(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Problems and questions in physics"

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," is something that no one ever reads. The 
plan also shortens the statement of a problem, relieving it of 
much reiterated information. 

A few words should be said concerning the use of the cal- 
culus notation. As the tendency of writers of elementary 
works in Physics seems to be towards a greater use of the 
language of the calculus, it is only appropriate that a fair 
number of problems should be inserted here which cannot be 
satisfactorily worked by other than calculus methods. Their 
number, however, is not large, and the usefulness of the book 
to students not prepared for them will be in nowise dimin- 
ished. It is believed that the number of problems is sufficiently 
large to enable the instructor to make an adequate selection for 
any class. 

Occasional questions not requiring numerical answers have 
been asked. These are purposely few in number, and are put 
in to indicate the general character of 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 profitably extended by the student. On the other 
hand, solutions and hints have been omitted in many cases 
where the student might perhaps expect to find them. It is 
felt that the methods preferred by the instructor in charge or 
suggested by the text in use should be used rather than those 
of the writers, since the general character of the course and the 
degree of the student's advancement may be thus considered. 
It is not expected that the student should work the problems 
without suggestion, and inability to do so in particular cases 
may indicate to both student and instructor just where some 
law or definition is not clearly understood. There are undoubt- 
edly obscurities in the text and errors in answers, and the 



PREFACE vii 

authors would esteem it a favor if readers would call attention 
to them. 

Some criticism may be incurred because of the use of mixed 
units. Many of the students who will use these problems are 
pursuing engineering courses. In such case they must of 
necessity use engineering units. The aim has been not so 
much to train them in the use of these units, an abundance 
of this training comes to them during their course, but to 
bring out the relation of the 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, in the domain of physical science, 
we step from the position of a mere observer of the phenomena 
around about us to that of an investigator, we seek the aid of a 
process known as measurement. Whether this process be sim- 
ple or complex, there is but one operation in it that is funda- 
mental, the determination of the value of one magnitude in 
terms of another of the same kind. We may content ourselves 
with the crudest approximation, as when we estimate moun- 
tain heights in terms of the highest peak of the range, or, we 
may make a comparison with the utmost scientific accuracy, 
using for such a purpose a quantity agreed upon among men as 
a standard or unit. In either case the result sought is a ratio ; 
namely, that existing between the magnitude and the chosen 
unit of like kind. This ratio is the measure of the given magni- 
tude, and the process by which it is found is called measurement. 
The accuracy with which measurements are made is governed 
largely by practical needs. It should, however, be borne in 
mind that the process is, at the best, an approximate one. Even 
the most exact measurements of physics must be regarded as 
attempts to determine numerical quantities whose true values 
must ever/ remain unknown. 

Units. It follows that the complete expression of a physical 
quantity, so far as its magnitude is concerned, involves two fac- 



2 PROBLEMS IN PHYSICS 

tors, one a concrete unit, the other a number or numeric. Thus 
if L be a unit of length, the measure or numerical value of a 

length /is n = , and the complete expression of the magni- 
tude of / is 



The product of numeric and unit is constant. Whether a debt 
be paid in dimes or in dollars, it is yet the same debt, but the 
number representing it in the one case is ten times that repre- 
senting it in the other. The unit and numeric, in other words, 
vary inversely. 

Fundamental and Derived Units. Consider the case in which 
the unit of length is taken as the foot, and the unit of area the 
square yard. Then a rectangular area a feet long and b feet 

wide is expressed as 

A=\ab sq. yd. 

And, in general, the area is given by 

A = kab, 

where k is a constant depending upon the units of length and 
area involved. If, however, it is agreed that the unit of area 
shall be the square foot, the value of k reduces to unity, and 

A=ab sq. ft. 

It thus appears that, in a system made up of arbitrarily 
chosen units, transformations call into use a number of pro- 
portionality constants, many of which will involve endless deci- 
mals, introducing into computations much unnecessary labor 
and liability of error. The earlier units were largely of this 
character. They were chosen to meet the needs of practical 
life at a time when simple and definite relations among them 
were not deemed essential. Thus the origin of the foot is obvi- 
ous, as is also its variation in different countries.* Further, 

* The Russian foot is 30.5 cm.; the Austrian foot, 31.6 cm. ; the Saxon foot, 
28.32 cm.; etc. 



INTRODUCTION 3 

derived units based on powers of the fundamental are not 
always convenient. The yard is a convenient length for the 
measurement of cloth, but the cubic yard is too large a volume 
for the grocer's needs. Yet the awkwardness of systems made 
up of grains, scruples, drams, and ounces, of links, rods, and 
chains, needs no comment. The metric system, now generally 
used by physicists, obviates these difficulties by making all 
change ratios multiples or sub-multiples of 10. All the complex 
units of physics are thus bound together by ties that may be 
easily manipulated. 

The system in common use is based on three arbitrarily 
chosen units. These are 

the centimeter, the T ^-g- part of the length of a certain plati- 

num bar kept in the Archives of Paris ; 
the gram, the YoVo P ai "t f a certain piece of platinum (the 

kilogram des Archives) which is intended to have the same 

mass as a cubic decimeter of water at the temperature of 

maximum density (3.9 C.) ; 
the second, the ^ art ^ ^ e mean 



These units of length, mass, and time, respectively, are known 
as the fundamental units of the C.G.S. system. Other units 
based upon them are called derived units. 

Another system, much less in use, is based on the same 
physical quantities, but the units of length and mass are of 
different value. They are 

the /<?#/, as a unit of length ; 
the pound, as a unit of mass ; 
the second, as above denned. 

These units are the basis of the 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 is found that the force between 
two magnetic poles varies as the product of their pole strengths 
and inversely as the square of the distance between them. 
That is, in air, 



Whence unit magnet pole is a pole of such strength that it 
repels an equal and like pole, placed I cm. away, with a force of 
one dyne. This unit of force, itself a derived unit, has already 
'been referred to. 

Dimensions and Dimensional Equations. Suppose that for 



INTRODUCTION 



5 



the unit of area in any system a square be taken one of whose 
sides is the unit of length, and let an area a contain n such 
units. That is, 

a = nA. 

Further suppose that it is desired to double the unit of length. 
The new unit of area built upon the changed unit of length is 
four times the old unit. In other words, the unit of area varies 
as the square of the unit of length, or it is said to be of two 
dimensions in length. To indicate this, the last equation may 
be written 

a = nL\ 

Let v be a concrete velocity such that a distance / is trav- 
ersed in time /. The numerical values of these quantities 
are found by dividing each by the appropriate unit. Let V, L, 
and T be these units. Then the numerical values are , , . 
We have then two numerical values of this velocity, viz., 



v , L 

v and T 



but these values are to be equal, which gives 

2-=L L 

V L ' t 
Writing the equation so as to separate the units' part, we have, 



Or, in words, the unit of velocity varies directly as the unit of 
length and inversely as the unit of time. That is, the dimen- 
sions of unit velocity are LT~ l . In passing to dimensional 
equations we may discard constant numerical factors, since the 
units, and therefore the dimensions, are not affected thereby. 



6 PROBLEMS IN PHYSICS 

So, the dimensions of the unit of acceleration are readily seen 
to be L T~ 2 ; of the unit of force, ML T~ 2 ; of the unit of work, 
ML*T 2 ; and so on. 

It becomes apparent at once that dimensional formulas show 
the powers of the fundamental units that enter into derived 
units. Hence dimensional equations are of much use in facili- 
tating change of units. 

EXAMPLE. The numerical value of the acceleration due to 
gravity, when the centimetre and second are used as units of 
length and time, is 980. Find the value in terms of the foot 
and minute. 

The dimensions of acceleration, it has been seen, are L 7^~ 2 . 
We have 



= 980 x .033 x 3600 ^- 
|_min. 

= II6424 ^ 

Lmin. I J 



That is to say, the acceleration due to gravity is 116424 ft. 
per minute per minute. 

Whenever problems involving change of units occur in the 
following collection, the student is strongly advised to work 
them in this way, until the processes become so familiar as not 
to need formal statement. 

The two members of every equation must reduce to the same 
dimensions, otherwise the equation is absurd. Or, what amounts 
to the same thing, every term of an equation is homogeneous 
with respect to each fundamental unit involved. The equation 
of the motion of a particle having uniform acceleration in the 
direction of motion is 



INTRODUCTION ; 

wherein / and a are of the dimensions L, 

b, a velocity, is of the dimensions LT~ l , 
and c, an acceleration, is of the dimensions LT~ Z . 

Thus each term of the expression for / is of the dimensions L 
of / itself. 

This gives a very convenient check upon our work in deriving 
such an equation. 

Mass and Weight. These words stand for two distinct phys- 
ical concepts. Thus, mass is quantity of matter, while weight 
is force. Physically, then, they are no more alike than length 
and time. Not infrequently the beginner fails to apprehend 
this fact. Confusion arises partly because masses are compared 
by comparing their weights, and partly because the same word 
is often used ambiguously to name both a unit of mass and a 
unit of force. 

If a point move over equal spaces in equal times, any con- 
stant distance corresponds to a constant time. Or, in other 
words, distance traversed and time vary in direct proportion. 
For example, when, in railroad parlance, two stations are said 
to be "four hours" apart, every one understands roughly what 
distance is meant. Now it is precisely this relation that exists 
between mass and weight, and it is largely because of their 
proportionality in any one locality that some license is admissi- 
ble in naming their units. 

Masses attract each other according to the fundamental law 
of gravitation. To the attraction between the earth and the 
bodies upon its surface the special name weight is given. The 
weight of a body, therefore, is the force with which it is drawn 
towards the earth, or with which the earth is drawn towards it. 
When two bodies are placed in opposite pans of a beam balance 
and do not destroy its equilibrium, they are said to be of equal 
weight. That is, the forces acting at the ends of the beam are 
equal. Further, by the law of proportionality, the bodies are of 



8 PROBLEMS IN PHYSICS 

equal mass, since we have for each force (or so much of it as 
may be due to the added mass), 

F= MS, 

wherein g is the acceleration with which the mass M would fall 
if released. The balance thus serves to determine equal masses, 
and it is evident that if the system were carried to any other 
locality the equilibrium would remain perfect, the masses re- 
maining unaltered and the weights varying with g. It is in this 
way that masses are compared through the agency of their 
weights. 

As to units of mass, there are two in common use : 

the pound, 
the gram,* 

each of which is the quantity of .matter in a certain carefully 
preserved piece of platinum. To obtain the weights of these 
masses we must multiply by the value of g appropriate to the 
system of which the unit is a fundamental, and to the locality 
at which the weight is desired. Thus the weight of a pound 
where g = 32.2, is 

W P = mg= i x 32.2 

= 32.2 units of force in the F.P.S. system 
= 32.2 poundals. 

The weight of a gram where g = 980 is 

W g mg = i x 980 

= 980 units of force in the C.G.S. system 
= 980 dynes. 

All this is clear enough. But unfortunately, perhaps, the terms 
pound and kilogram are used in such expressions as, "a body 
weighs 16 pounds" or "a weight of 12 kilograms." The 

* The original standard is the kilogram 1000 grams. 



INTRODUCTION 9 

pound and kilogram being units of mass, such usage, taken 
literally, is absurd. The expressions, however, are elliptical, 
their full meaning being "a body weighs the same as 16 pounds 
weigh," or "a weight equal to the local weight of 12 kilo- 
grams." Or, we may say, with equal correctness and greater 
brevity, " 16 pounds' weight" or " 12 kilograms' weight." So, 
a grocer is said to weigh out tea; but he does not sell weight 
he has no force for sale but mass. 

A still greater source of confusion arises from the fact that the 
engineer finds the poundal (-3^2 P oun d's weight)* and the dyne 
(QFO x ToVo kilogram's weight) too small for practical needs as 
units of force. The engineering unit of force among English- 
speaking people is the weight of a pound (called simply a 
pottnd), and among people using the metric system the weight 
of a kilogram (called simply a kilogram}. Since these units 
depend on the value of g, they are slightly variable, but the 
variation is so small as to be usually negligible for engineer- 
ing purposes. 

