing it 20 km. ?
235. On a perfectly level road it was found that the pull re
quired to keep a wagon moving uniformly was .01 of its weight.
What work is done in drawing a wagon weighing 2000 kg. a
distance of 3 km. ?
236. A man presses a tool on a grindstone with a force equal
to io kg. weight. The circumference of the stone is 3 m., the
coefficient of friction .2. How much work is done in one turn
of the crank ? (Neglecting friction of bearings, etc.)
Fig. 24.
237. If BC = . i AB y what mass at M 1 will draw M 2 up AB
without acceleration, neglecting friction ? What effect would
be observed if a greater mass were placed at M l ?
WORK AND ENERGY 65
238. State how you could apply the principle of work to
above case when there is friction.
239. Find the work done in drawing 120 kg. up an inclined
plane of base 4 m., height 3 m., //< = f^.
240. How much of the work is due to friction ?
241. A mass of 100 g. is moving in a circle of radius I m.,
and makes 10 revolutions per second. What is its kinetic
energy ? What would be its energy if the circle were half as
large ?
242. Five masses, 3, 8, 5, 7, and 1 1 g., are attached at dis
tances n, 7, 5, 8, 3 cm., respectively, from the centre of a
wheel making 20 revolutions per second. Find the kinetic en
ergy of each. How far from the center could the whole mass
be placed so that the energy would be the same ?
Fig. 25.
Let a constant force F be applied at a point r distant from O J_ OP r If
the rod OP^ is rigid, the work done in turning through an angle 0, since
P^P Z = rO, is FrB = force x displacement. So work done by a couple or
torque
= moment of couple (Fr) x angle turned through
= torque x angle turned through
= average torque x angle turned through
.when torque is not constant
= FrdO. [Where Fr =/(0)
243. A shaft s turns 120 times per minute. The radius of
the shaft is 2 cm. The distance from the center of the shaft to
the point where the mass is applied is 2 m. It requires a mass
F
66 PROBLEMS IN PHYSICS
of 80 kg. to hold the lever in equilibrium. Find the work done
in 5 min.
Fig. 26.
244. A mass of 80 g., moving with a velocity of 10 cm. per
second, has what kinetic energy ?
245. What is the kinetic energy of a bullet, mass 100 g.,
velocity 1 50 m. per second ?
246. A body of mass 60 g. has a velocity 40 cm. per second,
and an acceleration of 10 cm. per second per second. How
much kinetic energy will it acquire in the next second ? How
much the fifth second later ?
247. A body of mass 5 kg. is given an initial velocity of 20 m.
per second on smooth ice. If the total average resisting force
which it encounters is io 5 dynes, how far will it go before coming
to rest ? How much energy will it have when it has gone half
the distance ?
248. A ball of mass 4 kg., velocity 80 m. per second, penetrates
a bank of earth to a depth of 2 m. Find average resistance.
249. A ball of mass io g. enters a plank with a velocity of
io m. per second and leaves it with a velocity of 2 m. per second.
How much energy has it lost ?
250. If the plank is 20 cm. thick and all the work is expended
in piercing it, what is the average resistance ?
251. A bullet is fired vertically upward with an initial velocity
of 500 m. per second. What is its kinetic energy : (a) initially ?
(b) when halfway up ? (c) at its highest point ? (d) when half
way back ? What is its potential energy in each case ? What
is the sum of Ek and E P in each case ?
WORK AND ENERGY 67
252. A mass m falling freely acquires how much kinetic
energy per centimeter of its fall ? It loses how much potential
energy ?
253. Two balls of mass 100 and 200 kg. are attached to a
firm light rod. The distance between the centers of the balls
is i m. The system is thrown so that the center of gravity has
a velocity of 20 m. per second, and the system turns ten times
per second around this center. Find the kinetic energy of the
system.
254. Compare their energies of rotation about the center of
gravity of the system.
255. What is meant by the term "closed system " as applied
to energy ? Give examples.
256. State in words the relation between the work done on a
system by an external force and the rate of gain of energy by
the system and the losses by friction.
257. Trace the energy changes in a single vibration of a
pendulum : (i) When the air resistance may be neglected.
(2) When air resistance is taken into account.
258. Express in calculus notation the statement that the sum
of the potential and kinetic energy of the bob of a simple pendu
lum is constant.
259. A mass of 60 g. is vibrating in a straight line with
S.H.M. The length of the line is 4 cm., the periodic time is
2 sec. What is its average kinetic energy ?
260. The velocity of a bullet is decreased from 500 to 400 m.
per second by passing through an obstacle ; its mass is 100 g.
What energy has it lost ? What has become of that energy ?
261. Calculate (in ergs, and also in kilogrammeters) the
work necessary to discharge a bullet weighing 10 g., with a
velocity of 10,000 cm. per second.
262. If the potential energy of a stone of mass m and at a
height h . is entirely converted into kinetic energy, find the
68 PROBLEMS IN PHYSICS
velocity it must acquire. Would air friction increase or decrease
this velocity ?
263. If the stone were attached to a very flexible and exten
sible spring, what alteration of energy distribution would occur ?
264. A solid sphere of cast iron is rolling up an incline of
30, and at a certain instant its center has a velocity of 40 cm.
per second. Explain how to find how far it will ascend the
incline, neglecting friction of all kinds. Would the distance be
the same if it were sliding up the incline ?
265. If the sphere were hollow, would it acquire the same
velocity as the solid one in rolling the same distance down the
plane ?
266. What are the dimensions of power ? If the unit of
time were the minute, the unit of length the meter, how would
the unit of mass need to be altered that a given power should
be expressed by the same number ?
267. Define erg, joule, watt.
268. A constant force is applied to a body on a horizontal
plane. If the applied force is greater than the friction between
the body and the plane, why cannot an infinite velocity be
obtained ?
269. The mass of a car is 2000 kg. The resistance due to
friction is 12 io 4 dynes. A man pushes the car with a force
which would support a mass 90 kg. His maximum power is
746 io 6 ergs per second. How long can he continue to exert
his full force ?
When the component of force along the path of the point of application is
variable, we must find how its magnitude varies along this path and apply the
integral calculus to add up the elements of work.
When W = (* z Fdx,
i
where F must be expressed in terms of x, i.e. F = f(x).
WORK AND ENERGY 69
The cases of most interest are perhaps when
f(x) kx, \k a constant.
The first applies to cases of compression and stretching, as springs, etc. ;
the second to gravitation, electricity, and magnetism, etc.
270. When F= $x t find the work done in displacing a body
loom.
Jio 4 fr ^2no 4
5*aEri = f .io 8 ergs,
which is the same as taking half the sum of the initial and final
force, and multiplying by entire displacement.
20
271. When J F= , find work done in displacing the point
3C
of application from x = 20 to x 220.
r m Jr
W = \ 20
Could this result be obtained by taking \ (final force initial force) x dis
placement ?
272. A coiled spring is attached to a 50 kg. weight. What
work is done if the increase of length of the spring is 2 m.
when the weight is just lifted ?
273. If the pressure of a gas increases as its volume decreases,
show how work done in compression could be computed.
274. A horse is hitched to a loaded wagon by a long exten
sible spring. Does the work done by the horse in just starting
depend on the ease with which the spring is stretched ?
275. A bicycle rider moves up a grade against the wind.
Against what forces does he do work ? In what ways does he
expend energy ? From which of these expenditures can he get
a return of energy, and how ?
The general expression for work may be written W = ^Fds, where ds is
so short that F may be considered constant over its length. We may then
resolve both /^and ds along any three lines we please, as OX, OY, OZ.
PROBLEMS IN PHYSICS
Let jr, /, z, components of F, be X, Y, Z.
Let x, y, z, components of ds, be dx, dy, dz.
Then
W
= j* [Atf*
ds ds ds.
where X, Y, Z may depend on x, y, z.
When F is constant and along j, the formula reduces to \ Fds, as the
student may prove.
Fig. 27.
As an example we may take the work done by a couple in turning through
360. Taking the plane of xy as the plane in which the lever arm lies, we have
Ydy~\
By symmetry we see that the
Xdx = 
W=
= F circumference of O.
It is often convenient to use the law of the 'conservation of energy in the
solution of problems dealing with machines of various types. To do this,
we form an equation involving the element required; one member of the
equation representing all the work expended on the machine, the other all the
WORK AND ENERGY 71
work done by the machine. That is, equate the entire energy supplied to the
machine to the entire energy used, stored, and wasted.
Fig. 28.
The energy given to the machine may be used in various ways ; as,
(1) Lifting weights, etc. (visible and useful work).
(2) Overcoming friction (waste, transformed to heat).
(3) Strain of parts of machine (potential energy).
(4) Momentum of parts of machine (kinetic energy).
(5) Transformed to other forms, as electric, chemical, etc.
The complete analytical expression in case all of these are considered is
likely to be very complicated. We therefore simplify matters by neglecting
certain items of relatively small importance, yet it should be remembered that
in actual cases these may cause serious errors if neglected.
In most of the problems that follow, (3), (4), and (5) are neglected, and
unless otherwise stated, friction is also negligible.
The student should note careftilly that all forces which do not cause motion
are excluded, as they do no work.
276. Explain why a machine should be of sufficient rigidity
that the deformation of its parts should be extremely small.
277. Distinguish between the total energy of a system and
its available energy.
278. A railway train, in which the couplings between the
cars are heavy springs, begins to move, due to the work done
by the engine. State how the energy supplied is being distrib
uted while the train is acquiring speed.
?2 PROBLEMS IN PHYSICS
279. If the steam is shut off, from whence comes the energy
which keeps the train in motion ?
280. What becomes of the potential energy which we store
in a watchspring when we wind it ?
281. The pitch of a screw is .5 mm. A lever 40 cm. long is
used to turn it. A force equal to a weight of 20 kg. applied to
the lever will cause the screw to exert what force ?
282. Show that the screw is an example of the inclined
plane.
283. A lever is 2 m. long, the point of support 30 cm. from
the end. A force of io 8 dynes applied to the long arm will
give what force at the short arm ?
Consider the work in any displacement. Then
force applied x distance it moves = force exerted x distance moved.
Let the angle turned through = 0.
Distances are 170$ and 30$.
Work = io 8 1 70 = x 30 0.
.. x  1 / io 8 dynes.
284. The radius of the wheel of a copying press is 30 cm.
One turn lowers the plate .25 cm. Find the force exerted if
the applied force is enough to lift 20 kg.
285. In a hydrostatic press the distances moved by the
pistons are in the ratios of i to 1000. What is the force ratio ?
286. In an ordinary pump handle the long lever arm is 3 ft.,
the short one 6 in. What force applied to the longer will lift
40 kg. on the shorter ?
287. A system of gear wheels is used to raise weights.
When the first is turned 360 the last turns 60. The radius of
the first is four times that of the last. What is the force ratio ?
288. In the system of pulleys connected as shown in Fig. 29,
find the relation between w and W\ (a) by principle of work ;
WORK AND ENERGY
73
I
(b) by considering the tensions of the cords. Neglect the weight
of the pulleys.
289. In a system of eight movable pulleys connected as in
Fig. 29, find the weight which 20 kg. ,
would lift, neglecting the weight of
the pulleys and friction.
290. Find by the principle of work
the relation between w and W when
each pulley weighs / grams. It is
found by experiment that the values
of w computed above are too small
to explain this.
I 1 291. A system
of two movable
pulleys, as in Fig.
29, is of negligible
friction, and the
weight w is twice
as large as it should be for equilibrium.
What will be the acceleration of w ? of W}
292. In a system connected as in Fig.
30, find the relation between w and W\
(a) Neglecting weight of pulleys, (b) When
lower block weighs M grams.
293. Find the relation when there are n
pulleys above and n below. When there is
one more above than below.
W
Fig. 29.
X
In the wheel and axle we have, if connection is
rigid and the cord inextensible, light, and flexible,
Work done by falling of M l when angle turned
r ~ R'
(Weights are inversely as radii.)
74
PROBLEMS IN PHYSICS
For gear wheels we have the same principle. Let /?, R v and r be the
radii of the large wheel, the small wheel, and the axle of the small wheel.
M 2
M,
Fig. 31. Fig. 32.
If there is no slipping when R turns through an angle 0, R l turns through
an angle (9.
Work by M l = M l  RQ.
Workon^ = ^'^
294. If the axle of the wheel (Fig. 33) be 4 cm. in diameter,
the mean radius of the wheel 40 cm., the mass of the rim
800 g., the axle and spokes being small in comparison, the
mass M = 200 g, what will be the velocity of M when it
has fallen 4 m. ? Forming the energy equation we have, if v is
the velocity of M,
M ^__
Fig. 33. Fig. 34.
EI = % 200 v 2  + \ 800 [20 2/] 2 [Kinetic energy acquired.
Zip = 200 g 400, [Potential energy lost.
Equate and solve for v.
WORK AND ENERGY
75
M + M 1
295. A mass M is suspended by a flexible cord wound
around a heavy rimmed wheel. The radius of the wheel is R ;
the mass of the rim M r . What will be the velocity of M after
falling a distance Ji ? (Neglecting the spokes.)
Let v velocity required.
Every particle of the rim is moving with a velocity v.
Lost
296. In Fig. 35,
M= 8000 g.
M' = 200 g.
R= i m.
r= 2 cm.
The spring lies on a
frictionless shelf, and is
connected by flexible thread
to the axle. If M falls 2 m., discuss the energy changes in the
system : (i) Neglecting friction of all kinds.
(2) When friction is considered constant.
297. A weight W is carried through the point
P any number of times. Is its potential en P
ergy when at the point P any different at
successive times of passage ?
298. A crank C is turned, thereby " winding
C up " a spring s. Is the potential
energy of the crank dependent only
on its position ? Explain.
299. A strong rubber band is
stretched between two points on a
horizontal table A and B. If A
Fig 37< remain fixed and B is moved to B'
Fig. 35.
Fig. 36.
76 PROBLEMS IN PHYSICS
by any path such that the band is straight, show that the work
done depends only on AB' AB\
i.e. on the initial and final posi
tions of the ends.
300. If the band were drawn
around a peg at C, or made to
occupy any curved path between
A and B, upon what would the Fig . 38
work done depend ?
301. If the force law is m , find the work done in carrying
m' from r^ to r^.
Since the force is not constant, we must divide up the displacement into
very short elements, multiply each by the mean force for that element, and
add all these results together;
JUT mm' ,
i.e. dW =  dr,
W= rmm' * = mm' f  
or
T C GO T
If r = oo r =  since = o.
ftlftl I
 ,
r \
302. How much potential energy will I kg. have when it is
I m. above the sealevel, if we consider its potential energy as o
when at the sealevel ?
303. If g were constant and a surface were drawn everywhere
I m. from the sealevel, would i kg. placed in this surface have
a definite potential energy ? What would this surface be called ?
A stone falling freely would strike such a surface at what angle ?
304. Explain how the "potential " at a point differs from the
potential energy which a mass would have if placed at the
point.
305. If two masses attract each other according to the law
, what will be the force pulling them together when r is
infinite ?
' 306. In skating on smooth "level" ice, does one gain potential
energy ? In climbing an icy hill, is one's potential increased ?
WORK AND ENERGY 77
307. Are the horizontal floors of a building " equipotential "
surfaces ?
308. If the work done in carrying I kg. from the basement
to the first floor is called the potential of that floor, the distance
between the floors being uniform, what is the potential of the
fourth floor ?
309. If the potential of the first floor is 3io 8 , what work will
be required to carry 80 kg. from the first to the third floor ?
310. In a brick building perfectly built, do the horizontal
edges of the bricks lie in equipotential surfaces ? Given the
potential at the level of one layer, the mass of one brick in a
layer, the number of bricks in that layer, how find the work
in elevating the whole number ?
311. A man walks from a certain point along any path or up
hill and down, and returns to his startingpoint. What relation
exists between the work he has done against gravity, and the
work done by gravity on him ?
Does it follow that he has done no work ? Explain your
answer.
312. A man standing on a sloping roof has potential energy.
What hinders its transformation into kinetic energy ?
313. A body is drawn up a rough inclined plane. Against
what forces is work done ? State the relation between the
energy expended, the potential energy of the body at its highest
point, and the work done against friction.
314. How much work is done in taking 80 units of mass
from a place where the potential is 5 to one where potential
is i ? where the potential is 25 ?
315. A reservoir on a hill filled with water is said to have
what potential ? If connected by a pipe with the sealevel, in
what direction will water flow ?
316. When the potentials at two points very close together
are given, how can the force at that point be found ?
PROBLEMS IN PHYSICS
317. If the potential at points along a certain line is given
by V=f(x), find the force function.
318. If Vf(x) between two points x l and :r 2 , and the force
is constant, what condition does  satisfy between x^ and x^ ?
319. Two cylindrical reservoirs of the same capacity stand on
the same horizontal plane ; the height of one is four times the
height of the other. Which would you prefer to fill with water ?
320. When two reservoirs have the same depth of water and
one is larger than the other, compare the pressure exerted by
each at a given point to which each is connected by a pipe.
Compare the potential energy of the two.
321. If the values of a working force are taken as y, and the
distance moved as ,r,
what will the area of
the surface between two
ordinates, the curve, and
the axis of x mean ?
322. When will the
, i , i i SEA LEVEL
curve be a straight line : 
What will its slope
mean ?
flC
323. A constant force
acts on a mass subject
to friction, the force be
ing greater than the
friction. Draw the time Fig. 39.
velocity curve (initial velocity o). Discuss the curve, and ex
plain the meaning of its slope, area, etc.
324. A reservoir A is made below the sealevel. What can
you say of its potential (taking that of the sealevel as o) ? If A
and B are connected, is the potential of B altered (c closed) ? If
A and B are connected and c is opened, what potential changes
will occur ? (Fig. 39.)
FRICTION
325. Define friction. What do yon mean by sliding friction ?
326. What are the laws of sliding friction ?
327. State what you mean by the coefficient of friction.
328. If a body is "slippery," is the coefficient of friction
between it and other bodies large or small ?
329. Explain why it is difficult to walk up an icy hill.
330. Explain why rails are " sanded." Why is a violin bow
"resined"?
331. A certain force is required to move one surface over
another when the pressure between them is P. If P were
doubled, what force would be required ? if /JL were doubled and
P were unchanged ? if both P and //, were tripled ?
332. A mass of 80 kg. on a horizontal plane requires a force
equal to the weight of 1.6 kg. to keep it in uniform motion.
What is the coefficient of friction ?
333. The coefficient of friction between two surfaces is 0.14.
A pull of 20 kg. weight will overcome what pressure between
the surfaces ?
334. If the coefficient of friction is 0.2 between the driving
wheel of a locomotive and the rail, what must be the weight, in
tons, of the locomotive in order to exert a pull equal to 8.96 T. ?
335 The coefficient of friction between a block and a plane
is .3. At what angle should the plane be inclined that the
block may just slide down it when started ? What is the angle
named ?
79
80 PROBLEMS IN PHYSICS
336. For a certain plane and block, the coefficient of friction
is .2. What force applied parallel to the plane would just draw
the block up if it weighs 100 kg., and the plane is inclined 5
with the horizontal ?
337. L is a load drawing W along a horizontal plane by
means of a cord and pulley, as in Fig. 17.
If L = 8 kg., W = 40 kg., pulley friction o ; find /*.
If /i=.i8, IV =80 kg., pulley friction o ; find L.
If /LI = .3, L 10 kg., pulley friction o ; find W.
Supposing in each case that the system moves uniformly
when started.
338. Solve each part of the preceding example if the co
efficient for the pulley = .03.
339. If L were twice as large as specified in 337, find the
acceleration.
340. Draw a diagram showing the forces acting when one
body is slid uniformly over another.
341. The coefficient of friction between two surfaces is .2.
They are inclined at an angle of 60 with the horizontal. What
will be the acceleration ?
342. A mass of 40 kg. is placed on a plane inclined 50.
The coefficient of friction is .3. What force will be required to
draw the mass up the plane with an acceleration of 100 cm.
per second per second ?
343. If a series of observed values of L and J^were used as
coordinates, what kind of a line would result ?
344. If in determining //, by the horizontal plate method the
cord passing over the pulley is not parallel to block, show how
the correct value of JJL may be found.
345. Find the direction and magnitude of the least force
required to drag a heavy body up a rough inclined plane. What
is the result if the plane is horizontal ?
FRICTION 8 1
346. A block of weight W rests on a horizontal plane ; an
elastic spring is used to draw it along at a uniform rate. If
the angle at which the elongation of the spring is least is <, find
the coefficient of friction.
347. A force of 8 io 5 dynes acts for i min. on a mass of
i kg. sliding on horizontal surface. The velocity acquired was
3 io 4 . What was the coefficient of friction ?
348. A long plank lies on a nearly smooth inclined plane. A
man attempts to walk up the plank. What happens ?
PENDULUMS. MOMENTS OF INERTIA
349. Find the time of vibration of the following simple
pendulums : [g = 980] ; / = 16 cm., 32 cm., 36 cm., 9 cm.
350. A heavy sphere of small radius is suspended by a thread
5 m. long. How many times will it vibrate in an hour ?
351. What must be the ratio of the lengths of two simple
pendulums that one may make three vibrations while the other
makes four?
352. A seconds pendulum loses 8 sec. per day when carried
to another station. Compare the values of g at the two places.
353. A pendulum is carried upward with an acceleration
equal to g. What will be the effect on its period ?
What would be the effect if it moved downward
with the same acceleration ?
354. AC is a light rigid rod suspended at A. B
and C are two small heavy spheres attached to the
rod.
AB = 30 cm., AC 80 cm.
Mass of B, 20 g. ; mass of C, 50 g.
(a) Find the periodic time of each if the other
were absent.
(6) Find the periodic time of the system.
/V 2
The expression r = 2 TT\ becomes, in this case,
* MgR
30

(20 + 50)980 66 c
Taking the numerator and dividing it by the total mass, we
have" A" for this case.
Hence if R had been given, the actual masses need not be known.
82
PENDULUMS. MOMENTS OF INERTIA 83
355. In the system shown in Fig. 21, all lengths are measured
from 5.
Find (a) the Swr 2 ;
(b) the distance from 5 to center of gravity.
(c) the periodic time of the system.
(Neglect weight of the rod.)
356. Find the time of vibration of a compound pendulum
consisting of a uniform cylindrical rod 2 m. long, radius 2 cm.,
knife edges 40 cm. from end.
V^ fM ^2
, what do you mean by R ? Name two
o
values of R which could not be used in finding g.
358. Find the moment of inertia of a thin uniform rod :
(a) When the axis is _L to end of rod.
(b) When the axis is J_ to middle point.
becomes
ox^dx,
O
What is the relation between these two values and the center
of gravity of the rod ?
359. Find the moment of inertia of a thin rod whose density
increases uniformly from one end to the other :
(a) When axis is L to light end.
(b) When axis is _L to heavy end.
(Note that p = p Q + kx.)
360. What relation exists between the two values above and
the energy which the rod would have with a given angular
velocity in the two cases ?
361. Find the moment of inertia of a rectangular area, axis
through the center and in the plane of the figure parallel to
one side.
362. Find the moment of inertia of a thin circular plate, axis
any diameter.
8 4
PROBLEMS IN PHYSICS
363. Find the moment of inertia of a circular plate of uni
form density, axis through center and perpendicular to plane
of the circle.
364. Find the moment of inertia of a circular plate, axis
perpendicular to plane of circle and through its center, when
the density increases uniformly from the center outward.
365. Find the moment of inertia of a right circular cylinder,
axis through center and perpendicular to axis of the cylinder,
length of cylinder /.
Et*
ff$\ r
:.4,;HW4
R/
V
Fig. 41.
By direct integration we may consider the volume element as having a
base rdOdr, and a thickness dx.
Then
dm = prdrdOdx,
It may be observed that this result is the sum of two parts, the first the
same as Ex. 358 (), the second the same as Ex. 362. The energy of the
rotating cylinder is, in fact, made up of two parts, one due to the motion of
the center of gravity of each circular lamina, the other due to the rotation of
these laminae about their diameter with the same angular velocity as the axis
of the rod.
In all cases of finding moment of inertia, we have to express ^mr 2 as an
integral whose form and limits are determined by the problem in hand. It
should be remembered by the student in physics that energy of rotation is the
thing of real interest and importance rather than the particular mathematical
machinery involved.
ELASTICITY
366. Define elasticity of solids ; of fluids.
367. When is a body said to be highly elastic and when
inelastic ? To which of these classes does rubber belong ?
glass ?
368. State what is meant by the term stress. What is the
stress when 40 kg. rests on a horizontal surface 10 cm.
square ?
369. A vertical rod 4 sq. cm. crosssection sustains a weight
of 100 kg. What is the stress?
How would the stress be changed if the weight were doubled
and the crosssection halved ?
370. Define and illustrate the term strain.
371. A rod i m. long is stretched so that its length is
100.04 cm  What is the strain ?
372. A cube 20 cm. edge is compressed so that its volume
is 7995 c.c. What is the strain ?
373. What is meant by the term elastic limit?
374. What sort of a curve would represent Hooke's law ?
375. A series of weights are suspended by a wrought
iron wire. The ratio '"" is taken as * and
 for f a PP lied  asKg. 42shows the result. What
area of crosssection
does the straight portion OB represent ? What does the slope
85
86
PROBLEMS IN PHYSICS
of that portion mean ? Estimate the safe load. What does
the bend indicate ?
Fig. 42.
376. Define Young's Modulus. It was found that if the
elastic limit would permit so great an extension, it would
require a force of 17 IO 11 dynes per unit area of crosssection
to double the length of an iron rod. What was Young's
Modulus ?
377. Taking Young's Modulus for iron as 2 io 12 , find the
increase in length of an iron wire 3 m. long when stretched by
a force equal to the weight of 4.5 kg., the radius of the wire
being .5 mm.
378. What effect will stretching a wire have on its radius ?
379. A glass tube is stretched in the direction of its length,
would its capacity be changed, and if so in what way ?
380. A circular cylinder AB, Fig. 43, is rigidly clamped at
A, and a twist can be given to it by a wheel and weight as
shown. A series of pointers are fastened at points distant
l  t , ^, etc., from A.
o 4
" (a) If the wheel is turned 16, through what angle would each
pointer turn ?
OF Tin:
ELASTICITY
(b) If J/was 10 kg. in case (a), what would be the twist pro
duced by 25 kg. ?
(c) If M were as in case (a) and R were multiplied by 2,
how would the distortion compare with that in b ?
Fig. 43.
(d) If the length were half as great, compare the moments
required to turn the wheel through the same angle.
(e) If the radius of the cylinder were reduced onehalf, how
would the angles mentioned in a be altered if the length and
the moment of the applied force were unchanged ?
B
Fig. 44.
381. If A and B, Fig. 44, are the crosssections of two circu
lar cylinders of the same material and length, the free end
of each is twisted through the same angle 6.
88 PROBLEMS IN PHYSICS
Compare (a) the number of elements of area displaced.
(b) the mean displacement of these elementary areas.
(c) the mean return forces per unit area.
(d) the mean leverages for these return forces.
(e) the total torques or moments tending to restore
the cylinders to their former positions.
382. How does the torque vary with the length of the
cylinder ?
383. By reference to 380 and 381, find the moment of torsion
for a brass wire 3 m. long, 5 mm. radius, given the coefficient
of rigidity for brass = 38 io 10 .
Show that T= n , etc.
384. The moment of torsion of a wire 240 cm. long, radius
.7 mm. is 17.7. What force applied 2 cm. from its axis and
perpendicular to a radius would twist one end of a meter length
of this wire 360 ?
LIQUIDS AND GASES
385. Distinguish between a liquid and a gas.
386. State fully the reasoning by which the following con
clusions are reached :
(a) At any point in a liquid at rest the pressure is equal in all
directions.
(b) The pressure at any point on a submerged surface is
normal to that surface.
387. Show that the intensity of pressure in a homogeneous
heavy liquid varies directly as the depth.
388. Explain what is meant by a "head" of h feet of water,
a pressure of h cm. of mercury.
389. Express a pressure of 100 Ib. per square inch in kilo
grams per square meter.
390. Is it essential that a barometer tube be of uniform bore ?
391. A barometer tube inclined from the vertical by 5 reads
765 mm. Find the correct reading.
392. Compute the height of the " homogeneous atmosphere"
when the barometer stands at 740 mm.
393. Express in atmospheres the pressure existing at a depth
of 20 m. in sea water.
394. Find the pressure at a depth of 6 cm. in mercury sur
mounted by 4 cm. of water of unit density ; and this, again, by
12 cm. of oil of density .9, atmospheric pressure not considered.
395. Neglecting atmospheric pressure find the intensity of
pressure due to a head of 10.37 m  (34 ft.) of water; (a) in
grams weight, (b) in dynes.
89
90 PROBLEMS IN PHYSICS
396. Find in centimetres of mercury the pressure at a depth
of 20 m. in water of unit density, the barometer standing at
76 cm.
397. The pressure at the bottom of a lake is 3 times that at
a depth of 2 meters, what is the depth of the lake ?
398. At what depth in mercury will be found a pressure
equal to that existing in sea water at a depth of i km. ?
399. The air sustains a column of water 33 ft. (10.0 m.) high.
To what internal pressure is the tube of a syphon subjected at
a height of 30 ft. above the reservoir ?
400. Explain the action of an ordinary suction pump. What
is the maximum theoretical height to which water can be raised
by such a pump ?
401. A body of volume 24 cc. weighs in air at o and 760
mm. 16.142 grams. Correct the reading for the weight of dis
placed air, neglecting the air displacement of the weights.
402. Two liquids that do not mix are contained in a U tube,
the difference of level is 4 cm., and the distance between the
free surface of the heavier liquid and their common surface is
6 cm. Compare their densities.
403. A U tube 16 cm. high contains mercury to a height of
4 cm. ; how many centimeters of chloroform can now be poured
into one arm ?
404. Alcohol is poured into one arm of a U tube containing
mercury ; when equilibrium obtains it is found that the free
surface of the alcohol is 17 times as high as that of the mercury
above the common surface of the two liquids ; what is the den
sity of alcohol ?
405. Find the pressure on the upper surface of a horizontal
plane 12 cm. square when immersed to a depth of 30 cm. in a
solution of density .12.
HYDROSTATIC PRESSURE 91
On every square centimeter of the plane the pressure is the weight of a
column of the solution i sq. cm. in section and 30 cm. high plus the pressure
of the atmosphere on i sq. cm. of the free surface. This
gives as total pressure on one side of the plane, the ba
rometer reading 76 cm.
144 [(30 x 1.2) + (76 x 13.6)] =
The pressure on the under surface of the plane is equal and
opposite to this.
406. To what depth must the plane in the
last problem be sunk in order that the pres
sure on its upper surface may be double the atmospheric
pressure ?
407. A square of area 1.24 sq. m. has its upper edge in the
free surface of a body of water and its lower edge 80 cm. below
the free surface. Find the liquid pressure upon one side of it.
Note that here we have an intensity of pressure varying uniformly from
zero at the surface of the liquid to a maximum at the lower edge of the area.
We need to find the mean intensity of pressure.
408. By what law would the pressure on the area mentioned
in the last problem vary with its inclination to the free surface ?
409. Sketch the form of a dish such that the hydrostatic
pressure on its bottom shall be (a) greater than, (b) equal to,
and (c) less than, the weight of the contained liquid.
410. A hole 15 cm. square is punched in the hull of a sea
going vessel at a depth of 3.2 m. below the surface of the water.
Compute the force necessary to hold a board over the opening.
411. The water in a pond is confined by a dam of rectangular
surface. After heavy rains the water rises by J its normal
height, although still not overflowing the dam, the surface area
of the pond increases at the same time twofold. How does the
total pressure on the dam vary ?
412. Find the total pressure on a rectangular sluicegate 8 ft.
wide and 6 ft. deep when the water stands at a height of 5 ft.
92 PROBLEMS IN PHYSICS
413 Find the center of pressure of a rectangle whose upper
edge is in the free surface of the liquid.
The resultant pressure does not pass through the geometrical center of the
rectangle because the distribution of pressure is not uniform but varies as
the depth. Let b the breadth of the rectangle. Im _
agine the total fluid pressure on the right of the rectan
gle to be concentrated at a certain point distant x from
the surface. Then if we imagine equilibrium to still
exist, we must have the sum of the moments of the
various pressures about the upper edge as an axis = o. .
The pressure on a horizontal strip d/i wide and b long. ~ p.
is h.bdh. Its moment about the upper edge is hbdh.
Summing these moments, together with the moment of P. which is negative,
we have
p
= Px
bh*
 =fx
Remembering that P = (mean depth) x area,
we have finally x\h.
414. Find the center of pressure of a rectangle whose upper
edge is horizontal but submerged to a depth of Ji v
415. If the rectangle were inclined at an angle a to the sur
face of the liquid, would the center of pressure change ?
416. A right cone, vertex upward is filled with water. Show
that the resultant pressure on the curved surface is equal to
twice the weight of water in the cone and acts vertically
upward.
The volume of the cone is equal to \ the volume of a right cylinder of the
same base and altitude. If such a cylinder be placed over the cone, and the
space between it and the conical surface filled with water and the water inside
the cone removed, the pressure on the curved surface would remain unaltered.
Using this fact the proposition is readily proved.
417. The diameter of the small plunger of a hydrostatic press
is 8 cm. That of the large plunger is i m. The pressure ap
plied to the small plunger is 260 kg. What load is sustained
on the large plunger?
HYDROSTATIC PRESSURE 93
418. The diameters of the two plungers of a hydrostatic press
are 4 in. and 3 ft., both being circular. The smaller plunger is
worked by a lever whose arms are in the ratio 10: i. Find
the total load that can be lifted by a man exerting a force of
120 Ib.
SPECIFIC GRAVITY AND PRINCIPLE OF
ARCHIMEDES
419. A man can just lift a cylindrical jar when filled with
water. How many men would be required to lift the same jar
filled with a liquid of sp. gr. 12 ?
420. To what height could the jar be filled with mercury in
order that one man could just lift it ?
421. Why is it easier to swim in salt than in fresh water?
422. Explain why a balloon filled with hot air rises.
423. Four spheres of the same size are made of Pt, Pb,
Ni, and Al respectively. Compare their weights.
