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 class-room 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," something that no one ever reads. The
is

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
worked by other than calculus methods. Their
satisfactorily
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
number, and are put
in

in to indicate the general character of class-room 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 by the student. On the other
profitably extended
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 willuse 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 so-called 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, the domain of physical science,

in

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. Itfollows 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.

thus appears that, in a system made up of arbitrarily

It

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 the Austrian foot, 31.6 cm. the Saxon foot,
30.5 cm.; ;

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 All the complex

ratios multiples or sub-multiples of 10.

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 foot-pound-second (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 isfound 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
and
, L
v 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
passing to dimensional
. In

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.
becomes apparent at once that dimensional formulas show
It

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 ^ I
Lmin. 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

opposite pans in 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

WP = 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

Wg mg = x 980
i

= 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 "a body
as,

weighs 16 pounds" or "a weight of 12 kilograms." The

* The 1000 grams.
original standard is the kilogram
 INTRODUCTION 9

pound and kilogram being units of mass, such usage, taken

literally, is The expressions, however, are elliptical,
absurd.
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
" 16 "
brevity, 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 suppose that the piece of plati-
illustrating this last usage,
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
a
way 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 the mass of the body times the acceleration
by
which it would possess if allowed to fall freely. Taking g=$ 2. 2,
we write

* The accepted value of at Ithaca is 980, which corresponds to 32.15 in foot-

g
second units. 32.2 is commonly used, however. See Church's " Mechanics of
Engineering."

OP1
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
understanding thisquite cor- it is

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 foot-pound and the kilo-
gram-meter 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
~ x 64.4 = 24 Ib.
32-2
 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 kilogram-meters.

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 3
x io x 980 dynes,

and the work done is

W= Fl 12 x 2 x 980 x io6 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 the
logical objection to a system of units in
is

which the inch taken as the unit of length, the square rod as
is

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- = 543g r
I -

.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,
5
9.8 x io dynes.
i
kilogram's weight
- 9
2249 x io~ pound's weight.

UNITS OF WORK.

i
foot-pound = 7
1.35485 x io ergs
= 13825 gram-centimeters
= .138 kilogram-meters.
i
kilogram-meter =7.233 foot-pounds.
i
joule = io 7 ergs.
i watt-hour = 9
36 x io ergs.
i
horse-power-hour = 2
26856 x io joules.

UNITS OF POWER.

i
horse-power = 746 watts
= 7
746 x io ergs per second
= 33000 foot-pounds 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. = 7
4.2 x io ergs
= .4281 kilogram-meters.
Ib. through F. =
i
15
1.058 x io ergs
= 780.8 foot-pounds.

TABLE I

DENSITIES
SOLIDS

Aluminum 2.6 Asbestos . ,

2.4
Antimony 6.7 Chalk . . ,
2.3-3-2
Bismuth 9.8 Coal . . . .
1.4-1.8
Brass 8.4 Cork . . . .
I4--3
Copper 8.9 Glass, common 2.5-2.7
Gold 19.3 Glass, flint . .
3-3-5
Iron 7.8 Ice . . . .
.917
Lead 11.3 Iceland Spar .
2.75
Nickel 8.9 Ivory 1.9
Platinum 21.5 Marble . . .
2.7
Silver 10.5 Paraffine .87-. 9 i
Sodium 98 Quartz . . .
2.65
Oak
'

Tin .
7-3 . . . .
7-1
Zinc 7.1 Pine .
5

LIQUIDS, oC.
Alcohol . . . .806 Sea Water . . 1.026
Ether .... -736 Sulphuric Acid .,
!.8 5
Carbon Bisulphide 1.29 Nitric Acid . .
1.56
Glvcerine 1.27 Hydrochloric Acid 1.27
Mercury .
13-596 Oil of Turpentine .87

TABLE II

SPECIFIC HEATS OF SOLIDS

Aluminum .2122 Calcium ,1804

Bismuth 0298 Carbon, diamond ,1128
Brass 0940 Carbon, graphite ,1604
 INTRODUCTION

Carbon, charcoal .

Copper
i6 PROBLEMS IN PHYSICS

TABLE V
BOILING-POINTS 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 (gram-degree C.) = 4.2 x io 7
i
major calorie (kilogram-degree C.) = 4200 x io7
i
pound-degree Centigrade = 1905 x io 7
i
pound-degree Fahrenheit = 1058 x io7

TABLE VII
COEFFICIENTS OF LINEAR EXPANSION
Brass 180 }

Copper 170
Glass .
085
Gold 150
Iron 120
x io
Lead 280
Platinum 085
Silver 190
Tin 200
Zinc 290

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
C.G.S.
Conductivity.
Silver 100 1-3
Copper 74 0.99
Iron 12 0.16
Lead 8.5 o.n
Bismuth 1.8 O.O2
Ice 0.2 0.003
White Marble .... o.i 0.001
Glass 0.05 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 773-3 c.c.

* Everett.

TABLE X
RESISTANCE

Substance.
i8 PROBLEMS IN PHYSICS

TABLE XI
UNITS OF RESISTANCE
i true ohm unit of resistance, i B. A. unit = .9867 true ohms,
i
legal ohm = .9972 true ohms. i Siemen's unit = .9407 true ohms.

TABLE XII
SPECIFIC INDUCTIVE CAPACITIES

Air = i

Solids.
 INTRODUCTION

TABLE XIV
SOUND
VELOCITY OF SOUND IN METERS PER SECOND

Solids (20 C).

20 PROBLEMS IN PHYSICS

TABLE XVII
*
INDICES OF REFRACTION [Z? LINE]

Density. Index.

Glass (hard crown) .......... 2.486 .517

Glass (soft crown) .......... 2.55 145
Glass (light flint) .......... 3.206
.5

.574
Glass (dense flint) .......... 3-658 .622
Glass (extra dense flint) ........ 3-889 .65
Glass (double extra dense flint) ...... 4-429 .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 ........ 1-334

UNIAXIAL CRYSTALS
Ordinary Extraordinary
Index. Index.
......
Iceland Spar 1.6584 1.4864
Tourmaline ...... 1.6366 1.6193
Quartz ........ 1-5432 i-55 12

TABLE XVIII
NUMERICAL CONSTANTS
LOGARITHMS
= 2.7183 ........ Log 10 e = .434294
Log 10 -W = Log AT- .434294
e

Log e lV =Log lQ AT- 2.3025 85

i radian .... 57.2958
i ....... 01745 radians

Log10 Log10
^=3.14159 -497H9 ^ "
9-8696 ^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

1.4142 _.I5<>5I5 1.7321

I: V2 .7071 1.849485 I :
V3 .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 starting-point 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
then the resultant required.
is 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.