As illustrating this last usage, suppose that the piece of plati- 
num which the English people have agreed to call a pound were 
hitched to a spring balance and the whole arrangement carried 
to different points on the surface of the earth. The registry of 
the balance would evidently vary to a slight extent. The engi- 
neer says we will neglect this variation as being of negligible 
importance, and say that any agent which stretches the balance 
spring ten times as much as does the freely suspended pound 
mass is a force of 10 Ib. Let us suppose, then, that in this 
way a body is found to weigh 10 Ib., and let us inquire what 
the mass of this body is. By Newton's second law this force 
is measured by the mass of the body times the acceleration 
which it would possess if allowed to fall freely. Taking g=$ 2. 2, 
we write 

* The accepted value of g at Ithaca is 980, which corresponds to 32.15 in foot- 
second units. 32.2 is commonly used, however. See Church's " Mechanics of 
Engineering." 



OP 1 THK 



10 PROBLEMS IN PHYSICS 



10 = m x 32.2, 
whence, mass = = 



g 32.2 

This makes the mass of the body invariable, as it must be. 
To the unit mass in this system no name has been given, but it 
is readily seen to be the mass of a body weighing 32.2, or more 
generally g, pounds. With this understanding it is quite cor- 
rect to say that a body weighs G pounds, to speak of a pull or 
thrust of G pounds, a pressure of G pounds per square inch, etc. 

The pound and the kilogram are sometimes called gravita- 
tional units of force. Likewise the 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 
~ 32-2 



x 64.4 = 24 Ib. 



INTRODUCTION 1 1 

A force of 12 kg. is overcome through a distance of 20 m. 
Find the work done. We have 

W= Fl 

12 X 20 

= 240 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 x io 3 x 980 dynes, 
and the work done is 

W= Fl 12 x 2 x 980 x io 6 ergs. 

a. What two elements are necessary for the complete expres- 
sion of the magnitude of a physical quantity? Explain fully 
in what the process of measurement consists. 

b. What is the logical objection to a system of units in 
which the inch is taken as the unit of length, the square rod as 
the unit of area and the cubic metre as the unit of volume ? 

c. A certain surface is a units long and b units wide; the 
general expression for the area is 

A = kab. 
Under what conditions will the area be expressed as ab simply? 

d. If in the last example a and b are given in feet, what will 
be the value of k if the unit of area be taken as I square mile ? 

e. Explain what is meant by fundamental and derived units. 

f. Imagine the unit in which a definite magnitude is meas- 
ured to vary continuously. Plot values of the unit as abscissae 
and corresponding values of the numeric (or measure) as ordi- 
nates. Discuss the locus. 



12 PROBLEMS IN PHYSICS 

NOTE. Many examples involving change of units, use of dimensional 
equations and like matters are to be found further on in this book. It has 
seemed better to place such examples, with the exception of the few general 
ones above, where they may be used after the student is in some degree 
familiar with the ideas involved. 



UNITS OF LENGTH. 

NOTE. The student is advised to study the approximate values. They 
are of assistance in mental calculations, and are frequently sufficiently exact 
for problem work. 

Roughly 

approximate 

values. 

I in. = 2.54 cm 2*. 

i ft. = 30.48 cm 30^. 

i mi. = 160933 cm. 
= 1.6 km. 

i cm. = .394 in |. 

i cm. = .0328 ft T |fo. 

i m. = 39.37 in 40. 

i km. = .6214 mi f . 



UNITS OF AREA. 

i sq. in. = 6.45 sq. cm 
i sq. ft = 929.01 sq. cm. 
i sq. mi. = 25899 x io' 2 sq. m. 
I sq. cm. = .155 sq. in 

= .001076 sq. ft. 
i sq. m. = 3.861 x io~ 7 sq. mi. 



UNITS OF VOLUME. 

i cu. in. = 16.387 cu. cm. . . . i6\. 
i cu. ft. = 28316. cu. cm. 
i gal. = 4541. cu. cm. 

= 4. 54 litres 4$. 

i cu. cm. = .061 cu. in ^. 

= 3.532 x lo- 6 cu. ft. 



INTRODUCTION 



UNITS OF MASS. 

i ib. = 453.59 g- 

i oz. (av.) = 28.35 g- 

1 g- = I 543g r - 

.0353 oz. 

= .0022 Ib. 



UNITS OF FORCE. 

[g = 980 in all gravitational units.] 
i poundal = 13825 dynes, 

i gram's weight = 980 dynes, 

i pound's weight = 444518 dynes, 

i kilogram's weight 9.8 x io 5 dynes. 

- 2249 x io~ 9 pound's weight. 



UNITS OF WORK. 

i foot-pound = 1.35485 x io 7 ergs 

= 13825 gram-centimeters 
= .138 kilogram-meters. 

i kilogram-meter =7.233 foot-pounds. 

i joule = io 7 ergs. 

i watt-hour = 36 x io 9 ergs. 

i horse-power-hour = 26856 x io 2 joules. 



UNITS OF POWER. 

i horse-power = 746 watts 

= 746 x io 7 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. = 4.2 x io 7 ergs 

= .4281 kilogram-meters. 

Ib. through i F. = 1.058 x io 15 ergs 
= 780.8 foot-pounds. 



TABLE I 
DENSITIES 



SOLIDS 



Aluminum 2.6 

Antimony 6.7 

Bismuth 9.8 

Brass 8.4 

Copper 8.9 

Gold 19.3 

Iron 7.8 

Lead 11.3 

Nickel 8.9 

Platinum 21.5 

Silver 10.5 

Sodium 98 

Tin ' . 7-3 

Zinc 7.1 



Asbestos . , 

Chalk . . , 

Coal . . . . 

Cork . . . . 
Glass, common 

Glass, flint . . 

Ice . . . . 

Iceland Spar . 
Ivory 

Marble . . . 
Paraffine 

Quartz . . . 

Oak . . . . 
Pine . 



2.4 

2.3-3-2 
1.4-1.8 

I4--3 
2.5-2.7 

3-3-5 

.917 
2.75 
1.9 
2.7 

.87-. 9 i 
2.65 

7-1 

5 



Alcohol . . . 
Ether .... 
Carbon Bisulphide 
Glvcerine 



LIQUIDS, oC. 



.806 
-736 
1.29 
1.27 



Mercury . 13-596 Oil of Turpentine 



Sea Water . . 
Sulphuric Acid ., 
Nitric Acid . . 
Hydrochloric Acid 



1.026 
!.8 5 
1.56 
1.27 
.87 



TABLE II 
SPECIFIC HEATS OF SOLIDS 



Aluminum .2122 

Bismuth 0298 

Brass 0940 



Calcium 

Carbon, diamond 
Carbon, graphite 



,1804 
,1128 
,1604 



INTRODUCTION 



Carbon, charcoal . 
Copper 


1935 

OQ'2 7 


Gold 


O3l6 


Glass 


l8?7 


Ice 


rn/lO 


Iron 
Lead 


.1124. 
0315 



Magnesium 



2450 



Nickel ......... 1092 

Platinum ........ 0323 

Silver ......... 0559 

Tin .......... 0559 

Zinc . . ........ 0935 



TABLE III 

SPECIFIC HEATS OF LIQUIDS 

Alcohol 55 

Carbon Bisulphide . 24 

Ether 53 



SPECIFIC HEATS OF GASES AND VAPORS 

( Constant Pressure} 

Air 237 

Oxygen 217 

Hydrogen 3.4 

Nitrogen 244 

Steam . 



Marsh Gas 
Alcohol 



.48 

593 

453 



TABLE IV 



MELTING-POINTS AND HEATS OF LIQUEFACTION 



Aluminum 
Copper 

Glass . . 

Gold . , 

Ice . . , 

Iron . . 
Lead 



Melting- 
point. 

o 


Heat of 

Liquefaction. 
Calories. 




600 
1054 
1 100 




Mercury . . . 
Nickel . . . . 
Platinum . . . 


1045 

o 


80 


Silver 
Tin . . . . 


1600 




Zinc . . . . 


326 


5-4 





Melting- 
point. 

o 


Heat of 
Liquefaction. 
Calories. 


-40 


2.82 


1450 


4.64 


1775 


27.2 


954 


24.7 


230 


14.6 


412 


28.1 



i6 



PROBLEMS IN PHYSICS 



TABLE V 
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 io 7 
i pound-degree Centigrade = 1905 x io 7 

i pound-degree Fahrenheit = 1058 x io 7 



TABLE VII 

COEFFICIENTS OF LINEAR EXPANSION 

Brass 180 } 

Copper 170 

Glass . 085 

Gold 150 

Iron 120 

Lead 280 

Platinum 085 

Silver 190 

Tin 200 

Zinc 290 



x io 



COEFFICIENTS OF VOLUME EXPANSION 

Alcohol (mean o 78) 00104 

Mercury (mean o 100 C.) 000182 

Water (mean o 100) 000062 



INTRODUCTION 



TABLE VIII 
THERMAL CONDUCTIVITIES 

Relative 
Conductivity. 

Silver 100 

Copper 74 

Iron 12 

Lead 8.5 

Bismuth 1.8 

Ice 0.2 

White Marble .... o.i 
Glass 0.05 



C.G.S. 

1-3 
0.99 
0.16 
o.n 

O.O2 

0.003 

0.001 

0.0007 



TABLE IX 
COLLECTED DATA FOR DRY AIR * 

Expansion from o to 100 at constant pressure as 273 : 373 

Specific Heat at constant pressure 2375 

Specific Heat at constant volume 1691 

Standard barometric height 76 cm. 

Density at o and 76 cm 001293 

Volume i g. at o and 76 cm 773-3 c.c. 

* Everett. 



TABLE X 
RESISTANCE 



Substance. 


Specific Resistance. 


Temperature Coeffi- 
cient (0-100). 


Aluminum (annealed) .... 
Copper (annealed) 
Gold 


289 io~ 8 ohms 
160 io~ 8 ohms 
208 io~ 8 ohms 


388 io- 5 
^6c io~ 5 


Iron (pure) 


964 io~ 8 ohms 




Iron (telegraph wire) . ... 


1500 io~ 8 ohms 




Lead . . . 


1963 io~ 8 ohms 


187 io~ 5 


Mercury .... 


9434 io~ 8 ohms 


72 IO~ 5 


Platinum . 


898 io~ 8 ohms 




Silver . . . 


149 io~ 8 ohms 


377 


German Silver . ... 


2100 io~ 8 ohms 


4,4. to 6; IO~ 5 


Platinoid .... . . . 


3200- io- 8 ohms 


21 IO~ 5 


Mano^anin 


4700 io~ 8 ohms 


122 IO~ 5 









i8 



PROBLEMS IN PHYSICS 



TABLE XI 
UNITS OF RESISTANCE 



i true ohm unit of resistance, 
i legal ohm = .9972 true ohms. 



i B. A. unit = .9867 true ohms, 
i Siemen's unit = .9407 true ohms. 



TABLE XII 
SPECIFIC INDUCTIVE CAPACITIES 

Air = i 



Solids. 


K. 


Liquids. 


K. 


Glass 


A to 7 


Acetone 


21 8 


Gypsum 


r 6 


Alcohol 


2 r 


Ice 


3*** 

2 8C 


Aldehyde 


"O 

18 6 


Iceland Spar .... 


***3 

7-4. 


Benzine .... 


7 -2 


Marble 


64 


Carbon Disulphide 


2 


Mica. 


6 to 8 


Ether 


A 27 


Paraffine 


2 2 


Glvcerine 


c6 2 




4.1:4 


Oils 


3^'^ 
2.2 


Rosin 
Rubber, soft .... 
vulcanite 


2.55 
2.4 
2.7 


Petroleum .... 
Turpentine .... 
Water 


2.06 
2.23 

7c r 


Salt 


5.8 






Sandstone .... 
Shellac 


6.2 

9 


Gases. 