If of the same weight, compare their radii ; their volumes.
424. A gold and a silver coin are exactly similar in form
and of equal weight. What is the ratio of their volumes ?
425. Explain why the actual intensity of gravity need not be
known in finding specific gravity.
426. If a place could be found where g is o, could specific
gravity still be found, and if so, how ?
427. Suppose the space V in a liquid (Fig. 47) to contain
matter of steadily increasing density. At first
onetenth that of the liquid, and finally ten
times as dense. Show how the resultant force
should vary. Draw a curve using density as
&
x, and resultant force on Va.sy. Fig. 47.
428. A bottle filled with water weighs 172 g. ; the bottle
weighs 72 g. What will it weigh when filled with sulphuric
acid? Mercury? Oil of turpentine?
94
SPECIFIC GRAVITY 95
429. A cube of silver and one of gold are of equal size.
Compare their weights. If of equal weight, compare their edges.
430. A body in air weighs 40 g. ; immersed in water, it
weighs 30 g. Find its specific gravity.
431. A body weighing 80 g. and sp.gr. 4 is immersed in a
liquid sp. g. 2. How much weight does the body lose ?
432. A body of volume 8 c.c., sp. g. 6, is immersed in liquid
of sp. gr. 4. What is its loss of weight ?
433. What force would be required to hold a mass of 80 g.,
sp. gr. 5, under the surface of a liquid of sp. gr. 13.6?
434. A body weighed in water loses 25 g. ; weighed in a
liquid of unknown density it loses 50 g. Find density of the
liquid.
435. A body in air weighs 50 g. ; its sp. gr. is 8. When
weighed in a liquid, it loses 10 g. What is the specific gravity
of the liquid ?
436. A body immersed in one liquid loses 20 per cent of its
weight ; when immersed in a second liquid it loses 40 per cent
of its weight. Find the ratio of the specific gravities of the
liquids.
437. A sinker in water weighs 40 g., a block of wood in
air weighs 30 g. ; both in water weigh 20 g. Find specific
gravity of the wood. Draw the force system when both are
weighed in water.
438. A cork in air weighs 8 g. ; a sinker in water weighs
60 g. ; both in H 2 O weigh 28 g. Find the specific gravity of
the cork.
439. The specific gravity of a body is 4. What would be its
acceleration due to gravity when in water, neglecting friction ?
What if specific gravity were .4 ?
440. A body floating in water is placed under the receiver of
an air pump and the air is exhausted. Will the depth to which
the body sinks be altered ? Explain your answer fully.
96 PROBLEMS IN PHYSICS
441. A sinker, volume 80 c.c., sp. gr. 8, is fastened to a piece
of wood weighing 35 g. in air; both in water weigh 525 g.
What is the specific gravity of the wood?
442. Does specific gravity depend on the units of mass, etc.,
employed ?
443. A cork, sp. gr. .6 and volume 15 c.c., is attached to a
brass sinker, sp. gr. 8. What must be the volume of the brass
in order that the combination may just sink in water?
444. What must be the edge of a hollow brass cube I cm.
thick that will just float in water?
445. A sinker of lead, sp. gr. 11.3, is attached to a fish line
weighing .005 g. per centimeter and sp. gr. .1. What must be
the volume of the lead to pull 10 m. of the line under water?
446. A uniform rod weighted at the bottom is immersed suc
cessively in several liquids whose densities increase uniformly.
What will be the relation of the volumes immersed ?
447. A block of lead in air weighs 330 g. When suspended
in water it is found that the water and containing vessel gains
30 g. in weight. What is the specific gravity of lead ?
448. Eighty c.c. of lead, sp. gr. 11.3, 20 c.c. of cork, sp. gr. .2,
and 10 c.c. iron, sp. gr. 7.8., are fastened together. What would
they weigh in water ?
449. Compute the specific gravity of glass from the following
data :
Weight of bottle 20 g.
Weight of bottle and H 2 O 100 g.
Weight of powdered glass 1 5 g.
Weight of bottle containing glass and filled up with H 2 O . 1 10 g.
450. A specific gravity bottle is counterbalanced ; it is then
filled with water, and 19.66 g. more are needed to keep it bal
anced. When filled with alcohol only 15.46 g. are needed.
What is the specific gravity of alcohol ?
SPECIFIC GRAVITY 97
451. A hydrometer weighing 100 g. sinks to a certain mark
in water, and requires 20 g. additional to sink it to the same
mark in another liquid. What is the specific gravity of the
second ?
452. The specific gravity of a block of wood is .9. What
proportion of its volume will be under water when it floats ?
453. A block of wood, sp. gr. .7, is to be loaded with lead,
sp. gr. 11.4, so as to float with .9 of its volume immersed. What
weight of lead is required if the wood weighs i kg. : (i) When
the lead is on the top ? (2) When the lead is immersed ?
454. Show how to compute the specific gravity of a mixture
of two or more liquids when the volumes mixed and their specific
gravities are known :
(a) When new volume is the sum of the volumes of com
ponents.
(b) When there is a decrease of volume.
455. Two liquids which do not mix and of specific gravities
2 and 5 are placed in a beaker. A body of unknown specific
gravity is observed to sink until .3 of its volume was in the
lower liquid. What was its specific gravity ?
456. Eight parts by volume of a liquid whose sp. gr. is 6 are
mixed with five parts of a liquid sp. gr. 3. Find the specific
gravity of the mixture when there is no reduction of volume.
Find it when the total volume is reduced 5 per cent.
457. What is the difference between hydrometers of constant
immersion and those of variable immersion ?
458. Explain how each is used, giving an example.
459. A Nicholson's hydrometer weighs 100 g. and sinks to a
certain point in H 2 O when 40 g. are added. It sinks to the
same point in another liquid when 20 g. are added. Find
specific gravity of second liquid.
460. A long testtube with mercury in the bottom and of
uniform crosssection is used to determine the specific gravity
H
9 8
PROBLEMS IN PHYSICS
of a number of liquids lighter than water. Show how to cali
brate when the point to which it sinks in two liquids of known
specific gravity is given.
461. A piece of lead, volume 20 c.c., sp. gr. 1 1.4, is suspended
from one arm of a balance and is immersed in oil, sp. gr. .9.
From the other end an irregular mass of gold, sp. gr. 19.3, is
suspended in turpentine, sp. g. .8. What is the volume of the
gold if the beam remains horizontal ?
462. A brick, sp. gr. 2, is dropped into a vessel containing
mercury and water. Find its position of equilibrium.
463. Two equal cubes of oak and pine respectively are placed
in water. The edge of each is 20 cm. What height of each
will be above the surface?
464. A cylindrical rod of wood and iron is to be made so as
to just sink in water. Specific gravity of wood, .5 ; of iron, 7.5.
The length of the iron rod is 75 cm. How long must the
wood be ?
465. According to Boyle's law pv = k at constant tempera
ture. Give two definitions of k from a consideration of the
formula. Also show graphically the meaning of k.
466. A cylinder 24 in. long contains 2 cu. ft. of air at a
pressure of 15 Ib. per square inch.
The cylinder is slowly pushed in.
(a) Find the pressure at several points
of the stroke and lay them off as
ordinates, thus forming a pressure
volume curve with axis as shown.
Discuss this curve. (b) What is
the total pressure on the inner sur
face of the piston ?
467. Show that it follows from Fig ' 48 '
Boyle's law that the pressure of a gas at constant tempera
ture must be proportional to its density.
BOYLE'S LAW 99
468. Forty c.c. of air are enclosed in an inverted tube over
mercury. The difference of level is 50 cm. The tube is
depressed until the difference of level becomes 30 cm. What
is the volume of the enclosed air ?
469. A glass tube 60 cm. long and 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 wet the inside of the tube to a point
5 cm. below the top ; what is the depth of the ocean ?
470. An air bubble at the bottom of a pond 6 m. deep has a
volume of o. I c.c. Find its volume just as it reaches the
surface, the barometer showing 760 mm.
HEAT
TEMPERATURE
471. Define temperature. Is the sense of touch a reliable
measure of temperature ? Explain fully.
472. Bodies at different temperatures are sometimes said to
be at different thermal levels. What is meant ? Explain the
difference between temperature and quantity of heat.
473. What does a mercuryinglass thermometer really indi
cate ? How is such a thermometer graduated ?
474. How would you construct a thermometer to be "sen
sitive " ? to be " delicate " ?
475. What special advantages does mercury possess as a
thermometric substance ?
476. If the coefficient of cubical expansion of the liquid in a
thermometer is less than that of the envelope, what effect will
be produced on heating the thermometer ?
477. Reduce to Fahrenheit readings, the following Centi
grade temperatures: 45, 12, 20.
478. Reduce to Centigrade readings the following Fahrenheit
temperatures: 212, 72, 32, 30.
479. Plot Centigrade temperatures as abscissas and corre
sponding Fahrenheit readings as ordinates, and discuss the
locus. Also, take from the crosssection paper convenient
values, and construct a double thermometer scale ; i.e. one
which gives the temperatures in both systems.
100
EXPANSION OF SOLIDS IOI
480. At what temperature will both Fahrenheit and Centi
grade thermometers give the same reading ? What happens to
mercury at this temperature ?
481. The temperature of a given liquid is taken by both
Fahrenheit and Centigrade thermometers. The Fahrenheit
reading is found to be double the Centigrade reading. What
is the temperature of the liquid in degrees Centigrade ?
482. Define the coefficient of linear expansion and establish
the formula
l t = / (i + A/),
where l t is the length of a bar of given material at temperature
t, / its length at zero, and A the mean coefficient of expansion
for the material between o and t.
If a bar of given material be heated, it lengthens. Every unit of the
original length elongates for every degree rise of temperature an amount A.
This is the coefficient of linear expansion. Between narrow limits of tem
perature the elongation may be taken as proportional to the temperature rise.
The total elongation for a temperature rise of / degrees from zero must there
fore be / A/, which makes the new length
/, = / Q + / A/ = / (i + A/).
When t is large, / can no longer be taken as a linear function of the tem
perature, but is represented by
/, = / (i + A/ + A7 2 + ..).
483. Show that the true linear expansion coefficient at
temperature / is given by
idi
~l,7t
484. A platinum wire is 4 m. long at o ; find its length at
100.
We have / 100 = / (i + .000009*)
= 4 x 1.0009
= 4.0036 m.
485. Show that the value of X is independent of the unit of
length used, but depends upon the thermometric scale used.
102 PROBLEMS IN PHYSICS
486. A lead pipe has a length of 12.623 m. at 15 ; find its
length at o.
487. Why is platinum wire well adapted for use in the
" leading in " wires of a glow lamp, or in any circumstances
in which it needs to be fused into glass ?
488. A certain induction coil has 20,000 turns of copper wire
in its secondary coil. If climatic changes cause a rise of 40 in
its temperature, express the resulting expansion in turns of
mean length.
489. The length of a brass wire at 3 is 12 m. ; find its length
at 33.
In this example we might first find the length of the wire at zero degrees,
and then by resubstitution find the length at 33. A sufficiently accurate
result, however, is obtained by an approximation. We have
i +A/
whence the length at any other temperature t' is
// / * + A/'
It i t  ,
i + A/
= /,[i+A(/'/)],
very approximately when A is small. [See V.]
490. Assuming that 43 is the maximum temperature to
which steel rails, 10 m. long at o, are ever subjected during
the changing seasons, compute the space which should be left
between them when laid at 15.
491. Measurements are made at 25 upon a brass tube by a
steel meter scale, correct at o. The result is 6.426 m. Find
the length of the tube at o.
One should here consider that the result of these measurements is a
number which shows the ratio of the length of the tube to the length of the
scale at the temperature at which the measurements are made. Since the
length of the tube at zero is required, the number obtained is too large because
of the expansion of the thing measured and too small because of the expansion
"of the unit. The result sought will therefore be found by multiplying the
number by the ratio of the expansion factor of steel to the expansion factor
of brass.
EXPANSION OF SOLIDS
103
d
492. A brass rod is found to measure 100.019 cm. at 10 and
100.19 cm. at 100. Find the mean coefficient of linear ex
pansion of brass between 10 and 100.
The student should work this example first by the accurate method and
then by use of the approximate formula (see V.) and compare the results.
493. A platinum bar originally at 15 is placed in a glass
blower's furnace. The increase in length is .96 per cent. Find
the temperature of the furnace.
494. When it is desired that a point
p shall remain at a constant distance d
from a support, an arrangement built
on the principle shown in the figure
may be used. The rods a, a, and b are
of one metal and the rods c, c, are of
another. This principle is used in the
"gridiron" clock pendulum. Derive
the conditions for compensation.
495. A lever at A controls a distant
railway signal at B. If A and B are
connected by a rod, changes in temper
ature may cause a movement of the signal independent of any
motion of the lever. Devise a scheme by which this may be
avoided, the same rod being retained.
496. A clock which keeps correct time at 22 has a pendu
lum made of iron. If the temperature fall to 8, how many
seconds per day will the clock gain ?
NOTE. The time of vibration of a pendulum is proportional to the square
root of its length.
497. Show that if X be taken as the coefficient of linear ex
pansion of a given material, the coefficient of volume expansion
of the same material is approximately 3 X. [See V.]
498. A silver dish has a capacity of 1.026 1. at 75 ; at what
temperature will its capacity be just one liter?
P
Fig. 49.
104 PROBLEMS IN PHYSICS
499. A steel boiler has a surface area of 9.2 sq. m. at 6 ;
find the per cent increase in this area for a rise in temperature
of 80.
500. Find the mean coefficient of volume expansion of tin on
the Fahrenheit scale.
501. Explain how density varies with temperature, and show
that when t is small
& t = S,(i
and further that
NOTE. These results are obtained by approximate methods. [See V.]
502. The density at o of a specimen of wrought iron is 7.3,
and the density at o of a specimen of tin is 7.4 ; at what tem
perature will these two specimens have the same density ?
503. Distinguish between real and apparent expansion of
liquids. Show that the coefficient of real expansion of a liquid
is sensibly equal to the coefficient of apparent expansion to
gether with the coefficient of cubical expansion of the envelope.
504. The coefficient of apparent expansion of mercury in
glass is erVo > tne coefficient of real expansion of mercury is
^Vo Find the coefficient of volume expansion of glass.
505. A graduated glass tube contains 40 c.c. of mercury at o.
If the whole be heated to 32, what is the apparent volume of
the mercury ?
If glass and mercury had the same coefficient of expansion, the apparent
volume would remain unaltered. But taking the expansion coefficient of
mercury at 182 x io~ 6 and that of glass at 3 x 85 x io~ 7 , it is evident that
the volume of the mercury increases more rapidly than the volume of the tube.
This means that the apparent volume of the mercury will increase.
506. A glass flask holds 842 g. of mercury at o. How much
will overflow if the whole be heated to 100 ?
507. Taking the density of mercury at o at 13.6, calculate
the density at 200.
EXPANSION OF LIQUIDS 105
508. Taking the density of mercury at 60 as 13.45, find the
density at 100.
509. It is desired to study the true expansion of water. If
the proper amount of mercury be placed in a glass bulb, the
expansion of the mercury, for any rise of temperature, will
equal that of the bulb itself. The volume above the mercury
will thus remain constant, and may be filled with water. Any
observed increase in the volume of water must therefore be
its true expansion. What fraction of the volume of the bulb
at zero must be filled with mercury to secure this result ?
510. Describe the manner in which water behaves between
zero and 10.
511. The surface of a pond of water is observed to be just
freezing. Would you expect the water at the bottom of the
pond to be at the same temperature and density as that at
the top?
512. Describe the weight thermometer. The bulb of a ther
mometer contains 2.4 kg. of mercury at o. The whole is heated
to t t causing an overflow of 40 g. Required t.
Let M total mass of mercury.
m = overflow.
8 = density of mercury.
K = coefficient of expansion of glass.
a = coefficient of expansion of mercury.
Now the volume of the thermometer at o is
M
8'
which becomes, at /,
The mass of mercury filling the thermometer at / D is
M m,
its volume at o is
M m
106 PROBLEMS IN PHYSICS
and this volume expands at f to
Mm,
y(i+0.
But the volume of the expanded mercury is the same as that of the expanded
bulb, from which relation t is readily found.
513. A weight thermometer containing i kg. of mercury at
o is placed in an oil bath, and the mass of expelled mercury
is found to be 26.4 g. Find the temperature of the bath, the
coefficient of apparent expansion of mercury in glass being ggVo
514. What is the law of the expansion of the permanent
gases with rise of temperature ? Through what range of tem
perature must a mass of gas be heated, at constant pressure,
in order to double its volume ?
515. If Charles' law be assumed to hold true for all tempera
tures, what happens at 273 ? What is this temperature
called ? If temperatures be reckoned from this point, how is
the expression for the law modified ?
516. A mass of gas at 15 occupies 120 c.c. Find its volume
at 87, the pressure remaining constant.
We have according to Charles' law,
~ 120 x = 120 x 1.25
288
= 150 c.c.
517. Take volumes as ordinates and temperatures as abscis
sas, and give a graphical representation of Charles' law.
518. At what temperature will the volume of a given mass
of gas be three times what it is at 17 ?
.519. A volume of hydrogen at 11 measures 4 1. If the
temperature be raised, at constant pressure to 82, what is
the change in volume ?
EXPANSION OF GASES IO/
520. The temperature of a constant volume of gas is raised
from o to 91. Find the per cent increase in pressure.
521. Show that for a given mass of gas the quantity *= t or
pressure x volume
, is invariable.
absolute temperature'
522. Find the dimensions of the product/^.
523. Find the volume of 2 Ib. oxygen at a pressure of 3
atmospheres and temperature 27, the volume of I Ib. oxygen
at o and i atmosphere being 11.204 cu  ft
The volume at o and i atmosphere is
z/ 2 x 11.204 cu. ft.
If the gas is heated at constant pressure to 27, it expands by Charles' law to
v r = fff x 2 x 11.204 cu. ft.
Now if the pressure be increased threefold at constant temperature,
V" \ X f f f X 2 X 11.204 CU. ft.
= 8.2 CU. ft.
524. Find the numerical value of ^ for a mass of i g.
of air.
Now ^ = ^^o, where v is the volume of i g. at o and p Q is a pressure
i
of i atmosphere.
A = J 3596 x 76
in grams' weight per square centimeter
T = 273
.001293
c.c.
Therefore, ^ = '3596 x 76 =
r c 273 x .001293
i f) f l)
525. Compute the value of ~; for a gas s times heavier
than air, of which m grams are taken. Show that the value
of this constant depends on the quality and quantity of the
gas used.
526. The pressure on a given mass of gas is doubled, and
at the same time the temperature is raised from o to 91.
How is the volume affected ?
108 PROBLEMS IN PHYSICS
527. The pressure of a given mass of air is that due to
1 20 cm. of mercury, its volume is 1000 cu. cm., and temperature
15. If now the pressure be increased to 250 cm., the volume
becomes 300 c.c. ; what is the temperature ?
i)"V
528. Find the value of *=, where / is measured in pounds
per square foot, v in cubic feet, and T in Fahrenheit degrees.
529. For a certain mass of air ^= 58540. Find its volume
at o and 760 mm.
530. Show that the final temperature resulting from mixing
M grams of a substance of specific heat c and at a temperature
/"with m grams of water at a temperature t is
mt
Me + m
531. Solve the equation of 530 for the specific heat c, and ex
tend the problem to the case in which the thermal capacity of
the calorimeter is considered.
SUGGESTION. Some of the heat liberated by the hot body goes to
warm the calorimeter, which is assumed to be carried through the same tem
perature range as the water. This amount of heat is therefore M c c' (9 /),
where M e is the mass of the calorimeter, and c' the specific heat of the material
of which it is made.
532. How many minor calories are required to raise the tem
perature 3 kg. of copper from 16 to no ?
533. Equal masses of iron and aluminum cool through the
same range of temperature ; compare the amounts of heat lost.
534. Assuming no loss of heat, how much heat must be
imparted to 2 gal. of water, initially at 14, in order to raise it
to the boilingpoint?
535 Compare the thermal capacities of equal volumes of
gold and aluminum.
536. Three liters of water at 40 are mixed with two at 9 ;
what is the temperature of the mixture?
SPECIFIC HEAT AND CALORIMETRY 109
537. If one has available water at the boilingpoint and
water at 5, what amounts must he take of each in order to
form a mixture of 55 1. at a temperature of 20 ?
538. Into 12 kg. of water at 30 are dropped, at the same
instant, i kg. of copper at 100 arid 1.2 kg. of zinc at 60 ; find
the resultant temperature.
539. If a calorimeter be made of material of specific heat c' ,
and if it have a mass m' , the product m'c f is sometimes called
the water equivalent of the calorimeter. What justifies the use
of the term ?
540. A copper calorimeter weighs 62 g. ; what is its water
equivalent ?
541. In determining the water equivalent of a calorimeter
the following data are observed :
Weight of calorimeter 52.66 g.
Weight of calorimeter + cold water .... 302.71
Initial temperature n
Temperature of hot water ...... 80
Final temperature ........ 14.8
Total weight after addition of hot water . . . 317.61
Compute the water equivalent.
542. Compare the result obtained in the last problem with
the computed value, assuming the calorimeter to be made
entirely of copper.
543. A silver dish weighing 50 g. contains 500 g. of water at
1 6 ; a piece of silver weighing 65 g. is heated to 100 and then
plunged into the water; the resulting temperature is 16.50;
what is the specific heat of silver?
544. A mass of 200 g. of copper is heated to 100 and
placed in 100 g. of alcohol at 8 contained in a copper calorim
eter, whose mass is 25 g., and the temperature rises to 28.5.
Find the specific heat of alcohol.
110 PROBLEMS IN PHYSICS
545. An iron ball is heated to 100 and then dropped in
3 1. of water at 6, causing a rise of temperature of 2 ; what
is the diameter of the ball ?
5450. The specific heat of most substances is not a constant,
but is a function of the temperature. If the quantity of heat
necessary to raise one gram of a substance from o to / be
given by
Q, = at + bt* + a*,
show that the specific heat at a temperature t is
C=a + 2 bt + 3 cP,
and that the mean specific heat between f and tf is
C m = a + b (t + S) + c (/ 2 + tf + t") .
546. One starts with 100 g. of water at 10, and to this one
adds successive amounts of water from a reservoir maintained
always at 100. Express the temperature of the mixture as a
function of the amount of hot water added. Plot a curve
between amounts of water added (abscissas) and final tempera
tures (ordinates). Note the limit beyond which the curve has
no physical meaning.
547. Show from the equation for the final temperature in the
method of mixtures, that loci similar to that in the last problem
are hyperbolas. Discuss fully.
548. Define heat of fusion. What seemed to justify the term
latent heat ?
>
549. Taking temperatures as ordinates and quantities of heat
as abscissas, plot the relation between these quantities for the
case in which ice at 10 is converted into water at 90.
550. How many calories must be supplied to 15 kg. of ice at
o to completely melt it ?
551. How many grams of ice at o must be added to 1000 g.
of water at 30 to produce a final temperature of 5 ?
CHANGE OF STATE III
552. In a determination of the heat of fusion of ice, the fol
lowing data are observed :
Weight of calorimeter 71.5 g.
Water equivalent . . . . . . . . 8.5 g.
Weight of calorimeter and water 156 g.
Temperature of water . . .... 54
Temperature after ice is melted 32
Weight after addition of ice ...... 174.5 g.
Compute the heat of fusion of ice.
553. Required, the amount of heat necessary to raise 3 kg.
of lead at 10 to the meltingpoint, and then to melt it.
554. How many grams of lead could be melted by the heat
set free, when 160 g. of molten tin solidifies? Each substance
is supposed to be at its meltingpoint.
555. How much ice must be thrown into 6 kg. of water at
41 to produce a final temperature of 8 ?
556. Find the least quantity of water at o which, surround
ing a kilogram of solid mercury at its meltingpoint ( 40),
will just melt the mercury without altering the temperature of
either substance.
557. Find the ultimate common temperature of the ice and
mercury in the last problem.
558. What will be the result of mixing 12 kg. of snow at o
with the same mass of water at 20 ? What must the tempera
ture of the water be in order that the snow may entirely melt,
the mixture having a temperature of o ?
559. Show how the specific heat of a solid may be obtained
by the use of the ice calorimeter.
560. In a determination of the specific heat of iron a mass of
1 60 g. is heated to 100 and dropped in the calorimeter. The
mass of ice melted is 22.4 g. Compute the specific heat of the
sample.
112 PROBLEMS IN PHYSICS
561. A mass of 400 g. of copper is heated in an oil bath and
then placed in an ice calorimeter. The mass of ice melted is
150 g. Required the temperature of the bath.
562. It is desired to determine the specific heats of several
metals by the ice calorimeter. The samples chosen are of the
same mass and are heated to the same temperature, in a bath
of boiling water. What mass must be used in order that the
computation will be simplified to
mass of ice melted ,
?
100
563. Explain the action of freezing mixtures.
564. What is meant by regelation? In what substances
should we look for the phenomenon ?
565. Explain the making of snowballs, the formation of ice on
pavements, and the flow of glaciers, as phenomena of regelation.
566. Why is iron an excellent metal for casting ? Why are
coins stamped instead of being cast ?
567. Punched rifle bullets pursue a straighter course than do
cast bullets. What reason can be given for this ?
568. What property of wrought iron enables it to be readily
welded ? How does sealingwax behave when heated ?
569. What amount of heat must be supplied to 10 kg. of
water at 100 to convert it into steam at the same temperature ?
To convert I g. of water at 100 into steam at the same temperature requires
536 calories (heat of vaporization of water) . In this case we must have
H 536 x io 4 = 5360 calories.
570. Find the numerical value of the heat of vaporization of
water in terms (a) of pound and degree Centigrade units, (b) in
terms of pound and degree Fahrenheit units.
571. Explain why evaporation cools. If a few drops of ether
be placed on the bulb of a thermometer, an immediate lowering
of the mercury is observed ; but when the thermometer is dipped
in a bottle of the ether, no lowering is observed. Explain.
CHANGE OF STATE 113
572. A kettle contains 2 kg. of water at 40. How much
heat must be supplied in order to boil the water away?
573. A calorimeter contains 316 g. of water at 40. Steam
at 1 00 is passed into the water until the mass of water becomes
336 g. What is the temperature ?
The mass of steam condensed is
336 316 = 20 g.,
which yields the heat of vaporization,
20 x 536 calories.
Further, the 20 g. of condensed steam in cooling to the final temperature 6
yields
20 (100 0) calories.
The 316 g. of water originally in the calorimeter is raised from 40 to 0,
which means a gain of heat of
316 (0 40) calories.
Now equating the heat evolved in condensing and cooling to the heat
absorbed by the cool water, the unknown temperature 6 is readily found.
574. In a determination of the heat of vaporization of water
by passing steam into a calorimeter containing cold water, the
following data are obtained :
Weight of calorimeter . . . . 71.5 g.
Water equivalent of calorimeter . . . . 8.5 g.
Weight of calorimeter and water . . . 173 g.
Temperature of cold water 17
After passage of steam :
Weight of calorimeter and water . . . 181 g.
Temperature 41
Compute the heat of vaporization.
575. What is meant by the total heat of steam ?
576. What amount of steam at 100 must be passed into
1 6 kg. of water at o in which 4 kg. of ice are floating, in order
to raise the whole to 30 ?
577. Calculate the heat necessary to raise to the boiling
point, and to completely vaporize 120 g. of alcohol at 12.
114 PROBLEMS IN PHYSICS
578. What is meant by a saturated vapor ? Upon what does
the pressure of a saturated vapor depend ?
579. Some values from Regnault's determination of the max
imum pressure of water vapor are given below. Plot them.
Temperature Pressure
(abscissas). (ordinates).
o 0.46 cm.
10 0.91 cm.
20 1.74 cm.
30 3.15 cm.
40 . . . . . . 5.49 cm.
50 9.20 cm.
60 14.90 cm.
70 23.30 cm.
80 355 cm.
90 ... ... 52.50 cm.
100 76.00 cm.
580. Into a barometer tube in which the mercury stands at
760 mm. a few drops of water are introduced. (a) Explain
what happens. (&) If the temperature be 30, and there still
remain a little water on top of the mercury, what will be the
reading of the barometer ? (The height of the layer of water
is neglected.) (c) What are the effects of raising and of lower
ing the barometer tube, supposing the cistern to be deep
enough to admit of this ?
581. In a closed chamber saturated water vapor in contact
with its liquid exists at a pressure of 23.3 cm. What is the
temperature ? If means are provided for pumping out the
vapor, what happens ?
582. How define the boilingpoint of a liquid in terms of the
pressure of its saturated vapor, and the pressure upon its free
surface ?
583. How do the results compare with the rise of pressure
at constant volume of a gas such as air with increasing temper
ature ? What conclusion can be drawn as to the relative danger
TRANSMISSION OF HEAT 115
from explosion of steam and air engines working at the same
temperature ?
584. What is the maximum pressure of water vapor at 55?
585. At Quito, Ecuador, the mean barometer reading is
52.5 cm. What is the boilingpoint? How can cooking opera
tions requiring a temperature of 100 be carried on at this
altitude ?
586. Explain the action of (a) vacuum pans for converting
sap into sugar ; (b) of digesters for boiling substances at high
temperatures.
587. In a closed vessel is contained water which has cooled
so that ebullition has ceased. How may the water be made to
boil again without applying heat to the vessel ?
588. Give examples of the transference of heat by conduc
tion. Name several metals in order of their conducting powers.
What of the conductivity of liquids ?
589. A thermometer placed in contact with the different
bodies in a cold room shows no variation in temperature, yet
some of the bodies feel colder than others. Explain.
590. Why are woolen blankets equally good for keeping the
person warm in winter and for preserving ice in summer ?
591. Define the coefficient of thermal conductivity.
592. One side of a wall of indefinite extent is maintained
constantly at o, while the other side is maintained constantly
at t. Give reasoning to show that after a certain lapse of time
(a) the flow of heat across a section of the wall parallel to the
faces is the same as that across any similar section ; and (b) that
the rate of fall of temperature across the wall is uniform.
593. Show that the dimensions of k, thermal conductivity,
are, in thermal units, ML~ 1 T~\ Whence, given that the con
ductivity of silver in C.G.S. is 1.3, find the corresponding value
Il6 PROBLEMS IN PHYSICS
in terms of the pound, foot, and minute. Explain how it hap
pens that k thus measured is independent of the thermometric
unit.
594. What would be the thickness of a plate of iron that
would permit the same flow of heat as a plate of glass 0.3 cm.
thick, the areas and temperature difference between faces being
the same ?
595. What would be the disadvantages of a thermometer
whose bulb contained a very large amount of mercury ?
596. A coil of copper wire lowered over the flame of an alco
hol lamp will extinguish it. Explain.
597. What is the function of the wire gauze in a miner's
safety lamp ?
598. If 1,440,000 calories pass in i hr. through an iron plate
2 cm. thick and 500 sq. cm. in area, when the sides are kept
at o and 10, compute the thermal conductivity of iron.
599. The surface of a pond is coated with ice 18 cm. thick.
The temperature of the air is 12. Compute the amount of
heat passing upward through a surface of I sq. m. in I hr.
Be careful to use consistent units. If .003 be taken as the thermal conduc
tivity of ice, C.G.S. units must be used throughout.
600. The last problem is to be worked on the assumption that
the thickness of the ice does not increase sufficiently in one
hour to appreciably change the flow of heat. As a matter of
fact the ice is growing thicker at a rate proportional to the
extraction of heat from the water. Find the law by which the
thickness of ice increases with time, temperature remaining as
above stated.
601. What is meant by the transfer of heat by convection?
Which plays the greater part in the heating of a room, convec
tion or conduction ?
602. Explain the method of heating buildings by hot water.
TRANSFORMATION OF HEAT 1 17
603. Give examples of the modification of climate by ocean
convection currents.
604. What is meant by radiation? Draw a curve showing
the distribution of energy in the visible and nonvisible spectra.
605. What class of bodies are good reflectors of radiant heat ?
good absorbers ?
606. Explain how the specific heat of a substance may be
determined by the method of cooling.
607. Equal masses of water and alcohol cool successively
through the same range of temperature in the same dish in
times whose ratio is J^. Compute the specific heat of alcohol
for the range of temperature used in the experiment.
608. What is meant by the radiation constant of a calorim
eter ? How is it determined experimentally, and how is it used
in a specific heat determination by the method of mixtures.
609. What is meant by the term mechanical equivalent of
heat? Describe any method by which it has been determined.
610. Express 20 calories in ergs.
From Introduction, we take as the value of J, 4.2 x io 7 ergs. Hence
20 calories = 8.4 x io 8 ergs.
611. Show that the numerical value of J in gravitational
units varies as unit of temperature.
unit of length
612. To raise I gr. of water i C. requires 4.2 x io 7 ergs.
Find the number of footpounds required to raise I Ib. of water
i F.
613. In a certain machine the power wasted in friction is 21
kilogrammeters per hour. How much water per hour could
be heated from o to 100 by this amount of power?
614. With what speed should ice at o p be fired against an
impenetrable wall in order to be completely melted, assuming
that no heat is lost ?
n8
PROBLEMS IN PHYSICS
615. Why does the specific heat of a gas at constant pres
sure differ from the specific heat at constant volume ?
616. Describe an experiment to show that air is not cooled
by expansion if no external work is done. Is this result true
of all gases ?
617. A cubic meter of air at o
and 76 cm. pressure is contained in a
cylinder whose piston moves without
friction. If the air be heated to 100,
what is the external work done ?
Fig. 49 (a).
By the conditions of the problem, external
work is done against the pressure of the atmos
phere. This pressure is
p = 76 x 13.6 grams' weight per square centimeter.
Since the gas expands at constant pressure, the increase in volume is
100
Whence the work is
= x io 6 c.c.
100
pv = 76 x 13.6 x x io 6 gramcentimeters.
618. Compute the heat supplied to cause this expansion.
This is readily done by finding the mass of the air in the
cylinder and using the specific heat at constant pressure.
619. Compute the heat required to raise the temperature of
this mass of air at constant volume.