Theprocess 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 isresolved 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 = AB sin 0. 1

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
itslength 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
.
(.r-axis)

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
 24 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 given in magnitude and direction by n times
is

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 starting-point, 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
// know,
2,
//
3 , etc. (Fig. 6). We
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 co-ordinates, and
are usually chosen at right angles to each other. One is often
and the other the axis of y, and the lengths
called the axis of x,
measured along the .r-axis 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 co-ordinates of the

point which they determine.

" "
Self-registering instruments usually draw a curve by some
mechanical device. An example is the self-registering 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. curve such as the A

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 theobserved 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 time-temper-
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 quantity-temperature 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 29

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 co-ordinates 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 velocity-time curve ? the acceleration-time 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 and irregular. To find the temperature
less rapid

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 by taking a smaller number. The
this

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 one-half the sum of the first and last. Examples
of this will be found in problems on velocity, force, etc. Again,
when a curve drawn showing the relation between the two
is

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
 PROBLEMS IN PHYSICS

For, Area = fydx (Fig- 9)

= AB average height.

The student should be very careful in averaging quantities to

first find the actual values to be averaged. For example, the

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 text-books
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
3-45^

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."
^
<

n (H ~
Since [i ]
n
= i +n ( ) + ... 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 wrought-iron 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 [i +
3
.000 182] 3

= 203 [i + 3 .000182

+ Higher powers of small quantities.]

= 20 [1.000546]
3

= 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]
= F24 [i -.0008784]
= FiJ-9991216].

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.
Asanother 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 can be neglected z>. when the corner rectangle is

a/3 ;

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. Anexpress 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 and times as abscissas,

ordinates
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 ^r-axis ? 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 ?

Aparticle 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 calledone
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 the final speed of a body which, moving with a

is

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 = 2
\ at ,

space described in / i seconds = % a(t i)

2
;

whence space described in the /th second

= I at1 - \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)
/4th =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 m
has an acceleration of 40 cm. per second per
body
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 the character of the motion produced by a con-

is

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
5
a= 3.6 x io
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
= 4
392 x io dynes.
42 PROBLEMS IN PHYSICS

Show that the dyne is, roughly speaking, the weight of

62.
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 earthImagine 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 ?

Aside from any possible difference in value, would there

68.
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 to describe in that
it

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 43

In the fundamental relation

.we have F= 12 and a = 32.2;

whence M= units of mass.

32.2
The force required is

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 is sensibly constant in any one locality.
acceleration due to gravity
Problems bodies, therefore, come under the head of uniformly
in falling
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

VA - V B = gO,
that simply the velocity acquired by A before B was allowed to fall.
is,

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 jr-axis of + 0. The constant intercept MN represents their rela-

tive velocity.

Extend the foregoing problem to the case in wbich

79.
both and B have initial velocities, and discuss the conditions
A
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 VQ . 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
isthe 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, VQ 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 isprojected 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 starting-point. Find the distance traversed in time t.

Velocity at starting-point 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

= -03- b
7
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

# = ?^ii = 49 cm. per second per second

Grade - ^- = -~-
40 = i
= 5 per cent.
g 980 20

93. A body slides down the plane

OA. Showthat 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 426 cm. and its grade
is

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
N
without initial velocity, they will all reach B
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 per hour

at the rate of 8 mi. ;

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 ho-ur ;
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 a, b, c, and d of magnitudes 6, 8,

four velocities
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 projected upward at an angle of 30 to the

particle is

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. Aprojected with a velocity Fat an angle

body is 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, projected with horizontal and vertical velocities
if it is

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. Define angular velocity,

(a) (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- pressure will a man weighing 150 Ib. exert on

What
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 balloonwith a uniform acceleration of 4 m. per

rises

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.

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 must M
equal the upward velocity of /, and their accelera-
tions must be numerically equal. Let a be this com-
mon value.
M<
Consider the forces acting on M. These are: (i)
Fig. 16.
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 are readily T
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.)

Explain how the value of

143. 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 hanging over the edge of a table draws

of 20 g.
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 and m% are connected by

l
a string. m 1

hangs freely while m 2 rests on a plane inclined at an angle a to

the horizontal. the string passes over a small frictionless
If

pulley at the summit of the plane, find the resulting acceleration.

 VELOCITY, ACCELERATION, AND FORCE 53

Consider the forces acting on mr These are :

(i) itsweight m^g and (2)
the cord tension T. If be the f 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 athe square of their

particle. If

resultant is equal to three times their product, what is the angle

between the forces ?

At what angle must two

156. 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.

Let the angle between two forces of constant magni-

158.
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

two forces.
first

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 isapplied 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 40 g. rests on a plane inclined at 30. Find

of
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 is held from sliding down an inclined plane by

block
a cleat. two curves showing the variations of the pressure
Plot
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 driving a pile 6 in.
ft. 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 mid-point 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 cross-section, / Ib. weight per linear
foot. At a distance -| a from one end a load of Ib. weight P
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 is acted upon at its ends by two equal

straight bar
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 5-9+ 13-5 + 17-6 + 27-10 = sum .of mass-distance products,

9+5+6+10 = sum of masses.

.-. distance required 2

is -Vo - = 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 g (xis 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 xz
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 , i.e.

2
where
= k-x + p Q .

(c) Find x for an isosceles triangle.

6o PROBLEMS IN PHYSICS

198. Show directly from the definition of C.G. that its co-ordi-

*-pxdv
l

nates are given by three equations of the form x=

pdi>

Explain the distinction in meaning and use between the

199.
- ^mx
above expression for x and x =
f

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 co-ordinates of the

C.G. to the square, the circle, the rectangle, the triangle.

205. Two bodies, attracting each other with a force measured

m i* move toward each Where
by t
other. 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. io 10 ergs are expended in raising 100 kg.

98 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- io8 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. io8 ergs of work are required to move a body 8

4 io4
cm. What was the average force required ?
219. io 10 ergs of work have been expended in moving a
6
body against a resisting force of 3 io dynes. How far was it
5

moved ?

62
 WORK AND ENERGY 63

220. A stone of volume io3 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 io4 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 cross-section is 4 sq. m., the height is 6 m. Find the
work required to fill it.

225. Show work required to raise a system of bodies

that the
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 initialvalue 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 l
? M
 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_ OPr If

the rod OP^ is rigid, the work done in turning through an angle 0, since
P^PZ = 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 dynes, how far will it go before coming

5

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 half-way 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 EP 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 ?

The velocity of a bullet is decreased from 500 to 400 m.

260.

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 kilogram-meters) 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. stone were attached to a very flexible and exten-

If the

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. the sphere were hollow, would it acquire the same

If

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 applied to a body on a horizontal

constant force is

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 dynes. A man pushes the car with a force
4

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 itsmagnitude varies along this path and apply the
integral calculus to add up the elements of work.

-When W = (* Fdx, z

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.
4 4
fr ^-2-no
= 8
f .io ergs,
Jio 5-*aEr-|i-

which is the same as taking half the sum of the initial and final

force, and multiplying by entire displacement.