Hydrogen 


O QQO8 


Sulphur 


2 6q 


Vacuum 


w.yyyo 

o oo8c 


Wood 


,.wy 
2.QC 


Vapors 


w.yyo^j 

i.ooi to i.oi 











TABLE XIII 

PRACTICAL UNITS EXPRESSED IN C.G.S UNITS 
Let Kbe the velocity of light, about 3-io 10 cm. per sec. 





Electromagnetic 


Electrostatic. 
C.G.S. 


Practical. 


C.G.S. 


Quantity .... 


i coulomb 


I/IO 


I 7 / 10, i.e. 3-io 9 


Current . . . . 


i ampere 


I/IO 


V/io 3-io 9 


Potential . . 


i volt 


I0 8 


io 8 /y i/(3-io 2 ) 


Resistance .... 


i ohm 


I0 9 


io 9 /^ 2 i/(9-io u ) 


Capacity .... 


i farad 


I/IO 9 


F-/io 9 9-io n 


Self-induction . . . 


i henry 


I0 9 





INTRODUCTION 



TABLE XIV 

SOUND 
VELOCITY OF SOUND IN METERS PER SECOND 



Solids (20 C). 




Liquids (20 C.). 




Gases (o). 


Brass 


2/180 


Alcohol 


1 1 60 


Air 1^2 












Copper . . . 


3560 


Water . . . 


1440 


Illuminating Gas, 490 


Iron .... 


5'3o 


Petroleum . . 


1395 


Hydrogen. . . 1280 


Steel, cast . . 


5000 






Oxygen ... 317 



TABLE XV 
LIGHT 

Velocity of light | 2 99 86 kilometers per sec. Nearly 3 - 10*. 
( 186323 miles per sec. 

TABLE XVI 

WAVE-LENGTHS OF THE PRINCIPAL FRAUNHOFER LINES IN 
TENTH-METERS * 



Line. 


Wave-length. 


Line. 


Wave-length. 


A 


7594.059 


M 


$ 3727-763 


B 


6867.461 




\ 3727. 20 


C 


6563.054 


N 


3581.344 


A 


5896.154 





344LI35 


D, 


5890.182 


P 


3361.30 




(5270.533 


Q 


3286.87 


E 


^ 5270.448 


jg 


^ 3l8l.40 




(5269.722 




'3179-45 


F 


4861.496 


$i 


3100.779 


G 


( 4308.071 


s. 


3100.064 




\ 4307-904 


T 


53021.19! 


H 


3968.620 




\ 3020.759 


K 


3933-809 


U 


2947-993 


L 


3820.567 







* i tenth-meter = io~ 8 of a centimeter. 



20 



PROBLEMS IN PHYSICS 



TABLE XVII 

INDICES OF REFRACTION [Z? LINE] * 



Density. 



Index. 



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

Glass (soft crown) .......... 2.55 .5 145 

Glass (light flint) .......... 3.206 .574 

Glass (dense flint) .......... 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 e AT- .434294 
Log e lV =Log lQ AT- 2.3025 85 
i radian .... 57.2958 

i ....... 01745 radians 

Log 10 Log 10 

^=3.14159 -497H9 ^ " 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 



I: V2 



1.4142 
.7071 



_.I5<>5I5 
1.849485 



I : V3 



1.7321 

.5773 



1.761439 




II. DIRECTED QUANTITIES, VECTORS 

Many of the quantities considered in physics involve the idea 
of direction, and require the statement of two things before we 
can form any clear idea of them. First, we must state how 
large they are as compared with a thing of like kind taken as 
a unit ; second, in what direction they must be taken. The 
familiar idea of motion from one point to another may be con- 
sidered as typical of this class of quantities. Suppose one asks 
the way from one point in a city to another. The answer might 
be to go a certain distance north, then a certain distance west, 
etc. Or, if circumstances permit, he may be told to go a certain 
distance in a specified direction without turns. 

The answer is one based on the experience that we may go 
from one point to another either by a series of connected "steps" 
or courses such that they begin at the 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 is then the resultant required. Or we may form 
a parallelogram with one corner at A, and whose sides are i 

21 




22 



PROBLEMS IN PHYSICS 



and 2. The diagonal drawn from A is the equivalent step or 
resultant. 

The student should remember that the problem of finding the 
resultant of a given system of steps is perfectly definite, and 
only one solution can be found ; but the converse is not true, as 
a given step may be made up of any one of an endless number 
of step systems. 

The process of finding the resultant of a given system is 
often spoken of as the composition of steps; while that of replac- 
ing a single step by a system, usually two, is called the resolution 
of steps. 

The simplest, and, at the same time, the most useful case of 
resolution is when the step is resolved into two at right angles 




Fig. 2. 

to each other. Or the line is said to be projected on two rectan- 
gular axes X and Y. 

Then X component = AB' = AB cos 0, 

Y component = BB 1 = AB sin 0. 

The name vector (i.e. carrier) is usually applied to this class 
of quantities, and the resultant of a system of vectors is spoken 
of as the vector sum of the components. A thorough under- 
standing of the geometrical ideas involved in adding and resolv- 
ing vectors is of the greatest importance to the student in 
physics, and must be acquired before any real progress in the 



DIRECTED QUANTITIES, VECTORS 23 

subject is made. The following simple problems are added to 
assist the student toward this end. 

1. Which of the following quantities are vectors? Force; 
mass ; acceleration ; momentum ; energy ; volume ; velocity ; 
current ; weight ; time ; interest. 

2. Show by diagram the vector sum (i.e. the equivalent 
straight path) of the following set of paths : E. 4 mi. ; N. 2 mi. ; 
N.W. 3 mi. ; S.W. 5 mi. 

3. Draw the same set of paths in the reverse order; i.e. 
S.W. 5 mi. ; N.W. 3 mi. ; etc. 

4. When the vectors are not in the same plane, show how 
the vector sum is found. 

5. What is the vector sum of the length, breadth, and 
height of a room ? 

6. Two vectors at right angles to each other, of lengths 4 
and 3 respectively, have what vector sum or resultant ? If at 
60 ? 180? o? 

7. Six vectors equal in length are placed end to end so 
that the angle between each pair is 120. What is the vector 
sum ? 

8. Show that the order in which "steps" are taken in no 
way modifies the sum. 

A vector may be given in either of two ways, by its components or by 
its length and direction, or the angle it makes with a given line. In the fol- 
lowing examples the line of reference is the horizontal line drawn to the right 
(.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 



2 4 



PROBLEMS IN PHYSICS 



10. Draw the following vectors : 3, 90 ; 4, 180; 5, 190. 

11. Draw the vectors whose components at right angles to 
each other are 2 and 3, 4 and 6, 2 and 3. 

12. A vector 10 units in length makes an angle of 30 with 
one of two perpendicular lines. Find the component along 
each line. 

13. A given vector is to be resolved into two at right angles, 
such that one component is double the other. Find the angle 
which the longer must make with the given vector. 

14. Could the vector AB be considered as the vector sum 
of the set of short vectors parallel to the axes ? Those parallel 
to X are called what in calculus ? Those parallel to F? 

Y 




Fig. 3. 

15. Two vectors, a and b, are given, making angles l and 2 
with the reference line. Find the sum of their X components. 
Find the sum of their Y components. From these find the 
Y 




Fig. 4. 

' resultant of a and b. Reduce to the formula given in trigo- 
nometry for the cosine of an angle in terms of the sides. 



DIRECTED QUANTITIES, VECTORS 25 

16. Show that the resultant of two vectors may be found 
from the theorem in geometry : The square on any side of a 
triangle is equal to the sum of the squares on the other two 
sides twice the product, etc. 

17. n coplanar vectors are drawn from a common point. 
A polygon is formed by joining their extremities. Prove that 
the resultant is given in magnitude and direction by n times 
the vector joining the origin and the center* of gravity of the 
polygon. 

18. Test the above statement for two, three, and four 
vectors. 

19. If the vectors were so numerous that their ends formed 
a continuous curve, what method could be used to find the 
resultant ? 

* See 197. 



III. GRAPHIC METHODS 



It is frequently impossible to keep in mind the complete 
time history of variable phenomena, or to readily compare the 
values of quantities which alter with time or position. A 
clearer conception in such cases may often be obtained by some 
geometrical method of representing the relative values of two 
quantities at different times or places. Take, for example, the 
motion of a ball struck by a bat ; we may wish to compare any 
two of the various quantities which are involved in its motion. 
The height above the earth may be compared with the hori- 
zontal distance from the 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 // 2 , // 3 , etc. (Fig. 6). We know, 
however, that the ball took, in succession, every height between 
those indicated ; hence if we were to erect a perpendicular at 
every point between ^ and / 2 , and measure along each the 
corresponding height of the ball, the ends of these perpendic- 
ulars would form a continuous curve. This process is known 
as " plotting " the curve, and is of fundamental importance in 




Fig- 6. 

the study of physics. The two lines of reference from which 
distances are measured are called the axes of co-ordinates, and 
are usually chosen at right angles to each other. One is often 
called the axis of x, and the other the axis of y, and the lengths 
measured along the .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. A curve such as the 
following is the result (Fig. 7). Both time and temperature are 
continuous, and the curve is a fairly true picture of the time- 
temperature relation. 



28 



PROBLEMS IN PHYSICS 



In case we had observed the temperature at 2, 2.30, 3, 3.30, 
4, etc., and had no knowledge of intermediate temperatures, 
we would draw a continuous curve through the observed points, 
which would be more and more reliable as the time intervals 
were made smaller. In general, the more irregular the changes 
in the observed quantity, the shorter these intervals must be 
made to ensure that no sudden variation escapes notice. 




Fig. 7. 



We may expect in each case certain peculiarities in the 
curve, depending on the physical relations which determine 
it, and, conversely, any peculiarity, as a maximum or minimum, 
change of curvature, asymptote, etc., will usually have a physical 
meaning. For example, every change of temperature requires 
a certain time interval, so that no portion of the 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 



2 9 



required per gram for any temperature change may be read 
from the curve. 




EMPERATURE 



Fig. 8. 

Curves are used in physics for various purposes ; as, 

(a) To represent graphically general laws. 

Ex. Path of a projectile. Laws of falling bodies. 

(b) Asa record of results of observation of two related varying 

quantities. 

(c) For use in computation. As a sort of numerical map of 

simultaneous values. 

The student should not rest content with simply drawing the 
curve, but should endeavor to associate the changes or peculi- 
arities in form with the underlying physical conditions. If 
familiar with the methods of analytic geometry and calculus, he 
may apply these methods to their study. 

In particular, if the curve is a graphic representation of a 
general law, he should note whether all portions of the curve 
have an actual physical interpretation, whether the physical 
conditions indicated by certain portions of the curve could be 
realized ; if it cuts the axes, what the intercepts mean ; whether 
the direction of the tangent line at any point has a physical inter- 



30 PROBLEMS IN PHYSICS 

pretation ; does the area of a given portion represent some physi- 
cal quantity ; etc. When it is drawn from observed values, the 
relation between the 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 less rapid and irregular. To find the temperature 
which may fairly be taken to represent the temperature at that 
point during the hour, we would be obliged to add together a 
great number of observed temperatures and divide the result by 
this number. The greater the number added, the more nearly 
correct the average. We might also have required the average 
temperature at a given instant along a given line, over a given 
area or throughout a given volume. In all these cases we should 
take the sum of an indefinitely great number of separate values 
and divide by the time, length, area, or volume considered. We 
actually only approximate this by taking a smaller number. The 
actual addition of these quantities can in certain cases be avoided. 
As when the values to be averaged increase or decrease at a 
constant rate, the terms then form an arithmetic series, and the 
mean value is 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 is drawn showing the relation between the two 
variables, if by means of calculus or otherwise we are able to find 
the area ABB'A', we may divide this area by AB and get the 
average ordinate. 

31 



For, 



PROBLEMS IN PHYSICS 

Area = f ydx 

= AB average height. 