620. One liter of air at o is confined by a weightless piston
in a cylinder whose sectional area is i sq. dm. The pressure
of the atmosphere is 76 cm. The temperature of the gas is
raised to 273, thus increasing the volume to 2 1. Compute
the mechanical equivalent of heat. [Ratio of specific heat at
constant pressure to specific heat at constant volume =1.41.]
621. What is an isothermal line? an adiabatic line? Why
is the adiabatic line through any point of the pressurevolume
diagram steeper than the corresponding isothermal ?
TRANSFORMATION OF HEAT
622. Sketch an indicator diagram made up of two isother
mals crossed by two adiabatics. Discuss the four steps which
are made in carrying the working substance through this cycle.
623. Find the work done on the piston of a steam engine
after cutoff, i.e. after the entrance port of the cylinder is
closed, when the expansion is assumed to take place in accord
ance with Boyle's law, the back pressure being zero.
 X
j
Fig. 50.
Let the positions of the piston at different times be laid off along OX and
the corresponding pressures along OY. At E, when the piston has proceeded
a distance a, cutoff occurs, after which the pressure falls along BC. Our
problem is to find the work corresponding to the area BCDE.
If the area of the piston is A, the pressure upon it when it has proceeded
a distance x is pA. If it move under this pressure, a small distance dx, the
work done is
dw pAdx,
and the total work corresponding to a distance / a is
W  A ( l pdx.
Ja
But the condition that pv = constant gives
p'Aa = pAx,
so that
= Ap'a log, .
a
Note that Ap'a is the work done on the piston during admission
I2O PROBLEMS IN PHYSICS
624. Find an expression for the entire effective work of the
forward stroke of an engine working under the conditions above
named except that there is a constant back pressure (condenser
pressure) p c .
Note that the pressure of admission is constant, as is also the back pressure.
The work due to these pressures is readily calculated.
625. (a) Apply the results of the last problem to finding the
work per forward stroke when the numerical data are :
Area of piston = 100 sq. in.
Length of stroke = 14 in.
Boiler pressure = 60 Ib. per square inch.
Back pressure = 2.5 Ib. per square inch (actual).
Cutoff at T 3 T stroke.
If an ordinary steam gauge shows 60 Ib., the actual pressure is 60 f 14.7 Ib.
per square inch.
(b) The engine is doubleacting and makes 180 revolutions
per minute. Compute the horsepower.
626. As the result of an engine trial the data are :
Mean effective pressure from indicator card = 32.6 Ib. per square inch.
Area of piston = 64 sq. in.
Length of stroke 10 in.
Speed = 400 revolutions per minute.
The indicator diagrams being the same on both sides of the
piston, it is required to find the indicated horsepower.
627. Why are condensing engines more efficient than those
which exhaust into the air ?
628. A perfect engine takes steam from a boiler at 150 C, and
exhausts into a condenser at 30 C. Compute the efficiency.
629. If a compound marine engine consumes 2 Ib. of coal
per indicated horsepower every hour, what per cent of the
energy of the coal is being transformed into work in the
cylinder ? The heat value of I Ib. of coal may be taken at
"i 2,000 B.T.U. (pound, degree Fahrenheit units).
ELECTRICITY
STATIC ELECTRICITY
630. Two bodies are rubbed together and then separated.
It is found that they are electrified and have energy. What is
the source of this energy ?
631. Draw diagrams showing how an electric charge dis
tributes itself over the surface of a conductor. What funda
mental law of electrostatics explains this ?
632. Two unit quantities of electricity are placed 10 cm.
apart in air. What force will be exerted between them ?
633. A charge of +10 is 25 cm. from a charge of 40.
Find the force exerted between them.
634. The force between two charges is measured ; each charge
is then doubled. What will the force be if the distance is
unchanged ? How much must the distance between them be
altered that the force may be as before ?
635. The distance between two charges is 16. cm. One charge
is + 20. What must the other be in order that the force of
repulsion may be 2 dynes ?
636. Two charges q and q' are r cm. apart, q' is doubled, q
divided by 8, and r is altered so as to leave the force unchanged.
Find change in r,
637. Explain why light uncharged bodies are attracted when
a charged body is brought near them.
638. Explain fully how a goldleaf electroscope is charged
by induction. State briefly how the lines of force are dis
tributed at each step.
639. Define surface density.
121
122 PROBLEMS IN PHYSICS
640. A sphere of radius 20 cm. is charged with 400 units of
electricity. What is the surface density ?
641. The quantity on a sphere is increased fourfold. How
must the radius be changed that the surface density may be the
same ?
642. What is meant by a line of force ? a field of force ?
643. A charge of 80 units is placed at a point where the
strength of field is 100. What force will act on the charged
body ?
644. Would the presence of a field of electric force be
observed if no charged body were placed in it ?
645. Explain why an electrophorus may be used to obtain
a considerable quantity of electrification with only a small
initial charge.
646. An electrophorus (the lower plate) is charged. What
will be the nature of the electrification of a body charged by
means of it ?
647. In using an electrophorus we may divide the process
into four parts : (i) the approach of the metallic plate to the
charged one; (2) "grounding" the upper plate; (3) separating
the two ; (4) the discharge of metallic plate.
648. Draw diagrams showing the distribution of charge in
each case of Example 647.
649. Draw the lines of force in each case of Example 647.
650. Two equal light insulated conducting spheres are sus
pended so as to hang near together. One is charged positively.
Will it attract the other ? The second is grounded. Will the
force action be altered, and how ?
. 651. If instead of grounding the second they had been
brought in contact and then separated, what change in the
force action would be observed ?
POTENTIAL 123
652. Give numerical values to the charge and distance
between the centers of the spheres in the latter case, and find
the force action before and after contact.
653. Define electrical potential at a point. In what units is
it measured?
654. An isolated charge causes a potential of 25 at a point
near it. What would the potential be if the charge were in
creased fourfold ? if a charge opposite in sign, and twice as
large, were combined with the first ?
655. Show that for a single charge q the potential, at a dis
Q
tance r, is 
r
656. Find the potential at a point midway between A and B
in Fig. 5 1 ; between B and C.
Q=160 q'80
AB = i m., BC ' = 20 cm. @ ~B
Fig. 51.
657. How much work would be required to move a charge of
2 + units from a point on AB 10 cm. from A to one 10 cm.
from B} (In Fig. 51.)
658. A small sphere is charged with 40 + units. Draw the
distancepotential curve, taking the origin i cm. from center of
the sphere (r< i cm.). Draw the distanceforce curve in the
same way. Where do these curves intersect ? How might the
second be derived from the first ?
659. A conductor 20 cm. long is placed in an electrical field.
The potential at the points occupied by its ends would be 40
and 10 respectively, if the conductor were absent. How would
the potential of these points be altered by the introduction of
the conductor ?
660. What takes place on the conductor when it is moved
across the equipotential surfaces of the field ?
124 PROBLEMS IN PHYSICS
66 1. Two spheres of equal radii are suspended by silk
threads, and each is grounded. After the "ground" is broken
charged bodies are brought in the neighborhood, such that
the potentials at the points occupied by the center of the
spheres would be at potentials 10 and 10 respectively. What
changes would occur if the spheres were connected by a wire ?
662. A sphere of radius 10 is charged so that the surface
density is IOQ . What quantity is required ? What is the
4?r
potential of a point just outside the sphere? What is the
electric force at that point ? Would any of these quantities be
altered if the sphere were immersed in turpentine ? Explain.
663. What work is done in moving a charge of + 30 from a
point where V 40 to one where V= 100?
664. To move a charge of + 4 from V = 10 to V = + 10
will require how much work ?
665. A small sphere has a charge of 84 units. Draw six
equipotential surfaces; three having V< I, one V= i, two
666. Indicate, briefly, the change in these surfaces if a
charge of 4 were brought to a point 9 cm. from the first
sphere.
667. A charged sphere A is brought near to an insulated
conductor B.
Describe the electrical state of B (charge C B ^)
and potential) : Fig. 52.
(a) When A is placed near B.
(b) After grounding B.
(c) When B is again insulated and A removed.
. (d) When B is again insulated and A brought nearer
than before.
'668. Draw the lines of force in each case of Ex. 667.
669. Draw the equipotential surfaces of Ex. 667.
POTENTIAL 125
670. Two equal charges are 80 cm. apart. If each charge is
+ 40, what is the potential halfway between them ? What is
the force at that point ?
671. Indicate the difference between the electrical condition
at a point halfway between two charges when they are equal,
and when they are equal but opposite (i.e. force and potential
at the point).
672. A small charged sphere is lowered through an opening
into a spherical conducting shell. Draw the lines of force and
the equipotential surfaces in the following cases :
(a) When charged sphere is near center of the shell.
(b) When brought quite near one side of shell.
(c) After touching the inside of the shell.
673. Show that the potential inside a closed spherical shell
is constant. What conclusion concerning the electric force
within a shell follows from this ?
674. A straight line is drawn in any direction across the
lines of force and equipotential surfaces of a uniform field.
What is the meaning of the ratio of the difference in potential
between two points on the line to the distance between the
points ?
675. What is the meaning of the above ratio when the field
is not uniform? when the field is variable, but the distance
between the points is very small ?
676. Assuming that the charge on an isolated sphere acts on
a small charge just outside the sphere as though the entire
charge were placed at the center, show that the electric force
just outside is 4717) (p = surface density). Since independent of
radius of the sphere, what follows in regard to an infinite
charged plane ?
677. Can two equipotential surfaces intersect ? Can an
equipotential surface intersect itself ? Explain your answers.
126 PROBLEMS IN PHYSICS
678. Explain how an insulated conductor in the presence of
charged bodies remains an equipotential region.
679. A charged sphere is placed between two very large
conducting plates. Draw the lines of force and equipotential
surfaces.
680. What peculiarity of the distribution of the lines of force
indicates a strong field ? of the equipotential surfaces ?
681. Draw a curve showing the relation between the charge
and potential of an isolated conductor, using Q as x and V as y.
What does the slope of the line mean ? What does the area of
the curve mean ?
682. After Q has reached a certain value, a grounded con
ductor is placed near the first, and Q is again increased. What
changes in the Q V line would indicate this ?
683. When Q is stationary, and the second conductor is near,
they are both surrounded by paraffine and Q is again increased.
Show how the Q V line would differ from the preceding.
684. A conducting sphere A is charged and placed on an
insulating support at a great distance from all other conductors.
Another conductor, B (uncharged), is brought near A. Will
the charge on A be altered ? the distribution ? the potential
of A ? the force at neighboring points ? If the distribution of
force is altered, where would it be increased and where dimin
ished ? Answer the same questions if B were "grounded."
685. A straight line is drawn in any direction in a uniform
field. If the potential at each point of the line be taken as y,
and distances from a fixed point on the line as x, what kind of a
curve will be found ? What will the slope mean? What will
the slope be when the given line is drawn perpendicular to the
lines of force ? When will the slope be a maximum ?
686. Explain fully the difference between the electric force
at a point, and the electric potential at that point. What rela
tion is there between them ?
CAPACITY
127
687. Is potential a directed quantity or vector? Find the
dimensions of electric potential.
688. The difference of potential between two points is 500 ;
the distance between them is 40 cm. What is the average field
strength between them ?
689. The average field strength between two points is 50;
they are 2 m. apart. What is the difference of potential ?
690. Find the term C, V, or Q, omitted in the following table,
where C = capacity of a con
ductor ; V= the potential to
which it is raised ; Q = charge
required to give a potential V.
691. Find the energy in each
case of Example 690.
692. What is meant by the
term capacity as applied to a
conductor or condenser ?
Q
V
C
80
20
20
80
80
20
20
80
80
20
2O
80
693. A charge of 400 raises the potential of a sphere from
o to 100. What is its radius ?
694. Three spheres, capacities 4, 8, 12, respectively, are
charged to potentials 24, 16, and 8. What is the quantity on
each ? The spheres are connected by a wire of negligible
capacity. What will be the common potential ?
695. What energy is required to charge a sphere of radius
10 to a potential of 100 ? of radius 100 to a potential of 10 ?
to charge a sphere of radius 10 with a charge of 100?
696. The radius and charge on a sphere are each increased
threefold. How is the potential affected ? the energy ?
697. (a) Upon what does the electrical capacity of a con
ductor depend ? Explain why the capacity of a body is altered
by bringing a grounded conductor near, (b) If a body whose
capacity is 200 C.G.S. is charged to a potential of 4 (C.G.S.),
128 PROBLEMS IN PHYSICS
what is the quantity of electricity ? How much work is done
in charging the body ? (If formulas are needed derive them.)
(Winter, '96.)
698. A and B are two spheres, radius of each i cm. What
is the capacity of each* ?
699. A is given a charge of f 80. B is given a charge of
40. The distance between their centers is 50 cm. Locate a
point on the line joining their centers where V= o ; f 2 ; 3.
700. B is brought in contact with A and then replaced.
How would the charges be altered ? What change in potential
would occur at each of the points mentioned above ?
701. What do you mean by a condenser? Upon what does
the capacity of a condenser depend ?
NOTE. Unless otherwise stated, it will be assumed that one coating of a
condenser is grounded, i.e. V o.
702. State the analogy between electric condensers and
water reservoirs.
703. A condenser of capacity 1000 is charged with 500 units.
Half of this charge escapes. What proportion of the energy
has been lost ?
704. A quantity Q charges a condenser to a potential V.
What energy is stored ?
705. The area of the plates, the thickness of the dielectric
and its specific inductive capacity are each doubled. How will
its capacity be changed ?
706. Define specific inductive capacity.
707. A certain condenser when air is used as the dielectric
has a capacity of 400 ; when glass is substituted, the capacity is
found to be 2600. What is the specific inductive capacity of
the glass ?
708. The force action between two charged plates is found to
be onethird as great when shellac is between them as when air
is the dielectric. Find the specific inductive capacity of shellac.
CONDENSERS
129
709. Derive the formula for the capacity of a spherical con
denser : radii of conductors r and r 2 , specific inductive capacity
of dielectric k.
710. Derive the expression for the energy required to charge
a condenser in terms of Q and V\ in terms of Q and C; in
terms of C and V.
NOTE. dW = VdQ. But V is a function of Q, V Q.
711. Compare the energy required to charge two spherical
condensers to the same potential when the radii of the shells
of one are 20 cm. and 20.1 cm., sp. ind. cap. of dielectric 2, while
for the other these quantities are 40, 40.2, and 6.
712. A condenser of capacity 50 and charge 400 is connected
by a poor conductor to earth until its energy is reduced to one
sixteenth of its initial energy. What charge escapes ? How
much is the potential decreased ?
713. It is observed that the energies of discharge of two
jars charged from the same source to earth are as I to 9. Find
the ratio of their capacities.
714. A and B are two reservoirs of the same volume, but of
unequal height. P is a pump powerful enough to force water
to the top of A.
(a) Which would possess
the more potential energy
when filled ?
(b) Which would exert the
greater pressure when full ?
(c) The stopcock k is closed
when B is full, and A is filled,
k 1 is closed, and k is opened.
What change in energy dis Fig . 53 .
tribution occurs ?
(d) If the system were connected with a reservoir below the
130 PROBLEMS IN PHYSICS
level of the source from which the water is pumped, how would
the available energy be altered ?
State the analogous electrical problem for each case.
715. Draw a diagram of a charged Leyden jar when one
coating is grounded, showing the distribution of lines of force
and equipotential surfaces.
716. Two oppositely charged and insulated plates are placed
parallel to each other and near together. Explain why when
either is touched only a slight shock is received.
717. Would an increase of the distance apart change the
effect, and if so, how ?
718. What effect would an increase of the specific inductive
capacity of the medium between the plates have ?
719. There are three conducting spheres of equal radii. The
first is charged and brought in contact with the second, this
in turn brought in contact with the third. Find the energy
changes in each operation. How much energy is still stored
in the system ? How much was stored in the first sphere ?
What relation exists between these quantities ?
720. What would be the capacity of a plate condenser when
the area of each plate is I sq. m., the distance apart is .1 cm.,
the specific inductive capacity of the dielectric being 4 ?
721. How much energy is required to charge such a con
denser to a potential of 100 ?
722. In the discharge of a condenser what becomes of the
energy ? What experiments confirm your statement ?
723. How would you proceed in order to charge a Leyden
jar?
724. Find the energy of discharge of a condenser when the
plates are of potentials V l and V^ and the capacity is C.
725. There are three Leyden jars, A, B, and C, equal in
capacity, having their outer coatings connected to earth. A is
CONDENSERS 131
first charged. Its knob is then connected with the knob of B.
It is then disconnected from B and connected with C. Finally
the knobs of A, B, and C are connected. Find the energies
of the several discharges. (Larden.)
726. When are two or more condensers said to be connected
in " series " ? When in parallel or multiple ?
727. The inner plates of four similar condensers are joined,
and each outer plate is grounded. What is the ratio of the
capacity of the set to that of a single one ?
Compare : (a) The potential to which a given charge would
raise the system with that to which it would raise one. (b) The
energy required to raise the system to a given V with that
required for one ?
728. Four similar condensers are joined in series ; the outer
plate of the last is grounded, the inner plate of the first is
charged to a potential V. The capacity of each condenser is
C. What is the potential of each jar? What is the total
charge ? What is the entire energy stored ?
729. Two spheres, A and B, radii 5 and 2 respectively, and
charges +40 and 10 are joined by a wire of negligible
capacity. Find the capacity of the system ; the quantity on
each sphere ; the amount of electricity which has passed along
the wire ; the initial energy and the final energy.
CURRENT ELECTRICITY
730. State Ohm's law. For what kind of conductors and
under what conditions is it true ?
The units used in measuring current, electromotive force or potential
difference, and resistance are named the ampere, volt, and ohm respectively.
The relation of these to the C.G.S. system will be illustrated later (see p. 187).
Ohm's Law is not dependent on the units employed. Hence in any system
potential difference . .
/ = . In the practical system, current in amfieres
resistance
_ potential difference in volts
resistance in ohms
731. When potential difference = 80 volts, resistance = 40
ohms, what current will flow ? What quantity will pass each
crosssection of the wire in 5 min. (i coulomb = I ampere
second).
732. The terminals of a wire of 10 ohms' resistance are at
potentials + 40 and 40 respectively. What is the current
strength : when at 4 60 and 20 ? when at 80 and o ?
733. The potential at each end of a circuit is multiplied by
three. How must the resistance be changed that the current
may remain the same ?
734. A quantity of 200 coulombs is transferred along a wire
in 40 sec. What is the current strength ?
735. A current of strength 40 continues 2 min. What quan
tity passes ?
736. A and B are two charged conductors. V A is + , Vg is .
They are connected by a poor conductor. What changes of
potential will take place ?
132
OHM'S LAW
133
737. In the above case, if the charge on A is reduced 80 +
units and the charge on B is reduced 80 units, what is the
total quantity which has passed along the connecting wire?
738. If this transfer takes place in 5 sec., what is the aver
age current strength ?
739. Two bodies of different potential are joined by a moist
thread. It is observed that the change of potential is slow and
the current is small. Explain.
740. What do you mean by the resistance of a conductor ?
What effect does the resistance of a conductor joining two
points of constant difference of potential have ?
741. Find the terms omitted, /, potential difference, or R, in
the following table :
Potential
Difference.
j?
'
120
5
5
200
500
250
25
5
I! 5
20
340
17
35
7
400
50
2OOO
.0005
742. The terminals of a wire of resistance 60 ohms are kept
at potentials of 100 and 10 for 5 min. ; the terminal of lower
potential is then " grounded " and the potential of the other
reduced to 90; current flows again for 10 min. Compare the
quantities transferred.
743. If in the equation V = / R, we take each quantity in
turn as constant and the others as x and y, what loci would
be obtained ?
744. A uniform wire AB is kept at a uniform temperature,
and its ends at a constant difference of potential. Draw a
134 PROBLEMS IN PHYSICS
diagram showing the relation between the fall of potential and
length of the wire.
745. If in Example 744 V A V B = 100 volts, and V A = 200
volts, what will be the potential midway between A and B ?
at onefourth the distance from A to B ?
746. The electromotive force of a battery is 4 volts, and its
resistance is 6 ohms. The external resistance consists of four
pieces of wire in series ; their resistances are 10, 20, 30, and
40 ohms, respectively. Find
(a) the total current,
(b) the fall of potential along each wire,
(c) the difference of potential of the terminals of the battery.
747. Explain the difference between electromotive force and
difference of potential.
748. A Leclanche cell is connected in series with a low
resistance galvanometer. The deflection of the galvanometer
is observed to steadily decrease. Give two causes which may
explain this.
749. If the cell is shaken, the deflection rises to nearly its
former value. Explain.
750. (a) What is meant by polarization in the case of a
galvanic cell ? (b) Explain the action of some cell in which
polarization is prevented.
The relation between current, potential difference, and resistance through
out a circuit may often be best understood by a properly constructed diagram.
We may choose either of two ways, according to the end in view. We may
assume any potential we please as our arbitrary, o, since we are concerned
only with differences of potential. Then K may be plotted as y and R as x,
or we may use V as y and distances measured from an arbitrary point in cir
cuit as x. In case the circuit contains sources of electromotive force, we may
usually consider the rise of potential through them as sudden, and the line
becomes a broken one. If, however, the source of electromotive force is dis
tributed like the armature of a dynamo, the line in such places would be
curved. Potentialresistance curves are of considerable importance, and the
student is advised to study carefully the simpler cases explained below before
drawing those of more complicated circuits.
POTENTIAL DIAGRAMS
135
Take the case of a single cell, electromotive force 3 volts, internal resist
ance 6 ohms, external resistance 10 ohms. Starting at any point as /?, and
Fig. 55.
assuming MB as representing the potential at B, Ohm's law states that along
/?/?'
BC the potential falls uniformly, so that = /.
B' C
At C we may suppose an abrupt rise of potential taking place at the bound
ing surface of liquid and plate, then another uniform fall due to the resistance
of the cell, another rise at D 1 falling again
along DB to the value MB. Note that the
lines of fall are all parallel, which is equiva
lent to the statement that the current is the
same throughout the circuit. Suppose now
that the external resistance were increased, /
must decrease, and all of the sloping lines
would become more nearly parallel with OX.
But the vertical lines CC V and DD l are con
stant in length and independent of R ; it follows then, in order that C^D may
remain parallel to BC and D^B, either CC l must fall or DD^ rise, or both.
This is the same as saying that the difference of potential of the terminals of
a battery depends upon the external resistance, and approaches the electro
motive force of the cell as this resistance is increased.
When potential and distance from a fixed point are used as coordinates,
the lines of fall would not be uniform in slope, and the diagram would show
through what absolute lengths of the circuit the fall is greatest.
The relations between current, resistance, electromotive force, and potential
difference may often be better understood by reference to the flow of water in
pipes, in so far as the analogy between the two exists.
In Fig. 56 suppose P is a pump capable of forcing water to a height H w
connected to a tank 7", from which leads a straight pipe A } A 2 ~S; A^H y
A^Hfr etc., a series of vertical pipes opening from the main whereby the
136
PROBLEMS IN PHYSICS
pressure at each point can be measured ; 6" a stopcock whereby the flow in
the main can be checked. When S is open and the pump working, so that
H
L H 1
HI
"""..^
H,
H.
T
"""^.^
H 4
H
g
^223
,
*i i
^ 2 >
\3 ^
^4 /
^5 S
Fig. 5
6.
the current is steady, the pump will be unable to keep T full up to // , and it
will be found that the tops of the water columns will be in a straight line
751. What is the electrical analogue of :
(a) The friction of the pipe ?
(b) The friction of the pump ?
(c) The pressure at A 1 ?
(d) The difference between A^H^ and A^H^t
(e) The ratio,
pump pressure _ difference of pressure between A l and A 2 ?
total friction friction between A 1 and A z
(/) The height of line H^H^ vertically above 5 ?
(g) Current and quantity ?
(h) The changes which occur when 5 is slowly closed ?
752. Would the analogy hold if the pipe were bent ? if it
were enlarged at some point ?
753. State a case in flow of water analogous to cells in series ;
in multiple. Explain fully.
754. In the circuit shown (Fig. 55), a point in the external
resistance is "grounded." Draw the potentialresistance curve.
What change in your diagram would indicate a change in the
position of the ground ?
755. Determine what external resistance is required in the
v^ ^* v
f OF THB
f rjNrVEHi
OOF THB
'NTVKP'
POTENTIAL DIAGRAMS
137
circuit of Fig. 57 in order that the potential difference of A and
B may be I volt? i^ volts ? \ volt ?
Fig. 57.
756. If the resistances of the cells in Fig. 58 are very small,
draw the potentialresistance curve.
SUGGESTION. Each electromotive force causes a rise of V independent of
the other.
em/ = 3 em/ = 6
, JA B 1C
E rN
10
200
Fig. 58.
757. What is the potential difference between A and B,
Fig. 58 ?
758. The electromotive force of a battery is 5 ; when the
external resistance is 100, the potential difference at the termi
nals is 4. What is the internal resistance?
759. A circuit consists of three cells, in series; E.M.F.'s i,
2, 3 ; resistances 4, 5, 6, respectively. The external resistance
is 20 ohms. Draw the potentialresistance curve. What is the
potential difference between the negative plate of the first and
the positive plate of the last ?
138
PROBLEMS IN PHYSICS
760. In a conductor where the resistance increases as the
square of the distance from the end (decreasing crosssection),
draw a curve, using V as y, and distance from one end as x>
when the potential difference of ends remains constant.
EACH = 2
EACH r = 4
.40
c 1 ID E
E
Fig. 59.
761. Draw the potentialresistance curve for the circuit in
Fig. 59 :
(a) When "ground" is broken.
(b) When " grounded " as shown.
762. Each cell in Fig. 60 has an electromotive force of 2
volts, and a resistance of .4 ohms. Other resistances as shown.
All connecting wires (A 1 A, AB, etc.) are so large that their
resistance can be neglected. A is connected to the earth.
Fig. 60.
(a) Draw diagram to show the variation of potential along
A' CDS', (b} Compute the difference of potential between
C and D.
763. Name four things upon which the resistance of a wire
depends.
764. Two copper wires are of the same crosssection, but one
is twice as long as the other. Compare their resistances.
765. What do you mean by the resistance of wires in multiple
or parallel?
RESISTANCE 139
766. How is Ohm's law applied to find how multiple resist
ance depends on the resistance of the separate branches ?
767. The length of a wire is increased fourfold. How much
must its radius be changed that its resistance may be the same
as before ?
* . '
768. An iron wire of a certain length and crosssection has
a resistance of 40 ohms. What would be the resistance of an
iron wire ten times as long and onefifth the diameter of the
first ?
769. What would be the resistance of n equal resistances
joined in multiple? in series?
770. Thirty incandescent lamps, each R = 50 ohms, are
joined in multiple. Wliat is their combined resistance ?
771. Find the resistance between two points in a circuit
when they are joined by :
(a) Three wires in multiple, resistances 2, 5, 7, respectively.
(b) Three wires in series, resistances 2, 5, 7, respectively.
(c) Four wires in multiple, resistances 40, 20, 30, 50, respec
tively.
(d) Four wires in series, resistances 40, 30, 20, 50, respec
tively.
772. The resistance between two points in a circuit is 60
ohms. What must be placed in multiple with this to reduce
the resistance to 22 ohms ?
Fig. 61.
773. What is the resistance between A and B ? C and D ?
A and D? Fig. 61.
I I I 4 12
^ = \ = , or A R B = =3 ohms.
A X B 4 12 12' 4
140
PROBLEMS IN PHYSICS
774. A copper wire of length / is divided in the ratio of
3 to 5, and the pieces joined in multiple. What lengtJi of the
same wire might have been taken to get the same resistance ?
In dealing with a complex circuit it is well to compute each multiple resist
ance first, and then deal with
the set in series. 5
775. Find the total
resistance of the circuit,
Fig. 62. In this sys
tem we may compute
the resistance from A
to B, then from C to D,
finally add together all
the resistances in se
ries.
776. Find the total
resistance of the cir
cuit, Fig. 63. (Com
pute each multiple re
sistance first.)
B
Fig. 62.
Fig. 63.
777* Find the resistance of the system shown in Fig. 64.
778. AC and BD (Fig. 65) are two metal plates of o resist
ance. A and B are joined by a wire of
10000 ohms resistance. Find x^ so that
when placed in multiple with the first the
combined resistance is 1000. Then x^ so
that multiple resistance of the three is 100,
etc. 3
Fig. 65.
MULTIPLE RESISTANCE
141
779. Prove that the resistance of two wires in multiple is
always less than that of either.
780. Prove the following construction for computing multiple
resistance.
Lay off on O Y a length r^.
Lay off any line  to O Y a length r v
Fig. 66.
Join the upper end of each line with the lower of the other.
The ordinate of the intersection of these lines is the resistance
required.
For three or more resistances we may extend the construction as for r s .
By using crosssection paper the results may be quickly obtained.
Fig. 67.
781. Prove the following construction for resistances in
multiple. Take ;r=r 1 , / = r 2 ; join their extremities. Then
the resistance of r^ and r 2 in multiple is given by the coordinate
(y or x} of the intersection of this line, with a line drawn at an
angle of 45 with the axes.
This may be extended to any number of resistances in multiple, and easily
effected by the use of crosssection paper.
142
PROBLEMS IN PHYSICS
In the following problems it should be remembered that in dealing with
cells in multiple and in series we must be careful to consider both the electro
motive force and the resistance of the combination. It is assumed that the
cells are exactly alike, both in resistance and electromotive force, unless
otherwise stated.
The electromotive force of any number (?/) of cells in series
sum of electromotive forces.
The electromotive force of any number (;/) of cells in multiple
= electromotive force of one.
The resistance of any number () of cells in series
sum of resistances.
The resistance of any number (n) of cells in multiple is computed just
as any other multiple resistance.
782. Six cells, resistance of each 12 ohms, electromotive
force of each 2 volts are connected in series. Find combined
electromotive force and resistance. Find them when in mul
tiple.
Fig. 68.
783. A system of ten cells, electromotive force 3 volts, r
6 ohms, are connected as shown in Fig. 68. Find the electro
motive force and resistance.
784. A system of fifty cells, electromotive force I volt, r .4
ohms, are placed "ten in a row" (series), and the five rows
in multiple. What is the internal resistance of the battery?
the electromotive force ?
785. Find the current strength when each circuit (Examples
783 and 784) is closed by an external resistance of 200 ohms.
DIVIDED CIRCUITS
143
786. Given twentyfour cells, electromotive force 2 volts, r 4
ohms, external resistance 5 ohms. Separate 24 into its various
factors (as 2, 12; 3, 8 ; etc.); choose each factor in turn as the
number of cells in a row, and the other as the number of rows.
Compute the current strength in each case.
787. Do the same when external resistance = I ohm ;
200 ohms.
788. When two or more wires are joined in multiple, at each
junction they have a common potential. Hence by Ohm's law
the current through any wire will be the common potential
difference between A and B divided by the R of that wire.
789. Three wires in multiple (Fig. 69); potential difference
between A and B = 24 volts ; resist
ances as shown. What current flows
in each branch ? What is the total
current ?
790. The currents in two branches
of a divided circuit are as 4 to 12.
What is the ratio of their resistances?
791. In the circuit shown in Fig. 70, find
(a) The total electromotive force.
(b) The total resistance.
(c) The total current.
(d) The fall of potential between K and G.
40 15
Fig. 69.
HH
Fig. 70.
(e) The fall of potential between A and B ; C and D ;
E and F.
(/) The current in each branch between A and B.
State your reason for each step in the numerical work.
144 PROBLEMS IN PHYSICS
792. Twenty 5ovolt lamps, each requiring 1.2 amperes, are
connected as shown. The resistance of BB' and CO is nearly
o, that of AB + DC is i ohm.
A B R
D
Fig. 71.
Find (a) The resistance between B and C.
(b) The total current.
(c) Difference of potential between B and C.
(d) Difference of potential between A and D.
(e) The heat developed per minute in the lamps.
(/) What change takes place when five pairs of lamps
are turned off ?
(g) What objection would there be to shortcircuiting
one of each pair of lamps ?
793. A resistance of 80 ohms joins the terminals of a battery,
electromotive force 100, resistance 20. A shunt of 5 ohms is
placed around 20 ohms of the external resistance. What effect
will this have on the total current ? What effect on difference
of potential of the points where it is joined?
794. In what case will a shunt placed around a portion of
a circuit have no appreciable effect on the total current ?
795. State and explain Ohm's law. If the connections
and resistances of a cer
tain circuit are as shown
in Fig. 72, compute the
current flowing in each of
the two branches between 5
A and B. Each cell has
an electromotive force of
I volt and a resistance of
5 ohms. Fig. 72.
SHUNTS 145
796. The resistance between A and B is 100 ohms. What
resistance, x^ must be placed in shunt with A/vwwvv\B
this in order only .1 as much current will
flow along AB as before ? (V A potential LyVVVW
difference, V B to remain the same.) Find x^ #2
so as to reduce the current in AB to .01 of  /VVN/ \ / \'
its former value, etc. LAAAA/V^
By Ohm's law, x = 100 I R x l I x .
But /* = 9 I R . Fig. 73.
.. 100 I R = x^I R .
x l =  l ^ = 1 1 i ohms.
(Compare Example 778.)
797. Prove that when a shunt of resistance s is placed around
a wire of resistance r the current is r = total current.
s + r
Extend this to three or more resistances in multiple.
f
In general, 7 ri = \ **''*"''
798. A galvanometer of 1980 ohms resistance is "shunted""
by a wire r = 20. What proportion of the total current passes
through the galvanometer ?
799. The difference of potential between A and B, Fig. 63,
is to be measured by placing an instrument (voltmeter) in shunt
with the resistance between A and B. What change in this
difference of potential is caused by the insertion of the instru
ment ?
800. In the circuit of Fig. 70, what is the smallest resist
ance a voltmeter could have that when placed in shunt with
AB the difference of potential between A and B may be changed
only onehalf, of one per cent ?
801. The current between two points in a circuit is to be
measured by passing it through a measuring instrument (am
meter). Under what conditions is the current unaltered by the
introduction of the ammeter ?
146 PROBLEMS IN PHYSICS
802. In the circuit shown in Fig. 70, what is the largest
resistance which an ammeter could have and only alter the
current strength onehalf of one per cent ?