271. When F= 20
J
,
find work done in displacing the point
3C

of application from x = 20 to x 220.

rm Jr
W= \ 20

Could this result be obtained by taking \ (final force initial force) x dis-

placement ?

272. A coiled spring attached to a 50 kg. weight.

is 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. Wetherefore 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 2 m. long, the

is point of support 30 cm. from
the end. A force of io dynes 8
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.

In a hydrostatic press the distances moved by the

285.
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

(b) by considering the tensions of the cords. Neglect the weight

of the pulleys.

In a system of eight movable pulleys connected as in

289.
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 and w when W
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,

weight w
and
is twice
the I W
Fig. 29.

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 and W\ w
(a) Neglecting weight of pulleys, (b) When
lower block weighs grams. M
293. Find the relation when there are n

pulleys above and n below. When there is

one more above than below.

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.

M2

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 =M
l 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 when it M
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 + \ 800 [20 2/]

2 - 2
[Kinetic energy acquired.
Zip
= 200 g 400, [Potential energy lost.

Equate and solve for v.

 WORK AND ENERGY 75

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

What will be the velocity of
r
.after M
falling a distance Ji ?
(Neglecting the spokes.)
Let v velocity required.

Every particle of the rim is moving with a velocity v.

Lost

M+M 1

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
Fig. 35.

connected by flexible thread

to the axle. If M
discuss the energy changes in the
falls 2 m.,

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 Fig. 36.
"
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'

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 ?
If the force law is
m find the work done in carrying
301. ,

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;

i.e.
JUT = mm'
dW -
dr,
,

or W= r-mm' * = mm' f- - -
TC
If r = oo GO T
r = -- since =
ftlftl
,
I
o.
r\

302. How much potential energy will I

kg. have when it is

I m. above the sea-level, if we consider its potential energy as o

when at the sea-level ?

303. If g were constant and a surface were drawn everywhere

I m. from the
sea-level, 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 ?

'

In skating on smooth "level" ice, does one gain potential

306.
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 ?

8
309. If the potential of the first floor is 3-io ,
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 layer, the mass of one brick in a
one
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

hilland down, and returns to his starting-point. 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 isdrawn 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 sea-level, 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.the potential at points along a certain line

If 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

constant, what condition does between x^ and x^

-
is satisfy ?

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,
the axis of

What
mean
322.

?
,
When
curve be a straight line
will
x mean
will

its
i , i
?

slope
and

the
i
:
-
SEA LEVEL

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 sea-level. What can

you say of its potential (taking that of the sea-level 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.

" sanded."
330. Explain why rails are 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. ?

The coefficient of friction between a block and a plane

335-
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

co-ordinates, 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 81

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 8io 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.

fM ^2
V^ o ,
what do you mean by R ? Name two
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 = pQ + 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.
84 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 circleand 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 of inertia, we have to express ^mr2 as an
moment
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. cross-section sustains a weight

of 100 kg. What isthe stress?

How would the stress be changed if the weight were doubled

and the cross-section 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
* and
iron wire.

- for
f
area of cross-section
The
a PP lied - asKg.
ratio

42shows
is taken as

the result. What

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
11
require a force of 17- IO dynes per unit area of cross-section
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
- from A.
o
t -, ^, etc.,
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 were asM 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 one-half, how
would the angles mentioned in a be altered if the length and

the moment of the applied force were unchanged ?

Fig. 44.

381. If A
and B, Fig. 44, are the cross-sections 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) themean 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.
Find the pressure at a depth of 6 cm. in mercury sur-
394.
mounted by 4 cm. of water of unit density and this, again, by ;

12 cm. of oil of density .9, atmospheric pressure not considered.

Neglecting atmospheric pressure find the intensity of

395.
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 4 cm., and the distance
is 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. poured into one arm of a U tube containing

Alcohol is

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

lastproblem 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 confined by a dam of rectangular

in a pond is

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 on a rectangular sluice-gate 8 ft.

total pressure
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
p
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
= Px

- =fx
bh*

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 upward is filled with water. Show

right cone, vertex
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. such a cylinder be placed over the cone, and the
If

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
one-tenth that of the liquid, and finally ten
times as dense.
should vary.
Show how the resultant force
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. body A 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 HO2 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 specific gravity were .4 ?

if

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 H2O 100 g.

Weight of powdered glass 1

5 g.

Weight of bottle containing glass and filled up with H2O . 1 10 g.

450. A specific gravity bottle is counterbalanced ;

it is then
filledwith 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 howcompute the specific gravity

to 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 a liquid sp. gr. 3.
five parts of 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 2
H
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 test-tube with mercury in the bottom and of
uniform cross-section is used to determine the specific gravity
H
98 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 slowly pushed in.
is

(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 ?
Fig 48 '
467. Show that it follows from
'

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 mercury-in-glass 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. the coefficient of cubical expansion of the liquid in a

If

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 cross-section paper convenient
values, and construct a double thermometer scale i.e. one ;

which gives the temperatures in both systems.

100
 EXPANSION OF SOLIDS IOI

At what temperature will both Fahrenheit and Centi-

480.
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 ?

Define the coefficient of linear expansion and establish

482.
the formula
lt = / (i + A/),

where lt isthe 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
" wires of a glow lamp, or in any circumstances
leading in
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. have We

i +A/
whence the length at any other temperature t' is

//
It /
it
--
*

+
i
+ A/'

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.
Measurements are made at 25 upon a brass tube by a
491.
steel scale, correct at o.
meter 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
"ofthe 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

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 example first by the accurate method and
this
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 d
another. This principle is used in the
"gridiron" clock pendulum. Derive
the conditions for compensation.

495. A lever at A controls a distant

A and B P
railway signal at B. If are
Fig. 49.
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?
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
6 7
mercury at 182 x io~ and that of glass at 3 x 85 x io~ 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 /,

D
The mass of mercury filling the thermometer at / is

M m,
its volume at o is
M- m
106 PROBLEMS IN PHYSICS

and this volume expands at f to

M-m,
-y-(i+0.
But the volume of the expanded mercury is the same as that of the expanded
bulb, fromwhich 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 g-gVo-

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 273 ? What is this temperature

at
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
288
1.25

= 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 three-fold at constant temperature,

V" \ X fff 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, i
where v is the volume of i
g. at o and pQ is a pressure
of i
atmosphere.
A= J
3-596 x 76
in grams' weight per square centimeter
T = 273-
c.c.
.001293

Therefore, ^ r c
= '3-596 x 76
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.

at
529. For a certain mass
o and 760 mm.
of air
^= 58540. Find its volume

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 heat liberated by the hot body goes to
of the
warm the calorimeter, which assumed to be carried through the same tem-
is

perature range as the water. This amount of heat is therefore c c' (9 /), M
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 ?

Equal masses of iron and aluminum cool through the

533.
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 boiling-point?

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 boiling-point 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 ?