(Fig- 9) 



The student should be very careful in averaging quantities to 
first find the actual values to be averaged. For example, the 



V 



dx 

Fig. 9. 



average of a series of fractions is not the average of the numer- 
ators divided by the average of the denominators. The average 
of a series of quantities each the product of two factors will not 
be the product of the average value of each factor. 



V. APPROXIMATIONS 

The computation of results from physical data is often labori- 
ous, on account of the number of decimal places involved in the 
constants required. In many cases, however, we may diminish 
the work by the use of suitable methods and approximate 
formulae. Not only is the labor of computation increased by 
the retention of too many decimal places, but the results so 
obtained are actually misleading, in that they give an appearance 
of accuracy not warranted by the data. For example, any re- 
sult obtained by data accurate to one part in one hundred will 
not be accurate to any higher degree. 

Suppose that two sides of a rectangle have been measured by 
a metre bar divided to hundredths, and that the tenths of a divi- 
sion have been estimated, giving 4.258 and 6.543 . The last 
figure in each case is only approximate, and if the area is com- 
puted the result contains six decimal places, only three of which 
should in any case be retained. The labor of writing these 
superfluous figures may be easily avoided by using only those 
partial products giving the orders we wish to retain. We see 
that 4 units x .003 gives a product which we 4 . 2 5 g 
require, while .2 x .003 is of secondary impor- 
tance. The lowest partial products required are 
readily seen from the diagram, in which we 6 . 5 4 
"step down" one in the multiplicand as we Fi ?- 10 - 

"step up" one in the multiplier (Fig. 10), the arrows connecting 
the factors of the products required. 

The simplest arrangement of work is that given in 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." < 

Since [i ] n = i + n ( ) + n ( H ~ ^ ... for all values of 

n, whether positive, negative, integral, or fractional, and, when 
is small in comparison with unity, we may neglect 2 and all 



APPROXIMATIONS 35 

higher powers of , the approximate multiplier consists of 
i na. 

EXAMPLE. The edge of a 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 3 [i + .000 182] 3 
= 20 3 [i + 3 .000182 
+ Higher powers of small quantities.] 
= 20 3 [1.000546] 
= F [1.000546]. 

Had the volume at 24 C. been given and the volume at o 
been required, we have, in like manner, 



= V^ [i 3 24 -.0000 122] 
= F 24 [i -.0008784] 
= 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. 

As another example, consider the area of a rectangle of sides 
a and b when each side is slightly increased. 



36 PROBLEMS IN PHYSICS 

If a is increased by , and b is increased by /:?, 

the new area = (a 4- a) ( 4- /3) 

= # + tf/3 + ba + a/8 (Fig. 1 1) 
= ab + afi + a, 

when a/3 can be neglected ; z>. when the corner rectangle is 
very small in comparison with those on the sides. 

The student will be able to form approximate formulae similar 

to those given in many cases, 
and these, in connection with 
the various tables, will greatly 
reduce tiresome numerical 
computations which in them- 
j3 selves give no insight into 

physical laws and phenomena. 

In addition to these, a few points in connection with arrange- 
ment of work and notation may be useful. 

It is customary and convenient in expressing very large or 
very small numbers to write only the few figures actually 
observed or derived, and to indicate their position by a power of 
10 used as a multiplier ; as, 

45630000000 = 456.3 io 8 , 
.0000122 = 122 io~ 7 , etc. 

In every case where numerical work is required, spend a little 
time and thought in a general survey of the problem. 

Note in what order it is best to perform the various parts, 
whether factors can be cancelled or approximate values used. 
It is often best to write out the entire expression before any 
numerical work is done. Bear in mind that the understanding 
of the method and the facts involved is of primary importance, 
and numerical results are often only secondary. 



MECHANICS 



VELOCITY, ACCELERATION, AND FORCE 

23. Express a velocity of 22 mi. per hour in (a) feet per 
minute, (b) kilometers per hour, (c) centimeters per second. 

24. An express train leaves Albany at 10. 13 A.M., and arrives 
in Buffalo at 4.45 P.M. The distance is 297 miles. Find the 
average velocity of the train over this distance. 

25. Using velocities as ordinates and times as abscissas, 
draw a curve which might show the changes in velocity between 
any chosen time limits in a train's run. 'What is represented 
by the area included between the curve and the ^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 ? 

A particle has unit acceleration when it gains (or loses) unit velocity in 
unit time. The C.G.S. unit of velocity is a velocity of one centimeter per 
second. The corresponding unit of acceleration may therefore be called one 
centimeter per second per second. This is a somewhat cumbersome name, 
but it is conducive to clearness. 

33. Show that the general expression for acceleration is 



Take a as constant, integrate twice, and discuss the resulting 
equations. 

34. A body acquires in 4 sec. a velocity of 300 cm. per 
second. What is the value of its acceleration ? 

3-^ = 75 cm. per second per second. 

35. A train having a speed of 64 km. per hour is brought 
to rest under the action of brakes in a. distance of 510 m. 
What is the acceleration, if assumed to be constant ? 

36. What is the final speed of a body which, moving with a 
uniformly accelerated motion, covers 72 m. in 2 min., if 

(a) the initial speed = o, 

(b) the initial speed = 15 cm. per second. 



VELOCITY AND ACCELERATION 39 

37. Plot a curve showing the relation between distance 
passed over and time in the case of a body having a constant 
acceleration. What is shown by the pitch of such a curve at 
any given point ? 

38. Find the distance passed over in the /th second by 
a body having a uniformly accelerated motion. 

We have 

space described in t seconds = \ at 2 , 
space described in / i seconds = % a(t i) 2 ; 

whence space described in the /th second 

= I at 1 - \a(t- i) 2 
(*/-*> 

If the body has an initial velocity -z> , we have, obviously, 
space passed over in the /th second 



39. What are the ratios of the spaces passed over in succes- 
sive seconds by a body moving with a constant acceleration ? 

40. If a body starting from rest has an acceleration of 36 
cm. per second per second, over what distance will it pass in 
the seventh second ? 

41. A body has a uniform acceleration of 36 cm. per second 
per second. Initial velocity = o. 

(a) How far does it travel in 8 sec. ? 

(b) How far does it travel during the eighth second ? 

42. With an initial velocity of 14 cm. per second, how 
answer the preceding problem ? 

43. A train acquires 8 min. after starting a velocity of 64 
km. per hour. Assuming constant acceleration, what is the 
distance passed over in the fifth minute ? 

44. A body starting from rest with a constant acceleration 
passes over 18 km. the fourth hour. Find the acceleration. 



40 PROBLEMS IN PHYSICS 

a (2 x 4 - i) 
/ 4 th =18 ^-~ 

1 = 18, 

p er h our p er hour. 



45. A body starts from rest with a uniformly accelerated 
motion. In what second does it describe five times the distance 
described in the second second ? 

46. A and B are initially at the same point. If A move to 
the right with a uniform velocity of 6 km. per hour, and B 
move to the left with a uniform acceleration of 3 km. per hour 
per hour, what is the distance between them at the end of 
4 hr. ? 

47. Suppose in the preceding problem that at the expiration 
of the 4 hr. A turns and follows B with a uniform acceleration 
of 4 km. per hour per hour, how long before A overtakes B ? 

48. A body moving with uniform acceleration passes over 
distances of 13 and 23 m. in the seventh and twelfth minutes 
respectively. Find its initial velocity and its acceleration. 

49. A body starting from rest passes over 1.2 m. in the first 
second. The acceleration being uniform and the initial velocity 
zero, how long has it been in motion when it has acquired a 
velocity such that 6 m. are described in the last second of its 
motion ? 

50. A body m has an acceleration of 40 cm. per second per 
second ; a body n has an acceleration of 56 cm. per second. 
Provided both bodies start from the same origin at the same 
instant and travel (a) in the same direction, (b} in opposite direc- 
tions, how long before they will be 6 m. apart ? 

51. What definition of force is implied in Newton's first 
law ? What quantitative definition of force is embodied in New- 
ton's second law ? 

52. Discuss Newton's third law, giving one or more familiar 
examples. 



VELOCITY, ACCELERATION, AND FORCE 41 

53. Define the C.G.S. unit of force, the dyne. 

54. Define the dyne in terms of momentum and time. 

55. What is the character of the motion produced by a con- 
stant force acting on a given mass ? 

56. What constant force will give to a mass of 40 g. a 
velocity of 4.8 m. per sec. in 12 sec. ? 

57. A force of 30 dynes acts on a mass of 2 g. Find the 
velocity acquired in I sec. : 

30 = 2 a, 
a= 15. 

Find the velocity acquired in 6 sec. : 

v at = 6 x 15= 90 cm. per sec. 

58. Explain fully the difference between mass and weight. 

59. A body of 6 g. mass is moving with a velocity of 
3.6 km. per hour. Find the force in dynes that will bring it 
to rest in 5 sec. 

The application of a constant force to the body will produce a constant 
(negative) acceleration. Since the body is to lose all of its velocity in 5 sec., 
the rate of change of velocity, i.e. the acceleration is 

a= 3.6 x io 5 
36 x io' 2 x 5 

= 20. 

And the force necessary to produce this acceleration is 

f = ma = 6 x 20 
= 120 dynes. 

60. A mass of 500 g. moving at the rate of io m. per 
second is opposed by a force of 1000 dynes. How long must 
this force act in order to bring the body to rest ? 

61. A mass of 4 kg. falls freely. What is the value of the 
force acting upon it ? 

The acceleration due to gravity is 980 cm. per second per 

second. We have 

F = Ma 

= 4000 x 980 

= 392 x io 4 dynes. 



42 PROBLEMS IN PHYSICS 

62. Show that the dyne is, roughly speaking, the weight of 
i mg., and that the unit of force in the F.P.S. system (called 
the poundal) is the weight of -| oz. approximately. 

63. Engineers use the weight of a pound 2& the unit of force. 
Taking g as 32.2, what is the value of the unit of mass in this 
system ? 

64. Reduce a force of 2 kg. weight to dynes. 

65. Find the weight in dynes of a man who gives his weight 
as 140 Ib. 

66. What is the value of "the acceleration due to gravity" 
in terms of (a) the centimeter and second, (b) the foot and 
second, (c) the meter and minute ? 

67. Would any change occur in the weight of a ball if it 
were, carried to the center of the earth ? Imagine the ball to 
be in motion at the center of the earth ; is the same force 
required to stop it in a given time as would be required under 
the same conditions at the surface of the earth ? 

68. Aside from any possible difference in value, would there 
be any advantage in buying silver in Philadelphia and selling it 
in Berlin, provided weighings at both places were made with the 
same spring balance ? Explain your answer fully. 

69. A force equal to the weight of 2 kg. acts on a mass 
of 40 kg. for half a minute. Find the velocity acquired, and 
the space passed over in this time. 

70. A force equal to the weight of a kilogram acts on a 
body continuously for 10 sec., causing it to describe in that 
time a distance of 10 m. Find the mass of the body. 

71. The weight of a pound being taken as the unit of 
force (the engineer's unit, called by him simply a pound}, 
find the constant horizontal pull necessary to draw a block 
of 12 Ib. weight over a frictionless horizontal table, with an 
acceleration of 8.05 ft. per second per second. 



VELOCITY, ACCELERATION, AND FORCE 
In the fundamental relation 



43 



.we have 

whence 

The force required is 



F= 12 and a = 32.2; 

M = units of mass. 

32.2 



12 



F=Ma' = ---8.05 = 3 Ibs. weight. 

72. How far will a body fall from rest in five sec. ? What 
is its final velocity ? What is its mean velocity during this 
time ? 

The acceleration due to gravity is sensibly constant in any one locality. 
Problems in falling bodies, therefore, come under the head of uniformly 
accelerated motion, and the same formulas apply. 