803. APB and AQB (Fig. 74) are two conductors joined in
multiple. A and B are kept at different potentials. Draw the
potentialresistance diagram for each path from A to B.
Fig. 74.
804. If potential at A is 50 volts, and at B is 40 volts, what
range of potentials may be found along APB1 along AQB ?
805. If P and Q are two points of the same potential and
the key k is closed, would the distribution of the potential
be altered ?
806. When is it certain that if any point P is chosen on the
upper branch, a point of the same potential can be found on the
lower one ? Explain fully.
807. If a source of electromotive force were in any part of the
circuit between A and B, would it always be possible to find for
any potential along APB a corresponding point in AQB1
808. Find the relation between the resistances AP, PB, etc.,
when V P = V fi , in case of no electromotive force between A
and B. (Wheatstone's bridge.)
809. Show that the best arrangement of a given number
of cells is that which makes the external and internal resist
ances as nearly equal as possible.
nE [" = electromotive force of one cell.
/ =
nr [r = resistance of one cell.
m [/? = external resistance.
mnE [n = No. cells in a " row."
nr + mR \m No. of rows.
ARRANGEMENT OF CELLS 147
Since mn = number of cells, the numerator is constant.
.. /is a maximum when nr + mR is a minimum ;
i.e. nr f mR is a minimum by variation of n and m.
.. rdn + Rdm o.
But mn constant.
.. mdn \ndm = o.
Whence = or = R.
m n m
It does not follow that the two simultaneous equations mn = JV and
~ R have integer roots ; and as fractional parts of a cell are meaningless.
we must choose the two factors of A 7 " which make as nearly^? as possible.
810. Deduce from the statement of how to group for maxi
mum current a rule when the external resistance is very great ;
very small.
811. How would you group twentyfour cells, each r = 6,
E=3, R=i6, for a maximum current? ^=36? ^=9?
R = 25 ?
n6
m~ = l6 >
mn = 24.
Multiply these equations,
2 6 = 16 24,
n 2 16 . 4 or n = 8.
.. m = 3. [8 in a row, 3 rows.
812. Apply Kirchhoff s laws to the circuit shown in Fig. 75,
where electromotive force of the cell is E and the resistance of
cell and connecting wires is r.
These laws are often stated as fol
lows :
(1) If any number of conductors
meet in a point S/ = o ; or there is no
accumulation of electrification at the
point.
(2) In any complete circuit
In applying the first law, if we consider the current flowing toward A as
+ ,we must consider those from A as . While in the use of the second
148 PROBLEMS IN PHYSICS
law, if we start from A toward B, i.e. 'with the current, and call S l r l +, we
must, when returning along r 2 , take 7 2 r 2 as .
By (i) 7=7 1 + 7 2 +7 3 .
(2) 7^7^2 = 0,
I\ r \  7 3 r 3 = O,
7 2 r 2  7 3 r 3 = o,
7r + 7^ = E. [Where E = electromotive force of cell.]
From the first and second of (2) we may express 7 2 and 7 3 in terms of I v
r v r 2 , and r. y
Substituting in (i),
..
. for / and /
If R = equivalent resistance of r v r 2 , and r 3 ,
7^? + Ir = ,
7^ + fr = E,
7^ + Sr = ,
7 3 r 3 + 7r  .
Add last three and equate to three times the first. Solve for R, using 7j
above.
813. Find the distribution of current in a set of five un
equal resistances joined in multiple.
Fig. 76.
814. In the circuit of Fig. 76, show that
( _ r i r s ^  f r *  r * r * r <
where / = total current.
/ IPJ =/ 4 1+1* + ^  2l ,
V ri + r?) 4 V r 2 r s r, + rj a '
KIRCHHOFPS LAWS 149
815. The resistance of ADB is 10, of ACS is 40. Find the
current in AB.
Assuming direction of currents as indicated by the arrows,
7 L = 7 2 + / 3 ,
7 3 40 / 2 20 = 10,
7 L 10 + / 2 20 = 2.
Eliminating / x and 7 3 , we have / 2 = 7 V amperes. What does the nega
tive sign mean ? Solve when the arrow from A to B is reversed.
816. Find 7 2 when one cell is re . _
versed (Fig. 77).
817. What electromotive force
must be inserted in branch (i)
(Fig. 77), that no current shall pass
through (2) ?
Put / 2 = o. and the third equation = e.
Whence E = 2% volts.
8 1 8. Three cells, electromotive force E v E^ E& internal
resistances r lt r^ r 3 , are joined in multiple and the external
resistance is R. Find the total current. Test your answer by
reference to the case when the cells are alike.
819. Assume, in Example 8 1 8, E l = 2, E^ 4, E z = 6, r = 3,
r 2 = 6, r 8 = 12, ^ = 40. Find the current in amperes.
820. A and B are two points in a circuit which is carrying a
current of 10 amperes. A R B = 100 ohms. What work is done
in this portion of the circuit per minute? What becomes of
this energy ?
821. How much heat is developed per second in a portion of
:a circuit, potential difference of the ends 50 volts, and the
current 50 amperes ?
Current in amperes x potential difference in volts = energy in watts.
Heat per second in calories = wa s = watts x .24.
Or/f=/. V .24=72^ . .24. '"
150 PROBLEMS IN PHYSICS
822. The resistance of a conductor is doubled and the cur
rent halved. How is the heat developed affected ?
823. The current in a wire is multiplied by three. How
much must the resistance of the conductor be altered that the
loss by heat shall be unchanged ?
824. A current of 10 amperes develops 144. io 4 calories per
minute. What was the resistance ? What quantity passes per
minute ? What potential difference is required to maintain the
current ?
825. State clearly the meaning of the terms watt and joule.
Watts x time = ?
826. A current of 40 amperes flowing in a coil causes a
difference of potential of 20 volts between its terminals ;
(a) How much energy is consumed in i hour ?
(b] How much heat is developed ?
827. Four wires of equal length and diameter, but of differ
ent specific resistances, are joined in series. For example, soft
steel, copper, platinum, and silver are used. Find the ratios of
the heat developed in the wires.
828. Given mn similar cells, each E.M.F. = e> resistance of
each = r ; external resistance R. How must they be arranged
to secure the greatest heating effect ?
829. A wire of resistance 1000 ohms is found to develop
heat enough in io sec. to raise 24 kg. of water 10. What
current does the wire carry ? What difference of potential was
required to maintain it ?
830. If work is done by the current in addition to overcoming
resistance, would IE and I 2 R have the same value ? Explain.
831 Find the distribution of heat in the circuit shown in
Fig;. 72, when there is no back electromotive force.
832. When a given set of generators are connected so as to
give a maximum current through a given external resistance,
show that onehalf the total heat is developed in the generator.
TRANSMISSION OF ENERGY 151
833. Three copper wires of equal length, diameters .1 mm.,
.3 mm., .5 mm., respectively, are joined in multiple. The elec
tromotive force of the junctions is kept constant. Find the
ratio of the heats developed in the wires.
834. Why are large conductors usually used to transmit
electrical energy ? Why is copper used in many cases rather
than iron ? What determines which shall be used ?
835. Why is it desirable to transmit electrical energy at high
potential ?
836. Why is it desirable to transform a small current at high
potential to a larger current at lower potential at the point where
it is used ?
837. A current of 40 amperes is sent over a line of 10 ohms
resistance. What is the fall of potential in the line? If the
end of higher potential is at V= 1000, what energy per second
is delivered at the end of lower potential ? What is the heat
loss per second ? Answer the last two questions if V at the
higher end were 2000 volts.
838. The voltage at which a certain amount of power is
supplied to a line is doubled. What is the effect on the heat
losses ? How much could the length of the line be increased
and still have no more loss in the line than at the lower voltage ?
How might the crosssection of the wire be changed in order
that, the length remaining the same, the heat loss is the same
as at the lower voltage ?
839. What considerations limit the voltage used in practical
work ?
In order to compare resistances of various substances as well as to compute
the resistance of a conductor from its dimensions, it is convenient to know
the resistance of a cube of the substance of i cm. edge, at o. The actual
resistance depends somewhat on the purity and previous history of the speci
men, so the values given either refer to pure specimens, or are average values.
The resistance of such a cube is named the specific resistance of the material.
The statement that the specific resistance of copper is lyio 7 means that
152
PROBLEMS IN PHYSICS
i cm. length of a piece of copper i sq. cm. crosssection has a resistance of
.0000017 ohms at a temperature of o.
The values of specific resistance used are taken from Landolt and Born
stein's Physikalisch Chemische Tabellen.
To find the resistance of a copper wire 10 m. long, i sq. mm. crosssection
at o we have
17iQ 7 io 8
R = 5 = .17 ohm.
io~ 2
840. The specific resistance of silver is i5io~ 7 . Find the
resistance of a silver wire I ft. long and joVo i n  i n diameter.
841. A copper wire of known resistance is to be replaced by
a platinum wire of half the crosssection. What length must be
chosen to have the same resistance ?
842. Find the resistances of the following circular wires at o.
Material.
Length.
Radius.
Specific Resistance.
Hard steel
Soft steel
io m.
io m.
.5 mm.
.5 mm.
3i4io~ 7
I C7IO" 7
Copper
I km.
.2 mm.
I7IO" 7
Platinum
100 m.
2 mm
I 3 C I O~ 7
Silver
100 m.
2 mm
I CIO" 7
German silver ....
Carbon
100 m.
i m.
.2 mm.
.1 mm.
*3 !<
236 io~ 7
59350107
843. From the table of specific resistances above, compute
the resistances of wires i m. long and I sq. mm. crosssection
in each case.
844. A wire is drawn out into an extremely long circular
cone. If its radius at each point is a times the distance from
the end, and the specific resistance of the metal is 35 icr 7 , find
the resistance of the wire.
Form the expression for the resistance of a length dl and integrate.
As a first approximation, and between certain limits of temperature, the
change of resistance of a wire with temperature may be expressed as a certain
percentage of the resistance at o times the temperature above o. The state
ment that the temperature coefficient of copper is .00388 means that for each
degree a copper wire is heated above o, its resistance is increased the .oo388th
part of its resistance at o.
TEMPERATURE COEFFICIENTS 153
845. The resistance of a coil of copper wire at o is 1785
ohms. What will it be at 40 ?
The increase is .00388 40 1785.
^4o =I 785[i + .1552], etc.
846. The resistance of an iron wire at 20 C. is 1010.6 ohms.
The temperature coefficient is .0053. What is its resistance at
o ? 40 ? 80 ?
847. Taking the specific resistance of copper as 17 io~ 7 , and
temperature coefficient as 39 io~ 4 , &n& ^assuming this coefficient
as constant, at what temperature would copper have no re
sistance ?
848. The temperature coefficient of a certain iron wire is
53io~ 4 . A coil of the wire has a resistance of 2000 ohms at
25. What will be its resistance at 5 ? 45 ?
849. A coil of copper wire has a resistance of 2000 ohms at
16. What is the range of temperature through which it may
be used as a standard of resistance if the error must not exceed
onefourth of one per cent ?
850. The temperature coefficient for a certain Cu wire is
.0039; for a carbon filament it is .0003. How many ohms
of Cu resistance must be joined with a carbon filament of
100 ohms resistance so that the combined resistance may be
constant ?
851. Define the term electrochemical equivalent. State the
relation between the electrochemical equivalent and the chemical
equivalent.
852. The electrochemical equivalent of H is 1038 io~ 8 (for I
coulomb). The atomic weight of sodium is 23, its valence I.
Find the electrochemical equivalent of sodium.
853. A current of 2 amperes passes through a copper sul
phate solution for i hour. If the anode is a copper wire, how
much copper will be deposited on the cathode ?
154 PROBLEMS IN PHYSICS
854. Compute the following electrochemical equivalents :
Substance.
Atomic Weight.
Valence.
Electrochemical
Equivalent.
Hydrogen
I
104. io~ 7
Potassium
Gold
39i
Io6 2
Copperic salts ....
Copperous salts . . .
Lead
63.18
63.18
2O6.4
2
I
2
855. A deposit of 8.856 g. of copper is made by a current in
ij hours in a CuCuSO 4 Cu voltameter. What was the cur
rent strength ?
856. A copper and a silver voltameter are placed in series.
Find the ratio of the deposits formed.
857. Explain how you would arrange your apparatus in order
to "plate" an article with silver.
858. A magnetic needle free to turn is placed in a uniform
magnetic field. A new field at right angles to the first is then
developed. Show by diagram what position the needle will
assume. Does it depend on the pole strength or length of
the needle ? What would be the effect of reversing either field ?
both fields ?
859. A wire carrying current is stretched north and south.
The current flows from south to north. What position will a
compass needle take when held over the wire ? How will its
position alter as it is brought nearer the wire ? What position
would it take if placed under the wire ? if placed midway
between two such wires carrying equal currents in the same
direction ? if in opposite directions ? when between, but nearer
to one than to the other ?
860. A piece of wire i cm. in length is bent into a circu
lar arc of I cm. radius. A current of I ampere flows in the
conductor. What force would act on a + unit pole at the
GALVANOMETERS 1 5 5
center of the circle? What would be the field strength at
the center of the circle when,
(a) I=i ampere, one complete turn ?
(b) I=i ampere, n complete turns?
(c) I=\ ampere, n turns, radius = r?
Note that i ampere = ^ C.G.S. unit of current.
861. A circular coil of wire is placed in a north and south
plane with its axis horizontal. A current is sent through the
coil, flowing north on the upper side. What effect would the
current have on a freely suspended magnetic needle when
placed directly above the coil ? directly below ? in the same
plane and just north? south? at the center?
862. What would be the strength of the magnetic field at
the center of a coil of n turns, mean radius R, I = one ampere ?
From this derive the law of a tangent galvanometer, consisting
of one large coil and a short (?) needle at the center.
863. What do you mean by the term constant of a galvanome
ter? What is a tangent galvanometer? a sine galvanometer?
Is a galvanometer of necessity one or the other ?
864. Compute the current in each of the following cases,
where 7 = galvanometer constant, B = deflection in degrees :
Tangent galvanometer, / = 4.5, 8 = 25.
/ = 42.icr 6 , S = 20.
^
What would the currents be if the galvanometer were a
" sine " galvanometer ?
865. When is a galvanometer said to be sensitive ?
866. Explain how a sensitive galvanometer is constructed.
867. Explain how a given galvanometer may be made more
sensitive.
156 PROBLEMS IN PHYSICS
868. If 7=10 tan S, H=.i$, 72=10, 8 = 25, what
r
must be the radius of the coil if / = 2 amperes ? How would 8
be changed if H were reduced onehalf ?
869. A tangent galvanometer, 7 = 6 io~ 3 , 7? = 200 ohms, is
placed in shunt with a resistance of 50 ohms. A deflection of
70 is observed. Find the total current.
870. A piece of soft iron is placed near a tangent galvanome
ter. What effect will it have on the galvanometer constant :
(a) when placed in the same plane as the needle, and just north
or south of it ? (b) when in the same plane east or west ?
(c) when placed just below?
871. How would the action of the soft iron in Example 870
differ from that of a magnet ?
872. The 7 of a certain tangent galvanometer is 4io~ 3 ,
where H .145. What will 7 be when the galvanometer is
moved to a place where H = . 102 ?
873. A current of .2 amperes causes a deflection of 40 in
a tangent galvanometer where H = .2. What current would
give the same deflection where H is . I ?
874. The needle of a tangent galvanometer is observed to
make 40 complete vibrations in one minute. 7 at that point is
34*io~ 6 . When moved to another place it is found to make 25
complete vibrations in one minute. Find the constant in the
new position.
875. (a) Give a diagram showing the construction of a simple
type of tangent galvanometer. Explain in what position it must
be placed in measuring current, and derive formula, (b) State
the distinction between magnetic and diamagnetic substances.
Describe an experiment by which the behavior of each, when
placed in a magnetic field, can be shown.
BALLISTIC GALVANOMETER 157
876. A tangent galvanometer is connected in series with a
generator of constant electromotive force and a known resist
o
ance which can be varied. A series of resistances are inserted,
and corresponding deflections are observed. If these resistances
are taken as x, and tangents of deflection as y, what sort of
a curve will result ? Does the entire curve have a physical
meaning ?
877. How is the quantity of electricity measured when it
passes as an intense and variable current for a very short time.
(Examples, condenser discharges and induced currents.)
878. What is a ballistic galvanometer ? What is meant by
the term constant of a ballistic galvanometer?
From the theory of the ballistic galvanometer we find that
Q= I0 . r.^ sm i0 = smi0,
7T G
while from the magnetic pendulum
where H  horizontal component of earth's field.
T '= periodic time of magnet in that field.
G = "true" constant of the galvanometer (tan.).
M pole strength x distance between poles = ml.
K a moment of inertia.
= angle of maximum deflection.
T ff
We may write Q = 10 ^
[7 = tan. const.
879. Find the effect on Q of increasing the horizontal in
tensity in any given ratio. Compare this with the change in 7
due to the same increase in H.
158
PROBLEMS IN PHYSICS
880. The needle of a ballistic galvanometer is accidentally
dropped ; its pole strength is decreased. Will <2 be changed ? 7 ?
881. The constant of a ballistic galvanometer is .046 at a
certain place. What will the constant be where H is nine times
as great, if the needle is remagnetized and its magnetic moment
increased fourfold ?
882. The constant of a ballistic galvanometer at a point
where T is 4 sec. is .045. What will the constant be where
T = 2 sec. ?
883. A coil of 100 turns, mean radius 40 cm., is turned 180
about a diameter which is perpendicular to the lines of force of
a field of strength 10. The coil is connected with a ballistic
galvanometer, and a deflection of 20 is observed. Resistance
of the circuit 1 5 ohms. Find Q Q .
Fig. 78.
If ds is a current element so short that it may be regarded as straight, the
laws concerning the magnetic force due to ds at any point A may be stated as
follows :
(1) The force is _L to the plane APQ.
(2) The force is proportional to the length of ds.
(3) The force is inversely proportional to AP .
(4) The force is proportional to the " broadside " projection of ds, i.e. to
PR = PQ sin PQR = ds sin (9.
Summing up the last three of these four laws, we may say that
T? _ ^ ds sin
** K 9
r 2
[k depends on current strength and units used.]
FIELDS DUE TO CURRENTS 159
The field at the point A is then found by integrating this expression. In
order to perform the integration a relation between the variables must be
given, i.e. the shape and position, with reference to A, of the circuit must
be specified.
In the case of a very long straight wire, we have, if p = perpendicular
distance from A to the wire,
ds 2.k
For a wire of finite length 2 /, A in the plane perpendicular to its middle point,
the limits would be / and f /.
884. Find how long the wire must be in order that when p
is 5 cm. the field is within one per cent of that due to an
infinite wire with the same current.
885. The horizontal component of the earth's field at a
certain point in the Cornell Physical Laboratory is .145. At
what distance from a long straight wire carrying 10 C.G.S.
units of current would the field due to the current have the
same intensity? (Here ki.}
886. What current must flow in an infinite straight wire that
the magnetic field 10 cm. from the wire may exert a force on
unit pole equal to the weight of I g. ?
887. Find the field strength at the center of a square, if a
current passes around it.
888. Find the force exerted on a + unit pole placed at the
intersection of the diagonals of a rectangle, sides a and b, and
carrying a current /.
889. Apply the formula FA = k ds ' * in 6 to the case of a
circular wire of radius R, when A is taken in the line perpendicular
to the plane of the circle, and through its center. (Axis of coil.)
Show by diagram the direction of the force for each element,
and for the complete circle.
What is the force component along the axis ? Where is this
a maximum ?
MAGNETISM
890. State the law of attraction or repulsion between magnet
poles. Where do similar laws occur in physics ? Show how a
definition of unit magnet poles follows directly from the law.
891. Find the force in dynes between two unlike magnet
poles of strength 8 and 12 units respectively when the distance
between them is .04 m.
The force varies according to the law ^i.
a 2 "
8 T2
Expressing d in centimeters F = 6 dynes.
16
892. Two like magnet poles, of strengths 10 and 27 units
respectively, are separated by a distance of 30 mm. Find the
force in milligram's weight between them.
893. When two magnet poles are placed a distance apart of
I cm. the force between them is 12 dynes. How must the
distance be varied in order that the force may increase to 48
dynes ?
894. What is a magnetic field of force ? a magnetic line of
force ?
895. (a) Map the field of force around an ordinary bar mag
net. (&) Map the field around two magnets placed with their
like poles (supposed of equal strength) near each other and
their axes at right angles.
896. A bar magnet is laid on a horizontal plane with its
axis north and south, and its northseeking pole north. Draw
the resultant field, considering the earth's field as uniform.
160
MAGNETIC FIELDS l6l
897. In Example 896 find two points where the resultant
magnetic force is o. Where would these points be if the mag
net .were reversed ?
898. How does the distribution of lines of force due to a bar
magnet differ from that of electric lines due to + and in
duced charges on a cylindrical conductor?
899. A bar magnet is 40 cm. between the poles and pole
strength 100, what is the direction and intensity of the magnetic
force due to it at a point on the perpendicular to the line joining
the poles and 50 cm. from this line ?
900. Define strength of field. Find the force exerted on a
pole of 12 units placed in a field of strength 326.
901. What is the strength of the magnet pole which is urged
with a force of 2 mg. weight when placed in a field of strength
.42?
902. What position does a magnet take when placed in a
magnetic field (a) of which the lines of force are straight? (b) of
which they are curved ? Explain why the lines of force in a
magnetic field can never cross.
903. Show that the number of lines of force coming from a
pole of strength m is ^.trm.
904. The strength of a magnet pole is 72 units. Find the
strength of field at a point 3 cm. away from it, assuming the
other pole of the magnet to be so far away as to be of negligible
effect at the point considered.
905. What are consequent poles in a magnet? How may they
be produced ?
906. How may a long magnet be placed with reference to a
compass needle so that the needle is affected by one pole of the
magnet only ?
907. The angle of magnetic dip at Washington is 70 18',
and the value of H is .2026. Find the total strength of field.
162 PROBLEMS IN PHYSICS
908. The angle of dip at New York is 70 6', and the total
strength of field at that point is .61. Find the horizontal and
vertical components.
909. Why is the earth's field simply directive in its action on
a suspended magnet ?
910. Why does not an ordinary compass needle dip or tend
to dip ?
911. Define magnetic moment. Find the dimensions of mag
netic moment, and compute the moment of a magnet .13 m.
long, and of pole strength 42, the magnetization being assumed
uniform throughout the. length of the magnet.
912. A magnet having a moment M is broken into n equal
pieces of the same crosssection as the original magnet. What
is the magnetic moment of each piece ?
913. A magnet is placed in a uniform field of strength .362.
When the axis of the magnet is normal to the .direction of the
field, the couple acting on the magnet is 2172 dynecentimeters.
Find the magnetic moment.
914. A magnet 10 cm. long has a pole strength of 60. When
this magnet is placed in a field of strength .17, what is the
couple acting on it (a) if the axis of the magnet be at right
angles to the field ? (b) if the axis be inclined at 45 to the field?
The force acting on each pole of the magnet is equal to the strength of the
field x pole strength, i.e.,
F=Hm.
If the magnet lie at right angles to the field, this force is wholly effective in
turning the magnet. If the magnet be inclined to the field by an angle 9
the turning component of the force is less, being given by
F' = Hm sin 0,
and the moment of the effective couple is
= HMO
for small deflections.
The student should compare this result with the couple causing the vibra
tion of an ordinary pendulum, and draw conclusions as to the character of the
motion produced in each case. See 741, 742.
MAGNETOMETERS 163
915. Show that the magnetic moment of a uniformly magne
tized bar is proportional to the volume of the bar. Whence
define intensity of magnetization.
916. A bar magnet has a crosssection of 1.2 sq. cm., a length
of 12 cm., and a pole strength of 168. Assuming the magne
tization to be uniform throughout the magnet, compute the
intensity of magnetization.
917. Prove that the potential at a point distant r from a
magnet pole of strength m is .
918. In what units is magnetic potential measured ? Find
the potential of a point distant .6 m. from a magnet pole of
strength 72.
919. Find the work done in carrying a pole of strength 4
units from a point distant 5 cm. from a magnet pole of strength
100 units to a point distant 2 cm. from this pole.
920. Find the potential at a point 6 cm. distant from the
north pole of, and in line with the axis of, a bar magnet 10 cm.
long and of pole strength 80.
921. A point P is distant OP from the center of a small
magnet whose magnetic moment is M. Show that the potential
at P is 2 > where is the inclination of OP to the axis of
the magnet.
N
t
it
+m[^^F
m +ra
m
i
H i
I Fig. 79.
S
922. When the left hand magnet, Fig. 79, is so short com
pared with d that the lines joining their poles may be considered
164
PROBLEMS IN PHYSICS
as parallel with that joining their centers, what is the torque
exerted by the large magnet on the small one ?
Treat force action of each pair of poles separately. Then take moments
and add.
923. What torque is exerted by the earth's field ?
924. By means of the last two examples show that when
small magnet is in equilibrium
H 2d
[Where l=\ distance between poles of large magnet.
925. Explain why pole strength of small magnet need not be
known. Why could it not be reduced to zero and yet have
equation of Example 924 true ?
926. Show that when d is very great in comparison with /,
2 ml
tan
pm
927. If H= .24, d \ m., / = 10 cm.,
4> = 25.
What is the pole strength of the magnet ?
928. Prove that when magnets are placed
as in Fig. 80 [the length of the small mag
net being small compared with d~\
2ml
d
H
= X 2 4/ 2 ]
929. When / may be neglected, show that
2 ml
H
= d tan
+m
S w
rn
930. How do the results of Examples 926
and 929 compare. Explain why such a dif
ference should be expected.
931. If the large magnet were reversed, what change of
position would the small one experience ?
Fig. 80.
MAGNETIZATION 165
932. If the magnets were exactly alike, and each were sus
pended so as to be free to move, would each turn through the
same angle in Fig. 79 ? in Fig. 80 ?
933. Taking axes parallel and normal to the axis of a magnet,
plot curves showing (a) the variation of potential and (b) the
variation of magnetic force with distance along the axis. Dis
cuss the relation existing between these curves. (Only one
pole of the magnet is to be considered.)
934. Define magnetic induction (B), permeability (/j), and
susceptibility (K). Imagine a piece of soft iron placed in a
weak field. Further, imagine the field to gradually increase
in strength. Show by means of a curve the changes which
take place in the induction in the iron with the increase in the
field strength.
Such a curve is called a magnetization curve, and is of great practical value.
It is usually plotted with the induction B and the field strength H as co
ordinates. Obviously the ratio of any ordinate B to the corresponding
abscissa H is the permeability /x of the iron.
935 Which is the more easily magnetized, soft iron or steel?
Which retains the greater amount of magnetism when the mag
netizing force is removed ? Explain answers fully in accordance
with the molecular theory of magnetism.
936. Why is magnetism removed by heating ? Why are iron
rods subjected to tapping or jarring liable to become magnetized ?
937. An iron tube is driven into the earth in the Northern
Hemisphere. What would be its magnetic condition ?
938. What kind of iron would you choose for the construction
of permanent magnets ? of telegraph instruments ?
939. Show that B, H, and / are quantities of the same kind
or dimensions. What must therefore be true of //, and K?
940. Explain the principle of magnetic screening, as when
a galvanometer needle is protected by an iron screen.
166 PROBLEMS IN PHYSICS
941. A sample of transformer iron gives the following data.
Plot and discuss the magnetization curve.
H B
1.32 1324
2.0 3650
4.64 8800
/.I lOQSo
IO./3 12450
14.65 13320
19.42 13920
37.0 15032
49.8 15465
942. Compute the data requisite to plot a permeability curve,
using H and /* as coordinates.
943. Discuss the equation B = H 4 47r/, explaining the
meaning of each term.
944. A sample of iron shows /= 1226 for H = 40. Com
pute the susceptibility ; the induction ; the permeability.
945. Show that the force with which a magnet attracts its
keeper is
stating clearly the conditions that must be fulfilled in order that
this equation may be true.
946. It is found that when the poles of a certain magnet are
reduced in area the lifting power of the magnet is increased.
Why is this ?
947. A certain magnet having a pole face of area 4 sq. cm.
is found to sustain a maximum load of 2 kg. Find the induc
tion.
948. What is meant by the term magnetomotive force?
What is the magnetic analogue of Ohm's law ?
MAGNETIZATION 167
949. The field magnet of a dynamo is wound with 3200 turns
of wire. The normal field current is 820 milliamperes. What
is the number of ampere turns ?
950. A circular ring of iron has a crosssection of 8 sq. cm.
and a mean radius of 7.5 cm. What magnetomotive force
must be used to set up a total magnetic flux of 120000 lines ?
The permeability for this induction is 526.
951. If an air gap is cut in a magnetic circuit, how is the
magnetization curve affected ?
952. A current flowing in the turns of a short solenoid pro
duces a field of a given strength along the axis. When an iron
core is inserted, the value of H is changed. Why is this ?
953. A certain magnetic circuit has a crosssection of 36
sq. in. It is made of cast iron, showing a permeability of 71
for a magnetizing force of 127. Compute the total magnetic
flux (or induction).
954. A long solenoid is wound with 20 turns per cm. Com
pute the value of H along the middle of the solenoid, (a) when
no iron is present, (b) when iron giving the data of Problem 941
is present, the current in both cases being 5 amperes.
955. What is hysteresis ? What is represented by the area
of a hysteresis loop ?
956. A transformer core contains 3840 cu. cm. of iron. The
hysteresis loss is 16300 ergs per cycle per cubic centimeter.
If this transformer be supplied with an alternating current of
frequency 120 periods per second, what is the power (in watts)
lost in hysteresis ?
957. How does the energy spent in hysteresis appear ? What
is the effect of jarring on hysteresis ?
958. State clearly the meaning of the symbols in the formula
/ K
for the magnetic pendulum, T= 2 TT \ * .
168 PROBLEMS IN PHYSICS
959. Explain how the magnetic pendulum differs from the
gravitational pendulum. Would there be any objection to using
a magnetic pendulum for a clock ?
960. What must be the pole strength of a magnet, moment
of inertia 1800, distance between the poles 10 cm., that it may
make 20 complete vibrations in 4 min., where H = .145 ?
961. A large block of soft iron is placed beneath a horizontal
vibrating magnet. What will be the effect on T?
962. A magnet is set in vibration where H is .16, and T is
found to be 3 sec. When taken to another place, T' is found to
be 3.2 sec. Find H' .
963. Derive the equation T 2 TT "\ ^^r, explaining any
approximations or assumptions made.
964. If a magnet is struck several blows, what will be the prob
able effect on its time of vibration as a magnetic pendulum ?
965. A strip of lead is bound to a magnetic pendulum.
What is the effect on 7\ ?
In the study of the magnetic forces due to currents, of tendencies of con
ductors carrying current to move in a magnetic field, and of the direction of
induced currents, it will be found that the concept of lines of force is one of
great utility. Remembering that two magnets placed parallel, with their like
poles contiguous, will tend to separate, we see that if this action is to be
ascribed to a property of lines of magnetic force we should say that lines of
force parallel and in the same direction repel.
It will be found convenient to suppose that the characteristics of lines of
magnetic force are in part as follows : ^.~_^r^_^^
(a) Magnetic lines of force parallel and in same f,^' ~^<\
direction repel each other. l ;i
() Magnetic lines of force parallel and in oppo fill
site directions attract. vj^
(c) Magnetic lines of force are similar to tense, ^^^^^^^
elastic threads which first bend, and then ^3^^.
break when a conductor moves across i'^^ ~"A"~'
them. dV '*'
(d) These lines tend to shorten and also to pass /L^
through iron rather than air. '^^ g
These, together with the fact that when a cur r ^==
rent flows lines of magnetic force tend to form pig. 81.
FIELDS DUE TO CURRENTS
169
circles around it, are very useful in indicating the relations of currents to
varying fields, etc.
The direction of current and the positive direction of the lines of force due
to it are related to each other in the same way as are the direction of transla
tion of a righthanded screw, and the direction in which it is turned. Or, if
M
X
X X
X X X X X
X X X X X X
Fig. 82.
we imagine current to flow from the eye to a clockface, lines of force around
the current would be such that a 4 pole would go around it like the hands
of the clock or "clockwise. 11 If current pass down perpendicular to the paper
at A, the entire plane has lines directed as shown.
For convenience in diagram, we shall indicate that a line of force is coming
up through the paper by a , going down by a x . Thus, if current flows in
the line MN in the plane of the page, the magnetic lines are vertical circles
encircling MN clockwise, looking from M to N.
This is not suggested as the only way in which these relations may be
remembered, but as one found of considerable convenience in practice.
A few diagrams are added to show the application of these statements.
(1)
X X X X X
(3)
X X X X X X X
XXX XXXXX
xxxxxxxxx
(2)
Fig. 83.
(1) Two parallel currents in the same direction attract.
(2) Two parallel currents in opposite directions repel. Likewise for con
ductors inclined to each other.
(3) Two rectilinear currents perpendicular to each other. AB free to turn
about A. B moves to the left. Similarly, if CD is a circle and AB a radial
current.
(4) Current down perpendicular to plane of magnet. At A conductor and
magnet tend to approach ; at B to separate. (See Fig. 81.)
The property of magnetic lines of force assumed in (c} may be conveniently
used in determining the direction of induced currents. We might look at
PROBLEMS IN PHYSICS
the matter of relative motion of a conductor and lines of magnetic force some
what, as indicated by Fig. 84.
Let A be the intersection of a conductor with the plane of the paper, and
let the lines of force be parallel to this plane. When A is moving to the
right or the field moves to the left, we may consider the lines of force from
c d
MOTION
Fig. 84.
a to c as crowded together and stretched, d is stretched so far that lateral
compression is forcing it to encircle A, e has gone through the phases b, c, d,
and the points corresponding to P and Q of d have united as at s, leaving e r
encircling the wire. Current tends then to flow down, just as current would
flow to set up like lines or to
oppose the motion.
966. The case of an
east and west wire in
the earth's field is a
good example. If MN
and OP (Fig. 85) repre
sent two lines of the
earth's field, AB an east
and west wire, then, if
INDUCED CURRENTS
I/I
AB is moved up, the lines tend to encircle it as shown. Which
way does current tend to flow ? Does the current help or
oppose the motion ?