Into 12 kg. of water at 30 are dropped, at the same

538.
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
Weight of calorimeter + cold water .... 52.66 g.
302.71
n
Initial temperature
Temperature of hot water
Final temperature
......
........ 80
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 6a 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 25 g., and the temperature rises to 28.5.
is

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

Cm = a + b (t + S) + c (/2 + tf + t") .

546. One starts with 100 g. of 10, and to this one

water at

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

Temperature of water . . .... 156
54
g.

Temperature
Weight
after ice is

after addition of ice

melted
...... 32
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 melting-point, 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 melting-point.

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 melting-point ( 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 meant by regelation?

is 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 sealing-wax behave when heated ?
569. What amount of heat must be supplied to 10 kg. of
water at 100 to convert it same temperature ?
into steam at the

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
.

536H 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. Explainevaporation cools.

why 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.

Nowequating the heat evolved in condensing and cooling to the heat
absorbed by the cool water, the unknown temperature 6 is readily found.

In a determination of the heat of vaporization of water

574.
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. Somevalues 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 cm.
90
100
... ... 35-5
52.50 cm.
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 boiling-point 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 Ecuador, the mean barometer reading is
Quito,
52.5 cm. What is the boiling-point? 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? a curve showing Draw

the distribution of energy in the visible and non-visible spectra.

605. What class of bodies are good reflectors of radiant heat ?

good absorbers ?

Explain how the specific heat of a substance

606. may be
determined by the method of cooling.

Equal masses of water and alcohol cool successively

607.
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 itdetermined 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

unit of temperature.
units varies as
unit of length

612. To raise I
gr. of water i C. requires 4.2 x io 7 ergs.
Find the number of foot-pounds required to raise I Ib. of water
i F.

613.In a certain machine the power wasted in friction is 21

kilogram-meters per hour. How much water per hour could
be heated from o to 100 by this amount of power?

614. With what speed should fired against an

ice at o p be
impenetrable wall in order to be completely melted, assuming
that no heat is lost ?
n8 PROBLEMS IN PHYSICS

615. does the specific heat of a gas at constant pres-

Why
sure differ from the specific heat at constant volume ?

Describe an experiment to show that air is not cooled

616.

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 ?

By the conditions of the problem, external

work is done against the pressure of the atmos- 49
Fig. (a).

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
= x io 6 c.c.

Whence the work is

100
pv = 76 x 13.6 x x io 6 gram-centimeters.

618. Compute the heat supplied to cause this expansion.

This readily done by finding the mass of the air in the
is

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

confined by a weightless piston
is

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

isthe adiabatic line through any point of the pressure-volume

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 cut-off, i.e. after the entrance portthe cylinder is
of

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, cut-off 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 Ja( pdx.
l

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).
Cut-off at T3T 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 double-acting and makes 180 revolutions

per minute. Compute the horse-power.

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 horse-power.

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 horse-power 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 lightuncharged bodies are attracted when

a charged body is brought near them.
638. Explain fully how a gold-leaf 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 ?

In using an electrophorus we may divide the process

647.
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 grounding the second they had been
of

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 a charge opposite in sign, and twice as
? if

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. @- '

Fig. 51.
~B
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

distance-potential curve, taking the origin i cm. from center of
the sphere(r< i cm.). Draw the distance-force 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 ends would be 40

its

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
radii are suspended by silk
equal
threads, and each grounded. After
is 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
IOQ
density is 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 I, one V< 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 placed near B.

is

(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 half-way between them ? What is

the force at that point ?

671. Indicate the difference between the electrical condition

at a point half-way 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 force and equipotential surfaces of a uniform field.
of
What the meaning of the ratio of the difference in potential
is

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 placed near the first, and Q is again increased.
is 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.

V
Show how the Q 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. the potential at each point of the line be taken as y,
If

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
Q
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 ?
128 PROBLEMS IN PHYSICS

what the quantity of electricity ?

is 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.

B is brought in contact with A and then replaced.

700.
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 one-third 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 r2 specific inductive
capacity
,

of dielectric k.

Derive the expression for the energy required to charge

710.
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.

A
condenser of capacity 50 and charge 400 is connected
712.
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. observed that the energies of discharge of two

It is

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 ?

Which would exert the

(b)

greater pressure when full ?

(c) The stop-cock 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 I
sq. m., the distance apart is .1 cm.,
is

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 Vl and V^ and the capacity is C.
725. There are three Leyden A, B, and C, equal
jars, 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 isthe 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 named the ampere, volt, and ohm respectively.
resistance are
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
difference
/ = potential .
. .

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
cross-section 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. VA 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.
134 PROBLEMS IN PHYSICS

diagram showing the relation between the fall of potential and

length of the wire.

745. If in Example 744 VA V B = 100 volts, and VA = 200

volts, what will be the potential midway between A and B ?

at one-fourth 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 between current, potential difference, and resistance through-
relation
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. Potential-resistance curves are of considerable importance, and the
student isadvised 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

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 1
D
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 Suppose now
circuit.

that the external resistance were increased, /

must decrease, and all of the sloping lines
would become more nearly parallel with OX.
Fig. 55.
But the vertical lines CCV and l
DD
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 CCl 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 co-ordinates,
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 w H
connected to a tank 7", from which leads a straight pipe A A 2 -~S; A^Hy
}

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 stop-cock whereby the flow in
;

the main can be checked. When S is open and the pump working, so that

H
 v
v^ ^-*
OF THB
f
f OOF
'NTVKP'
rjNrVEHi

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 potential-resistance curve.
SUGGESTION. Each electromotive force causes a rise of V independent of
the other.

em/ = 3 em/ = 6

,
JA B 1C
E
r-N 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 potential-resistance 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 cross-section),
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
1

c ID E

Fig. 59.

761. Draw the potential-resistance 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 A, 1
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 cross-section, 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 Ohm's law

applied to find how multiple resist-
is

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 cross-section has

a resistance of 40 ohms. What would be the resistance of an
iron wire ten times as long and one-fifth 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
-^--
A XB
= --\
12
=
12'
, or A RB = =3 ohms.
4 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

Find the total B

775.
resistance of the circuit,

Fig. 62. In this sys-

tem we may compute
the resistance from A
Fig. 62.
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-
Fig. 63.
sistance first.)

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 arejoined 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

Prove that the resistance of two wires in multiple

779. is

always less than that of either.

780. Prove the following construction for computing multiple
resistance.

Lay off on OY a length r^.

Lay off any line ||

to OY a length rv

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 rs .

By using cross-section paper the results may be quickly obtained.

Fig. 67.