73. The Washington monument is 169 m. high. In what 
time will a stone fall from top to bottom ? 

74. What velocity does a body acquire in falling through a 
distance of 100 m. ? 

75. From what height must a body fall to acquire a velocity 
equal to that of an express train making 96 km. per hour ? 

76. A stone dropped from the top of a building strikes the 
ground in 3 sec. What is the height of the building ? 

77. A pebble thrown vertically downward from the top of a 
tower with a velocity of 3 m. per second, strikes the earth in 4 
sec. What is the height of the tower ? 

78. Show that if two bodies A 
and B be let fall a time interval 
of 6 apart, As velocity relative to 
B is constant. 

After a time /, A has acquired the velocity 



But B has now been falling a time / and 
has acquired the velocity 




Fig. 12. 



44 PROBLEMS IN PHYSICS 

Their relative velocity is therefore 

V A - V B = gO, 

that is, simply the velocity acquired by A before B was allowed to fall. 
Graphically A's velocity is represented by the line OA drawn at a pitch g- Ws 
velocity is represented by BC drawn at the same pitch but having an inter- 
cept on the jr-axis of + 0. The constant intercept MN represents their rela- 
tive velocity. 

79. Extend the foregoing problem to the case in wbich 
both A and B have initial velocities, and discuss the conditions 
under which their relative velocity may be +, o, or . 

80. A body is thrown vertically upward with a velocity V Q . 
Find an expression for its velocity at any time /. 

The student should here remember that the conditions differ from those of 
a body thrown downward with an initial velocity only in the direction of this 
velocity. In time / the body acquires the velocity gt irrespective of its initial 
velocity. If we count velocity upward as positive, we must have then 

v VQ gt. 

81. A body is projected upward with a velocity of 30 m. 
per second. Find its velocity after 2 sec. ; after 4 sec. 

82. A body is projected upward with a velocity V Q . When 
will it reach a given height // ? 

The equation of this motion is 

* = /-*** 

Its solution gives two roots which, if real, are both positive. The smaller root 
is the time required to reach a height h during the ascent. The greater one is 
the time required to reach the same height during the descent. If the roots 
are imaginary, V Q is not great enough to carry the body to the height h. The 
student will readily interpret the case in which the roots are equal. 

83. A body is projected vertically upward with a velocity of 
24 m. per second. When will it reach a height of 10 m. ? 

84. Show that when a body is thrown upward it has, at a 
height h, numerically the same velocity, whether the body be 
rising or falling. 

85. A body is projected upward with a velocity of 20 m. per 
second. How high will it rise before beginning to descend ? 



^\\ BRA/; 

" OF THB 

fVERS*** 
VELOCITY, ACCELERATION, AND FORCE 45 

86. A ball is thrown upward with a velocity of 20 m. per 
second. How long before it will cease to rise ? How long 
before it returns to the hand ? 

87. The velocity of a body varies as the square of the time. 
If in 2 seconds after starting it has acquired a velocity of 40 
cm. per second, how far will it go in 5 sec. ? 

88. The velocity of a particle varies as its distance from 
the 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 



7 = -03- b 

Fig. 13. 

Obviously, grade sin <. 

90. The pitch of a plane is .75. With what acceleration 
would a body slide down its surface ? 

a =-sinec = 980 -f = 588. 

91. Which is the steeper, a 6 per cent grade or a 6 per 
cent pitch f 

92. A body sliding down an inclined plane describes in the 
third second of its motion a distance of 122.5 cm - Find the 
grade. . 



46 PROBLEMS IN PHYSICS 

, 

2 

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

^ 40 i 

Grade - - = -~- = = 5 per cent. 
g 980 20 

93. A body slides down the plane 
OA. Show that the velocity acquired 
on reaching A is the same as that 
which would be acquired in a free fall 
through the distance OH. 

94. A heavy particle slides from rest H 
down an inclined plane whose length is 

4 m. and whose height is 1.2 m. What is the velocity of the 
particle on reaching the ground ? What is the time of fall ? 

95. A man can just lift 150 Ib. What mass can he drag at 
a uniform rate up a frictionless grade of 7.5 per cent ? 




100 
x = 2000 Ib. 

96. A body slides down a plane 2.1 m. long in 3 sec.; to 
slide down another plane of the same height requires 5 sec. 
What is the length of the latter plane? 

97. A body slides freely down an inclined plane. The dis- 
tances passed over in successive seconds are in what ratio ? 
(Compare with 40.) 

98. A board is 4.95 m. long. To what angle must it be 
tipped in order that a body shall slide the full length in 3 sec. ? 

99. The height of an inclined plane is 426 cm. and its grade 
is 30 per cent. With what initial velocity must a particle be 
projected upward along the plane in order to come to rest just 
at the summit ? 




VELOCITY, ACCELERATION, AND FORCE 47 

100. A number of planes have lengths 
and inclinations equal to the chords OA, 
OB, etc. Show that if a number of parti- 
cles are allowed to slide down these planes, 
all starting from O at the same instant and 
without initial velocity, they will all reach B N 
the circumference in the same time. 

101. A point and a line lie in a vertical 

plane. Find the line of quickest descent from the point to 
the line. 

102. A freight train is moving at the rate of 8 mi. per hour ; 
a train man runs over the cars towards the rear of the train, a 
distance of 220 ft., in 30 sec. What is his speed relative to the 
surface of the earth ? 

103. Two trains of the same length are running with the 
same velocity on parallel tracks, but in opposite directions. 
Their combined length is 800 ft., and they pass each other in 
6 sec. What is the velocity of the trains relative to the track ? 

104. A and B are at one corner of a square. They desire to 
reach the diagonally opposite corner at the same instant. A 
chooses the diagonal path, while B follows around two sides. 
(a) What ratio must exist between the magnitudes of their 
velocities ? (It is assumed that these magnitudes are constant.) 

105. The component of a ship's velocity in an easterly direc- 
tion is 7.2 mi. per 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 four velocities a, b, c, and d of magnitudes 6, 8, 
12, and 20 units respectively. The angle between a and b is 
30, that between b and c is 15, and that between c and d\s 80. 
Find by resolving these velocities along any two rectangular 
axes their resultant in direction and magnitude. (See Intro- 
duction.) 

112. A man starts to row across a stream at a velocity of 4.4 
mi. per hour. If the velocity of the current at all points be 
3 mi. per hour, at what angle to either bank must he make his 
course in order to land at a point directly opposite that from 
which he started ? If there were no current, at what speed 
should he row directly across in order to make the trip in the 
same time as under the foregoing conditions ? 

113. A point is moving along a straight line with an accelera- 
tion of 22 cm. per second per second. Find the acceleration of 
the point in directions 30, 90, and 180 from this line. 

114. A particle is projected upward at an angle of 45 to the 
horizontal with a velocity of 120 m. per second. In what time 
will it reach its greatest height ? 

SUGGESTION. When the body reaches its greatest height, the vertical 
component of its velocity must be zero. Hence find the vertical component 
of the initial velocity, and divide by the loss of velocity per second; that is, 
find the time required for the body to lose all of its initial velocity in a vertical 
direction. 



VELOCITY, ACCELERATION, AND FORCE 49 

115. A particle is projected upward at an angle of 30 to the 
horizontal with a velocity of 70 m. per second. Find the time 
of flight, i.e. the time elapsing before the particle again reaches 
the horizontal. 

116. A body is projected with a velocity Fat an angle a. 
Find the horizontal distance (the range] described. 

Without considering the nature of the path, the range is readily obtained by 
multiplying the horizontal velocity, which is constant, by the time of flight. 

117. For a given initial velocity, show that the range is a 
maximum when the body is projected at an angle of 45. 

118. A body is projected at a given angle a to the horizontal. 
If the initial velocity be doubled, how does the range vary ? 

119. Show that any two complementary angles of projection 
give the same range. 

120. Find the greatest height to which a body will rise and 
its range, if it is projected with horizontal and vertical velocities 
of 40 and 80 m. per second. 

121. A body is thrown horizontally from the top of a tower 
100 ft. high with a velocity of 200 ft. per second. Find 

(a) the time of flight, 

(b) the range, 

(c) the velocity with which the body strikes the ground, 

(d) the angle at which it strikes the horizontal. 

122. Find the equation of the path of a projectile, and show 
that the trajectory is a parabola. 

123. Find an expression for the angle at which a particle 
must be projected with a velocity of given magnitude in order 
that it shall pass through a given point in the plane of the 
motion. What indicates that the given point is out of range ? 

124. (a) Define angular velocity, (b) Find the angular ve- 
locity of a wheel making 1000 revolutions per minute. 

In engineering practice it is common to express rate of rotation in revolu- 
tions per minute. In these units the angular velocity would be simply 1000. 
But in physics the velocity would be taken in radians per second. 



50 PROBLEMS IN PHYSICS 

125. Compare the angular and linear velocities of two points 
distant I and 2 m. respectively from the center of a wheel mak- 
ing 40 revolutions per minute. 

126. What are the dimensions of angular velocity ? 

127. A wheel makes i revolution in .5 sec. What is its 
angular velocity ? 

128. Express the angular velocity of the rotation of the earth 
on its axis in radians per second. 

radians per second. 



24 x 3600 

129. What is the linear velocity of a point on the surface 
of earth at 60 north latitude ? (Rotation alone considered. 
Mean radius of earth 6366.8 km.) 

130. A pinion having 16 teeth is geared to another having 
66 teeth. Compare the angular velocities. 

131. The driving wheel of a locomotive is 1.5 m. in diame- 
ter. If the wheel makes 250 revolutions per minute, what is 
the mean linear velocity of a point on the periphery? What is 
the velocity of the point when it is vertically above the axis of 
rotation ? When it is vertically below ? 

132. A freely falling body acquires a momentum of 12,054 
C.G.S. units in 3 sec. What is its mass ? 

133. The velocities of two bodies are as 6:4, and their 
momenta are as 9 : 2. What is the ratio of their masses ? 

6 m _ 9 . 
4 m' ~ 2 ' 



= 
m' 12 



134. The mass of a gun is 4 tons. If a shot of mass 20 Ib. 
-be fired with an initial velocity of 1000 ft. per second, what is 
the initial velocity of the recoil ? 



VELOCITY, ACCELERATION, AND FORCE 



135- What pressure will a man weighing 150 Ib. exert on 
the floor of an elevator descending with an acceleration of 
4 ft. per sec. per sec. ? Explain the sensation of being lifted 
which one has in an elevator suddenly arrested in its descent. 

136. A balloon rises with a uniform acceleration of 4 m. per 
second per second, carrying with it a spring balance upon the 
hook of which is hung a ball of 7.35 kg. weight, (a) What is 
the reading of the balance in kilograms' weight ? (b) What 
reading would the balance show if the balloon were descending 
with the acceleration named ? 

137. Two masses M and m are connected by an inextensible 
string passing over a smooth peg. Neglect- 
ing the mass of the string, find : (a) the 

acceleration of M and m, and (b) the ten- 
sion of the string. 



M< 



Fig. 16. 



Since the string is without mass, and since it does 
not stretch, it has the same tension T at every point in 
its length. Further, the downward velocity of M must 
equal the upward velocity of /, and their accelera- 
tions must be numerically equal. Let a be this com- 
mon value. 

Consider the forces acting on M. These are: (i) 
the weight of M downwards, and (2) the tension T 
upwards. And there are no others. Hence we write 

Mg - T = Ma. 

Again, considering the forces acting on ;//, we arrive at a similar relation, 
and, from the two equations thus found, the values of a and T are readily 
deduced. 

138. Show that the value of a found above is independent of 
the unit in which M and m are measured. Can the .same be 
proved of 7\ ? 

139. If the masses M and m are equal, what kind of motion 
is possible ? What is the value of the tension 7\ ? 

140. Two masses are connected by a weightless cord hanging 
over a smooth peg ; the sum of the masses is twice their differ- 
ence. Find the common acceleration. 



52 PROBLEMS IN PHYSICS 

141. Show that, in order to derive the expression for the 
acceleration in 137, it is not necessary to consider the tension 
in the cord. 