967. Draw the diagram when the wire is falling.
968. A telegraph wire is stretched east and west. The
direction of the earth's field is 75 with the horizontal. Show
by diagram the direction of the induced currents
(a) When it falls vertically downward.
(b) When it is raised vertically.
Show also in what direction to move it in order to get a
maximum current ; a minimum current.
969. Two parallel wires are placed as in Fig. 86. When
the key k is closed, what takes
place in the other wire ? If the
wires moved apart with a velocity
equal to that of light, would the
same effect be observed ?
We may consider circular lines of mag
netic force as springing out from the first
wire. Their radii increasing at what rate ? Fi  86 '
970. The north pole of a magnet passes through the bottom
of a cup C. Mercury covers the bottom, and a wire suspended
\
1
\
1
\
t
\
\ /
\
Fig. 87. Fig. 87 (a).
vertically above N dips below the surface of the mercury. If
1/2 PROBLEMS IN PHYSICS
current flows from A to B, show that B will move away from
and rotate around N.
Consider the projection of the lines of force due to the magnet on the
surface of the mercury. (See Fig. 87.)
971. Extend to the case of a flexible conductor.
The student should apply this method to cases of action of magnetic fields
described in textbooks or observed in lectures.
972. A solenoid is placed with its axis north and south ; its
terminals are connected with a galvanometer. When a piece
of soft iron is thrust into or drawn from the coil, an induced
current is observed. Explain. Would the effect be increased
or diminished if the axis of the solenoid were east and west ?
973. A small piece of soft iron is suspended near a magnet
by a thread. Explain the position it will take by reference
to (d).
974. Explain why a solenoid tends to shorten when current
is passed through it.
975. Explain the effect of a copper box surrounding a vibra
ting magnetic needle.
976. A metal plate is revolved between the pole piece of an
electromagnet. It is observed that it is harder to maintain its
motion when current is passing through the coils of the magnet.
Explain this. What becomes of the energy used in turning
the plate ? Does the magnet tend to move ?
977. Show in what direction a magnet may move with refer
ence to a fixed wire in order that no electromotive force may
be set up in the wire.
978. In the figure of Example 1000, in what direction must
the coil turn that current may flow from A to D ?
979. A solenoid is wound so that it looks like a righthanded
screw. An iron core is placed in it and you are required to
make a given end a northseeking pole. Give a diagram show
ing the direction of the current.
MAGNETIC FIELDS
173
980. Two points of different electrical potential are joined by
(a) a straight wire,
(b) a coil of wire,
(c) a coil of wire with a soft iron core,
(d) a coil of wire with a permanent magnet as a core.
Indicate the differences in the magnetic fields produced in
these cases.
981. (a) A wire perpendicular to the plane of the paper
carries current downward. Indicate form and direction of the
lines of magnetic force, (b) A parallel wire carrying current
in the same direction is brought near. How is the field
altered ? What action takes place between the wires ?
982. (a) Define permeability. (b) Draw the lines of force
for the magnetic fields , IRON \
shown in diagrams, Fig.
88. (c) What is the power s I I N
of energy in the case of an x~7<r?OC7C7\7C" > v
induced current produced ( [ )[ J[ J f ]( ]( } / COIL WITH
by motion in a magnetic
field ? (Winter, '96.)
983. Find the force act
ing on a pole of 60 units'
CURRENT
+
o
strength at a distance of Fig  88 
5 cm. from an infinitely long straight conductor carrying a cur
rent of 5 amperes.
984. To reduce the force in the foregoing case by onehalf,
where must the pole be moved ?
985. A bar magnet is allowed to drop vertically through a
closed loop of wire. What are the directions of the induced
currents ?
986. A certain wire is moved through a magnetic field so as
to cut io 9 magnetic lines of force in 2 sec. What is the average
electromotive force induced ?
The E.M.F. induced is proportional to the rate of cutting. To reduce the
result to practical units (volts), divide by io 8 .
OF TftK
UNIVERSITY
174 PROBLEMS IN PHYSICS
987. A wire 30 cm. long is moved through a field of strength
6000 lines per sq. cm. at the rate of 10 m. per second. Find
the induced electromotive force in volts.
988. A centimeter length of a straight wire is placed at right
angles to the lines of force of a uniform magnetic field, i C.G.S.
unit of current flows through the wire. The strength of the
magnetic field is 1000. What force acts on the wire ? If the
current is ten times as great, the field onetenth as strong, and
the wire I m. long, what force would act ?
989. If a wire I m. long, current of 100 amperes, is placed
horizontally at an angle of 30 with a uniform horizontal field,
what force acts on the wire if the field strength is 1000? In
what direction does it act ?
990. A flat loop of wire of resistance .001 ohm, and area
i sq. m., rests on a horizontal table. If the loop be picked up
and turned over, what is the total quantity of electricity set
in motion ?
991. Would it make any difference in the quantity if the
loop were turned slowly or quickly ?
992. How can a straight wire be moved in a magnetic field,
and yet have no electromotive force developed in it ?
993. If a closed loop of wire be moved without change of
plane through a magnetic field of uniform strength, will any
current flow in it ? Will any electromotive force be developed
in it ?
994. A wire 2 m. long, and lying horizontally east and west,
is allowed to fall freely, (a) Find the value of the induced
electromotive force at the end of 3 sec. (b) Find the mean
value of the induced electromotive force during a fall of 5 sec.
(c) Find the time elapsing before the electromotive force shall
be just i volt.
995. AA' and BB' are a pair of copper rails, so large that their
resistance may be neglected in comparison with that of the rest
INDUCTION
175
of the circuit. 5 is a wire of resistance I ohm, sliding without
friction over the rails, and at right angles to them. Resistance
of galvanometer circuit, 3 ohms. If the rails are in a field of
3000 lines per sq. cm., the direction of the field being upward,
[ ,1,1
B'
'B
Fig. 89.
normal to the plane of the rails, and the distance between the
rails be 40 cm., find :
(a) The velocity required to develop an electromotive force
in S of i volt.
(b) The direction of this electromotive force when the motion
is in the direction indicated.
(c) The current in the circuit when k is closed.
(d) The work done in the circuit.
(e) The force necessary to propel 5 at this velocity.
996. Show that the quantity of electricity set in motion by
any displacement of the slider is independent of the velocity
with which that displacement takes place.
997. If the velocity of the slider were doubled, what would
be true of the work done in the circuit ?
998. If the galvanometer were replaced by a cell developing an
electromotive force of I volt, and having a resistance of 3 ohms,
in what direction and with what velocity would the slider move ?
999. How can the slider and rails of Problem 995 be used to
show that the dimensions of resistance in the electromagnetic
system are those of a velocity ?
1000. A rectangular loop of wire .1 m. wide and .2 m. long
rotates uniformly at a speed of 1200 revolutions per minute in a
1 7 6
PROBLEMS IN PHYSICS
field of 4000 lines per square centimeter. Find the average
value of the electromotive force
induced.
Since all that is desired is the aver
age value of the induced electromotive
force, we have only to find the total
change in the number of lines thread
ing the loop per revolution, and divide
this by the time of one revolution.
1001. With the direction of
field and of rotation as indicated,
what is the direction of the in
duced electromotive force ?
1002. When such a coil ro
tates in a uniform field, to what Fig. 90.
are the instantaneous values of electromotive force propor
tional ?
1003. If a loop of wire rotating in a magnetic field form part
of a closed circuit, the resulting current is an alternating one.
Sketch and describe a device by which the current may be
caused to flow always in the same direction in the external
circuit.
W
Fig. 91.
1004. A wire w is caused to rotate around the north pole of
a magnet by means of a cord on a pulley. Contact is made in
INDUCTION 177
the mercury cups a, a', the closed circuit being aa'g. The
strength of pole is 72. The wire is caused to rotate with a
speed of 600 revolutions per minute. The resistance of the
circuit is .01 ohm. What is the current in amperes ?
Would current flow if the wire extended the entire length of
the magnet ?
1005. If the wire were fixed and the magnet were placed on
a pivot so as to be free to turn about its axis, what would happen
when current is passed through the wire ?
1006. A Faraday disc has a radius of 15 cm. It rotates with
a speed of 2400 revolutions per minute in a field normal to the
disc of average density 2000 lines per square centimeter. Com
pute the electromotive force of the machine.
Fig. 92.
1007. What essential differences are found in the following
types of dynamos : (a) magneto, (b) series, (c) shunt, (d) com
pound ?
1008. What type of dynamo is best adapted to incandescent
lighting ?
1009. Which would suffer most from a short circuit, a shunt
or a series dynamo ?
1010. What is meant by residual magnetism ? What impor
tant part does it play in the operation of dynamos ?
ion. A certain serieswound dynamo refuses to generate.
The connections of the field coils are reversed, when the
machine immediately " picks up." Explain. Would reversing
the direction of rotation have the same effect?
1 78 PROBLEMS IN PHYSICS
1012. A bipolar dynamo has upon the surface of its arma
ture 480 conductors ; and the armature rotates with a speed
of 1 200 revolutions per minute in a total magnetic flux of
1250000 lines. Compute the electromotive force of the
machine.
1013. What difference exists between the ring (Gramme)
and drum armature windings ?
1014. A ring armature of 320 turns rotates with a speed of
1800, while a drum armature of 240 turns rotates with a speed
of 1 200. The field being the same for both armatures, compare
the E.M.F. developed.
1015. Arc lights are usually run in series. Does the arma
ture of an arclighting dynamo need to be wound with fine or
coarse wire ? Is a high degree of insulation necessary ? Are
few or many turns of wire required ?
1016. Glow lamps are run in parallel. Answer the questions
of the last problem, with reference to a dynamo for incandes
cent lighting.
1017. In what three ways may the electromotive force of
a dynamo be increased ?
1018. What fixes the maximum current output of a dynamo ?
1019. What should be the characteristic features of a dynamo
designed for electric welding ?
1020. The field circuit of a dynamo has the form shown in
Fig. 93. It is required to find the number of ampere turns
needed on the field limbs to set up in the air gap a magnetic
density of 6000 lines per square centimeter. Concerning this
machine the following data are known :
Diameter of armature core 25 cm.
Length of armature core 36 cm.
Mean length of magnetic circuit in field (i.e.
dotted line abed} 145 cm.
DYNAMO FIELD
179
Permeability of armature iron for a magnetic
density of 6000 1120
Coefficient of magnetic leakage for this type of
circuit 1.5
Permeability of field iron for a magnetic density
of 1.5 x 6000 2250
Depth of double air gap 0.72 cm.
The work done in carrying a + unit magnet pole around the path indicated
by the dotted line is
10
where S is the number of turns of wire on the
field, and i the current in them. Considering
the magnetic circuit as made up of three sepa
rate parts, in each of which the value of H is
assumed to be constant, we have
rm
rrn
10 . J
the subscripts a, g, and/ referring to the arma
ture, air gap, and field, respectively.
Taking the computations in the order indi
cated, we have
/? a 6000
rl a = = ,
Pa I I 20'
6000
Fig. 93.
and
25 = 134
I 1 20
For air, /x = i,
hence H g l g = 6000 x 0.72 4320.
Now in every dynamo there is a certain amount of stray field, or waste
magnetic flux, which forms closed loops by various paths outside the air gap.
The amount of stray field is readily found for different types of machines by
total magnetic flux .
experiment. The ratio useful ma * netic flux is called the coefficient of mag
netic leakage. The induction to be provided for in the field is, therefore,
kB a = 1.5 x 6000 = 9000,
and we have
HJ f 
145 = 580,
2250
= 134 4 4320 + 580
= 5034
l8o PROBLEiMS IN PHYSICS
The requisite number of ampere turns is therefore
St = = 4000 nearly.
i. 26
The student should note that in the foregoing method certain assumptions
are made which are not rigorously true. The method, however, gives results
which meet all the requirements of practical dynamo design.
1021. The armature of this dynamo has upon its surface 184
conductors, and it makes 1200 revolutions per minute. Com
pute the electromotive force.
Since the pole pieces are not likely to cover more than 80 per cent of the
armature, the magnetic density may be taken, as in the preceding case, as the
same in air gap and armature. The crosssection of the armatnre is
25 x 36 900 sq. cm.
The total number of lines is therefore
900 x 6000 = 54 x io 5 .
The total electromotive force developed is
NCn
~^'
where ./Vis the total flux, C the number of conductors on the armature, and ;/
the number of revolutions per second. This gives
54 x io x 184 x 20 = 2Qo yol nearl
IO 8
1022. It is found that over and above friction a certain
amount of power is required to turn the armature of a dynamo
when the machine is on open circuit. To what two causes is
this waste of power due? How may it be diminished ?
1023. What is meant by a characteristic curve? A series
machine gives the following data. Plot it, using current on
the Jfaxis.
Potential Difference. Current.
2.6 O
10.3 4
31.4 io
435 14
52.3 20
56.1 25
60 34
62 45
DYNAMO EFFICIENCY l8l
1024. This machine would work unsatisfactorily below 40
volts. Why ?
1025. Suppose a line to be drawn from any point on the
characteristic to the origin. What is indicated by its pitch ?
1026. The product of the coordinates of any point on the
curve is taken. What is shown by this product ?
1027. The data in the first column are potential differences
at the terminals. Given that the internal resistance of the
machine is .2 ohm, how may the total electromotive force be
found ?
1028. When the circuit of a series machine is closed through
a given resistance, why do not the current and electromotive
force continue to increase indefinitely ?
1029. What is the general shape of a shunt characteristic ?
What would be the characteristic of a perfectly "compounded"
dynamo ?
1030. What is meant by the gross efficiency of a dynamo?
the net efficiency ? the electrical efficiency ?
These terms are defined by the ratios :
~ Jv . total electrical energy developed
Gross efficiency = = = : = ^ ^
total mechanical energy supplied
Net efficiency
_ useful electrical energy developed
~ total mechanical energy supplied
, . . ... useful electrical energy
Electrical efficiency = = = : &
total electrical energy
Since every machine has some internal resistance, the electrical efficiency
can never reach 100 per cent.
1031. A certain dynamo develops electric power to the
amount of 10 kilowatts. If the gross efficiency of the machine
is 85 per cent, how many horsepower must be furnished to
drive it ?
1032. The internal resistance of a series dynamo is .2 ohm.
The machine develops a maximum current of 40 amperes at an
available potential difference of 100 volts. What is the electrical
efficiency ?
182 PROBLEMS IN PHYSICS
1033. The net efficiency of a certain dynamo is 70 per cent ;
the gross efficiency is 84 per cent. What is the electrical
efficiency of the machine ?
1034. A. certain dynamo requires 8 kilowatts when driven at
full capacity. The net efficiency being 82 per cent under these
conditions, and the pressure at the terminals being 105 volts,
what is the maximum current output ?
1035. A shunt dynamo has a field resistance of 70 ohms, and
an armature resistance of .022 ohm. When running at full
load the machine develops 80 amperes at an available potential
difference of no volts. What is the electrical efficiency of the
machine ?
1036. A house is to be lighted with 40 glow lamps, each re
quiring. 5 ampere and no volts. Allowing for a loss of 4 per
cent in the mains, a net efficiency in the dynamo of 84 per cent,
and a reserve power in the engine of 15 per cent more than
that actually required to run the lamps, what should be the
horsepower of the engine installed ?
1037. What determines the practical limit of longdistance
transmission of power ?
1038. When current is supplied to a directcurrent dynamo it
runs as a motor. Explain by reference to Problem 995.
1039. An ammeter is introduced into a motor circuit. The
current is found to be stronger when the armature is held still
than when it is allowed to run. Explain.
1040. If the wheels of a street car were securely locked, the
controller could not safely be turned so as to let maximum cur
rent flow. Why ?
1041. A wire i m. long, carrying a current of 20 amperes, is
held in a uniform field of 6000 lines per square centimeter.
Find the restraining force.
To obtain the force in dynes, the current must be reduced to C.G.S. units,
i.e. must be divided by 10.
MOTORS 183
1042. If the field of a motor be strengthened, will it run faster
or slower, other conditions remaining unaltered ?
1043. Assuming that the energy absorbed by a motor appears
in two ways only, namely, as useful work and as heat due to
resistance, show that the motor does maximum work when the
counter electromotive force is onehalf the impressed electro
motive force.
Let E be the constant impressed electromotive force, z the current, r the
internal resistance of the motor, and e the counter electromotive force. We
have, according to the foregoing assumption, total power absorbed =Ei=
ei \ir, whence useful power = <w = Eii' 2 r. i being the only variable in the
righthand number, we have merely to find the value of i to give maximum iv.
1044. Show that it follows from the foregoing that the effi
ciency of a motor doing maximum useful work is but 50 per cent.
1045. Under what conditions will a motor run at maximum
efficiency ?
1046. A serieswound motor has a resistance of .2 ohm.
When supplied with 5 amperes at a potential difference of no
volts, what is the energy wasted in heating ? Of the energy
not wasted in heating 92 per cent is used in overcoming the
torque due to friction hysteresis and eddy currents. What is
the net efficiency of the motor ?
1047. A motor is supplied with a current of 15 amperes at a
pressure of no volts. The power developed at the pulley is
i. 8 1 horsepower. Compute the net efficiency of the motor.
1048. If two armatures were mounted on the same shaft, would
it be possible to use one as motor and the other as a dynamo ?
What would such an arrangement be called, and what uses might
it have ?
1049. ( a ) What is meant by the period of an alternating cur
rent ? (b) A small 8pole alternator makes 1800 revolutions per
minute. What is the periodicity of the current developed ?
() Eight poles, alternately north and south, give 4 complete periods per
revolution ; hence the periodicity, or frequency,
4. x 1800
= 120.
60
1 84 PROBLEMS IN PHYSICS
1050. Find the mean value of an harmonic or sine electro
motive force.
Instantaneous values being given by
E e sin a,
we should have as the mean e
E\ sin ado.
Jo
da
which is readily integrated.
The mean value of an harmonic current is similarly found from the expres
sion * = /since.
NOTE. In the treatment of alternating currents it is usually justifiable to
consider them as harmonic even though they depart somewhat from the sine
law. In the following problems the current is assumed to be a simple sine
function of the time.
1051. The maximum value of an alternating current is 120
amperes. What is the mean value ?
1052. What is the maximum value of an alternating current
that will cause the same quantity to flow across any crosssection
of a conductor in a given time as does a direct current of 63.6
amperes ?
1053. An alternating current has a maximum value of /. What
is the value of the direct current that will develop the same heat
in any given resistance ?
By Joule's law the heat developed is proportional jointly to the square of
the current and to the resistance of the circuit. If the current be a varying
one, the heat is proportional to the mean square. We therefore have to find
the value of
(
Jo
which is the mean square of a current whose maximum value is /.
The " square root of the mean square " of an alternating current is called
its virtual value, and is of great importance.
1054. The virtual value of an alternating current is 35.3
amperes. What is its maximum value ? its mean value ?
SELFINDUCTION 185
1055. Which will develop the greater amount of heat in a
given circuit, a direct current of 50 amperes, or an alternating
current whose mean value is 50 amperes ?
1056. What is meant by selfinduction ? Give two definitions
of the coefficient of selfinduction. Define the henry.
1057. The field magnet of a shunt dynamo consists of an iron
core wrapped with a great many turns of fine wire. If a cur
rent be sent through such a field for an instant by striking the
proper wires across one another, only a slight spark is observed ;
but if the current be allowed to flow for a second and then the
circuit be broken, a heavy spark is obtained. Explain.
1058. If a current of 2.1 amperes flowing in a coil of 100
turns set up through that coil a magnetic flux of .084 x io 8
lines, what is the coefficient of selfinduction of the coil, assum
ing the coil to contain no iron ?
If the circuit were broken, the wire composing it would be cut
by 100 x .084 x io 8 lines. The change in the current is 2.1
amperes. Therefore the inductance of the circuit is
100 X .084 X 10" = h
2.1 X IO 8
1059. An harmonic current of 20 amperes (virtual value) is
flowing in a given circuit. If the frequency be 120 periods per
second and L .06 henry, what is the electromotive force of
selfinduction ?
1060.* If the resistance of the foregoing circuit be 2.4 ohms,
what is the value of the electromotive force impressed on the
circuit ?
1061. Find the impedance of a coil having a resistance of 40
ohms and an inductance of .6 henry. Frequency of the alter
nating current 120.
1062. The resistance of a given coil is 8 ohms, inductance,
.3 henry. Compute the angle of lag for an alternating current
of frequency 100.
1 86 PROBLEMS IN PHYSICS
1063. The current in a coil is 40 amperes ; the potential dif
ference around the terminals of the coil is 102 volts. The angle
of lag is found to be 36. Compute the power.
1064. Show by a diagram what is meant by the lagging of an
alternating current behind the impressed electromotive force.
1065. To obtain the power spent in a circuit in which a direct
current of constant value is flowing, it suffices to take the
product ei. Explain why this is usually incorrect in the case
of an alternating current.
1066. An alternating current of frequency 120 periods per
second is passing through a straight wire of negligible induct
ance. When the wire is coiled around an iron core, the current
is observed to fall off 40 per cent. The resistance of the wire
being 6 ohms, what is the inductance of the coil ?
1067. What are the essential features of a transformer, and
what advantages arise from its use ?
1068. In what four ways is energy wasted in a transformer ?
1069. The ratio of the primary and secondary turns of a
.transformer is 20 : i. If at full load, the primary power is 4000
watts and the primary current 2 amperes. What are the values
of the secondary E.M.F. and current, the efficiency of the
transformer being 90 per cent ?
1070. What is necessary that an ordinary alternator may run
as a motor ?
1071. What is meant by a rotating magnetic field? How
may it be produced ?
1072. A magnetic field whose instantaneous strength is
given by the equation
b = 6000 sin wt
is combined at right angles with another of strength
' = 5000 sin (wt  \
Find the magnitude of the resultant field.
MAGNETIC AND ELECTRICAL UNITS 187
1073. What are the important differences between synchron
ous motors and induction motors ?
Magnetic and Electrical Units. We have seen how from
the arbitrarily chosen units of mass, length, and time a con
venient and consistent system of mechanical units is built up.
From the same fundamentals, and in a similar way, the units
necessary for electrical and magnetic measurements may be
derived. In every case the definition of the unit is based on a
physical law or a deduction from a physical law. It is evident
that more than one unit might easily be chosen according as
different physical phenomena were made the basis of the selec
tion. Thus two distinct C.G.S. systems of electrical units
have arisen. One, the electrostatic system, is based on the
definition of unit quantity of electrification as defined from the
experimentally proved relation between the magnitudes of
electric charges and the force, in air, between them. This
relation is
Now since unit length is a fundamental, and unit force has
been already chosen, it is consistent to say that unit quantity is
such a quantity that acting on an equal quantity at unit dis
tance will repel it with a force of one dyne. Unit quantity is
thus made to depend directly upon the units of force and dis
tance. To ascertain the way in which the fundamentals are
involved in any measurements of quantity we must pass to
dimensions ; thus,
O 2
unit force = ML T~ 2 = J^
whence Q = M*L*T~\
Unit current is said to flow in a circuit when unit quantity is
conveyed in unit time. This makes the dimensions of current
1 88 PROBLEMS IN PHYSICS
PROBLEM. Suppose that the unit of time were increased
threefold, and the unit of length were doubled. How would
the C.G.S. electrostatic unit of current be affected?
Making these changes in the fundamentals, we have for the
new unit of current
That is, the new unit is smaller than the old, the ratio being
TWO'
Hence a given current would appear to be $$ times as
great.
The other system is called the C.G.S. electromagnetic system.
The primary definition is that of unit current, based on the
action between an electric current and a magnetpole in its
vicinity. It is known, as the result of experiment, that a
magnetpole placed at the center of a loop of wire carrying cur
rent is urged along the axis of the loop, i.e. at right angles to
the plane of the loop, with a force which varies as the current,
the strength of the magnetpole, and the length of the wire
directly, and as the square of the radius of the loop inversely.
That is,
7 2 Trrm
r = A  5  >
f=K>*
If 7 be such that when m is a unit, magnetpole and r is
unity, the force is 2 TT dynes, then
7= A'.
And if it be agreed to call this current unit current, then any
current thereafter is given by
( OTNIVERSr
DIMENSIONS OF UNITS 189
The dimensions of unit current are
force x distance
strength of pole
The quantity conveyed by unit current in unit time is taken
as unit quantity. The dimensions of unit quantity are
Unlike the unit of quantity in the electrostatic system, this
unit is independent of the unit of time.
Unit difference of potential exists between two points in an
electric conductor when one erg of work is done in transferring
unit quantity from one point to the other. If Q units be trans
ferred through a difference of potential A V, the work done is
Unit difference of potential is, therefore, measured by
work
, and its dimensions are
quantity
Other dimensions in both systems are left as problems for the student.
Their derivation involves the application of the general rule : Ascertain the
relation which the quantities have been found to bear to each other, and
hence to the fundamental quantities. Discard numerical quantities as not
affecting dimensions.
For the practical purposes of electrical measurement the
C.G.S. electromagnetic units are found to be of inconvenient
magnitude. Multiples and submultiples of them have been
adopted by electricians in conference as better adapted to every
day measurements. Their names and values in C.G.S. electro
magnetic units are :
IQO PROBLEMS IN PHYSICS
the ohm = io 9 C.G.S. units of resistance.
the volt = io 8 " " u electromotive force.
the ampere = io~ J " " u current.
the coulomb = lo" 1 " " " quantity.
the farad  io~ 9 " " " capacity.
the microfarad = io~ 15 " " " capacity.
the joule = io 7 " " work (ergs).
the watt = io 7 " " " power.
1074. Find the conversion factor required to change potential
in electromagnetic units to footpound units.
1075. What must be taken as the unit of force in order that
currents measured in electromagnetic units may appear four
times as large as now ?
1076. Show that the unit of resistance is independent of the
unit of mass chosen.
1077. A current measured in electromagnetic units is rep
resented by 25. What number would represent the same cur
rent if the footpoundsecond units were used ?
1078. Find the conversion factor required to change the
capacity of a condenser computed when the inch is taken as
the unit of length, and in electrostatic units to farads.
1079. The magnetic moment of a magnet in C.G.S. units is
1000. What would it be in footpoundsecond units*?
VIBRATIONS
1080. What is meant by a vibratory motion? Does the bob
of a pendulum have such motion ? Does the balance wheel of
a watch have such motion ? State any examples of vibration
which occur to you.
1081. In what ways do the motions of different particles along
a clock pendulum differ? In what respects are their motions
alike ?
1082. What kind of motion does the end of the minute hand
of a clock have ? How does its motion differ from that of the
hour hand ? the second hand ?
1083. Compare the angular velocities of the hour, minute, and
second hands of a clock.
1084. An elastic ball is dropped and allowed to bound and
rebound from the floor until it comes to rest. Is the motion
vibratory ? Draw the time and height curve. Draw the time
and velocity curve approximately. Explain any peculiarities of
these curves. (See falling bodies.)
1085. C anc l E are tw balls in circular and elliptic grooves
on a horizontal table. OP is a rod turning about the common
center of the ellipse and circle with a uniform angular velocity,
and pushing the balls around. Compare the linear velocities of
the two balls at AA', BB' , etc. Compare the average linear
velocity of E with the velocity of C. The periodic time of C
is 40 sec. What is that of E ? Is the motion of the balls
vibratory ? (See Fig. 94.)
191
1 9 2
PROBLEMS IN PHYSICS
1086. If OA', Fig. 94, is very
small, what kind of motion will the
ball moving in the ellipse approach ?
1087. How does the motion of
the piston of an engine differ from
that of a point in the flywheel ?
1088. A man walks at a uni
form rate in a circular track ABCD.
Another man starts from A at the
same time, and walks along the
diameter AC, so that the line join
ing them is always perpendicular Fig 94.
to AC. What kind of motion will the second man
Where will he walk the fastest ? The first goes clear
in 20 min. What is his angular velocity ? What
periodic time of the second man ? Fig. 95.
B
have ?
around
is the
Fig. 96.
1089. If P 1 P 2 = P 2 P& does M 1 M 2 = M 2 M 3 ? The time re
quired for the first to move from P 1 to P z is the same as from
P 2 to P s , and equals that for the second to go from M 1 to M z
or M 2 to M 3 . How has the motion of the second man changed
in going from M l to M z ? Fig. 96.
If P moves uniformly in a circle of radius #, and M is the
foot of the perpendicular dropped from P on a diameter OA,
SIMPLE HARMONIC MOTION 193
we have from trigonometry OM= a cos <. Making all measure
ments from OA, and calling CD the angle turned through in i
sec., we have < = wt.
Then displacement of M from center is
OM = x a cos &)/.
The period is the same as that of P ;
^ 2 TT 2 TT
i.e. T= or ft) = .
27T .
. '. x = a cos  /.
1090. When, i.e., for what values of / is x a maximum ? a
minimum ? How does the velocity of M vary ?
1091. Draw a curve with time as x and distance from O as y.
Draw the corresponding timevelocity curve. Draw the corre
sponding timeacceleration curve.
1092. Define simple harmonic motion and give several ex
amples.
1093. Is S.H.M. a vibratory motion? Give an example of
a vibratory motion which is not simple harmonic.
1094. A body has S.H.M. in a straight line. The expression
for this motion is y = 6 sin 15 A Draw to scale the representa
tive circle. Find the periodic time ; the amplitude. Find the
velocity when t = 3 sec.
1095. The displacement of a particle is given by j = 8 cos 20 1.
What is the maximum displacement ? What is the maximum
velocity ? What is the acceleration when y = 4 ? What is the
periodic time ?
1096. If the angular velocity were doubled, how would the
quantities in question be altered ?
1097. A body of mass m vibrates with S.H.M. in a straight
line. Find its average kinetic energy.
WAVES
In the study of wave motion, the student should bear in mind that all wave
motions have certain similarities, and the examples given are mainly for the
purpose of calling attention to these. It is by no means true that the actual
motion of drops of water in the passage of a water wave are like the motion
of air particles during the passage of a sound wave, yet the ideas of wave
length, periodic time, velocity of propagation, amplitude, relation between
the time required for a single particle to go through one complete series of
its motions, and the distance moved by any 'and every wave element, etc.,
are common to both and enter into the consideration of every type of wave
motion.
1098. A stone is dropped vertically into a pond of still water.
It is observed that when ten circular crests have started outward,
the outer one has a radius of 6 m. What is the wave length ?
If 40 sec. are required for the outer crest to acquire a radius
of 5 m., what is the period ?
1099. If a vertical section is made through the center of the
wave system described above, draw the curve of intersection
with the surface approximately. Would this curve change in
form from instant to instant ? Would it change in position ?
noo. A system of water waves X = i m., v 4m., is moving
across a lake parallel to a row of fine wires 25 cm. apart. These
wires, starting at a certain point, are numbered o, I, 2, 3, 4, 5.
etc. At a given instant a crest is observed at the wire marked o.
State (i) At which other wires crests would be found.
(2) At which other wires hollows or troughs would be
found.
(3) At which other wires the water is at its natural
level.
(4) At which other wires the water is at its natural
level, but falling.
194
WAVES 195
1 10 1. When crests are observed at two wires 4 m. apart, how
many crests would there be between them ? How many troughs ?
1 1 02. Suppose that each individual particle moves in a circle,
how many times would a particle go around its circle while a
crest was traveling 20 m. ?
1103. A system of water waves is moving across a lake. The
wave length is 5 m. The velocity of propagation is 6 m. per
second. A crest is observed at a stake at a given instant.
Where will that crest be in 10 sec. ? Where was it 20 sec.
before? At the instant when the crest is at the stake men
tioned, what was the condition at a stake 10 m. back? 15 m.
back ? \6\ m. back? 17^ m. back ? i8 m. back ?
1104. Two exactly similar wave systems are moving in oppo
site directions. Show by diagram how "nodes" and "loops"
will be formed.
NOTE. The student can easily trace or copy a sine curve on a card, and
then cut out a pattern so as to readily draw two like curves. Then compound
them by the ordinary method. Now move one ^ A to the right and the other
the same distance to the left, and again compound them. Move each again,
etc. It will be found that certain points will be permanently at rest and
others vibrate with greater or less amplitude.
1105. Distinguish clearly between a progressive and a station
ary wave system. Show how a stationary system may be pro
duced.
1106. A system of progressive waves is moving in a straight
line. The wave length and velocity of propagation is known
and the complete history of the motion of one particle is given.
What can be inferred from this ?
Fig. 97.
1107. A wave motion of simple harmonic type is propagated
along OX (Fig. 97). The wave length is X, the velocity of
196 PROBLEMS IN PHYSICS
propagation is v. The circle at the left is called the circle
of reference, which means that as P moves around the circle
with uniform angular velocity the line PM, varying harmoni
cally, is a representative of the actual motion of every disturbed
particle of the medium. How far will the wave travel through
the medium while P goes once around the circle ?
1108. Show that T= = , where T is the common pen
to v
odic time.
1 109. What relation is there between the angle turned through
by p and the distance traversed by every portion of the wave
disturbance in that time ?
i no. Use this relation to modify y = a sin wt so as to express
a progressive wave disturbance of simple harmonic type.
mi. Show that y a sin (a>t + otf ')
= a sin [vt f x\. \x = vt 1 .
A
1 1 12. Show that if the displacement at 5 is
y = a sin (vt + x),
\
it is identical with that which was at the origin  sec. before.
1113. The displacement at 5 is now given by
y=a sin (vt + x}.
A,
What will represent it when it reaches R, a distance / beyond ?
What was it represented by when at' a point /units back of 5?
1114. If y 4 sin [10 / + 5 x\ is the expression for a progres
PROPAGATION OF WAVES 197
sive wave, what is the periodic time ? the wave length ? the
velocity of propagation ?
1115. Waves of length 2 m. pass a certain point. It is ob
served that four pass per second. Write the expression for
their motion.
1 1 16. From the equation y = a sin (vt + x), we see that as
X ^
/ increases so that t' t= T= , y takes all values between
v
+ a and a. While if t is constant, that is, at any instant of
time, all possible values for y may be found by varying x from
x to x H X. What fact does this express ?
1117. How does the energy distribution of a progressive wave
system differ from that of a stationary system ?