781. Prove the following construction for resistances in

multiple. Take ;r=r1 / ,

= r2 ; join their extremities. Then
the resistance of r^ and r2 in multiple is given by the co-ordinate
(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 cross-section paper.
142 PROBLEMS IN PHYSICS

In the following problems it should be remembered that in

dealing with
and in series we must be careful to consider both the electro-
cells in multiple

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 twenty-four electromotive force 2 volts, r 4

cells,

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.
Fig. 69.
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

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

Twenty 5ovolt lamps, each requiring 1.2 amperes, are

792.
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 and C. B

(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 short-circuiting

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 IR xl
Ix .

But /*= 9 IR .
Fig. 73.
.-. 100 IR = x^IR .

x = -^ = 1
l
-
l
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, 7ri = \ **''*"''

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 measured by placing an instrument (voltmeter) in shunt
to be
with the resistance between A and B. What change in this
difference of potential is caused by the insertion of the instru-
ment ?

In the circuit of Fig. 70, what is the smallest resist-

800.
ance a voltmeter could have that when placed in shunt with
AB the difference of potential between A and B may be changed

only one-half, 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 one-half 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

potential-resistance diagram for each path from A to B.

Fig. 74.

804. If potential at Ais 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 VP = V in case of no electromotive force between A
fi
,

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 twenty-four cells, each r = 6,

E=3, R=i6, for a maximum current? ^=36? ^=9?

R= 25 ?

n-6
= l6
-m~ >

mn = 24.
Multiply these equations,
2
-6 = 16- 24,
n2 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 as A

+ ,we must consider those from as A
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 r +, we
l l

must, when returning along r 2 take 72 r2 as ,

.

By (i) 7=7 1
+ 72 +7 3
.

(2) 7^-7^2 = 0,

I\ r \
- 73 r3 = O,

72 r2 - 73 r3 = o,

7r + 7^ = E. [Where E= electromotive force of cell.]

From the first and second of (2) we may express 72 and 73 in terms of Iv
r v r2 and r.y
,

Substituting in (i),

.. . for / and /

If R= equivalent resistance of r v r2 and r3 , ,

7^? + Ir = ,

7^ + fr = E,
7^ + Sr = ,

73 r3 + 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

ri rs
/
( _ =/4 1+1* +^
^ - f r* -
r* r* r
<

-
V I--P-J
,

ri + r?) r r + rja2l '

V 2 s r,
where / = total current.
 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,

7L = 72 + /3 ,

73 40 /2 20 = 10,

7L 10 + /2 20 = 2.

Eliminating / and 73 we
x ,
have /2 = 7V amperes. What does the nega-
tive sign mean ? Solve when the arrow from A to B is reversed.

816. Find 72 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 Ev E^ E& internal

resistances rlt r^ r3 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 = 6,

z
r = 3,
r2 = 6, r8 = 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 B = 100 ohms. R 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. io4 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. work is done by the current in addition to overcoming

If

resistance, would IE and I R have the same value ?

2
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 one-half the total heat is developed in the generator.
 TRANSMISSION OF ENERGY 151

Three copper wires of equal length, diameters .1 mm.,

833.
.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 largeconductors 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 desirable to transform a small current at high

is it

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 than at the lower voltage ?
loss in the line

How might the cross-section 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.
7
The statement that the specific resistance of copper is ly-io- means that
152 PROBLEMS IN PHYSICS

i cm.
length of a piece of copper i sq. cm. cross-section 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. cross-section
at o we have
17-iQ- 7 io 8
R= io~ 2
5 = .17 ohm.

840. The specific resistance of silver is i5-io~

7
. Find the
resistance of a silver wire I ft. long and -j-oVo
i n - i n diameter.

841. A
copper wire of known resistance is to be replaced by
a platinum wire of half the cross-section. What length must be
chosen to have the same resistance ?

842. Find the resistances of the following circular wires at o.

Material.
 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

53-io~
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
the range of temperature through which it may
is

be used as a standard of resistance if the error must not exceed

one-fourth 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.

The electrochemical equivalent of

852. H is
8
1038- io~ (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.
 GALVANOMETERS 1
55

center of the circle? What would be the field strength at

the center of the circle when,

I=i ampere, one complete turn

(a) ?

(b) I=i ampere, n complete turns?

I=\ ampere, n turns, radius = r?

(c)

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 one-half ?

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 have on the galvanometer constant
it :

(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 4-io~

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 = .2. What current would H
give the same deflection where is I ? H .

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. Atangent 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,
7TG
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 f-f
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 is nine times H
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 QQ .

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 may be stated as A
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 .

The " broadside "

(4) force is proportional to the 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?
**
_K^ ds sin
r92

[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 field10 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 /.

ds '
* in 6
889. Apply the formula FA =k 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. 2"

a
8 T2
Expressing d in centimeters F = 6 dynes.
16

892. Two like magnet and 27 units

poles, of strengths 10

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 north-seeking 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 ofH 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 cross-section 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 dyne-centimeters.
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 ofan 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 cross-section 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 magnetic potential measured ? Find

is

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.
Find the potential at a point 6 cm. distant from the
920.
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

-it
+m[^^F -m +ra

m
i

H i

Fig. 79.
I

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~\
d
2ml = 2 2
|X 4-/
H ]

929. When / may be neglected, show that

2 ml
=d tan
H
930. How
do the results of Examples 926
+m S wrn
and 929 compare. Explain why such a dif-
ference should be expected. Fig. 80.

931. the large magnet were reversed, what

If change of

position would the small one experience ?

 MAGNETIZATION 165

If the magnets were exactly alike, and each were sus-

932.
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 (a) the variation of potential and (b) the
showing
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 curveis called a magnetization curve, and is of

great practical value.
It is usually plotted with the induction B
and the field strength as co- H
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 co-ordinates.

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 cross-section 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 cross-section 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 along H
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

T= / K*
for the magnetic pendulum, 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 struck several blows, what will be the prob-

is

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^
lines of force are similar to tense,
(c) Magnetic ^^^^-^-^-^
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

r
These, together with the fact that when a cur- -^==
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 and the positive direction of the lines of force due
direction of current
to it same way as are the direction of transla-
are related to each other in the
tion of a right-handed 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 clock-face, lines of force around
the current would be such that a 4- pole would go around it like the hands
11
of the clock or "clockwise. 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 offorce is coming
up through the paper by a , going down by a x .
Thus, if current flows in
the line MN the plane of the page, the magnetic
in 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
XXX XXXXX
xxxxxxxxx
(3)

X X X X X X X

(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 conductor and A

magnet tend to approach at to separate. (See Fig. 81.)
;
B
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 Abe 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 crowded together and stretched, d is stretched so far that lateral

c as

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 magnet passes through the bottom

pole of a
of a cup C. Mercury covers the bottom, and a wire suspended

\
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 text-books 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
It is observed that it is harder to maintain its
electromagnet.
motion when current passing through the coils of the magnet.
is

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 right-handed
screw. An iron core is placed in it and you are required to
make a given end a north-seeking 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.

s N
88. (c) What is the power I I

of energy in the case of an

x~7<r?OC7C7\7C">v
induced current produced ( [ )[ J[ J f ]( ]( } /
COIL WITH
CURRENT
by motion in a magnetic
field ?
(Winter, '96.)
+
983. Find the force
ing on a pole of 60 units'
strength at a distance of
act-

Fig - 88 -
o
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 one-half,
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 io9 magnetic lines of force in 2 sec. What is the average
electromotive force induced ?