142. A cord passing over a frictionless pulley has fastened to 
its ends masses of 5 and 10 kg. respectively. Find the pull on 
the hook sustaining the pulley when the masses are in motion. 
(Neglect weight of pulley itself.) 

143. Explain how the value of g may be determined by 
Atwood's machine. 

144. One has weights aggregating 10 kg. ; it is required to 
divide the total into two parts such that when connected by a 
string passing over a pulley, the whole will have an acceleration 
\ that due to a free fall. 

145. A mass m is drawn horizontally along a smooth table by 
a cord passing over a small fric- 
tionless pulley and attached to 

a mass M. Find expressions for 

the acceleration of both masses ^ 

and the tension in the cord. 




146. In the last problem what 

must be the ratio of M to m in Fi s- 17 - 

order to produce an acceleration equal to f that of a freely 

falling body ? 

147. A mass of 20 g. hanging over the edge of a table draws 
a mass of 84 g. along the horizontal surface. Assuming no 
friction, find the tension in the cord. In what time will the 
second mass traverse the length of the table if this latter is 
3 m. long ? 

148. Two masses m l and m% are connected by a string. m 1 
hangs freely while m 2 rests on a plane inclined at an angle a to 
the horizontal. If the string passes over a small frictionless 
pulley at the summit of the plane, find the resulting acceleration. 



VELOCITY, ACCELERATION, AND FORCE 53 

Consider the forces acting on m r These are : (i) its weight m^g and (2) 
the cord tension T. If f be the common acceleration, we must have 



So, the forces acting on m z are the resolved part of its weight acting along the 




Fig. 18. 

plane and the cord tension. This gives another and similar equation in 
which f and T are unknown. By eliminating these quantities are readily 
found. 

149. Show that when a = 90, the results are identical with 
those obtained in 142 ; also that when a = o, the results are 
identical with those in 145. 

150. In order to pull a mass of 1000 kg. up an incline of 30, 
a rope and pulley are used as in 148. Neglecting all friction, 
compute the tension in the rope when a mass is used sufficient 
to cause an acceleration of 0.4 m. per second per second. 

151. Find the resultant of two forces of 6 and 9 kg. weight : 

(1) Acting in the same straight line and in the same direction. 

(2) Acting in the same straight line but in opposite directions. 

(3) Acting at angles of 30, 45, 90, 120, and 150. 

152. A force is inclined 36 to the horizontal. What is the 
ratio of its vertical to its horizontal component ? 

153- Three concurrent forces of 8, 30, and 12 kg. weight are 
inclined to the horizontal by angles of 32, 60, and 143 respec- 
tively. Find the horizontal and vertical components of their 
resultant. 

154. Two forces acting at an angle of 60 have a resultant 
equal to 2V3 dynes. If one of the forces be 2 dynes, find the 
other force. 



54 PROBLEMS IN PHYSICS 

155. Two equal forces act on a particle. If the square of their 
resultant is equal to three times their product, what is the angle 
between the forces ? 

156. At what angle must two forces act so that their resultant 
may equal each of them ? 

157. Find the angle which shall make the resultant of two 
forces of constant magnitude a maximum. 

158. Let the angle between two forces of constant magni- 
tude increase continuously from o to TT. Discuss the variation 
of the angle between the resultant and one of the forces. 

159. Show that when three forces in the same plane are in 
equilibrium their lines of action meet in a point. 

160. Show that when three forces are in equilibrium each 
force is proportional to the sine of the angle between the other 
two (Lami's theorem). 

161. Find by graphic construction the resultant of four 
forces of 3, 7, 5, and 12 Ib. weight acting on a particle, and 
represented in direction by the successive sides of a square. 

162. Two forces of 3 and 4 units respectively are balanced 
by a third force of "N/37 units. Find the angle between the 
first two forces. 

163. A mass of 4 kg. is suspended at the middle of a cord 
whose two halves make an angle of 30 with the horizontal. 
What is the tension in the cord ? (Mass of cord neglected.) 
The mass remaining the same, how may the tension in the cord 
be varied ? Discuss the law of variation. 

164. A weight of 14 kg. hangs at the end of a string ; a 
force is applied horizontally deflecting the string 30 from the 
vertical. What is the value of this force and what the tension 
in the string ? 




VELOCITY, ACCELERATION, AND FORCE 55 

165. A string connecting two equal 
masses hangs over three smooth, equi- 
distant pegs. Neglecting the weight 
of the string, find the resultant pressure 
on each peg. 

166. Why is a long line desirable in 
towing a canal boat ? To pull a canal 
boat at a uniform rate requires a force 

Fig. 19. 

in the direction of motion of P Ib. 

weight. If the rope make an angle of 10 with the line of 
motion, and if the weight of the rope be neglected, what pull 
must the horses exert ? 

167. A body of weight 30 kg. is suspended by two strings 
of lengths 5 and 12 m., attached to two points in the same hor- 
izontal line whose distance apart is 13 m. Find the tensions in 
the strings. 

168. A mass of 40 g. rests on a plane inclined at 30. Find 
in grams' weight the force parallel to the plane : (i) neces- 
sary to hold it there, (2) necessary to draw it uniformly up the 
plane, (3) necessary to cause an acceleration of 30 cm. per sec- 
ond per second up the plane. 

169. A block having a mass of 100 g. is prevented from 
sliding down an inclined plane by means of a cleat. Find the 
inclination of the plane which will make the pressure on the 
plane equal that on the cleat, and give the numerical value of 
their sum. 

170. A block is held from sliding down an inclined plane by 
a cleat. Plot two curves showing the variations of the pressure 
exerted by the block (i) on the plane and (2) on the cleat, with 
variations of the angle of the plane. 

171. Determine analytically the angle for which the sum of 
the cleat pressure and plane pressure is a maximum. 




56 PROBLEMS IN PHYSICS 

172. A ball is held at rest on an 
inclined plane of given angle a by means 
of a cord. Find the cord tension when 
the angle between the cord and plane 
is 6. For what value of 6 is this tension 
a minimum ? 

173. The upper end of a ladder rests 
against a smooth vertical wall ; the lower 
end on a smooth horizontal floor, slip- 
ping being prevented by means of a p . 2Q 
cleat. The ladder is of uniform cross- 
section, weighs 100 lb., and is inclined at 60 to the hori- 
zontal. Find the reactions of the different surfaces against 
which the ladder rests. 

174. When a person sits in a hammock the tension on either 
sustaining hook is greater than the person's weight. Explain. 
Does the tension increase or decrease as the hammock is made 
more nearly horizontal ? 

175. A string hanging over a pulley has at one end a mass of 
10 kg. and at the other masses of 8 kg. and 4 kg. When the 
system has been in motion for 5 sec., the 4 kg. mass is re- 
moved. Find how much farther the weights go before coming 
to rest. 

176. The ram of a pile driver weighs 500 lb. It is allowed 
to fall 20 ft. driving a pile 6 in. Find the value of the resist- 
ance, assuming it to be uniform. 

[Consider the acceleration needed to bring the body to rest in the given 
distance.] 

177. Show graphically how to find the resultant of two 
parallel forces, (a) when the forces are like, and (b) when the 
forces are unlike. 

178. Apply the graphical construction to the case of two 
equal, unlike forces and interpret the result. 



VELOCITY, ACCELERATION, AND FORCE 57 

179. A man carries a bundle at the end of a stick placed over 
his shoulder. If he varies the distance between his hand and 
his shoulder, how does the pressure on his shoulder change ? 

180. The resultant of two like parallel forces is 16 kg. weight 
and its point of application is 6 cm. from that of the larger 
force, which is 10 kg. weight. Find the distance of the smaller 
force from the resultant. 

181. Equal weights hang from the corners of a triangle which 
is itself without weight. Find the point at which the triangle 
must be supported in order to lie horizontally. 

SUGGESTION. The forces at the corners are all equal and parallel. The 
resultant of any two must act at the 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 P Ib. weight 
is placed. Find the reactions of the piers. 

184. What is a couple and what is the moment of a couple ? 

185. Show that the algebraic sum of the moments of the two 
forces forming a couple about any point in their plane is 
constant. 

1 86. One of the forces of a couple is 60 dynes ; the distance 
between the forces is 0.3 m. Find the moment of the couple. 

187. A straight bar is acted upon at its ends by two equal 
and parallel but opposite forces of 12 kg. weight each. The 
bar makes an angle of 45 with the direction of the forces and 
is 3 m. long. Find the moment of the resulting couple. 



CENTER OF INERTIA (OR MASS) (OR GRAVITY; 

1 88. Two equal weights are connected by a light, stiff rod. 
Find the center of inertia. 

189. How would the center of inertia be moved if one of the 
weights were doubled-? if both were multiplied by three ? 

190. Three weights,, 4, 5, and 7, are joined by stiff weightless 
rods. Find the center of mass of the system. 

191. What is the center of gravity of a triangle ? a square ? a 
parallelogram ? a trapezoid ? Test your answers with pieces of 
cardboard. 

192. The diagonals of a square are drawn, and one of the tri- 
angles resulting is removed. Find the center of gravity of 
the remaining figure. 

193. Two lines are found on a surface such that the surface 
will " balance " about each. What point is determined by their 
intersection ? 

194. Four masses are supposed concentrated at the points A, 
B, C, D\ masses 9, 5, 6, 10, respectively. The lengths OA, AB, 
BC, CD are 5, 8, 4, 10, respectively. Find the distance of the 
center of mass of the system from the point O. 

A o D 

O 




^i in __?!__ 

Fig. 21. 

We have 5-9+ 13-5 + 17-6 + 27-10 = sum .of mass-distance products, 
9+5+6+10 = sum of masses. 

.-. distance required is -Vo 2 - = 16+. 

58 



CENTER OF INERTIA 59 

The distance from O to the center of gravity may be found from an equa- 
tion expressing the fact that about that point the sum of the moments of the 
couples due to gravity is zero. 

Let x : = distance required. 

Then lever arm for gravity action on A is ~x 5. 

Whence moment of couple due to A is gQc 5)9, 
couple due to B is g(x 13)5, 
couple due to Cis^(^ 17)6, 
couple due to D is g (x 27)10. 

Sum equals o. .-. 30 .r = 482, ~x = 16+, as before. 

195. A body is suspended by a flexible cord. What position 
will the center of gravity assume ? Explain. 

196. Explain the connection between the center of gravity 
of a body and its stability. 

197. Express the fact of no resultant couple about the center 
of gravity in the notation of the calculus. 



xx 



35 dx x z 

Fig. 22. 

When the body is linear or is symmetrical about a line. 

Let x= the distance of C.G. from O, 

x = the distance of any mass element from O, 
dx= the length of element. 
Then pdx mass element, 

x ~x lever arm. 
.-. mom. of couple = pdx(x ~x)g> 

Sum of mom. = I 2 pdx(x ~x) = o. [By def. of C.G. 

Jx r 



(a) Find ~x for a uniform rod of length /. 

(b) Find x for a rod where p increases from x^ to ;r 2 , i.e. where 
= 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- 

l pxdv 
nates are given by three equations of the form x = *- 

pdi> 

199. Explain the distinction in meaning and use between the 

f - ^mx 

above expression for x and x = 

200. Find C.G. of a cone of revolution. 




Fig. 23 

Take dv as a slice || to base. Then 



rjp**dx 

201. Find C.G. of a sector of a circle. 

202. Find C.G. of a segment of a circle. 

203. Find C.G. of an arc of a circle. 

204. Apply the general formula for the 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 
by m i* t move toward each other. Where will they meet ? 

206. Show that the momentum of any system of bodies, each 
of which has motion of translation only, is the same as the 
momentum of the sum of the masses moving with the velocity 
of the center of gravity of the system. 



CENTER OF INERTIA 6l 

207. Two masses are joined by a rigid rod ; the system is 
thrown in the air so that it whirls. What will be its center of 
rotation ? 

208. Two spheres glide freely on a light, rigid rod, and are 
joined by a spiral spring sliding freely on the rod ; the system 
is thrown so that the rod has an initial angular velocity <w . 
Discuss the relative position of the two spheres with reference 
to the center of gravity of the system. 