1118. Two progressive wave systems, wave lengths 2 : 3, are
compounded. Sketch approximately the resultant in various
phase relations.
1119. What do you mean by the terms like phase, opposite
phase, retardation of (2 n + i ) ?
1 1 20. Two wave systems of equal frequency are compounded.
Sketch approximately the resultant wave form in the following
cases :
(a) When the phases are alike and amplitudes equal.
(b) When the phases are alike and amplitudes are as I : 2.
(c) When the phase difference is 45, and amplitudes are
as 1:2.
(d) When the phase difference is 90, and amplitudes equal.
(e) When the phase difference is 180, and amplitudes equal.
(/) When the phase difference is 180, and amplitudes i : 2.
1 1 21. The displacement of a point is given by y l + y%> where
j/ 2 = A 2 cos (tot
Find the resultant displacement, and discuss the expression
obtained.
SOUND
1 122. If a sounding body were in the air, and at a considerable
distance from the earth, what would be the form of the wave
front if the temperature were uniform ? What would be the
direction of motion of those air particles in the same vertical
line as the source of sound ? the same horizontal line ? in a
line at an angle of 30 with the vertical ?
1123. If the velocity of sound in air is different in different
directions, how would the wave form be altered ?
Fig. 98
Suppose the air in an indefinitely long tube disturbed by the motion of the
piston, connected as shown in Fig. 98. Let the wheel be imagined to make
one revolution at a uniform angular velocity in the onehundredth part of a
second. When the piston reaches B, assume that the air at P is .just about
to be disturbed. Remembering that the disturbance will travel down the
tube at a uniform velocity, draw diagrams showing the state of the air in the
tube when crank pin is at i, 2, 3, 4, indicating,
(a) the points of greatest, least, and average pressure,
(b) the places of greatest and least displacement,
(c) the places of greatest and least velocity of particles of air.
1124. How far would the wave travel in I sec. if AP = 8$ cm. ?
NOTE. The distance AB has been neglected in comparison with AP.
1125. How far from A would the space of undisturbed air
extend at the end of I sec., if the wheel made only one revolu
tion ? What is the wave length ?
198
SOUND WAVES 199
1126. Describe the condition of the air in tube at the end of
onetwentieth of a second, if the wheel made just two revolu
tions and stopped.
1127. In the tube described above, consider the history of a
single lamina of air at the point P when piston makes just one
vibration. Draw a curve, using time in one fourhundredth of a
second as x, and (a) velocity of lamina as y ; (b) displacement
of lamina as y ; (c) density of lamina as y.
1128. How far does the wave travel when crank pin moves
through an angle of 30? 60 ? 90? 180? 270? What part
of a wave length in each case ?
1129. Consider two points in the tube a distance x apart, the
velocity and displacement of the first given at a time t. How
long before the second will acquire that velocity and displace
ment ? Through what angle will crank pin move in that time ?
1130. The velocity of sound at o C. = 33240 cm. per second.
Find the velocity when temperature is 25 C.
1131. Show that if V, = V Vi f .003665 t, velocity increases
nearly 60 cm. per second for i rise in temperature.
1132. The report of a cannon is heard 10 sec. after the flash
is seen. The temperature of the air is 20 C. How far was
the observer from the gun ?
1133. How much is the wave length of the air wave sent out
by a 256 fork altered by a rise of temperature from o to 20 ?
1134. A whistle giving 1000 vibrations per second is 156.20 m.
distant. How many complete waves between it and the obser
ver ? Temperature o C.
1135. The flash of a gun is seen, and 20 sec. later the report
is heard. The distance is known to be 6932 m. What was the
temperature ?
1136. Show that *y has the same dimensions as a velocity.
200 PROBLEMS IN PHYSICS
1137. Apply the formula to the case of iron, taking the value
of Young's modulus as 18 10" ; density 7.67.
1138. Find the ratio of the velocity of sound in brass to that
in iron.
1139. A string makes 256 complete vibrations per second.
When the velocity of sound is 34600 cm. per second, what is
the wave length of the sound ?
1140. If the temperature of the air were increased, what quan
tities referred to in Example 1139 would be altered?
1141. A tuningfork makes 1024 vibrations in a second; the
wave length of the sound in air is found to be 32 cm. Find the
velocity of sound.
1142. Name three ways in which musical sounds differ, and
explain the cause of differences.
1143. Define pitch ; timbre or character.
1144. Explain the connection between the pattern developed
in the "Chladni" plates and the character of the sound produced.
1145. Explain what is meant by the term tempered scale.
What is a musical interval ?
1146. Taking 256 as C, find the frequency of the notes of
the major scale, (a) Natural scale ; (b] when equally tempered.
STRINGS
stretching force
Formula: = "
2 length \ mass per unit length '
Since mass per unit length = area of crosssection x density ;
{A
F
area of crosssection density
~f
[T = force per unit area of crosssection.
NOTE. The mode of vibration considered above is the fundamental. The
string may vibrate in any integer multiple of this number, or in combinations
of such multiples.
1147. Under certain conditions of tension and length a string
makes 256 complete vibrations a second. How many would it
make if its length were doubled ? if its tension were doubled ?
if its mass were doubled without making it less flexible ?
1148. It is required to raise the pitch of a certain string
from C to D ; i.e. so that it shall make 9 vibrations in the same
time now required for 8. In what ways might this be done ?
Explain.
1149. A string making 400 vibrations per second has its
length and stretching force each divided by 4, and its mass per
unit length multiplied by 4. What effect on the pitch if the
string is made no less flexible ?
1150. A wire, I m. of which weighs I g. and is 80 cm. long,
is made to vibrate in unison with fork n = 128. What force is
used to stretch it ?
201
OF THB
TJNIVERSITY
202 PROBLEMS IN PHYSICS
1151. Why is the base string of a guitar wound with fine
wire ? If the wire makes each centimeter of the string four
times as heavy, how will the number of vibrations be altered ?
What objection is there to lowering the pitch by increasing the
radius of the string ?
1152. Explain why it is often more desirable to shorten all
the strings on a banjo by means of a clamp in order to raise
the pitch rather than to increase the tension of the strings.
1153. Draw a diagram to scale, showing the relative positions
of the frets on a fingerboard to produce the major scale.
1154. Explain how the violin illustrates the laws of transverse
vibrations of strings.
1155. What length of steel wire, mass of i m. = .98 g., stretch
ing force weight of 9 kg. (^=980), will make 256 complete
vibrations per second ?
.0098
r _ I J9 ' 98 ' I0 *
"512^ 98.10*
= J
512
1156. Two steel wires, mass of I m., respectively .98 and .45,
are stretched side by side. The length of the larger is observed
to be twothirds that of smaller. Compare the forces stretch
ing them ; (a) when in unison ; (&) when the smaller gives the
octave of the larger.
1157. What proportional lengths of the two wires above must
be taken such that when stretched with equal forces they will
vibrate in unison ?
1158. What proportional stretching forces will make the fre
quency of the smaller fourthirds that of the larger, their lengths
being equal ?
1159. Show that the expression for n is consistent with the
laws of motion.
VIBRATION OF STRINGS
203
1160. Show that each form of equation given above is of
proper dimensions.
1161. Two strings are carefully tuned so as to vibrate in uni
son in the fundamental. Will their overtones be harmonious ?
1162. A long string is stretched between two rigid posts ; a
small portion is distorted as shown in diagram. When sud
Fig. 99.
denly released it is found that triangular portion retains its
shape and moves along the cord at a uniform velocity. Draw
diagrams showing what happens at B.
1163. A uniform stretched wire is distorted as shown, A and
B being rigidly fixed. The distorted portion retains its form
and moves along the cord at a uniform velocity. Draw diagrams
showing reflection at D.
1164. Two like distortions are moving in opposite directions,
and with the same velocity along a string as shown. Draw a
series of diagrams showing their positions at several successive
short intervals of time. Explain why the point (P) midway
between 3 and 4 remains at rest (Fig. 101).
Fig. 101.
1165. Show by diagram how a string may vibrate in various
modes at the same time.
STRINGS GENERAL
It is shown in books on acoustics that the equation of motion for an elastic
string executing small free vibrations about a position of equilibrium is
where
W^r&i (Fig. 102)
m = mass per unit length,
F stretching force,
y = displacement of a point x distant from the origin,
at a time /
Fig. 102.
(1) Show that the equation is of consistent dimensions.
(2) Writing the equation in the form
m
m
show by substituting that a possible relation between y, a, x,
and / is
y = A s\i\px cospat. [A independent of x, y, t.
(3) If the string is fastened at the point x = o and also at the
point x = / (i.e. at those points y o for all values of /), find the
least value of /.
SUGGESTION. Sin// = o. Hence what set of values may pi have.
204
VIBRATION OF STRINGS 205
(4) Any part of the string between x = o and x = /, in other
words, any point of the string free to move, will have what kind
of motion ?
(5) If / == y, what is the frequency ?
(6) What other frequencies may occur ? What are the tones
due to these called? Is "A" the same for all of these fre
quencies ?
(7) Does the solution given correspond to a displacement
when / = o, or to an initial velocity ?
(8) Show that
IB sin px sin pat
C cos/;trcos/tf/
D cos/^r sin pat^
each satisfy the original equation, and that the sum of any
number of such terms is also a solution.
(9) Would the last two be consistent with a fixed point
at x = o ?
(10) If y B s'mflx sin pat is a consistent solution, and the
point x = were touched lightly, what would happen ?
1166. Draw diagrams showing places of maximum and of
minimum pressure changes in an open pipe : (a) when vibrating
in its fundamental mode ; (b) for the first overtone ; (c) for the
third overtone.
1 167. Do the same for maximum and minimum displacements.
1168. Draw similar diagrams for a closed tube.
1169. An open pipe is vibrating in its fundamental mode; a
hole in its side large enough to allow considerable air to pass in
or out is suddenly opened. If the hole is at the middle of the
tube, what effect will be produced ?
1170. If the end of the pipe in Example 1 169 is closed and the
hole left open, what differences will be observed ?
206 PROBLEMS IN PHYSICS
1171. Distinguish between "flue" and "reed" pipes, and
name instruments of each class.
1172. A closed organ pipe is 60 cm. long. What is the wave
length of its fundamental ?
1173. What is the wave length of its first overtone ?
1174. What is the wave length of the fourth overtone ?
1175. When the velocity of sound in air is 34800 cm., what is
the number of vibrations per second in each of the above cases ?
1176. Would increase of temperature change the pitch of an
organ pipe ?
1177. An open tube is 100 cm. long. Find the wave length
and frequency when the velocity of sound is 34000 cm. per
second.
1178. What is the wave length and frequency of its first three
overtones ?
1179. A fork making 332 vibrations per second is fixed in
front of a cylindrical tube, and the length adjusted to resonance
when temperature is o. How much must the length be
altered to resound at 25 ?
1180. A closed pipe is made just long enough to reinforce a
fork at its mouth, frequency of the fork 64. What must be the
frequencies of the next four forks of higher pitch which it will
also reinforce ?
1181. What would they be if tube were open ?
1182. A whistle making 4000 vibrations per second is moved
slowly away from a wall. What is the first position of reinforce
ment ? the second ?
1183. How far will the whistle be from the wall when there
are four nodes between it and the wall, and the sound is re
inforced ?
1184. How many beats per second will be heard when two
forks make 250 and 255 vibrations per second respectively ?
INTERFERENCE 2O/
1185. How could you determine, if 6 beats per second were
heard, which fork was the higher in pitch ?
1186. Show by diagram how the wave giving beats is made
up of two differing slightly in frequency and wave length.
1187. Explain the fluctuations in the intensity of sound from
a tuningfork when it is rotated near the ear.
1188. What are the conditions in order that two sound waves
may produce silence at a point ?
1189. If the scale in Konig's apparatus for the determination
of the velocity of sound in air is 40 cm., what would be the
lowest pitch which could be used as a source ? For what pitch
would there be found just three points where the flame was
stationary ?
1190. A tuningfork making 3000 vibrations per second is
slowly moved away from a wall. The velocity of sound is 34000
cm. per second. How far from the wall to the first point of
resonance ? to the second ? to the thirteenth ?
1191. Is there any difference in quality of sounds from open
and closed pipes of the same fundamental pitch ? If so, explain
the cause.
1192. Three shortest possible tubes containing respectively
air, oxygen, and hydrogen, velocities of sound, 33200, 31700,
126900, resound to a fork giving 1000 vibrations per second.
What are their lengths ?
1193. A locomotive whistle makes 1000 vibrations per second.
When moving 50 km. per hour, what will be the alteration in
pitch when approaching the observer? when receding? Tem
perature of air o C.
1194. A locomotive whistle makes 3000 vibrations per second.
Find the apparent number of vibrations :
(a) When approaching the station at the rate of 100 km. per
hour.
208 PROBLEMS IN PHYSICS
(b) When at rest and the observer is approaching the train at
the same rate.
(c) When they are moving away from each other each at the
rate of 100 km. per hour.
1195. Draw a diagram showing the effect of motion of the
source relative to the air upon the wave length in air.
1196. Indicate clearly the difference between motion of the
source when observer is at rest and motion of observer when
source is at rest.
LIGHT
REFLECTION
1197. State the laws of reflection of light.
1198. Show how reflection is explained on the wave theory.
1199. If a mirror were perfect, could it be seen ?
1200. Indicate how the form of a reflected wave front may be
found when the form of the incident wave and of the reflecting
surface is known.
1201. An object is placed in front of a plane mirror. Show
by diagram the path of the rays by which the image is seen.
What relation is there between the size of the object and the
size of the image ?
1202. A plane mirror is used to reflect a beam of parallel light.
The mirror is turned 10. Through what angle is the reflected
beam turned ? Give diagram.
1203. Show that the image formed by a plane mirror appears
to be as far back of the mirror as the object is in front.
1204. Show how spherical waves reflected at a plane surface
have their curvature reversed.
1205. Two mirrors are placed at an angle of 90, with a candle
between them. How many images will be seen ? Locate them.
1206. If a wave after reflection is to converge to a point,
what must be the wave form ?
1207. Two mirrors are inclined at any angle, and a luminous
point is placed between them. Show that all the images are on
p 209
210
PROBLEMS IN PHYSICS
a circle, and determine its radius and center. Show how to
find the angular position of each image.
1208. Two plane mirrors are placed parallel to each other, and
50 cm. apart. An object is placed 20 cm. from one of them.
Show how the images will be spaced. Draw the path of the
rays by which the fourth image on one side is seen.
1209. Explain why it is difficult to read the image of a printed
page in a plane mirror.
1210. A printed sheet is laid on a table between two parallel,
vertical, plane mirrors. Which of the images are easily read ?
1 21 1. A train of mirrors are placed vertical, and inclined to
each other. Given the angle of incidence on the first, and the
angle between the planes of each of the mirrors, find the devia
tion after successive reflection from each.
1212. The walls of a rectangular room are plane mirrors. A
candle is placed at any point in the room, and a person standing
at a given point, with his eye
in the same horizontal plane
as the candle, wishes to ob
serve it by rays reflected in
succession from each of the
walls. Find the point at which
he must look. Find the ap
parent distance of the image
seen.
Fig. 103.
Notation
used i
(Fig. 103).
C . .
MN .
A . .
P . .
Q
CA =R
P 1 . .
F . .
in problems relating to spherical mirrors
center of curvature.
aperture of mirror.
vertex of mirror. .
luminous point.
point of incidence.
radius of curvature.
intersection of reflected ray and PA.
principal focus.
CURVED MIRRORS 211
Lengths to the right from A are taken + .
AP' = p 1 = image distance = P' Q approximately.
AP p = object distance = PQ approximately.
AF =f= principal focal distance.
1213. Derive the formula showing the relation between/,/',
and R.
1214. What is meant by the term principal focus f
1215. The radius of a concave spherical mirror is 20 cm. The
sun's rays fall normally on a small portion of its surface. How
far from the mirror will the image of the sun be formed ?
1216. If R = 20 cm., find /' when / = 40 cm. ; 35 ; 25 ; 20 ;
15; 12; 10; 8; 5.
For which values of/ above will a real image be formed ?
1217. If the object is an arrow 5 cm. high, find the size of the
image in each of the cases of Example 1206. (Size refers to
linear dimensions.)
1218. Construct the image as formed by a concave mirror
when / > R, /</ < R, / </ When is it real ? when virtual ?
when larger than the object ? when smaller ?
1219. Show by diagram that if the aperture of a concave
mirror is large the image formed will be distorted.
1220. With a given concave mirror where must an object be
placed so that the image may be real and twice as large as the
object ? virtual and three times as large as the object ?
1221. What must be the radius of a concave spherical mirror
that an image of an object 20 ft. from a screen may be projected
on the screen and be magnified three times, the object being
placed between the mirror and the screen ?
1222. Show how to find the position and size of the image
formed by a convex mirror : (i) geometrically, (2) analytically.
1223. Derive the formula for a convex mirror, stating clearly
the approximations made.
212 PROBLEMS IN PHYSICS
1224. A convex mirror R = 80 cm. is placed 30 cm. from a
candle flame. Where will the image appear to be ? Construct
it. Find its size if the flame is I in. high.
1225. An object is moved from a point very near a convex
mirror to a great distance away from it. How far does the
image move ? How would its size change ?
1226. The radius of curvature of a concave mirror is 9 cm. ;
an object is 10 cm. in front of it. If the mirror is flattened out,
i.e. if r increases to oo , trace the changes in size and position
of the image, neglecting the decrease of/.
1227. The radius of curvature = 100 cm. The object is 90
cm. from the mirror and is moving outward with a velocity of
10 cm. per second. How fast is the image moving and in which
direction ?
1228. A luminous point is placed at the focus of a parabolic
mirror. Find the path of the reflected rays. Find the form of
the wave front.
1229. Can a very small element of any wave surface be con
sidered as spherical? If so, what would the center of the sphere
mean ? What surface would the center of the sphere trace as
the surface element moved over the surface of the wave ?
1230. State the laws of refraction. Show by diagram what
you mean by the terms used in stating the law.
1231. Derive the "sine law" from consideration of the velo
city of propagation of waves in the two media.
1232. If the velocity of light is altered in passing from one
medium to another, does the frequency change ? Does the wave
length change ?
1233. Does the index of refraction vary with the wave
length ?
1234. Show by diagram the path of a ray when passing from
water to air at angles of incidence less than the critical angle ;
just at this angle.
REFRACTION 213
1235. What is the critical angle for glass to air, index
a u =  3  ?
r'ff 2 *
1236. If the angle of incidence is observed to be 20 and of
refraction 15, find the index of refraction from each substance
to the other.
1237. If the angle of incidence is 40 and the index is J, find
the angle of refraction.
1238. A beam of light falls on the surface of still water at
an angle of 15 with the vertical. Find its direction in the
water, index > w = . Illustrate by a diagram drawn to scale.
1239. If the angle of incidence is 45 ; 60 ; 75 ; find the
direction in the water.
1240. If the angle of incidence is 45 in passing from water
to air, what is the direction in air ?
1241. Light is incident at an angle of 50 in water and passes
into air. Find path of ray.
1242. If the direction of a ray is reversed so that it passes
from water to air, what will be the index?
1243. A ray passes from water to air, angle of incidence 15.
Find direction in air.
1244. Does the critical angle depend on wave length? If
so, which wave lengths would you expect to have the greater
critical angle ?
1245. The velocity of light in air is approximately 3.jo 10 cm.
per second. What is its velocity in water, //, = ^ ? What in
glass, (J, = f ? in CS 2 , p = 1.63 ?
1246. How much longer would it take light to reach the
earth from the sun if the space were filled with water, neglect
ing the difference in velocity in air and vacuo ? Mean distance
earth to sun, 148. io 6 km.
1247. A plate of glass is immersed in water with its surface
horizontal. Light is incident at an angle of 60 on the surface
of the water. Find its direction in the glass, a /* w = , a p ff = f .
214 PROBLEMS IN PHYSICS
1248. The index from air to glass is 1.5. The index from
air to CS 2 is 1.6. Find the index from glass to CS 2 .
1249. A beam of monochromatic light is divided ; one part is
sent through i m. of water, the other part through an air path,
so that there may be no relative retardation. What is the air
path required ?
1250. Light is incident at an angle of 30 on a parallel plate
of glass 3 cm. thick. Draw the path of the ray. How much is
the beam displaced in passing through the plate, JJL =  ?
1251. An observer estimates the depth of a pond, looking
vertically downward, as 30 ft. What is the depth ?
1252. If he looked from water at an object 30 ft. above the
surface, how far above the surface would it appear to be ?
1253. A fish is 8 ft. below the surface of the water. A
man shoots at the place where the fish appears to be, holding
his gun at an angle of 45 with the surface of the water. Does
the bullet pass above or below the fish ? (Neglect any change
of direction of bullet.)
1254. Show by diagram how a straight stick held partly in
water at an angle of 60 appears to a person in the air. How
would it appear if the eye were under water ?
1255. Under what circumstances may light be propagated in
curved rather than straight lines ?
1256. Explain how the sun may be seen after it has passed
below the horizon.
1257. Prove that if A is the refracting angle of a prism, //.
the index of refraction, S the angle of minimum deviation,
sin k(A + &)
^ = sin 1 A
1258. IfA= 60, B = 53, find p.
1259. When A = 60, /* = , find 8.
When A = 30, /* = , find 8.
REFRACTION 215
1260. Compare the minimum deviation produced by a 30
water prism and that of a similar crownglass prism.
1261. A clear block of ice has a cavity in the form of tri
angular prism. The index from air to ice is 1.5. If the cavity
is filled with air, show the path of a ray of light through it; if
filled with a substance such that the index from ice to it were 1.6.
1262. A glass prism, index 1.5, refracting angle 60, is placed
in the path of a beam of monochromatic light. Draw a curve,
using angles of incidence as abscissas and angles of deviation as
ordinates.
1263. Show by diagram the path of a beam of monochromatic
light passing through a glass prism placed in air ; when placed
in water.
1264. Show the path when white light is used.
1265. What three kinds of spectra? Explain the occurrence
of dark lines in a spectrum. ('82.)
1266. Describe the experiment of the reversal of the sodium
lines. What inference is drawn from this experiment ? What
are the three classes of spectra, and to what does each owe its
origin ? ('88.)
1267. Show by diagram why a slit is used as a source of light
when a spectrum is required.
1268. Explain how deviation can be obtained without disper
sion.
THE LENS
Refraction at a spherical surface.
Let AQ\>z very small compared with sphere of radius
P be source of light,
P l apparent source to an eye is second medium,
PQ p = PA, Z.PQC =2,
( DENSE
3*^
n ^> TP; ,
^__p
C. PiR 5
\ PR
Fig. 104.
The A />Cg and P^CQ have a common angle C.
sin / ft R
sin 6"
sin C
P
P\
Law of
PR
sinr
sin /
P\ R
; LL ^ 1
sinr
PiR' P
i.e.
or
R P, P
(A) may be used to derive the formula for a lens if care is taken to
observe :
(1) The index from first medium to the second is the reciprocal of the
index from second to first.
(2) Distances to right are + , to left .
216
LM JL V
LENSES
217
(3) The thickness of the lens may be neglected.
(4) p l should be eliminated between the expressions for refraction in
and out.
For example, the biconvex lens, radii R v R 2 (A) becomes
 i /, i .
_ /A i f Since / 1 is the virtual
 7T   TI  1~ OUt.
R. 2 p p l source.
. / _ !\r_L _i__L I _ JL _ 1. ["Multiply second by /A and
) \R l R. 2 \~ ' p' p L add the equations.
If p' is negative, we have a real image or the light converges, and,
changing the signs,
1269. A convex lens is placed between a source of light and
a screen so as to give an image of the source on the screen.
How many such positions for the lens may be found ? Compare
the sizes of the image and object in each case.
1270. A double convex lens, the ratio of whose radii is 6 to i,
is used as a condenser for a magic lantern. When the light is
at a distance of 2 in., the emerging rays are parallel. What
are the radii, the material of the lens being crown glass ? ('78. )
1271. A candle is / cm. from a wall. A converging lens
forms an image on the wall; when moved a distance d it also
/2 _ ,J%
forms an image. Prove that f = 
4/
1272. In a lens where SB construct the image of an
/ / /
object placed between lens and F; when placed beyond F.
1273. Write a rule for the construction of images in case of
spherical lenses and mirrors.
1274. The focal length of a converging lens is 3 m. Find
the distance from the lens (assumed thin) to the image in each
of the following positions of the object : 4 m. ; 5 m. ; 8 m. ;
10 m. ; 20 m. ; i km. ; 3 m. ; 2 m. ; i m. ; 5 cm.
218 PROBLEMS IN PHYSICS
1275. Show by construction the position and size of the
image when /= i m. ; / = 3m.; /= 2m.; / = .5 m.
1276. In the derivation of the formulae for lenses, what
assumptions are made which are only approximately correct ?
1277. What do you mean by a converging lens? by a diverg
ing lens ?
1278. Assuming that a biconvex lens gives a real image,
construct it, and assuming that the lens is thin, prove that
 H  =  by use of similar triangles.
* P . size of image /'
Show also that : . . . & =
size of object /
1279. By means of the formula A,
Find the formula for a biconcave lens.
Find the formula for a planoconvex lens.
Find the formula for a planoconcave lens.
Find the formula for a concavoconvex lens.
1280. Find the focal length of a biconvex lens of crown
glass, fj, = f , *\ = r 2 = 30 cm.
1281. A lens of focal length 25 in air, >, = f. What will
be the focal length in water, > w = .
1282. A planoconvex lens is to be made of glass, index 1.6,
so as to form a real image of an object placed 2 cm. from it, and
magnify it three times. What must be the radius of curvature ?
1283. Find the optical center for several lenses, as biconvex
of equal radii, planoconvex, etc.
1284. If q and q' are the distances of object and image from
the principal focus, show that qq' =/ 2 .
1285. The radii of curvature of a biconvex lens are 30 and
32 cm. The focal length is 31 cm. What is the index of
the glass ?
1286. If yu = f , and the radii of curvature of the biconvex
lens are equal, find /.
LENSES 219
1287. Show by diagram what you mean by chromatic aberra
tion of a lens.
1288. Distinguish between chromatic and spherical aber
ration.
1289. What is meant by achromatism? How construct an
achromatic lens ? (Spring '79.)
1290. If values of  and are taken as coordinates, what
kind of a curve will be found ? Interpret its intercepts.
1291. If corresponding values of / and /' are measured
along two rectangular lines, and p lt //, / 2 , / 2 ', etc., are joined
by straight lines, show that all of these lines intersect in a
point, the coordinates of which are x=y = F. (A practical
fact.)
1292. If a series of observed values of / and p 1 are taken as
abscissas and ordinates, what kind of a curve will be found ?
1293. To what does the other branch of the curve correspond ?
1294. A small object is placed slightly beyond the principal
focus of a biconvex lens. The image formed is viewed through
a biconvex lens placed nearer to the image than the principal
focal distance. What is such an arrangement called ? Draw a
diagram showing formation of the image seen, and find the
ratio of its height to that of the object.
1295. Draw diagrams showing what is meant by "short"
sight or myopia. What form of lens is needed to correct
myopic vision ?
1296. What is meant by "long" sight, and how may it be
corrected ?
1297. A person is unable to see clearly objects 30 cm. from
the eye. Give two possible explanations of this.
1298. Indicate by diagram how inability to decrease the
radius of curvature of the crystalline lens would affect vision.
What kind of glasses would be needed ?
INTERFERENCE
1299. What must be the relation between the elements of
two light waves in order that interference may be possible ?
1300. Explain three general methods by which interference
may be obtained.
1301. Find the effective retardation of a ray of light reflected
from B over one reflected from C. Fig. 105.
E
Fig. 105.
Consider parallel rays incident at A and C such that the ray refracted at A,
reflected at B, and refracted at C proceeds along the same path CE as the ray
reflected at C. When 2 strikes the surface, the phase is the same as at D in i .
Draw CB' perpendicular to AB. Then i travels from D to C, while 2 travels
from A to B' .
Apparent retardation is B' B + BC.
Extend AB to C', making BC' = BC.
Then BB' + CB = 8,
CC' = 26.
.. 8 = 2 e cos r.
.. 8 = 2 fie cos r.
[Retardation due to glass path.
[Equivalent retardation in air.
But one reflection is with change of phase.
.. effective retardation,
8 = 2 fie cos r + .
It follows that if white light is reflected as shown in the figure, light of wave
length A. will be a minimum when 2 fie cos r = n\. (n any integer.)
220
LENSES 221
1302. What is the least thickness of crown glass, index ,
which will give interference for sodium light when r = 45 ?
1303. What thickness of a film, index , would retard light
of wave length 7610" three wave lengths ?
1304. Explain the changing colored bands seen when white
light is reflected from a soapbubble film stretched vertically.
1305. What shape would the bands have if the film was
attached to a ring held horizontally ?
1306. White light falls on a thin wedgeshaped film of air
and is reflected from each surface. It is observed that no light
of wave length X appears to come from a line parallel to the
edge of the wedge and 2 mm. from the edge. Show the position
of the next three lines of the same color.
1307. Explain the production of color in the soapbubble.
How can the wave length of light be measured ? Derive the
formula. Give diagram of apparatus used in projecting these
colors on a screen. ('88.)
1308. Derive the formula for "Newton's rings."
p = "Y & sec r> (2n+i) for bright ring.
p = ^/R sec r n\ for dark ring.
1309. If red light X = 76 io~ 6 is used and R = 9 cm., r = 45,
find the radii of the first four bright rings.
1310. What would be the ratio of the radii of rings of the
same order for X = 76 io~ 6 and X = 52 io~ 6 ?
1311. Find the general expression for the width of the rings
for a given wave length. Do they increase or decrease in width
as r is increased ?
DIFFRACTION
1312. Explain why the shadow of a twig cast by an arc light
on a frosty pane of glass is often fringed with color.
1313. A slit in a piece of cardboard is held close to the eye
and parallel to the filament of an incandescent lamp. Explain
the colored fringes observed. Are the colors pure spectral
colors ?
1314. White light diverging from a narrow slit falls on two
parallel narrow slits very close together. Show how the ap
pearance on a screen beyond the apertures depends on the
wave length considered and on the distance between the two
parallel slits.
1315. Light from a small source is divided and passes by
two paths of slightly different length to a screen. Explain
briefly the difference in the phenomena observed when the
light is white and when it is monochromatic.
1316. Parallel rays
of white light fall nor
mally on a transmis
sion grating and the
diffracted rays are
brought to a focus by
a lens. Show by dia
gram how spectra are
formed and derive the
formula (Fig. 106).
1317. Two gratings
are placed one above
222
Fig. 106
DIFFRACTION 223
the other in a horizontal beam of white light from a vertical
slit. If one has twice as many lines per centimeter as the
other, how will the spectra differ ?
1318. If d= icr 3 , x=59.icr 6 , find 6 1  # 2 ; 3 .
1319. For a certain wave length and grating, # 3 = 6 ; for a
different wave length, # 2 = 6. Find the ratio of the two wave
lengths and explain overlapping spectra.
1320. Show from the expression = sin # n how the length
of the spectrum will change with d.
1321. Sunlight passing through a narrow slit falls normally
on a transmission grating 800 lines per centimeter. The spectra
are focused on a screen 10 m. from the grating. Find the
position and length of the first spectrum.
1322. Light of wave length 589 io~ 7 passes through the slit
and falls on a grating G, Fig. 107. An eye placed just back of
the grating observes a series of images of the slit, as S v 5 2>
5 3 , etc. Explain how these images are formed.
If d^ = 5 cm. and / = 80 cm., find the number of lines per
centimeter in the grating.
ts"
S'
s
S 2
Fig. 107.
1323. How do the spectra formed by diffraction differ from
those formed by refraction ?
1324. What assumptions are made in the derivation of the
formula for a grating which are only approximately true ?
1325. Derive the formula for a reflection grating if the angle
of incidence = i and the grating space = d.
1326. Show by diagram the formation of the first spectrum
by a reflection grating.
TABLES
[In these tables the admirable arrangement made use of
in Bottomley's FourFigure Mathematical Tables has been
followed.]