The E.M.F. induced proportional to the rate of cutting.

is 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 one-tenth 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 ?

Would it make any difference

991. 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. a closed loop of wire be moved without change of

If

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 so large that their
rails,

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
be 40 cm., find
rails :

(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 76 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. a loop of wire rotating in a magnetic field form part

If

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.

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. the wire were fixed and the magnet were placed on
If

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 density2000 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 series-wound 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 arc-lighting 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. required to
It is find the number of ampere turns

needed on the limbs to set up in the air gap a magnetic

field

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
rm rrn

10

where S is the number of turns of wire on the

and i the current in them. Considering
field,
the magnetic circuit asmade up of three sepa-
rate parts, in each ofwhich the value of is H
assumed to be constant, we have

J
10 .

the subscripts a, g, and/ referring to the arma-

ture, air gap, and respectively.
field,

Taking the computations in the order indi-

cated, we have
rl a =
/? a
= 6000 ,
Fig. 93.
Pa I I 20'

6000
and = 134-
I 1 20 -25
For air, /x = i,

hence H l
g g
= 6000 x 0.72 4320.
Now in every dynamo a certain amount of stray field, or waste
there is

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 is called the coefficient of mag-

useful ma netic flux
netic leakage. The induction to be provided for in the field is, therefore,

kBa = 1.5 x 6000 = 9000,

and we have HJf - 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 cross-section 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 ison 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 Jf-axis.
Potential Difference. Current.

2.6 O
10.3 4
31.4 io
43-5 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 co-ordinates 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 :

~ total electrical energy developed ^

Gross Jv-
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 horse-power 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 is70 per cent ;

the gross efficiency is 84 per cent. What is the electrical

efficiency of the machine ?

1034. requires 8 kilowatts when driven at

A. certain dynamo
full capacity. net The
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 Allowing no volts.4 per for a loss of

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
horse-power of the engine installed ?

1037. What determines the practical limit of long-distance

transmission of power ?

1038. When current supplied to a direct-current dynamo

is 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 for-ce.
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 one-half 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 -\-i-r, whence useful power = <w = Eii' r.
2
i being the only variable in the

right-hand number, we have merely to find the value of i to give maximum iv.

1044. Show
follows from the foregoing that the effi-
that it

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 series-wound 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 horse-power. Compute the net efficiency of the motor.
1048. If two armatures were mounted on the same shaft, would
itbe 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) small A
8-pole 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\Jo sin ado.

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.

The maximum value of an

1051. 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 cross-section
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 mean square of a current whose maximum value is /.
the
" "
The square root of the mean square of an alternating current is called
its virtual value, and is of great importance.

The virtual value of an alternating current is

1054. 35.3

amperes. What is its maximum value ? its mean value ?

 SELF-INDUCTION 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 self-induction ? Give two definitions

of the coefficient of self-induction. 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. a current of 2.1 amperes flowing in a coil of 100

If

turns set up through that coil a magnetic flux of .084 x io 8

lines, what is the coefficient of self-induction 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 lines. The change in the current is 2.1

8

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. the frequency be 120 periods per

If

second and L .06 henry, what is the electromotive force of

self-induction ?

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. the primary power is 4000
If at full load,

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,

= ML T~ 2 = O
2
unit force
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. electro-magnetic system.
The primary definition is that of unit current, based on the
action between an electric current and a magnet-pole in its

vicinity. It is known, as the result of experiment, that a

magnet-pole 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 magnet-pole, and the length of the wire

directly, and as the square of the radius of the loop inversely.

That is,

r= A --
7 2 Trrm
5 >

f=K>-*-

If 7 be such that when m is a unit, magnet-pole 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 sub-multiples 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 foot-pound 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 foot-pound-second 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 foot-pound-second units*?

 VIBRATIONS
1080. What
meant by a vibratory motion? Does the bob
is

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.

In what ways do the motions of different particles along

1081.
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 ancl 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
92 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 fly-wheel ?

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 have ?

Where will he walk the fastest ? The

goes clear around
first

in 20 min. What is his angular velocity ? What is the

periodic time of the second man ? Fig. 95.
B

Fig. 96.

1089. If P P = P P&
1 2 2
does MM = MM
1 2 2 3
? The time re-

quired for the first to move from P 1

to P z
is the same as from
P2 to Ps and equals that for the second to go from
,
1
to z M M
or M2 to 3
M
How has
. the motion of the second man changed
in going from l
to z
?M M
Fig. 96.
If P moves uniformly in a circle of radius #, and is the M
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 ;

i.e.
^ 2
T= TT
or ft) = 2 TT
.

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 ?

109-1. Draw a curve with time as x and distance from O as y.

Draw the corresponding time-velocity curve. Draw the corre-
sponding time-acceleration 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
dropped vertically into a pond of still water.
is

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. length and velocity of propagation is known
The wave
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= to

= -,
v
where T is the common pen-
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-

11 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 A,
(vt + x}.

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= v ,
y takes all values between

+ 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 the temperature were uniform ?
if 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. the velocity of sound in air is different in different

If

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 one-hundredth 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

one-twentieth 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 when piston makes just oneP
vibration. Draw a curve, using time in one four-hundredth 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 when crank pin moves
travel

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 V, = that if 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 muchthe wave length of the air wave sent out
is

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. the temperature of the air were increased, what quan-

If

tities referred to in Example 1139 would be altered?

1141. A tuning-fork 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 cross-section x density ;

F
{-Aarea of cross-section density
~f

[T = force per unit area of cross-section.

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.
A string making 400 vibrations per second has its
1149.
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, m. of which weighs I

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 ? the wire makes each centimeter of the string four
If

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 finger-board 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 _ J9 98 I0
I ' '

"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 two-thirds 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 four-thirds 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 found that triangular portion retains its

it is

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

W^r&i (Fig. 102)

where 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) part of the string between x

Any = o and x = /, in other

words, any point of the string free to move, will have what kind
of motion ?

(5) If / == what is the frequency ?

y,

(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

sin px sin pat
C cos/;trcos/tf/
IBD 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 vibrating in its fundamental mode; a

is

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

169 is closed and the 1

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 making 332 vibrations per second is fixed in

fork
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 mouth, frequency of the fork 64. What must be the
its

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 tuning-fork 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. the scale in Konig's apparatus for the determination

If

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. Atuning-fork 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 thesame 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 placed in front of a plane mirror.

object is 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 placed at any point in
is 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
 CURVED MIRRORS 211

Lengths to the right from A are taken + .

AP' = p = image distance = P' Q approximately.