WORK AND ENERGY 

209. A constant force of 20 dynes moves a body 100 cm. 
What work is done ? 

210. A force of 9000 dynes is exerted constantly on a body, 
and moves it 4 m. per second. How much work is done in 

1 min. ? 

211. How much work is required to lift I kg. from the sea 
level to a height of I m. where g = 980 ? 3 kg. ? 8 kg. ? 

212. How much work is required to raise i kg. 2m.? 

2 kg. 5 m. ? 

213. What work is required to raise 80 kg. 3 m. against 
gravity ? 10 m. ? 

214. Raising 80 kg. 8 m. is equivalent to raising 40 kg. how 
many meters ? To lifting what mass 5 m. ? 

215. 98 io 10 ergs are expended in raising 100 kg. How high 
were they raised ? 

216. A force of 40 dynes is applied at an angle of 60 to the 
path along which the point of application moves. What work 
will be done when the point is moved 1000 cm. ? 

217. 8- io 8 ergs of work are required to move a body 400 m. 
in a straight line. What force is required if applied at an angle 
of 10 with the path ? of 20 ? of 30 ? of 80 ? 

218. 4 io 8 ergs of work are required to move a body 8 io 4 
cm. What was the average force required ? 

219. 6 io 10 ergs of work have been expended in moving a 
body against a resisting force of 3 io 5 dynes. How far was it 
moved ? 

62 



WORK AND ENERGY 63 

220. A stone of volume io 3 c.c., sp. gr. 2.6, is raised from 
the bottom of a lake to the surface, a distance of 20 m. Find 
the work done. See Ex. 422. 

221. Find the work done in forcing a block of wood, volume 
8 io 4 c.c., sp. gr. .7, to the bottom of a tank of water 4 m. deep. 
What if tank were filled with mercury ? 

222. Show that if gravity be the only resisting force, the 
work done on a given mass in raising it a given height is in- 
dependent of the path. Or that the force required always 
decreases in the same ratio as the path increases. 

223. Show why it is easier to draw a load up an inclined 
plane than lift it vertically, neglecting friction. What element 
is decreased ? What increased ? 

224. A vertical tank having its base in a horizontal plane is 
to be filled with water from a source in that plane. The area 
of the cross-section is 4 sq. m., the height is 6 m. Find the 
work required to fill it. 

225. Show that the work required to raise a system of bodies 
each to a certain height is the same as the work required to 
raise the entire mass to a height equal to that through which 
the center of gravity of the system is raised. 

226. A body is raised 80 m. against a force which constantly 
increases. The initial value of the force is 40 dynes, its final 
value 460 dynes. If the force increased uniformly with the 
distance moved, how much work was done ? 

227. In an ordinary swing is the force required to displace 
the swing constant ? If not, how could the work be computed ? 

228. A uniform rod io m. long, and mass per centimeter 
length 5 kg., is drawn vertically upward a height of io m. 
How much work is done? How much work would be'required 
to raise the rod from a horizontal to a vertical position ? 

NOTE. Consider the average height of elements of mass. 



64 



PROBLEMS IN PHYSICS 



229. A plank 4 m. long is hinged at one end. The plank 
is raised so as to make an angle of 45 with the horizontal. 
What work is done ? (Mass of I cm. of plank 9 kg.) 

230. Express work in terms of mass, acceleration, and dis- 
tance. 

231. If the unit of time were taken as 2 sec., how would the 
unit of work be altered ? 

232. Show that power = force x velocity. What does the 
statement mean when the velocity is changing ? In what units 
must force and velocity be measured so that power may be 
expressed in ergs per second ? 

233. In what two general ways is the energy of a railway 
locomotive expended while the train is acquiring velocity ? 

234. The force required to overcome the friction of a wagon 
on a certain road is 2 - io 10 dynes. How much work is done in 
drawing it 20 km. ? 

235. On a perfectly level road it was found that the pull re- 
quired to keep a wagon moving uniformly was .01 of its weight. 
What work is done in drawing a wagon weighing 2000 kg. a 
distance of 3 km. ? 

236. A man presses a tool on a grindstone with a force equal 
to io kg. weight. The circumference of the stone is 3 m., the 
coefficient of friction .2. How much work is done in one turn 
of the crank ? (Neglecting friction of bearings, etc.) 




Fig. 24. 



237. If BC = . i AB y what mass at M 1 will draw M 2 up AB 
without acceleration, neglecting friction ? What effect would 
be observed if a greater mass were placed at M l ? 



WORK AND ENERGY 65 

238. State how you could apply the principle of work to 
above case when there is friction. 

239. Find the work done in drawing 120 kg. up an inclined 
plane of base 4 m., height 3 m., //< = -f^. 

240. How much of the work is due to friction ? 

241. A mass of 100 g. is moving in a circle of radius I m., 
and makes 10 revolutions per second. What is its kinetic 
energy ? What would be its energy if the circle were half as 
large ? 

242. Five masses, 3, 8, 5, 7, and 1 1 g., are attached at dis- 
tances n, 7, 5, 8, 3 cm., respectively, from the centre of a 
wheel making 20 revolutions per second. Find the kinetic en- 
ergy of each. How far from the center could the whole mass 
be placed so that the energy would be the same ? 




Fig. 25. 

Let a constant force F be applied at a point r distant from O J_ OP r If 
the rod OP^ is rigid, the work done in turning through an angle 0, since 
P^P Z = rO, is FrB = force x displacement. So work done by a couple or 
torque 

= moment of couple (Fr) x angle turned through 
= torque x angle turned through 
= average torque x angle turned through 
.when torque is not constant 

= FrdO. [Where Fr =/(0) 

243. A shaft s turns 120 times per minute. The radius of 
the shaft is 2 cm. The distance from the center of the shaft to 
the point where the mass is applied is 2 m. It requires a mass 

F 



66 PROBLEMS IN PHYSICS 

of 80 kg. to hold the lever in equilibrium. Find the work done 
in 5 min. 



Fig. 26. 

244. A mass of 80 g., moving with a velocity of 10 cm. per 
second, has what kinetic energy ? 

245. What is the kinetic energy of a bullet, mass 100 g., 
velocity 1 50 m. per second ? 

246. A body of mass 60 g. has a velocity 40 cm. per second, 
and an acceleration of 10 cm. per second per second. How 
much kinetic energy will it acquire in the next second ? How 
much the fifth second later ? 

247. A body of mass 5 kg. is given an initial velocity of 20 m. 
per second on smooth ice. If the total average resisting force 
which it encounters is io 5 dynes, how far will it go before coming 
to rest ? How much energy will it have when it has gone half 
the distance ? 

248. A ball of mass 4 kg., velocity 80 m. per second, penetrates 
a bank of earth to a depth of 2 m. Find average resistance. 

249. A ball of mass io g. enters a plank with a velocity of 
io m. per second and leaves it with a velocity of 2 m. per second. 
How much energy has it lost ? 

250. If the plank is 20 cm. thick and all the work is expended 
in piercing it, what is the average resistance ? 

251. A bullet is fired vertically upward with an initial velocity 
of 500 m. per second. What is its kinetic energy : (a) initially ? 
(b) when 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 E P in each case ? 



WORK AND ENERGY 67 

252. A mass m falling freely acquires how much kinetic 
energy per centimeter of its fall ? It loses how much potential 
energy ? 

253. Two balls of mass 100 and 200 kg. are attached to a 
firm light rod. The distance between the centers of the balls 
is i m. The system is thrown so that the center of gravity has 
a velocity of 20 m. per second, and the system turns ten times 
per second around this center. Find the kinetic energy of the 
system. 

254. Compare their energies of rotation about the center of 
gravity of the system. 

255. What is meant by the term "closed system " as applied 
to energy ? Give examples. 

256. State in words the relation between the work done on a 
system by an external force and the rate of gain of energy by 
the system and the losses by friction. 

257. Trace the energy changes in a single vibration of a 
pendulum : (i) When the air resistance may be neglected. 
(2) When air resistance is taken into account. 

258. Express in calculus notation the statement that the sum 
of the potential and kinetic energy of the bob of a simple pendu- 
lum is constant. 

259. A mass of 60 g. is vibrating in a straight line with 
S.H.M. The length of the line is 4 cm., the periodic time is 
2 sec. What is its average kinetic energy ? 

260. The velocity of a bullet is decreased from 500 to 400 m. 
per second by passing through an obstacle ; its mass is 100 g. 
What energy has it lost ? What has become of that energy ? 

261. Calculate (in ergs, and also in 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. If the stone were attached to a very flexible and exten- 
sible spring, what alteration of energy distribution would occur ? 

264. A solid sphere of cast iron is rolling up an incline of 
30, and at a certain instant its center has a velocity of 40 cm. 
per second. Explain how to find how far it will ascend the 
incline, neglecting friction of all kinds. Would the distance be 
the same if it were sliding up the incline ? 

265. If the sphere were hollow, would it acquire the same 
velocity as the solid one in rolling the same distance down the 
plane ? 

266. What are the dimensions of power ? If the unit of 
time were the minute, the unit of length the meter, how would 
the unit of mass need to be altered that a given power should 
be expressed by the same number ? 

267. Define erg, joule, watt. 

268. A constant force is applied to a body on a horizontal 
plane. If the applied force is greater than the friction between 
the body and the plane, why cannot an infinite velocity be 
obtained ? 

269. The mass of a car is 2000 kg. The resistance due to 
friction is 12- io 4 dynes. A man pushes the car with a force 
which would support a mass 90 kg. His maximum power is 
746 io 6 ergs per second. How long can he continue to exert 
his full force ? 

When the component of force along the path of the point of application is 
variable, we must find how its magnitude varies along this path and apply the 
integral calculus to add up the elements of work. 

-When W = (* z Fdx, 

i 

where F must be expressed in terms of x, i.e. F = f(x). 



WORK AND ENERGY 69 

The cases of most interest are perhaps when 

f(x) kx, \k a constant. 



The first applies to cases of compression and stretching, as springs, etc. ; 
the second to gravitation, electricity, and magnetism, etc. 

270. When F= $x t find the work done in displacing a body 
loom. 

Jio 4 fr ^-2-no 4 

5-*aEr-|i- = f .io 8 ergs, 

which is the same as taking half the sum of the initial and final 
force, and multiplying by entire displacement. 

20 

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

3C 

of application from x = 20 to x 220. 

r m Jr 
W = \ 20 



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

272. A coiled spring is attached to a 50 kg. weight. What 
work is done if the increase of length of the spring is 2 m. 
when the weight is just lifted ? 

273. If the pressure of a gas increases as its volume decreases, 
show how work done in compression could be computed. 

274. A horse is hitched to a loaded wagon by a long exten- 
sible spring. Does the work done by the horse in just starting 
depend on the ease with which the spring is stretched ? 

275. A bicycle rider moves up a grade against the wind. 
Against what forces does he do work ? In what ways does he 
expend energy ? From which of these expenditures can he get 
a return of energy, and how ? 

The general expression for work may be written W = ^Fds, where ds is 
so short that F may be considered constant over its length. We may then 
resolve both /^and ds along any three lines we please, as OX, OY, OZ. 



PROBLEMS IN PHYSICS 



Let jr, /, z, components of F, be X, Y, Z. 
Let x, y, z, components of ds, be dx, dy, dz. 



Then 



W 



= j* [Atf* 



ds ds ds. 
where X, Y, Z may depend on x, y, z. 

When F is constant and along j, the formula reduces to \ Fds, as the 
student may prove. 





Fig. 27. 

As an example we may take the work done by a couple in turning through 
360. Taking the plane of xy as the plane in which the lever arm lies, we have 



Ydy~\ 



By symmetry we see that the 



Xdx = - 



W= 



= F- circumference of O. 



It is often convenient to use the law of the 'conservation of energy in the 
solution of problems dealing with machines of various types. To do this, 
we form an equation involving the element required; one member of the 
equation representing all the work expended on the machine, the other all the 



WORK AND ENERGY 71 

work done by the machine. That is, equate the entire energy supplied to the 
machine to the entire energy used, stored, and wasted. 