226
LOGARITHMS
O
1
2
3
4
5
6
7
8
9
123
456
789
10
oooo
0043
0086
0128
0170
0212
0253
0294
0334 0374
4 8 12
I 7 21 25
29 33 37
11
12
13
0414
0792
1139
0453
0828
"73
0492
0864
1206
0531
0899
1239
0569
0934
1271
0607
0969
1303
0645
1004
1335
0682
1038
1367
0719
1072
1399
755
1106
H30
4811
3 7 10
3 6 10
15 J 9 23
14 17 21
13 I 6 19
26 30 34
24 28 31
23 26 29
14
15
16
1461
1761
2041
1492
1790
2068
1523
1818
2095
1553
1847
2122
1584
1875
2148
1614
1903
2175
1644
I93 1
22OI
1673
1959
2227
1703
1987
2253
1732
2014
2279
369
368
3 5 8
12 15 18
II 14 17
ii 13 16
IO 12 15
9 12 14
9 ii 13
21 24 27
20 22 25
18 21 24
17
18
19
2304
2553
2788
2330
2577
2810
2355
2601
2833
2380
2625
2856
2405
2648
2878
2430
26 7 2
2900
2455
2695
2923
2480
2718
2945
2504
2742
2967
2529
2765
2989
2 57
2 5 7
247
17 20 22
16 19 21
16 18 20
20
3010
3032
3054
3075
3096
3 Il8
3139
3160
3181
3201
246
8 ii 13
15 17 19
21
22
23
3222
3424
3617
3243
3444
3636
3263
3464
3655
3284
3483
3674
3304
3502
3692
3324
3522
37"
3345
3729
3365
3560
3747
3385
3579
3766
3404
3598
3784
246
246
24 6
8 IO 12
8 10 12
7 9 ii
14 16 18
14 15 17
13 15 17
24
25
26
3802
3979
415
3820
3997
4166
3838
4014
4183
3856
4031
4200
3874
4048
4216
3892
4065
4232
3909
4082
4249
3927
4099
4265
3945
4116
4281
3962
4U3
4298
2 4 5
2 3 5
2 3 5
7 9 ii
7 9 10
7 8 10
12 14 16
12 14 15
27
28
29
43H
4472
4624
433
4639
4346
4502
4654
4362
45 l8
4669
4378
4533
4683
4393
4548
4698
4409
45 6 4
4713
4425
4579
4728
4440
4594
4742
445 6
4609
4757
2 3 5
2 3 5
i 3 4
689
689
679
II 13 I 4
II 12 14
IO 12 11
30
33
477i
4786
4800
4814
4829
4843
4857
4871
4886
4900
i 3 4
679
10 ii 13
4914
5i85
4928
5065
5198
4942
5079
5211
4955
5092
5224
4969
5105
5237
4983
5"9
525
4997
5132
5263
5011
5H5
5276
5024
5 ! 59
5289
5038
5172
5302
3 4
3 4
3 4
678
I I I
10 II 12
9 II 12
9 IO 12
34
35
36
5315
5441
5563
5328
5453
5575
5340
5465
5587
5353
5478
5599
5366
5490
5611
5378
55 02
5623
539i
55H
5635
5403
5527
5647
5539
5658
5428
5551
5670
3 4
2 4
2 4
568
5 6 7
5 6 7
9 10 ii
9 10 ii
8 10 ii
37
38
39
~40~
5682
5798
59H
5 6 94
5809
5922
5705
5821
5933
5717
5832
5944
5729
5843
5955
5740
5855
5966
5752
5866
5977
5763
5877
5988
5999
5786
5899
6010
2 3
2 3
2 3
5 6 7
5 6 7
4 5 7
8 9 10
8 9 10
8 9 10
6021
6031
6042
6o53
6064
6075
6085
6096
6107
6117
2 3
4 5 6
8 9 10
41
42
43
6128
6232
6335
6138
6243
6345
6149
6253
6355
6160
6263
6365
6170
6274
6 375
6180
6284
6385
6191
6294
^395
6493
6590
6684
6201
6304
6405
6212
6314
6415
6222
6325
6425
2 3
2 3
2 3
4 5 6
4 5 6
4 5 6
7 8 9
7 8 9
7 8 9
44
45
46
48
49 i
6435
6532
6628
6444
6542
6637
6 454
6$
6464
6561
6656
6474
6571
6665
6484
6580
6675
6503
6599
6693
6513
6609
6702
6522
6618
6712
2 3
2 3
2 3
4 5 6
456
4 5 6
7 8 9
7 8 9
7 7 8
6721
6812
6902
6730
6821
6911
6739
6830
692O
6749
6839
6928
6758
6848
6937
6767
6857
6946
6776
6866
6955
6785
6875
6964
6794
6884
6972
6803
6893
6981
2 3
2 3
I 2 3
4 5 5
4 4 5
4 4 5
678
678
678
50
~5T
52
53
6990
6998
7007
7016
7024
733
7042
7050
7059
7067
i 2 3
3 4 5
678
7076
7160
7243
7084
7168
7251
7093
7177
7259
7101
7185
7267
7110
7!93
7275
7118
7202
7284
7126
7210
7292
7135
7218
7300
7H3
7226
7308
7152
7235
I 2 3
I 2 2
I 2 2
3 4 5
3 4 5
3 4 5
678
677
667
54
7324
7332
734
7348
735 6
7364
7372
738o
7388
7396
I 2 2
3 4 5
667
LOGARITHMS
227
O
7404
1
2
3
4
5
6
7
8
7466
9
123
456
789
5 6 7
55
7412
74i9
7427
7435
7443
745i
7459
7474
I 2 2
345
56
57
58
7482
7559
7 6 34
7490
7566
7642
7497
7574
7649
755
7582
7657
7513
7589
7664
7520
7597
7672
7528
760^
7679
7536
7612
7686
7543
7619
7694
755i
7627
7701
2 2
2 2
345
3 4 5
344
5 6 7
5 6 7
5 6 7
5 6 7
5 6 6
5 6 6
59
60
61
7709
7782
7*53
7716
7789
7860
7723
7796
7868
773i
7803
7875
7738
7810
7882
7745
7818
7889
7752
7825
7896
7760
7832
7903
7767
7839
7910
7774
7846
7917
I 2
I 2
344
3 44
344
62
63
64
7924
7993
8062
793i
8000
8069
7938
8007
8075
7945
8014
8082
7952
8021
8089
7959
8028
8096
7966
8035
8102
8169
7973
8041
8109
7980
8048
8116
7987
8055
8122
I 2
I 2
I 2
334
334
334
5 6 6
5 5 6
5 5 6
65
8129
8136
8142
8149
8156
8162
8176
8182
8189
I I 2
334
5 5 6
66
67
68
8195
8261
8325
8388
8451
8513
8202
8267
8331
8209
8274
8338
8215
8280
8344
8222
8287
8351
8228
8293
8357
8235
8299
8363
8241
8306
8370
8248
8312
8376
8254
8319
8382
I 2
I 2
I 2
334
334
334
5 5 6
5 5 6
4 5 6
4 5 6
4 5 6
4 5 5
4 5 5
455
4 5 5
69
70
71
^T2~
73
74
8395
8457
8519
8401
8463
8525
8407
8470
8531
8414
8476
8537
8420
8482
8543
8426
8488
8549
8432
8494
8555
8439
8500
8561
8445
8506
8567
I 2
[ 2
I 2
234
234
234
8573
8633
8692
8579
8639
8698
8585
8645
8704
859i
8651
8710
8597
8657
8716
8603
8663
8722
8609
8669
8727
8615
8675
8733
8621
8681
8739
8627
8686
8745
I 2
I 2
I 2
234
234
234
75
^6~
77
78
8751
8756
8762
8768
8774
8779
8785
8791
8797
8802
I 2
2 33
233
233
233
4 5 5
4 5 5
445
445
8808
8865
8921
8814
8871
8927
8820
8876
8932
8825
8882
8938
8831
8887
8943
8837
8893
8949
8842
8899
8954
8848
8904
8960
8854
8910
8965
8859
8915
8971
I 2
I 2
I 2
79
80
81
18976
9031
9085
8982
9036
9090
8987
9042
9096
8993
9047
9101
8998
9053
9106
9004
9058
9112
9009
9063
9117
9015
9069
9122
9020
9074
9128
9025
9079
9133
I 2
I 2
I 2
233
2 3 3
233
445
445
4 4 5
82
83
84
9138
9191
9243
9H3
9196
9248
9149
9201
9253
9154
9206
9258
9159
9212
9263
9165
9217
9269
9170
9222
9274
9175
9227
9279
9180
9232
9284
9186
9238
9289
I 2
I 2
I 2
233
233
233
445
4 4 5
445
85
9294
9299
9304
9309
9315
9320
9325
9330
9335
9340
I 2
233
4 4 5
445
344
344
86
87
88
9345
9395
9445
935
9400
945
9355
9405
9455
9360
9410
9460
9365
9415
9465
937
9420
9469
9375
9425
9474
938o
943
9479
9385
9435
9484
9390
9440
9489
I 2
o
o
233
223
223
89
90
91
9494
9542
9590
9499
9547
9595
954
955 2
9600
959
9557
9605
9513
9562
9609
95i8
9566
9614
9523
957i
9619
9528
957 6
9624
9533
958i
9628
9538
9586
9633
o
o
223
2 2 3
223
344
344
344
92
93
94
9638
9685
9731
9643
9689
9736
9647
9694
9741
9652
9699
9745
9657
973
975
9661
9708
9754
9666
97*3
9759
9671
9717
9763
9675
9722!
9768
9680
9727
9773
223
223
223
344
344
344
95
~96~
97
98
9777
9782
9786
9791
9795
9800
9805
9809
9814
9818
I
223
344
344
344
344
9823
9868
9912
9827
9872
9917
9832
9877
9921
9836
9881
9926
9841
9886
9930
9845
9890
9934
9850
9894
9939
9854
9899
9943
9859
9903
9948
9863
9908
995 2
I
I
D I
223
223
223
99
995 6
9961
9965
9969
9974
9978
9983
9987
9991
9996
D I I
223
334
228
NATURAL SINES
O'
6'
12' 18'
24'
3O'
36'
42' 48'  54'
1 2 3 4 5
~~F
2
3
~4~
5
6
~T~
8
9
oooo
0017
0035 0052
0070
0087
0105
0122
0140
OI 57
369
12 I 5
0175
0349
0523
0192
0366
0541
0209 0227
0384 0401
0558 i 0576
0244
0419
0593
0262
0436
0610
0279
0454
0628
0297
0471
0645
0314
0488
o663
0332
0506
0680
369
369
369
12 I 5
12 I 5
12 I 5
0698
0872
1045
7 J 5
0889
1063
0732
0906
1080
0750
0924
1097
0767
0941
i"5
0785
0958
1132
0802
0976
1149
0819
0993
1167
0837
IOII
1184
0854
1028
I2OI
369
369
369
12 I 5
12 14
12 14
1219
1392
1564
1236
1409
1582
1253
1426
1599
1271
1444
1616
1288
1461
1633
1305
1478
1650
1323
H95
1668
1340
1513
1685
1357
153
1702
1374
1547
1719
369
369
369
12 14
12 14
12 14
10
"IT
12
13
1736
1754
1771
1788
1805
1822
1840
1857
1874
1891
369
12 14
1908
2079
2250
1925
2096
2267
1942
2113
2284
1959
2130
2300
1977
2147
2317
1994
2164
2334
2OII
2181
2351
2028
2198
2368
2045
2215
2385
2062
2232
2402
3 6 9
3 6 9
3 6 8
II I 4
II 14
II 14
14
15
16
2419
2588
2756
2436
2605
2773
2453
2622
2790
2470
2639
2807
2487
2656
2823
2504
2672
2840
2521
2689
2857
2538
2706
2874
2554
2723
2890
257i
2740
2907
3 6 8
368
368
II 14
II 14
II 14
17
18
19
2924
3090
3256
2940
3*07
3272
2957
3123
3289
2974
3140
3305
2990
3156
3322
3007
3i73
3338
3024
3190
3355
3040
3206
337 1
3057
3223
3387
3074
3239
3404
368
368
3 5 8
II 14
II 14
II 14
20
3420
3437
3453
3469
3486  3502
35i8
3535
355i
35 6 7
3 5 8
II 14
21
22
23
~24~
25
26
3584
3746
3907
3600
3762
3923
3616
3778
3939
3633
3795
3955
3 6 49
3811
397 1
3665
3827
3987
3681
3843
4003
3697
3859
4019
37*4
3875
4035
3730
3891
405 i
3 5 8
3 5 8
3 5 8
II 14
II 14
II 14
4067
4226
4384
4083
4242
4399
4099
4258
4415
4"5
4274
443i
4131
4289
4446
4H7
435
4462
4163
432i
4478
4179
4337
4493
4195
4352
459
4210
4368
4524
3 5 8
3 5 8
3 5 8
II I 3
II I 3
10 13
27
28
29
4540
4695
4848
4555
4710
4863
457 1
4726
4879
4586
474i
4894
4602
4756
4909
4617
4772
4924
4633
4787
4939
4648
4802
4955
4664
4818
4970
4679
4833
4985
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
CO CO CO CO
CO tO H* O
5000
5015
5030
5045
5060
5075
5090
5105
5120
5135
3 5 8
10 13
5 ! 5
5 2 99
5446
5165
53H
546i
5180
5329
5476
5195
5344
5490
5210
5358
555
5225
5373
5519
5240
5388
5534
5255
5402
5548
5270
5417
5563
5284
5432
5577
2 5 7
257
2 5 7
IO 12
IO 12
10 12
34
35
36
5592
5736
5878
5606
5750
5892
5621
5764
5906
5635
5779
5920
5650
5793
5934
5664
5807
5948
5678
5821
5962
5 6 93
5835
5976
577
5850
5990
572i
5864
6004
2 57
2 5 7
257
IO 12
10 12
9 12
37
38
39
lib"
6018
! 6l 57
6293
6032
6170
6307
6046
6184
6320
6060
6198
6334
6074
6211
6347
6088
6225
6361
6101
6239
6 374
6115
6252
6388
6129 6143
6266 6280
6401 6414
2 5 7
2 5 7
247
9 12
9 ii
9 ii
6428
6441
6455
6468
6481
6494
6508
6521
6534 6547
247
9 ii
41
42
43
6561
6691
6820
6574
6704
6833
6587
6717
6845
6600
6730
6858
6613
6743
6871
6626
6756
6884
6639
6769
6896
6652
6782
6909
6665
6794
6921
6678
6807
6934
247
2 4 6
246
9 ii
9 ii
8 ii
44
6947
6959
6972
6984
6997
7009
7022
7034 7046
759
246
8 10
NATURAL SINES
229
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
123
4 5
45
7071
7083
7096
7108
7120
7i33
7*45
7i57
7169
7181
246
8 10
46
47
48
7'93
73H
743i
7206
7325
7443
7218
7337
7455
7230
7349
7466
7242
736i
7478
7254
7373
7490
7266
7385
750i
7278
7396
75i3
7290
7408
7524
7302
7420
7536
246
2 4 6
246
8 10
8 10
8 10
49
50
51
7547
7660
7771
7558
7672
7782
757
7683
7793
7581
7694
7804
7593
7705
7815
7604
7716
7826
7 6l 5
7727
7837
7627
7738
7848
7638
7749
7859
7649
7760
7869
2 4 6
2 4 6
2 4 5
8 9
7 9
7 9
52
53
54
7880
7986
8090
7891
7997
8100
7902
8007
8111
7912
8018
8121
7923
8028
8131
7934
8039
8141
7944
8049
8151
7955
8059
8161
7965
8070
8171
7976
8080
8181
2 4 5
2 3 5
235
7 9
7 9
7 8
55
8192
8202
8211
8221
8231
8241
8251
8261
8271
8281
2 3 5
7 8
56
57
58
8290
8387
8480
8300
8396
8490
8310
8406
8499
8320
8415
8508
8329
8425
8517
8339
8434
8526
8348
8443
8536
8358
8453
8545
8368
8462
8554
8377
8471
8563
2 3 5
2 3 5
235
6 8
6 8
6 8
59
60
61
8572
8660
8746
8581
8669
8755
8590
8678
8763
8599
8686
8771
8607
8695
8780
8616
8704
8788
8625
8712
8796
8634
8721
8805
8643
8729
8813
8652
8738
8821
3 4
3 4
3 4
6 7
a ?
62
63
64
8829
8910
8988
8838
8918
8996
8846
8926
9003
8854
8934
9011
8862
8942
9018
8870
8949
9026
8878
8957
9033
8886
8965
9041
8894
8973
9048
8902
8980
9056
3 4
3 4
3 4
1 i
5 6
65
9063
9070
9078
9085
9092
9100
9107
9114
9121
9128
2 4
5 6
66
67
68
9135
9205
9272
9H3
9212
9278
915
9219
9285
9157
9225
9291
9164
9232
9298
9171
9239
9304
9178
9245
93"
9184
9252
9317
9191
9259
9323
9198
9265
9330
2 3
2 3
2 3
5 6
4 6
4 5
69
70
71
9336
9397
9455
9342
9403
9461
9348
9409
9466
9354
9415
9472
9361
9421
9478
9367
9426
9483
9373
9432
9489
9379
9438
9494
9385
9444
9500
939i
9449
955
2 3
2 3
2 3
4 5
4 5
4 5
72
73
74
95 11
9563
9613
95 l6
9568
9617
9521
9573
9622
95 2 7
9578
9627
9532
9583
9632
9537
9588
9636
9542
9593
9641
9548
9598
9646
9553
9603
9650
9558
9608
9655
2 3
2 2
2 2
4 4
3 4
3 4
75
9659
9664
9668
9673
9677
9681
9686
9690
9694
9699
I 2
3 4
76
77
78
973
9744
978i
9707
9748
9785
9711
9751
9789
9715
9755
9792
9720
9759
9796
9724
9763
9799
9728
9767
9803
9732
977
9806
9736
9774
9810
9740
9778
9813
2
2
2
3 3
3 3
2 3
79
80
81
9816
9848
9877
9820
9851
9880
9823
9854
9882
9826
9857
9885
9829
9860
9888
9833
9863
9890
9836
9866
9893
9839
9869
9895
9842
9871
9898
9845
9874
9900
I 2
2 3
2 2
2 2
82
83
84
9903
9925
9945
9905
9928
9947
9907
993
9949
9910
9932
995 *
9912
9934
995 2
9914
9936
9954
9917
9938
995 6
9919
9940
9957
9921
9942
9959
9923
9943
9960
o
2 2
I 2
I I
85
9962
9963
9965
9966
9968
9969
9971
9972
9973
9974
001
I I
86
87
88
9976
9986
9994
9977
9987
9995
9978
9988
9995
9979
9989
9996
9980
9990
9996
9981
9990
9997
9982
9991
9997
9983
9992
9997
9984
9993
9998
9985
9993
9998
I
000
o o o
I I
I I
O O
89
9998
9999
9999
9999
9999
I 000
nearly.
I 'OCX)
nearly.
I'OOO
nearly.
rooo
nearly.
I'OOO
nearly.
o o o
230
NATURAL COSINES
O'
6'
12'
18'
24'
30'
36'
42'
48'
54'
123
4 5
rooo
I'OOO
nearly.
rooo
nearly.
rooo
nearly.
rooo
nearly.
9999
9999
9999
9999
9999
000
1
2
3
9998
9994
9986
9998
9993
9985
9998
9993
9984
9997
9992
9983
9997
9991
9982
9997
9990
9981
9996
9990
9980
9996
9989
9979
9995
9988
9978
9995
9987
9977
O O O
O O O
I
O O
I I
I I
4
5
6
9976
9962
9945
9974
9960
9943
9973
9959
9942
9972
9957
9940
997 i
995 6
9938
9969
9954
9936
9968
9952
9934
9966
995 i
9932
9965
9949
9930
9963
9947
9928
O O
I
O I
I 2
I 2
7
8
9
9925
9903
9877
9923
9900
9874
9921
9898
9871
9919
9895
9869
9917
9893
9866
9914
9890
9863
9912
9888
9860
9910
9885
9857
9907
9882
9854
9905
9880
9851
I
O I
O I
2 2
2 2
2 2
10
9848
9845
9842
9839
9836
9833
9829
9826
9823
9820
I I 2
2 3
11
12
13
9816
9781
9744
9813
9778
9740
9810
9774
9736
9806
9770
9732
9803
9767
9728
9799
9763
9724
9796
9759
9720
9792
9755
97'5
9789
975 1
9711
9785
9748
9707
112
I I 2
112
2 3
3 3
3 3
14
15
16
973
9659
9613
9699
9655
9608
9694
9650
9603
9690
9646
9598
9686
9641
9593
9681
9636
9588
9677
9632
9583
9673
9627
9578
9668
9622
9573
9664
9617
9568
I I 2
I 2 2
122
3 4
3 4
3 4
17
18
19
9563
95 11
9455
9558
9505
9449
9553
9500
9444
9548
9494
9438
9542
9489
9432
9537
9483
9426
9532
9478
9421
9527
9472
9415
952i
9466
9409
95i6
9461
9403
I 2 3
I 2 3
I 2 3
4 4
4 5
4 5
20
9397
9391
9385
9379
9373
9367
936i
9354
9348
9342
I 2 3
4 5
21
22
23
9336
9272
9205
9330
9265
9198
9323
9259
9191
93 ! 7
9252
9184
93"
9245
9178
934
9239
9171
9298
9232
9164
9291
9225
9157
9285
9219
9150
9278
9212
9M3
I 2 3
I 2 3
I 2 3
4 5
4 6
5 6
24
25
26
9135
9063
9128
9056
8980
9121
9048
8973
9114
9041
8965
9107
9033
8957
9100
9026
8949
9092
9018
8942
9085
9011
8934
9078
9003
8926
9070
8996
8918
I 2 4
i 3 4
i 3 4
5 6
\ I
27
28
29
8910
8829
8746
8902
8821
8738
8894
8813
8729
8886
8805
8721
8878
8796
8712
8870
8788
8704
8862
8780
8695
8854
8771
8686
8846
8763
8678
8838
8755
8669
i 3 4
i 3 4
i 3 4
I I
6 7
30
8660
8652
8643
8634
8625
8616
8607
8599
8590
8581
i 3 4
6 7
31
32
33
8572
8480
8387
8563
8471
8377
8462
8368
8545
8453
8358
8536
8443
8348
8526
8434
8339
8517
8425
8329
8508
8415
8320
8499
8406
8310
8490
8396
8300
2 3 5
235
2 3 5
6 8
6 8
6 8
34
35
36
8290
8192
8090
8281
8181
8080
8271
8171
8070
8261
8161
8059
8251
8151
8049
8241
8141
8039
8231
8131
8028
8221
8121
8018
8211
8m
8007
8202
8100
7997
2 3 5
2 3 5
235
7 8
7 8
7 9
37
38
39
7986
7880
7771
7976
7869
7760
7965
7859
7749
7955
7848
7738
7944
7837
7727
7934
7826
7716
7923
7815
7705
7912
7804
7694
7902
7793
7683
7891
7782
7672
245
2 4 I
246
7 9
7 9
7 9
40
7660
7649
7638
7627
7 6l 5
7604
7593
758i
757
7559
246
8 9
41
42
43
7547
743i
73H
7536
7420
7302
7524
7408
7290
7513
7396
7278
75 01
7385
7266
7490
7373
7254
7478
736i
7242
7466
7349
7230
7455
7337
7218
7443
7325
7206
246
2 4 6
246
8 10
8 10
8 10
44
7193
7181
7169
7157
7H5
7133
7120
7108
7096
7083
246
8 10
N.B. Numbers in differencecolumns to be subtracted, not added.
NATURAL COSINES
2 3 I
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
123
4 5
45
7071
759
7046
7034
7022
7009
6997
6984
6972
6959
246
8 10
46
47
48
^49"
50
51
6947
6820
6691
6934
6807
6678
6921
6794
6605
6909
6782
6652
6896
6769
6639
6884
6756
6626
6871
6743
6613
6858
6730
6600
6845
6717
6587
6833
6704
6574
246
246
2 4 7
8 ii
9 ii
9 ii
6561
6428
6293
6547
6414
6280
6534
6401
6266
6521
6388
6252
6508
6374
6239
6494
6361
6225
6481
6347
6211
6468
6334
6198
6455
6320
6184
6441
6307
6170
2 4 7
2 4 7
2 5 7
9 ii
9 ii
9 ii
52
53
54
6018
5878
6i43
6004
5864
6129
5990
5850
6115
5976
5835
6101
5962
5821
6088
5948
5807
6074
5934
5793
6060
5920
5779
6046
5906
5764
6032
5892
575
2 5 7
2 5 7
2 5 7
9 12
9 12
9 12
55
5736
572i
5707
5 6 93
5678
5664
565
5635
5621
5606
2 5 7
10 12
56
57
58
5592
5446
5299
5577
5432
5284
5563
5417
5270
5548
5402
5255
5534
5388
5240
5373
5225
555
5358
5210
5490
5344
5476
5329
5180
53H
5165
2 5 7
2 5 7
2 57
IO 12
IO 12
IO 12
59
60
61
5150
5000
4848
5135
4985
4833
5120
4970
4818
5105
4955
4802
5090
4939
4787
575
4924
477 2
5060
4909
4756
5045
4894
4879
4726
5oi5
4863
4710
3 5 8
3 5 8
3 5 8
10 13
10 13
10 13
62
63
64
4695
4540
4384
4679
4524
4368
4664
459
4352
4648
4493
4337
4633
4478
4321
4617
4462
435
4602
4446
4289
4586
443i
4274
457i
4415
4258
4555
4399
4242
3 5 8
3 5 8
3 5 8
10 13
10 13
II 13
65
4226
4210
4195
4179
4163
4147
4131
4"5
4099
4083
3 5 8
II 13
66
67
68
4067
3907
3746
4051
3891
3730
4035
3875
37H
4019
3859
3697
4003
3843
3681
3987
3827
3665
397i
3811
3649
3955
3795
3633
3939
3778
3616
3923
3762
3600
3 5 8
3 5 8
3 5 8
II 14
II 14
II 14
69
70
71
3584
3420
3256
35 6 7
3404
3239
3387
3223
3535
337i
3206
3355
3190
35 2
3338
3173
3486
3322
3156
3469
3305
3453
3289
3123
3437
3272
3107
3 5 8
3 5 8
368
II 14
II 14
II 14
72
73
74
3090
2924
2756
3074
2907
2740
3057
2890
2723
3040
2874
2706
3024
2857
2689
3007
2840
2672
2990
2823
2656
2974
2807
2639
2957
2790
2622
2940
2773
2605
368
368
368
II 14
II 14
II 14
2588
2571
2554
2538
2521
2504
2487
2470
2453
2436
3 6 8
II 14
76
77
78
80
81
2419
2250
2079
2402
2233
2062
2385
2215
2045
2368
2198
2028
235 1
2181
201 I
2334
2164
1994
2317
2147
1977
2300
2130
1959
2284
2113
1942
2267
2096
1925
368
369
369
II 14
II 14
II 14
1908
1736
1891
1719
1547
1874
1702
1530
i857
1685
1513
1840
1668
1495
1822
1650
H78
1805
1633
1461
1788
1616
1444
1771
1599
1426
1754
1582
1409
369
369
369
12 14
12 14
12 14
82
83
84
1392
1219
1045
1374
1201
1028
1357
1184
ion
1340
1167
0993
1323
"49
0976
1305
1132
0958
1288
0941
1271
1097
0924
1253
1080
0906
1236
1063
0889
369
369
369
12 14
12 14
12 14
85
0872
0854
0837
0819
0802
0785
0767
0750
0732
0715
369
12 I 5
CO t> 00 O
oo oo oo oo
0698
0349
0680
0506
0332
o663
0488
0314
0645
0471
0297
0628
454
0279
0610
0436
0262
593
0419
0244
0576
0401
0227
0558
0384
0209
0366
0192
369
3 6 9
369
12 15
12 15
12 I 5
0175
I57
140
OI22
0105
0087
0070
0052
0035
0017
369
12 I 5
N.B,  Numbers in differencecolumns to be subtracted, not added.
232
NATURAL TANGENTS
0'
6'
12'
18'
24'
30'
36'
42'
48'
54'
123
4 5
oooo
0017
0035
0052
0070
0087
0105
OI22
0140
OI 57
369
12 14
1
2
3
0175
0349
0524
0192
0367
0542
0209
0384
0559
0227
0402
0577
0244
0419
0594
0262
0437
0612
0279
0454
0629
0297
0472
0647
03H
0489
0664
0332
0507
0682
369
369
369
12 I 5
12 I 5
12 I 5
4
5
6
0699
0875
1051
0717
0892
1069
0734
0910
1086
0752
0928
1104
0769
0945
1122
0787
0963
"39
0805
0981
H57
0822
99 8
"75
0840
1016
1192
0857
io33
I2IO
369
369
369
12 I 5
12 I 5
12 I 5
7
8
9
1228
1405
1584
1246
1423
1602
1263
1441
1620
1281
H59
1638
1299
H77
1655
1317
H95
1673
1334
1512
1691
1352
1530
1709
1370
1548
1727
1388
1566
1745
369
369
369
12 I 5
12 I 5
12 I 5
10
1763
1781
1799
1817
1835
1853
1871
1890
1908
1926
369
12 I 5
11
12
13
1944
2126
2309
1962
2144
2327
1980
2162
2345
1998
2180
2364
2016
2199
2382
2035
2217
2401
2053
2235
2419
2071
2254
2438
2089
2272
2456
2IO7
2290
2475
369
369
369
12 15
12 I 5
12 I 5
14
15
16
2493
2679
2867
2512
2698
2886
2530
2717
2905
2549
2736
2924
2568
2754
2943
2586
2773
2962
2605
2792
2981
2623
28ll
3OOO
2642
2830
3019
2661
28 4 9
3038
369
369
369
12 16
13 16
13 16
17
18
19
3057
3249
'3443
3076
3269
3463
3096
3288
3482
3"5
3307
35 2
3134
3327
3522
3i53
3346
354i
3172
3365
35 61
3191
3385
3581
3211
3404
3600
3230
3424
3620
3 6 10
3 6 10
3 6 10
13 16
13 16
13 17
20
3640
3659
3679
3699
37'9
3739
3759
3779
3799
3819
3 7 I0
13 17
21
22
23
3839
4040
!'4245
3859
4061
4265
3879
4081
4286
3899
4101
4307
3919
4122
4327
3939
4142
4348
3959
4163
4369
3979
4183
4390
4000
4204
4411
4O2O
4224
4431
3 7 I0
3 7 10
3 7 I0
13 17
14 17
14 17
24
25
26
i'4452
4663
4877
4473
4684
4899
4494
4706
4921
45'5
4727
4942
4536
4748
4964
4557
477
4986
4578
479i
5008
4599
4813
5029
4621
4834
5Q5 1
4642
4856
573
4 7 10
4 7 ii
4 7 ii
14 18
14 18
15 18
27
28
29
'595
5317
'5543
5"7
5340
5566
5U9
5362
5589
5161
5384
5612
5184
5407
5635
5206
5430
5658
5228
5452
5681
5250
5475
574
5272
5498
5727
5295
5520
575
4 7 ii
4 8 ii
4 8 12
15 18
15 J 9
15 19
30
'5774
5797
5820
5844
5867
5890
59H
5938
596i
5985
4 8 12
16 20
31
32
33
6009
6249
6494
6032
6273
6519
6056
6297
6544
6080
6322
6569
6104
6346
6594
6128
6371
6619
6152
6395
6644
6176
6420
6669
6200
6445
6694
6224
6469
6720
4 8 12
4 8 12
4 8 13
1 6 20
16 20
17 21
34
35
36
6745
7002
7265
6771
7028
7292
6796
754
73 J 9
6822
7080
7346
6847
7107
7373
6873
7133
7400
6899
7159
7427
6924
7186
7454
6950
7212
7481
6976
7239
7508
4 9 13
4 9 13
5 9 H
17 21
18 22
18 23
37
38
39
7536
:gi
7563
7841
8127
7590
7869
8156
7618
7898
8185
7646
7926
8214
7673
7954
8243
7701
7983
8273
7729
8012
8302
7757
8040
8332
7785
8069
8361
5 9 H
5 10 14
5 I0 J 5
18 23
19 24
20 24
40
8391
8421
8451
8481
8511
8541
857i
8601
8632
8662
5 I0 *5
20 25
41
42
43
86 93
9004
9325
8724
9036
9358
8754
9067
939i
8785
9099
9424
8816
9131
9457
8847
9163
9490
8878
9195
9523
8910
9228
9556
8941
9260
9590
8972
9293
9623
5 IO *6
5 ii 16
6 ii 17
21 26
21 27
22 28
44
9657
9691
97 2 5
9759
9793
9827
9861
9896
9930
9965
6 ii 17
23 2 9
NATURAL TANGENTS
233
45
O'
6' 12'
18'
24' 30'
36'
42'
48'
54'
123
4 5
I'OOOO
0035 0070 0105
0141 0176
O2 1 2
0247
0283
0319
6 12 18
24 30
46
47
48
10355
10724
11106
0392
0761
"45
0428' 0464
0799 0837
1184 1224
0501
0875
1263
0538
0913
1303
0575
095 l
1343
0612
0990
1383
0649
1028
1423
0686
1067
1463
6 12 18
6 13 19
7 13 20
25 3i
25 32
26 33
49
50
51
~52~
53
54
11504
11918
12349
1544
1960
2393
1585
2OO 2
2437
1626
2045
2482
1667
2088
2527
1708
2131
2572
175
2174
2617
1792
2218
2662
1833
2261
2708
1875
2305
2 753
7 H 21
7 14 22
8 15 23
28 34
29 36
30 38
12799
13270
i'3764
2846
3319
38H
2892
3367
3865
2938
34i6
3916
2985
3465
3968
3032
35*4
4019
3079
35 6 4
4071
3127
3613
4124
3i75
3663
4176
3222
37 J 3
4229
8 16 23
8 16 25
9 17 26
3i 39
33 4i
34 43
55 !
14281
4335
4388
4442
4496
455
4605
4659
4715
477
9 18 27
36 45
56
57
58
14826
15399
16003
4882
5458
6066
4938
5517
6128
4994
5577
6191
5051
5637
6255
5108
5697
6319
5 l66
5757
6383
5224
5818
6447
5282
5880
6512
5340
594i
6577
10 19 29
10 20 30
II 21 32
38 48
40 5
43 53
59
60
61
~62~
63
64
16643
17321
18040
6709 6775
7391 7461
8115 8190
6842
7532
8265
6909 6977
7603! 7675
8341! 8418
745
7747
8495
7"3
7820
8572
7182
7893
8650
7251
7966
8728
ii 23 34
12 24 36
13 26 38
45 5 6
48 60
5 1 64
18807
19626
20503
8887 8967
9711 9797
0594 0686
9047
9883
0778
9128
9970
0872
9210
0057
0965
9292
0145
1060
9375
0233
"55
9458
0323
1251
9542
0413
1348
14 27 41
15 29 44
16 31 47
55 68
58 73
63 78
65
21445
1543 1642
1742
1842
1943
2 45
2148
2251
2355
'7 34 5 1
68 85
66
67
68
22460
23559
2475 i
2566 2673
3673 3789
4876 5002
2781
3906
5 I2 9
2889
4023
5257
2998
4142
5386
3109
4262
5517
3220
4383
5649
3332
454
5782
3445
4627
59i6
18 37 55
20 40 60
22 43 65
74 92
79 99
87 1 08
69 i
70 1
71
~72~
73
74
26051
2 '7475
29042
6187 6325
7625 7776
9208! 9375
6464
7929
9544
6605
8083
9714
6746
8239
9887
6889
8397
0061
734
8556
0237
7179
8716
0415
7326
8878
595
24 47 71
26 52 78
29 58 87
95 II8
104 130
"5 M4
30777
32709
34874
0961
2914
5105
1146
3122
5339
1334
3332
5576
1524
3544
5816
1716
3759
6059
1910
3977
6305
2106
4197
6554
2305
4420
6806
2506
4646
7062
32 64 96
36 72 108
41 82 122
129 161
144 180
162 203
75
3732I
7583
7848
8118
8391
8667
8947
9232
9520
9812
46 94 139
i 86 232
76
77
78
40108
433I5
47046
0408
3662
7453
0713
4015
7867
1022
4374
8288
1335
4737
8716
'653
5 I0 7
9152
3955
9758
6912
1976
5483
9594
4486
0405
7920
2303
5864
0045
2635
6252
0504
2972
6646
0970
53 107 i 60
62 124 186
73 146 219
214 267
248 310
292 365
79
80
81
5^446
5'67i3
63138
1929
7297
3859
2422
7894
4596
2924
8502
5350
3435
9124
6122
5026
Fo66
8548
5578
1742
9395
6140
2432
0264
87 175 262
35 437
Differencecolumns
cease to be useful, owing
to the rapidity with
which the value of the
tangent changes.