1

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. the object is an arrow 5 cm. high, find the size of the
If

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. placed 30 cm. from a

is

candle flame. Where will the image appear to be ? Construct

it. Find its size if the flame is I in. high.

1225. objectAn 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 any wave surface be con-

of
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
r'ff
= -3- ?
2
*

1236.the angle of incidence is observed to be 20 and of

If

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. the angle of incidence

If is 45 ;
60 ; 75 ;
find the

direction in the water.

1240. the angle of incidence

If 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 ?

10
1245. The velocity of light in air is
approximately 3.jo cm.
per second. What is its velocity in water, //,
=^ ? What in

glass,
= f in CS 2
(J,
? , 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. io6 km.

1247. A plate of glass is immersed in water with its surface

horizontal. Light incident at an angle of 60 on the surface
is

of the water. Find its direction in the glass, a /*w a

p f
= ,
ff
= .
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. Anobserver 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 crown-glass 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 apparent source to an eye
l
is second medium,

PQ p = PA, Z.PQC =2,

( DENSE

3-*^
n
^>- TP; ,
 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 Rv R 2 (A) becomes
- i /, i .

-R.- -_ TI --
7T
/A i
1~ OUt.
f Since / 1
is the virtual
2 p p l source.

. / _ !\r_L _i__L _ JL _ 1. I
'
["Multiply
second by /A and
)
\-Rl 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.

converging lens A
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

Show by construction the position and size

1275. 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 plano-convex lens.
Find the formula for a plano-concave lens.
Find the formula for a concavo-convex lens.

1280. Find the focal length of a biconvex lens of crown

glass, fj,
=f *\,
= r2 = 30 cm.
1281. A lens of focal length 25 in air, >, = f.
What will

be the focal length in water, >w = |.

1282. A plano-convex 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, plano-convex, 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 co-ordinates, 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 //, /2 / 2 ', etc., are joined
lt ,

by straight lines, show that all of these lines intersect in a

point, the co-ordinates of which are x=y = F. (A practical

fact.)

1292. If a series of observed values of / and p are taken as

1

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.

Fig. 105.

Consider parallel rays incident at A

and C such that the ray refracted at A,
B, and refracted at C proceeds along the same path CE as the ray
reflected at
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. [Retardation due to glass path.
.. 8 = 2 fie cos r. [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 76-10" three wave lengths ?

1304. Explain the changing colored bands seen when white

light is reflected from a soap-bubble 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 wedge-shaped 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.

Explain the production of color in the soap-bubble.

1307.
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. red light X = 76- io~ 6 is used and

If 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 = 6
76- io~ 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 divided and passes by

is

two paths length to a screen.

of slightly different 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

106
are placed one above Fig.

222
 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
.

For a certain wave length and grating, # 3 = 6

1319. for a ;

differentwave 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.
7
1322.Light of wave length 589- io~ 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 52>
53 , Explain how these images are formed.
etc.

If d^ =
5 cm. and /
= 80 cm., find the number of lines per
centimeter in the grating.
t-s"

-S'

S2
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 Four-Figure Mathematical Tables has been
followed.]
226 LOGARITHMS
LOGARITHMS 227
228 NATURAL SINES
NATURAL SINES 229
230 NATURAL COSINES
NATURAL COSINES 23 I
232 NATURAL TANGENTS
 NATURAL TANGENTS 233

45|
234 NATURAL COTANGENTS
NATURAL COTANGENTS 235
ANSWERS
238 PROBLEMS IN PHYSICS [Exs. 124-272

124. () .1000 radians per min.

Exs. 281-452] ANSWERS 239

281.
240 PROBLEMS IN PHYSICS [Exs. 453-729

453. (i) 286 gr. wt.; 313.5 gr. wt.

Exs. 731-99] ANSWERS 241

731. 7=2 amperes; 600 coulombs. 841.

732. 8 amperes.
733. A" = 3*.
734. amperes.
5
735. 4800 coulombs.
742. (a) 450; (J) 900.
746. (a) .0377 amperes; () .377, .754,
1.131, i. 808; (V) 3.77 volts.
755. 8 ohms.
'

758. Ri =
25 ohms.
759. 2.58 volts.
4
762. Tg volts.

767. Radius doubled.

768. i oooo ohms.
770. 1.66 ohms.
771. (a) 1.19; (*) 14? (07-795 W
140.
772. A = 2 35.26 ohms.
773. 3; if; 4^ ohms.
774. Length = \l.
775. 2531 ohms.
776.
777. 27 ohms.
778. x= mi. ii ohms.

781. Take intersection of line -- 1-

r\ r%
= i with x = y.
782. 7i = i2; n 72 series; = 2;
r\ = 2 multiple.
783. E= 12 volts; A* = io| ohms.
784. .8 ohms.
789. 6; 3; 4; 13-
790. 3:1.
791. (0) 30 volts; () 59 ohms; (c)
.508+ amperes; (af) 16.3 volts;

0) 5.54 (^ to 7?).
792. (#) 8.332 ohms; () 12 amperes;
(V) 100 volts; 0) 111.96 volts.
795. .028 amperes in branch 10.
798. i%.
820. 6 x io 5 joules.
824. 1008 ohms; 10080 volts; 600 cou-
lombs passing per min.
826. 28.8- io5 joules; 6.9. io 5 calories.
829. 1= 10.04 amperes; 10040 volts.
837. 7V = 64000 ergs; I*R = 16000
ergs.
840. 45.11 ohms.
2 42 PROBLEMS IN PHYSICS [Exs. 994-1244

4 4
994. O) 8.5-IO- volts; (b\ ) 8. 38- io-

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 io5 dynes.
1046. Net eff.
91.2%.

1094. .419; 6; 77.4.

1095. 8; 160; 1600; .314.
1101. 2 crests, 3 troughs.
1102. 20.
1114. (I) .6283; (2) 1.25
1115. y= a sin TT [8 1 -f x~\ .

1124. 332m.
1128. (i) 27.7m.; (2) 55.
x
1129.
V
1130. 34740 cm.
1132. 3444m.
1133. X 20 =X 1.036.
1134. 500 waves.
1135. 23. 7 C.
1139. 135.2 cm.
1140. Velocity and wave
creased.
1141. 328m.
1147. 128; 362.1; 181.
1148. (i) Make string f c

length; (2) incr<

by the factor 1.26+.
1150. 42 io5 dynes.
1-4.
1156. = 0098] 6g .

Fz l_9 .0045 J

or
* n^-i
F* |_9 -0045 J

1157. 2s=
/2

'1158. 5=
Exs. 1245-1322] ANSWERS 543

1245. 225 io 8 ; 200- io 8 ; 185

-
io 8 . 1270. Taking 1.530 as index, 1.24 in.

1246. 165 sec. and 7.44 in.