82
83
84
~85~
7II54
8*1443
9'5 144
2066
2636
9677
3002
3863
9845
3962
5126
1002
4947
6427
1020
5958
7769
1039
6996
9152
1058
8062
0579
1078
9158
2052
1099
0285
3572
IT20
"43
1166
U'9 1
I2I6
1243
1271
1300
1330
1362
I395
86
87
88
1430
1908
2864
1467
1974
3014
1506
2045
3182
I5 4 6
2 1 2O
3369
1589
2202
358o
1635
2290
3819
1683
2386
4092
1734
2490
4407
1789
2603
4774
1846
2727
5208
89
5729
6366
7162
8185
95'49
1146
1432
1910
2865
573o
234
NATURAL COTANGENTS
O'
&
12'
18'
24' 3O' ! 36'
42'
48' 54'
Differencecolumns
not useful here, owing
to the rapidity with
which the value of the
cotangent changes.
Inf.
5730
2865
1910
1432 1 14619549
8185
7162 6366
31823014
2045 1974
1506 1467
1
2
3
5729
2864
1908
5208
2727
1846
4774
2603
1789
4407
2490
I734
409238193580
2386 2290 2202
1683 1635 1589
3369
2 1 2O
1546
4
5
6
1430
"'43
95I44
I395
IT2O
3572
1362
1099
2052
1330
1078
579
I3OO 1271 1124;:
1058 1039 I0'20
9152! 7769 6427
I2I6
IO'O2
5126
1191 n66
9845 9677
3863 2636
7
8
9
81443
r"54
63138
0285
0264
2432
9158
9395
1742
8062
8548
1066
6996 5958
7920 6912
0405J 9758
4947
6122
9124
3962
5350
8502
3002 2066
4596 3859
7894 7297
10
56713
6140
5578
5026
4486
3955
3435
2924
2422 1929
123
4 5
11
12
13
5^446
47046
433I5
0970
6646
2972
0504
6252
2635
0045
5864
2303
9594
5483
1976
9152
5107
1653
8716
4737
1335
8288
4374
1022
78671 7453
4015; 3662
0713 0408
74 148 222
63 125 188
53 107 160
296 370
252 314
214 267
14
15
16
40108
37321
3^874
98l2
7062
4646
9520
6806
4420
9232
6 554
4197
8947
6305
3977
8667
6059
3759
8391
5816
3544
8118
5576
3332
7848
5339
3122
7583
5 I0 5
2914
46 93 139
41 82 122
36 72 108
i 86 232
163 204
144 i 80
17
18
19
32709
3*0777
29042
2506
595
8878
2305
0415
8716
2106
0237
8556
1910
0061
8397
1716
9887
8239
5 2 4
97H
8083
1334
9544
7929
1146; 0961
9375 ' 9208
7776 7625
32 64 96
29 58 87
26 52 78
129 161
"5 !44
104 130
20
2 '7475
7326
7179
734
6889 6746
6605
6464
6325) 6187
24 47 7 1
95 II8
21
22
23
26051
24751
23559
5916
4627
3445
5782
454
3332
5649
4383
3220
5517
4262
3109
5386
4142
2998
5257
4023
2889
5 I2 9
3906
2781
50021 4876
3789! 3673
2673; 2566
22 43 65
20 40 60
18 37 55
87 108
79 99
74 92
24
25
26
22460
21445
20503
2355
1348
0413
2251
1251
0323
2148
"55
0233
2045 j 1943
1060 0965
0145 0057
1842
0872
997
1742
0778
9883
16421 1543
0686' 0594
9797! 9711
17 34 5 1
16 31 47
15 29 44
68 85
63 78
58 73
27
28
29
9626
8807
8040
9542
8728
7966
945 s
8650
7893
9375
8572
7820
9292
8495
7747
9210
8418
7 6 75
9128
8341
7603
9047
8265
7532
8967
8190
7461
8887
8115
739i
14 27 41
i3 26 38
12 24 36
55 68
51 64
48 60
30
7321
7251
7182
7"3
7045
6977
6909
6842
6775
6709
" 23 34
45 S 6
31
32
33
6643
6003
'5399
6 577
594i
5340
6512
5880
5282
6447
5818
5224
6383
5757
5166
6319
5697
5108
6255
5637
5051
6191
5577
4994
6128
4938
6066
ml
II 21 3 2
10 20 30
10 19 29
43 53
40 5
38 48
34
35
36
4826
4281
3764
477
4229
3713
4715
4176
3663
4659
4124
3613
4605
4071
3564
4550
4019
35 '4
4496
3968
3465
4442
39i6
34i6
4388
3865
3367
4335
3814
3319
9 18 27
9 17 26
8 16 25
36 45
34 43
33 4i
37
38
39
3270
2799
2349
3222
2753
2305
3175
2708
2261
3127
2662
2218
3079
2617
2174
3032
2572
2131
2985
2527
2088
2938
2482
2045
2892
2437
2OO2
2846
2393
1960
8 16 23
8 15 23
7 14 22
3i 39
30 38
29 36
40
1918
1875
1833
1792
1750
1708
1667
1626
1585
1544
7 H 21
28 34
41
42
43
i54
1106
10724
1463
1067
0686
H23
1028
0649
1383
0990
0612
1343
095 i
0575
1303
0913
0538
1263
0875
0501
1224
0837
0464
1184
0799
0428
"45
0761
0392
7 13 20
6 13 19
6 12 .18
26 33
25 32
25 3i
44
i'0355
0319
0283
0247
O2I2
0176
0141
0105
0070
0035
6 12 18
24 30
N.B. Numbers in differencecolumns to be subtracted, not added.
NATURAL COTANGENTS
235
0'
6'
12
18'
24'
30'
36'
42'
48'
54'
123
4 5
45
ro
09965
0993009896
09861
09827
09793
09759
09725
09691
6 ii 17
23 29
46
47
48
9657
9325
9004
9623
9293
8972
9590
9260
8941
9556
9228
8910
9523
9195
8878
^57^
8273
7983
9490
9163
8847
9457
9UI
8816
9424
9099
8785
939i
9067
8754
9358
9036
8724
6 ii 17
5 ii 16
5 10 16
22 28
21 27
21 26
49
50
51
8693
8391
8098
8662
8361
8069
8632
8332
8040
8601
8302
8012
8541
8243
7954
8511
8214
7926
8481
8185
7898
8451
8156
7869
8421
8127
7841
5 i J 5
5 10 15
5 I0 M
20 25
20 24
19 24
52
53
54
78i3
7536
7265
7785
7508
7239
7757
748i
7212
7729
7454
7186
7701
7427
7 1 S9
7 6 73
7400
7!33
7646
7373
7107
7618
7346
7080
7590
73i9
754
75 6 3
7292
7028
5 9 H
5 9 H
4 9 13
18 23
18 23
18 22
55
7002
6976
6950
6924
6899
6873
6847
6822
6796
6771
4 9 13
17 21
56
57
58
6745
6494
6249
6720
6469
6224
6694
6445
6200
6669
6420
6176
6644
6395
6152
6619
637 1
6128
6594
6346
6104
6569
6322
6080
6544
6297
6056
6519
6273
6032
4 8 13
4 8 12
4 8 12
17 21
16 20
16 20
59
60
61
6009
'5774
'5543
5985
575
5520
596i
5727
5498
5938
574
5475
59H
5681
5452
5890
5658
5430
5867
5635
5407
5 8 44
5612
5384
5820
5589
S3 62
5797
5566
5340
4 8 12
4 8 12
4 8 ii
16 20
15 19
15 *9
62
63
64
5317
5095
4877
5295
573
4856
5272
505 1
4834
5250
5029
4813
5228
5008
479i
5206
4986
477
5184
4964
4748
5161
4942
4727
5'39
492i
4706
5"7
4899
4684
4 7 ii
4 7 ii
4 7 ii
15 18
15 18
14 18
65
4663
4642
4621
4599
4578
4557
4536
45'5
4494
4473
4 7 10
14 18
66
67
68
*445 2
4245
4040
443i
4224
4020
4411
4204
4000
4390
4183
3979
4369
4163
3959
4348
4142
3939
4327
4122
3919
4307
4101
3899
4286
4081
3879
4265
4061
3859
3 7 10
3 7 I0
3 7 I0
14 17
14 17
13 17
69
70
71
3839
3640
'3443
3819
3620
3424
3799
3600
3404
3779
358i
3385
3759
356i
3365
3739
354i
3346
3719
3522
3327
3699
35 2
3307
3679
3482
3288
3659
3463
3269
3 7 10
3 6 10
3 6 10
13 17
13 17
13 16
72
73
74
3249
3057
2867
323
3038
2849
3211
3019
2830
3 J 9i
3000
2811
3172
2981
2792
3 : 53
2962
2773
3134
2943
2754
3H5
2924
2736
3096
2905
2717
3076
2886
2698
3 6 10
369
369
13 16
13 16
13 16
75
2679
2661
2642
2623
2605
2586
2568
2549
2530
2512
369
12 16
76
77
78
2493
2309
2126
2475
2290
2107
2456
2272
2089
2438
2254
2071
2419
2235
2053
2401
2217
2035
2382
2199
2016
2364
2180
1998
2345
2162
1980
2327
2144
1962
369
369
369
12 I 5
12 15
12 I 5
79
80
81
1944
1763
1584
1926
1745
1566
1908
1727
1548
1890
1709
1530
1871
1691
1512
1853
1673
H95
1835
1655
H77
1817
1638
M59
1799
1620
1441
1781
1602
1423
3 6 9
369
369
12 15
12 I 5
12 I 5
82
83
84
1405
1228
1051
1388
1210
I0 33
137
1192
1016
1352
"75
0998
1334
H57
0981
1317
H39
0963
1299
1122
0945
1281
1104
0928
1263
1086
0910
1246
1069
0892
3 6 9
369
3 6 9
12 15
12 I 5
12 15
85
0875
0857
0840
0822
0805
0787
0769
0752
0734
0717
369
12 15
86
87
88
~89~
0699
0524
0349
0682
0507
0332
0664
0489
3H
0647
0472
0297
0629
0454
0279
0612
0437
0262
0594
0419
0244
0577
0402
0227
0559
0384
0209
0542
0367
0192
3 6 9
369
369
12 15
12 15
12 15
0175
OI 57
0140
0122
0105
0087
0070
0052
0035
0017
3 6 9
12 14
N.B. Numbers in differencecolumns to be subtracted, not added.
ANSWERS
66.
Last part, 35280.
5. V/ 2 + P f /fc 2 , dir. cosines l\b\h.
6. S ,0 = tanif. (90 o
69.
First, 1470; second, 22050 cm.
i or  i. (180.)
70.
49kg.
7. (o.)
72.
122.5 m 5 2 4'5 m  P er sec 
7. o.
73.
5.87 sec.
12. 5,8.66.
74.
4427+.
23. O) 1936; (^)"35405;
75.
36.3.
(0 9835
76.
44.1 m.
24. 45+ mi. per hr.
77.
90.4 m.
26. 40 mi. per hr.
81.
10.4 m. up; 9.2 m. down.
27. 96.56.
83.
0.5 sec., nearly.
29. Area 2. x io, \_x = instantaneous
85.
20.4 m.
length of side.
86.
2.04 sec.; 4.08 sec.
Volume 3 x^ io, [x = instantaneous
87.
416?.
length of side.
94.
485 cm. per sec.; 0.5 sec.
30. 1162 m. If t = oC.
96.
583
33. See Introduction I, "Dimensions."
98.
655'
34. 75 cm. per sec. 2
99.
913.8 cm. per sec.
35. 4015.7 km. per hr. 2
102.
264 ft. per min.
36. 1 20 cm. per sec.
103.
66f ft. per sec.
39. i, 3, 5, ... (2  i).
104.
V2:l.
40. 234 cm.
105.
8.54 mi. per hr.; 5725i' E. of S.
41. O) 1152 cm. ; () 270 cm.
106.
7.071 mi. per hr.
42. (a) 1264 cm. ; () 284 cm.
107.
36.56 km. per hr.
44. ^ km. per hr. 2
108*
5 1. 96m. per min.; 30 m. per min.; o.
45. 8th sec.
109.
26.4 ft. per min.
46. 48 km.
110.
17.39; 12.30; 4.658; 2.
47. 27.5 hrs.
111.
30.53+, 71 with "a," nearly.
48. V = o; a = 2.
112.
47.1; 3.219.
49. 3 sec.
113.
19.05 ; o ; 22.
50. () 8.66 sec. ; () 3.54 sec.
114.
8.659 sec.
56. 1600 dynes.
115.
7.14 sec.
60. 500 sec.
120.
326.53 m.; 653.1 m.
64. 196 io 4 dynes.
121.
(I) 2.49 sec.; (2) 498.4 ft.; (3)
65. 623 io 5 dynes.
215 ft.; (4) 2i5o', nearly.
237
238
PROBLEMS IN PHYSICS
[Exs. 124272
124. () .1000 radians per min.
1 L r 9 , on P0 r
i * IXT+XIXZ+XI JH [*2 + ri]
fK\ 2
125. Angular velocity alike; linear as
I : 2.
0) \h.
127. 4 TT radians per sec.
209. 2000 ergs.
129. 523.6 mi. per hr.(\vhen r =4000 mi.).
210. 2i6.io.
130. 33:8.
211. 98 io 6 ; 294. io 6 .
131. 25 m.; 39f m. ; o.
213. 2352 io 7 .
132. 4.1 grams.
214. i6m.
134. 2.5 ft. per sec.
215. loo m.
135. 131+ Ib.
216. 34640.
136. 10.35 k  w t ; 435 kg. wt.
2Yf 2OOOO
137 T Mm . a ATm
COS IO
* M+ m" M + m
218. 5io 3 .
139. Uniform motion; 7^ = gM.
219. 2I0 5 .
140. \g.
220. 32  io 5 gr. cm.
142. 130! io 5 dynes.
221. 96 io 5 gr. cm.
144. 5.625,4.375.
224. 72 io 3 kg. m.
145 a M & T mM S
226. 2 io 6 ergs.
M + m ' M + m
228. 4910"; 24.5 io".
146. M=.
230. W=mal.
m 3
231. as large. .. Numeric 4 times as
150. 53 io 4 dynes.
great.
151. 15; 3; 14.5; 13.9; 10.82; 7.93;
234. 4  IO IG ergs.
4.84.
235. 588  io 10 ergs; 6 io 4 kg. m.
152. 0.7265.
236. 588  io 6 ergs.
153. 12.2; 37.4.
237. MI = Y 1 ^ MZ', kinetic energy will
154. 2.
be acquired by the system.
155. 60.
239. 41552 io 6 ergs.
156. 120.
240. 6272  io 6 .
157. o.
241. 197392 io 4 ; 49348 io 4 .
162. 60.
242. [11267 io 5 total energy]; 6.47cm.
163. 4 kg. wt.
243. 591 io" ergs approx.
164. 7921.4 dynes; 15843 dynes.
244. 4000 ergs.
166. /'sec. 10 Ib. wt.
245. H25I0 7 .
167. 11.5; 27.7..
246. 27 io 3 ; 51 . io 3 .
168. 20; 20; 21.22.
248. 64 io" dynes.
169. 45 inch
249. 48  io 5 ergs.
Algebraic sum = 282.8 gr. wt.
250. 24 . io 4 .
175. 911+ cm.
251. O) 25 io 7 m.; (b} 625 io 6 m.;
176. 20000 Ib. wt.
(tr) o; (d} 625 io 6 m.
180. io cm.
252. 980 . m. ergs.
183. \ ap + f P ; l ap + f P.
260. 45 io 9 ergs.
186. 3600.
261. 5 io 8 ergs; 5.1 kg. m.
187. 50.9 [kg. cm.].
262. Vzgh.
197. O) L t
271. 0.199.
272. 49  io 8 ergs.
Exs. 281452]
ANSWERS
239
281.
10053 kg wt.
351.
iV
285.
IOOO: I.
352.
Ratio 1.000046.
286.
6 kg. wt.
353.
7 i  T
287.
1:24.
V~2
289.
1 60 kg.
354.
(a} .875; 1.43; (<*) 1.253
302.
98 io 6 .
355.
309.
48  1012.
356.
1.718 sec.
314.
 320; 1600.
358.
\Ml' 2 ', ^Ml.
318.
^ 2 v _
359.
(a) po/ 8 + \kl*\
dx 2 "
(fr} x o / 3 4 * kfi
332.
.02.
Jbfvl
333.
142+.
362.
onefourth mass X square of
334.
44.8 tons.
4
radius.
335.
336.
l6 4 2'.
28.62.
363.
^ onehalf mass X square of
2
337.
(I) .2.
radius.
339.
Equate resultant force to (J/+Z),
368.
392  io 3 .
and solve for a.
369.
245  io 5 , increased fourfold.
341.
g [sin 60 /A cos 60].
371.
4 io~ 4 .
342.
2656 io*.
372.
625 io~ 6 .
349.
.8 sec. approx. 1= 16;
376.
17 io".
1.14 sec. approx. / 32.
377.
.26 cm.
350.
802+.
384.
II354
LIQUIDS AND GASES
393.
395.
396.
398.
402.
About 3 A.
96.4 gr. wt.
123 approx.
93.5 meters.
5:3.
417. 40560 kg.
418. 97200 Ibs.
419. 12.
420. of its height.
136
423. 21.5:11.3:8.9:2.6.
46.5:88.5:112:383.
3.6:4.45:4.96:726.
424. V,:V S = . 5 3 5 .
428. 2 1426; 159.
19.3; 2.66:2.15.
429.
430.
431.
257;
10.5:
4.
40.
432. 32.
433. 137.6 gr. wt.
434. 2.
435. 1.6.
436. i : 2.
437. .6.
438. .2.
439. 735; 1470.
441. .5.
443. f.
445. 4.37.
446. Inversely as the densities.
447. ii.
448. 876.
449. 3.
450. .79.
451. 1.2.
452. .9.
240
PROBLEMS IN PHYSICS
[Exs. 453729
453. (i) 286 gr. wt.; 313.5 gr. wt.
459. f.
455. 2.9 [Note that 5 = ~.
461. 1 1.3 c.c.
462. 1 8% approx. in Hg.
456. 4.84; 5.09.
464. 975 cm.
HEAT
477. 113; 536;  4
535. As 3 : 55 nearly.
478. 100; 22.2; o; 344.
536. 27.6.
480. 40.
537. 12.7 and 42.3 liters.
481. 160.
540. 5.78 grams.
486. 12.618 m.
541. 5.6 grams.
488. The increase in length is equiva
542. 4.91 grams.
lent to 13.6 added terms.
543. .06.
492. 189 x io~ 7 .
544. .62.
493. 1129.
545. 3.29 cm.
504. 3 T Ho
613. 4.9 grams.
505. 40.197 c.c.
614. 81363 cm. per sec.
507. 13.11.
618. 30618.75 calories.
508. 13.35.
619. 21851.7 calories.
509. T %V
620. 4.189 x i o 7 ergs.
513. 176.25.
532. 26226.
624. W = Ap'a [i + log ^ J  ApJ.
533. As n: 21.
625. (a) 4386.3 ft.lbs. () 47.85 H.P.
534. 781052.
626. 42 H.P.
ELECTRICITY AND MAGNETISM
632. F= .01 dynes repulsive.
693. 4 cm.
633. F= .64 dynes attractive.
695. (a) 50000; () 5000; (<:) 500.
634. /'=4/; r' = zr.
697. 1600 ergs.
635. q = 25.6.
703. Loss .
636. r' = 2r.
707. JP = 6.5.
640. Surface density =
47T
711. Energy = i : 6.
712. V and Q reduced initial values.
643. 8000 dynes.
719. energy remains.
654. O) V = 4 V; V = V.
100000
657. 42.
7T
662. Q = loooo; V = looo; force =
721. 15.9 io 7 ergs.
100.
725. \ W. each jar.
663. Work = 1800 ergs.
729. Cap. = 7; change in large sphere
664. 80 ergs.
= 21.43; small sphere 8.57; en
670. V = 2; /=o.
ergy over wire 18.57 units; initial
688. /= 12.5.
energy = 185 ; final energy =
689. V = loooo.
64.3; final potential = 4.29.
Exs. 73199]
ANSWERS
241
731.
732.
733.
734.
735.
742.
746.
755.
758. '
759.
762.
767.
768.
770.
771.
772.
773.
774.
775.
776.
777.
778.
781.
782.
783.
784.
789.
790.
791.
792.
795.
798.
820.
824.
826.
829.
837.
840.
7=2 amperes; 600 coulombs.
8 amperes.
A" = 3*.
5 amperes.
4800 coulombs.
(a) 450; (J) 900.
(a) .0377 amperes; () .377, .754,
1.131, i. 808; (V) 3.77 volts.
8 ohms.
Ri = 25 ohms.
2.58 volts.
T 4 g volts.
Radius doubled.
i oooo ohms.
1.66 ohms.
(a) 1.19; (*) 14? (07795 W
140.
A 2 = 35.26 ohms.
3; if; 4^ ohms.
Length = \l.
2531 ohms.
27 ohms.
x= mi. ii ohms.
Take intersection of line  1
r\ r%
= i with x = y.
7i = i2; n 72 series; = 2;
r\ = 2 multiple.
E = 12 volts; A* = io ohms.
.8 ohms.
6; 3; 4; 13
3:1.
(0) 30 volts; () 59 ohms; (c)
.508+ amperes; (af) 16.3 volts;
0) 5.54 (^ to 7?).
(#) 8.332 ohms; () 12 amperes;
(V) 100 volts; 0) 111.96 volts.
.028 amperes in branch 10.
i%.
6 x io 5 joules.
1008 ohms; 10080 volts; 600 cou
lombs passing per min.
28.8 io 5 joules; 6.9. io 5 calories.
1= 10.04 amperes; 10040 volts.
7V = 64000 ergs; I*R = 16000
ergs.
45.11 ohms.
841.
.126 L.
846.
A 40 = 1021.2; A'so = 1042.4.
847.
256 C.
848.
2187.2 ohms.
850.
7.7 ohms at o.
853.
2.362 g. of copper.
855.
5 amperes.
864.
1.9017 amperes; .026 amperes;
14.36 io~ 6 amperes.
868.
(#) radius =157 cm.; () 5 = 470.
869.
Total current = .0838 amperes.
872.
7 ' = 2.81 io 3 .
873.
.1 amperes.
874.
7o' = 13.3 io~ 6 .
881.
7 = 138.
882.
7 = .0225.
883.
7 = 38.6 io 5 .
884.
/= ip.
885.
138 cm.
886.
490 amperes.
887.
k
Force = 4\/2
a
899.
Force  to bar equals .29 dynes.
900.
3912 dynes.
901.
M = 4.66 C.G.S. units.
904.
77=8.
907,
77= .208; V = .534.
911.
M 546.
913.
M ' 6000.
914.
(a) 102; () 72.114.
916.
140.
918.
1.2.
919.
1 20 ergs.
920.
V  8.33
944.
k= 30.65; = 15438;
fj. = 386.
947.
B = 3508.
949.
2524.
950.
I343.
954.
(#) 77 = 125.7 P er sc l' cm
956.
751.1 watts.
960.
M =. 214.765.
962.
.141.
983.
6 dynes.
987.
1.8 volts.
988.
1000 dynes; 100000 dynes.
989.
5 io 5 dynes _L to field.
990.
13.76 io 7 coulombs.
2 4 2
PROBLEMS IN PHYSICS
[Exs. 9941244
994. O) 8.5IO 4 volts; (b\
volts; (r) 297.7 sec 
1000. .32 volts.
1031. 15.77 H.P.
1032. Electrical eff. 92.6%.
1033. Electrical eff. 83.3 %.
1034. 62.5 amperes.
1035. 96.5%.
1036. 4.5 H.P.
1041. 12 io 5 dynes.
1046. Net eff. 91.2%.
1094.
1095.
1101.
1102.
1114.
1115.
1124.
1128.
1129.
1130.
1132.
1133.
1134.
1135.
1139.
1140.
1141.
1147.
1148.
1150.
1156.
or
.419; 6; 77.4.
8; 160; 1600; .314.
2 crests, 3 troughs.
20.
(I) .6283; (2) 1.25
y = a sin TT [8 1 f x~\ .
332m.
(i) 27.7m.; (2) 55.
x
V
34740 cm.
3444m.
X 20 = X 1.036.
500 waves.
23. 7 C.
135.2 cm.
Velocity and wave
creased.
328m.
128; 362.1; 181.
(i) Make string f c
length; (2) incr<
by the factor 1.26+.
42 io 5 dynes.
= 14. 0098] 6g .
Fz l_9 .0045 J
* n^i
F* _9 0045 J
1157. 2s=
/2
'1158. 5 =
) 8. 38 io 4
1047. 77%.
1051. 76.44 amperes.
1052. 100 amperes.
1054. Max. 7=49.93 amperes; mean
value = 31.3 amperes.
1059. 843; 904.3.
1061. Imp. = 454.
1062. = 8 7 3 4 '.
1063. 3300 watts.
1066. Z = .024 henrys.
1069. loo volts; 36 amperes.
SOUND AND LIGHT
1172. 240 cm.
4
1173. 80 cm.
T
1174. 26.6cm.
1175. 145; 435; 1305.
; (3) 2.
1177. 120 cm.; 170.
1179. At o length 50 cm.; at 25
length 52.2 cm.
U (3) 83
1180. 192; 320; 448; 576.
1181. 128; 192; 256; 320.
1182. 2.1 cm.; 6.2 cm. from wall.
1183. 18.7 cm.
1184. 5 beats.
1189. n 208; n = 1040.
1190. 2.8; 8.5; 70.8 cm.
1192. 8.3 cm.; 7.9 cm.; 31.9 cm.
1215. io cm.
length in
1216. 13.3 cm.; 14; 16.7 etc.
1217. 1.7 cm.; 2.0 cm.; 3.3 cm. etc.
1220. f /?; \R.
1221. JP=i 5 ft.
' its former
r>
1225. ; from natural size to zero.
ase tension
2
.
1227. 15.6 cm. per. sec. toward.
1235. 4i49'
l
1236. 1.3214; .7567.
>.
1237. 7 437'
1238. Angle of refraction = III2 ; .
1239. 322'; 403o'; 46^25 '.
1240. 7032'.
1243. 20 11'.
1244. Yes; critical angle increases with
= 1.225*
increase of wave length.
Exs. 12451322]
ANSWERS
543
1245. 225 io 8 ; 200 io 8 ; 185  io 8 .
1246. 165 sec.
1247. Angle of refraction in glass
= 2 5 4 0'.
1248. 1.07
1249. 1.33 rn.
1250. .582 cm.
1251. 40 ft.
1252. 40 ft.
1253. Above.
1258. v. = 1.668.
1259. 2338'; io22'.
1260. For yellow light taking index of
crown glass as 1.530,
1270.
1274.
1280.
1281.
1282.
1285.
1286.
1302.
1303.
1309.
1310.
1318.
1319.
1322.
Taking 1.530 as index, 1.24 in.
and 7.44 in.
12 m.; 7.5 m.; 4.8 m. etc.
30 cm.
100 cm.
.9 cm.
/*=i.5.
/= radius.
3i2io 7 cm.
76 . io~ 6 sec.
22 io~ 3 ; 38 io 3 .etc.
1.21 : i.
323'; 647'; io!2'.
2:3.
1059.
INDEX
ACCELERATION, 38, 39, 40, 41.
Approximations, 33.
Archimedes' principle, 94.
Atmospheric pressure, 89.
Averages, 31.
BAROMETER, 89.
Batteries, 137, 138, 139, 142, 143.
best arrangement of, 146, 147.
Boiling points, table of, 16.
Boyle's law, 98.
CALORIE, 108.
Calorimeters, 109, no, in, 112, 113.
Capacity, electrical, 124.
specific inductive, 18, 128.
thermal, 108, 109.
Cells, best arrangement of, 147.
grouping of, 142, 143.
Center of inertia, 58, 61.
of mass, 58, 61.
of gravity, 58, 61.
Coefficient, of expansion, 101.
cubical, 103, 104, 105.
of gases, 106, 107, 108.
Coefficients of expansion, 1 6.
Condensers, 128, 129.
Conductivities, thermal, 17.
Critical angle, 213.
Current alternating, 183, 184, 1 86.
Current electricity, 132.
DENSITIES, tables of, 14.
Diffraction grating, 222, 223.
Dimensions, 5, 187190.
Dimensional equations, 5.
Doppler's principle, 207.
Dynamo, 179183.
characteristic curve, 180, 181.
efficiency, 181, 182.
ENERGY, of charge, 127.
of discharge, 130, 131.
kinetic, 67.
of rotation, 67, 74.
transformation to potential, 7578.
Elastic limit, 85.
Elasticity, 85.
Electric force, 123.
Electrochemical equivalent, 153, 154.
Electromagnetic units, 18, 187190.
attraction, repulsion, 169172.
induction, 169182.
Electromotive force, 142.
of induction, 173178.
Expansion coefficients, 16.
FALL, of potential in a wire, 135.
of potential and electromotive force,
135
Farad, 18.
Faraday's disc, 177.
Fields offeree, electric, 124, 125.
magnetic, 161, 169, 173.
Force, 4043.
systems, 5357.
Friction, 79.
angle of, 79, 80.
coefficient of, 80.
GALVANOMETER, 155, 156.
Ballistic, 157.
Gases, 89 et seq.
Graphic methods, 27, 30.
245
246
INDEX
HEAT, 100.
specific, 108.
specific variation of, no.
of fusion, no 1 1 2.
of vaporization, 112, 113.
in electric circuit, 149151, 183.
Heats, of liquefaction, table of, 15.
of vaporization, table of, 1 6.
Hydrometers, 97.
Hydrostatic pressure, 9092.
press, 92, 93.
Hysteresis, 167.
INDICATOR diagram, 119.
Indices of Refraction, 20.
KILOGRAM, 8, 9.
Kirchhoff s law, 147149.
LENSES, 216.
images by, 217, 218.
curvature of, 217219.
Light, reflection of, 209.
velocity of, 213, 214.
refraction of, 212215.
interference of, 220223.
diffraction of, 222, 223.
Lines offeree, 122.
magnetic, action of, 168170.
Liquids, and gases, 89.
pressure, 8991.
MAGNETIC, field, due to currents, 159.
induction, 165.
Magnetism, 161.
Magnetization curve, 165, 166.
Magnetometer, 163, 164.
Mass and weight, 7.
Measurement, I.
Mechanical equivalent of heat, 14.
Melting points, table of, 15.
Mirrors, 209.
plane, 209, 210.
concave, 210, 211.
convex, 211, 212.
Moment of inertia, 8284.
Motor, 183.
Multiple resistance, 139141.
graphic methods, 141.
NEWTON'S rings, 221.
OHMS, various, 18.
Ohm's law, 132 et seq.
Overtones, 205, 206.
PENDULUM, gravity, 82.
magnetic, 167, 168.
Potential, diagrams, 135138.
gravitational, 7578.
Prism, 214, 215.
Pressure, of atmosphere, 89.
of gases, 98, 99, 106108.
of liquids, 89, 92.
center of, of liquids, 92.
Projectiles, 49.
Pulleys, systems of, 73, 74.
REFRACTION, index of, 212215.
indices of, 20.
law of, 212.
Resistance, multiple, series, 138, 149.
specific table of, 17.
temperature coefficients of, 17.
units of, 1 8.
SELFINDUCTION, 185.
Shunts, 143145.
Simple harmonic motion, 191193.
Sound, 198.
musical, 200.
velocity of, 19, 199.
Specific gravity, 9497.
gravity bottle, 96.
heats, 14, 15.
inductive capacity, table of, 18.
resistance, 152.
Spectra, 215, 222, 223.
Static electricity, 121.
Strain, 85, 86.
Stress, 85, 86.
TEMPERATURE, 100.
Thermometer, 100.
scales, 101.
INDEX
247
Thermometer weight, 105, 106.
Thin plates, 220, 221.
Torsion, 87, 88.
moment of, 88.
Transformer, 186.
Transmission of energy, electric, 151.
UNITS, i.
C. G. S. and practical, 190.
electrical, magnetic, 187.
fundamental and derived, 2.
of area, 12.
of force, 13.
of heat, 1 6.
of length, 12.
of mass, 13.
of power, 13.
of resistance, 18.
of stress, 13.
of volume, 12.
of work, 13.
practical, in C.G.S., 18.
transformation of, 190.
VAPOR, pressure, 114, 118, 119.
volume, 114, 1 1 8, 119.
Vectors, 21.
addition of, 21, 22.
examples on, 23, 25.
Velocity, of light, 19.
of sound, 19, 199.
of sound, temperature, effect on, 199.
Vibration, 191.
columns of air, 205207.
elliptic, 192.
strings, 201204.
WAVE length of sound, 200.
of light, 221, 223.
Wave lengths of light, table of, 19.
Waves, 194, 195.
phase, 197.
progressive, 196.
retardation of, 197.
sound, 198.
Wheel and a^cle, 74, 75.
Work, by torque, 65.
constant force, 62.
general expression for, 69, 70.
principle of, applied to machines, 71.
variable force, 63, 69.
YOUNG'S modulus, 86.
WITH NUMEROUS ILLUSTRATIONS.
THE ELEMENTS OF PHYSICS.
BY
EDWARD L. NICHOLS, B.S., Ph.D.,
Professor of Physics in Cornell University,
AND
WILLIAM S. FRANKLIN, M.S.,
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