1247. Angle of refraction in glass 1274. 12 m.; 7.5 m.; 4.8 m. etc.
= 2 5 4 0'. 1280. 30 cm.
1248. 1.07 1281. 100 cm.
1249. 1.33 rn. 1282. .9 cm.
1250. .582 cm. 1285. /*=i.5.
1251. 40 ft. 1286. /= radius.
1252. 40 ft. 1302. 7
3i2-io- cm.
1253. Above. 1303. 76 .
io~6 sec.
1258. v.
= 1.668. 1309. 22- io~ 3 ; 38 io- 3 -.-etc.
1259. 2338'; io22'. 1310. 1.21 : i.

1260. For yellow light taking index of 1318. 323'; 647'; io!2'.
crown glass as 1.530, 1319. 2:3.
1322. 1059.
 INDEX

ACCELERATION, 38, 39, 40, 41. Dynamo, 179-183.

Approximations, 33. characteristic curve, 180, 181.
Archimedes' principle, 94. efficiency, 181, 182.

Atmospheric pressure, 89.

Averages, 31. ENERGY, of charge, 127.
of discharge, 130, 131.
BAROMETER, 89. kinetic, 67.
Batteries, 137, 138, 139, 142, 143. of rotation, 67, 74.
best arrangement of, 146, 147. transformation to potential, 75-78.

Boiling points, table of, 16. Elastic limit, 85.

Boyle's law, 98. Elasticity, 85.

Electric force, 123.

CALORIE, 108. Electrochemical equivalent, 153, 154.

Calorimeters, 109, no, in, 112, 113. Electromagnetic units, 18, 187-190.
Capacity, electrical, 124. attraction, repulsion, 169-172.

specific inductive, 18, 128. induction, 169-182.

thermal, 108, 109. Electromotive force, 142.
Cells, best arrangement of, 147. of induction, 173-178.

grouping of, 142, 143. Expansion coefficients, 16.

Center of inertia, 58, 61.
of mass, 58, 61. FALL, of potential in a wire, 135.
of gravity, 58, 61. of potential and electromotive force,
Coefficient, of expansion, 101. 135-
cubical, 103, 104, 105. Farad, 18.
of gases, 106, 107, 108. Faraday's disc, 177.
Coefficients of expansion, 1 6. Fields offeree, electric, 124, 125.
Condensers, 128, 129. magnetic, 161, 169, 173.
Conductivities, thermal, 17. Force, 40-43.
Critical angle, 213. systems, 53-57.
Current alternating, 183, 184, 1 86. Friction, 79.
Current electricity, 132. angle of, 79, 80.
coefficient of, 80.

DENSITIES, tables of, 14.

Diffraction grating, 222, 223. GALVANOMETER, 155, 156.
Dimensions, 5, 187-190. Ballistic, 157.
Dimensional equations, 5. Gases, 89 et seq.

Doppler's principle, 207. Graphic methods, 27, 30.

245
246 INDEX

HEAT, 100. Multiple resistance, 139-141.

specific, 108. graphic methods, 141.
specific variation of, no.
of fusion, no- 1 1 2. NEWTON'S rings, 221.
of vaporization, 112, 113.
in electric circuit, 149-151, 183. OHMS, various, 18.

Heats, of liquefaction, table of, 15. Ohm's law, 132 et seq.

of vaporization, table of, 1 6. Overtones, 205, 206.
Hydrometers, 97.
Hydrostatic pressure, 90-92. PENDULUM, gravity, 82.

press, 92, 93. magnetic, 167, 168.

Hysteresis, 167. Potential, diagrams, 135-138.
gravitational, 75-78.

INDICATOR diagram, Prism, 214, 215.

119.
Indices of Refraction, 20. Pressure, of atmosphere, 89.
of gases, 98, 99, 106-108.
of liquids, 89, 92.
KILOGRAM, 8, 9.
center of, of liquids, 92.
Kirchhoff s law, 147-149.
Projectiles, 49.

Pulleys, systems of, 73, 74.

LENSES, 216.
images by, 217, 218.
REFRACTION, index of, 212-215.
curvature of, 217-219. indices of, 20.
Light, reflection of, 209. law 212.
of,
velocity of, 213, 214.
Resistance, multiple, series, 138, 149.
refraction of, 212-215.
specific table of, 17.
interference of, 220-223.
temperature coefficients of, 17.
diffraction of, 222, 223.
units of, 1 8.
Lines offeree, 122.
magnetic, action of, 168-170. SELF-INDUCTION, 185.
Liquids, and gases, 89. Shunts, 143-145.
pressure, 89-91.
Simple harmonic motion, 191-193.
Sound, 198.
MAGNETIC, field, due to currents, 159. musical, 200.
induction, 165. velocity of, 19, 199.
Magnetism, 161. Specific gravity, 94-97.
Magnetization curve, 165, 166. gravity bottle, 96.
Magnetometer, 163, 164. heats, 14, 15.
Mass and weight, 7. inductive capacity, table of, 18.

Measurement, I. resistance, 152.

Mechanical equivalent of heat, 14. Spectra, 215, 222, 223.
Melting points, table of, 15. Static electricity, 121.
Mirrors, 209. Strain, 85, 86.
plane, 209, 210. Stress, 85, 86.
concave, 210, 211.
convex, 211, 212. TEMPERATURE, 100.
Moment of inertia, 82-84. Thermometer, 100.
Motor, 183. scales, 101.
 INDEX 247

Thermometer weight, 105, 106. Vectors, 21.

Thin plates, 220, 221. addition of, 21, 22.

Torsion, 87, 88. examples on, 23, 25.

moment of, 88. Velocity, of light, 19.
Transformer, 186. of sound, 19, 199.
Transmission of energy, electric, 151. of sound, temperature, effect on, 199.
Vibration, 191.
columns of air, 205-207.
UNITS, i.
elliptic, 192.
C. G. S. and practical, 190.
strings, 201-204.
electrical, magnetic, 187.
fundamental and derived, 2.

of area, 12.
WAVE length of sound, 200.
of light, 221, 223.
of force, 13.
of heat, 1 6.
Wave lengths of light, table of, 19.

of length, 12. Waves, 194, 195.

of mass, 13. phase, 197.

of power, 13. progressive, 196.

retardation of, 197.
of resistance, 18.
of stress, 13. sound, 198.
Wheel and a^cle, 74, 75.
of volume, 12.
of work, 13. Work, by torque, 65.
constant force, 62.
practical, in C.G.S., 18.
transformation general expression for, 69, 70.
of, 190.
principle of, applied to machines, 71.
variable force, 63, 69.

VAPOR, pressure, 114, 118, 119.

volume, 114, 1 1 8, 119. 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.,

Professor of Physics and Electrical Engineering at the Iowa Agricultural College, Ames, la.

I. Mechanics and Heat.

II. and Magnetism.
Electricity
{Vol. III. Sound and Light.

Volumes I. and II. now ready. Price $1.50 net, each.

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