Fig. 28. 

The energy given to the machine may be used in various ways ; as, 

(1) Lifting weights, etc. (visible and useful work). 

(2) Overcoming friction (waste, transformed to heat). 

(3) Strain of parts of machine (potential energy). 

(4) Momentum of parts of machine (kinetic energy). 

(5) Transformed to other forms, as electric, chemical, etc. 

The complete analytical expression in case all of these are considered is 
likely to be very complicated. We therefore simplify matters by neglecting 
certain items of relatively small importance, yet it should be remembered that 
in actual cases these may cause serious errors if neglected. 

In most of the problems that follow, (3), (4), and (5) are neglected, and 
unless otherwise stated, friction is also negligible. 

The student should note careftilly that all forces which do not cause motion 
are excluded, as they do no work. 

276. Explain why a machine should be of sufficient rigidity 
that the deformation of its parts should be extremely small. 

277. Distinguish between the total energy of a system and 
its available energy. 

278. A railway train, in which the couplings between the 
cars are heavy springs, begins to move, due to the work done 
by the engine. State how the energy supplied is being distrib- 
uted while the train is acquiring speed. 



?2 PROBLEMS IN PHYSICS 

279. If the steam is shut off, from whence comes the energy 
which keeps the train in motion ? 

280. What becomes of the potential energy which we store 
in a watchspring when we wind it ? 

281. The pitch of a screw is .5 mm. A lever 40 cm. long is 
used to turn it. A force equal to a weight of 20 kg. applied to 
the lever will cause the screw to exert what force ? 

282. Show that the screw is an example of the inclined 
plane. 

283. A lever is 2 m. long, the point of support 30 cm. from 
the end. A force of io 8 dynes applied to the long arm will 
give what force at the short arm ? 

Consider the work in any displacement. Then 
force applied x distance it moves = force exerted x distance moved. 

Let the angle turned through = 0. 
Distances are 170$ and 30$. 
Work = io 8 1 70 = x 30 0. 

.-. x - 1 / io 8 dynes. 

284. The radius of the wheel of a copying press is 30 cm. 
One turn lowers the plate .25 cm. Find the force exerted if 
the applied force is enough to lift 20 kg. 

285. In a hydrostatic press the distances moved by the 
pistons are in the ratios of i to 1000. What is the force ratio ? 

286. In an ordinary pump handle the long lever arm is 3 ft., 
the short one 6 in. What force applied to the longer will lift 
40 kg. on the shorter ? 

287. A system of gear wheels is used to raise weights. 
When the first is turned 360 the last turns 60. The radius of 
the first is four times that of the last. What is the force ratio ? 

288. In the system of pulleys connected as shown in Fig. 29, 
find the relation between w and W\ (a) by principle of work ; 



WORK AND ENERGY 



73 



I 



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

289. In a system of eight movable pulleys connected as in 

Fig. 29, find the weight which 20 kg. , 

would lift, neglecting the weight of 

the pulleys and friction. 

290. Find by the principle of work 
the relation between w and W when 
each pulley weighs / grams. It is 
found by experiment that the values 
of w computed above are too small 

to explain this. 

I 1 291. A system 

of two movable 
pulleys, as in Fig. 

29, is of negligible 
friction, and the 
weight w is twice 

as large as it should be for equilibrium. 
What will be the acceleration of w ? of W} 

292. In a system connected as in Fig. 

30, find the relation between w and W\ 
(a) Neglecting weight of pulleys, (b) When 
lower block weighs M grams. 

293. Find the relation when there are n 
pulleys above and n below. When there is 
one more above than below. 



W 

Fig. 29. 



X 



In the wheel and axle we have, if connection is 
rigid and the cord inextensible, light, and flexible, 
Work done by falling of M l when angle turned 



r ~ R' 

(Weights are inversely as radii.) 



74 



PROBLEMS IN PHYSICS 



For gear wheels we have the same principle. Let /?, R v and r be the 
radii of the large wheel, the small wheel, and the axle of the small wheel. 





M 2 



M, 

Fig. 31. Fig. 32. 

If there is no slipping when R turns through an angle 0, R l turns through 



an angle (9. 



Work by M l = M l - RQ. 
Workon^ = ^'^ 



294. If the axle of the wheel (Fig. 33) be 4 cm. in diameter, 
the mean radius of the wheel 40 cm., the mass of the rim 
800 g., the axle and spokes being small in comparison, the 
mass M = 200 g, what will be the velocity of M when it 
has fallen 4 m. ? Forming the energy equation we have, if v is 
the velocity of M, 





M ^__ 

Fig. 33. Fig. 34. 

EI = % 200 v 2 - + \ 800 [20 2/] 2 [Kinetic energy acquired. 
Zip = 200 g 400, [Potential energy lost. 

Equate and solve for v. 



WORK AND ENERGY 



75 



M + M 1 



295. A mass M is suspended by a flexible cord wound 
around a heavy rimmed wheel. The radius of the wheel is R ; 
the mass of the rim M r . What will be the velocity of M after 
falling a distance Ji ? (Neglecting the spokes.) 

Let v velocity required. 

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



Lost 

296. In Fig. 35, 
M= 8000 g. 
M' = 200 g. 
R= i m. 
r= 2 cm. 



The spring lies on a 
frictionless shelf, and is 
connected by flexible thread 

to the axle. If M falls 2 m., discuss the energy changes in the 
system : (i) Neglecting friction of all kinds. 
(2) When friction is considered constant. 

297. A weight W is carried through the point 

P any number of times. Is its potential en- P 
ergy when at the point P any different at 
successive times of passage ? 

298. A crank C is turned, thereby " winding 

C up " a spring s. Is the potential 
energy of the crank dependent only 
on its position ? Explain. 

299. A strong rubber band is 
stretched between two points on a 
horizontal table A and B. If A 
Fig 37< remain fixed and B is moved to B' 




Fig. 35. 




Fig. 36. 




76 PROBLEMS IN PHYSICS 

by any path such that the band is straight, show that the work 
done depends only on AB' AB\ 
i.e. on the initial and final posi- 
tions of the ends. 

300. If the band were drawn 
around a peg at C, or made to 
occupy any curved path between 

A and B, upon what would the Fig . 38 

work done depend ? 

301. If the force law is m , find the work done in carrying 
m' from r^ to r^. 

Since the force is not constant, we must divide up the displacement into 
very short elements, multiply each by the mean force for that element, and 
add all these results together; 

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




W= r-mm' * = mm' f- - - 



or 



T C GO T 

If r = oo r = -- since = o. 



ftlftl I 

-- , 
r \ 



302. How much potential energy will I kg. have when it is 
I m. above the 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 ? 

' 306. In skating on smooth "level" ice, does one gain potential 
energy ? In climbing an icy hill, is one's potential increased ? 



WORK AND ENERGY 77 

307. Are the horizontal floors of a building " equipotential " 
surfaces ? 

308. If the work done in carrying I kg. from the basement 
to the first floor is called the potential of that floor, the distance 
between the floors being uniform, what is the potential of the 
fourth floor ? 

309. If the potential of the first floor is 3-io 8 , what work will 
be required to carry 80 kg. from the first to the third floor ? 

310. In a brick building perfectly built, do the horizontal 
edges of the bricks lie in equipotential surfaces ? Given the 
potential at the level of one layer, the mass of one brick in a 
layer, the number of bricks in that layer, how find the work 
in elevating the whole number ? 

311. A man walks from a certain point along any path or up 
hill and down, and returns to his 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 is drawn up a rough inclined plane. Against 
what forces is work done ? State the relation between the 
energy expended, the potential energy of the body at its highest 
point, and the work done against friction. 

314. How much work is done in taking 80 units of mass 
from a place where the potential is 5 to one where potential 
is i ? where the potential is 25 ? 

315. A reservoir on a hill filled with water is said to have 
what potential ? If connected by a pipe with the 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. If the potential at points along a certain line is given 
by V=f(x), find the force function. 

318. If Vf(x) between two points x l and :r 2 , and the force 



is constant, what condition does - satisfy between x^ and x^ ? 

319. Two cylindrical reservoirs of the same capacity stand on 
the same horizontal plane ; the height of one is four times the 
height of the other. Which would you prefer to fill with water ? 

320. When two reservoirs have the same depth of water and 
one is larger than the other, compare the pressure exerted by 
each at a given point to which each is connected by a pipe. 
Compare the potential energy of the two. 

321. If the values of a working force are taken as y, and the 
distance moved as ,r, 

what will the area of 
the surface between two 
ordinates, the curve, and 
the axis of x mean ? 

322. When will the 

, i , i i SEA LEVEL 

curve be a straight line : - 
What will its slope 
mean ? 



flC 



323. A constant force 
acts on a mass subject 
to friction, the force be- 
ing greater than the 

friction. Draw the time- Fig. 39. 

velocity curve (initial velocity o). Discuss the curve, and ex- 
plain the meaning of its slope, area, etc. 

324. A reservoir A is made below the 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. 

330. Explain why rails are " sanded." Why is a violin bow 
"resined"? 

331. A certain force is required to move one surface over 
another when the pressure between them is P. If P were 
doubled, what force would be required ? if /JL were doubled and 
P were unchanged ? if both P and //, were tripled ? 

332. A mass of 80 kg. on a horizontal plane requires a force 
equal to the weight of 1.6 kg. to keep it in uniform motion. 
What is the coefficient of friction ? 

333. The coefficient of friction between two surfaces is 0.14. 
A pull of 20 kg. weight will overcome what pressure between 
the surfaces ? 

334. If the coefficient of friction is 0.2 between the driving 
wheel of a locomotive and the rail, what must be the weight, in 
tons, of the locomotive in order to exert a pull equal to 8.96 T. ? 

335- The coefficient of friction between a block and a plane 
is .3. At what angle should the plane be inclined that the 
block may just slide down it when started ? What is the angle 
named ? 

79 



80 PROBLEMS IN PHYSICS 

336. For a certain plane and block, the coefficient of friction 
is .2. What force applied parallel to the plane would just draw 
the block up if it weighs 100 kg., and the plane is inclined 5 
with the horizontal ? 

337. L is a load drawing W along a horizontal plane by 
means of a cord and pulley, as in Fig. 17. 

If L = 8 kg., W = 40 kg., pulley friction o ; find /*. 
If /i=.i8, IV =80 kg., pulley friction o ; find L. 
If /LI = .3, L 10 kg., pulley friction o ; find W. 

Supposing in each case that the system moves uniformly 
when started. 

338. Solve each part of the preceding example if the co- 
efficient for the pulley = .03. 

339. If L were twice as large as specified in 337, find the 
acceleration. 

340. Draw a diagram showing the forces acting when one 
body is slid uniformly over another. 

341. The coefficient of friction between two surfaces is .2. 
They are inclined at an angle of 60 with the horizontal. What 
will be the acceleration ? 

342. A mass of 40 kg. is placed on a plane inclined 50. 
The coefficient of friction is .3. What force will be required to 
draw the mass up the plane with an acceleration of 100 cm. 
per second per second ? 

343. If a series of observed values of L and J^were used as 
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 8 1 

346. A block of weight W rests on a horizontal plane ; an 
elastic spring is used to draw it along at a uniform rate. If 
the angle at which the elongation of the spring is least is <, find 
the coefficient of friction. 

347. A force of 8 io 5 dynes acts for i min. on a mass of 
i kg. sliding on horizontal surface. The velocity acquired was 
3 io 4 . What was the coefficient of friction ? 

348. A long plank lies on a nearly smooth inclined plane. A 
man attempts to walk up the plank. What happens ? 



PENDULUMS. MOMENTS OF INERTIA 

349. Find the time of vibration of the following simple 
pendulums : [g = 980] ; / = 16 cm., 32 cm., 36 cm., 9 cm. 

350. A heavy sphere of small radius is suspended by a thread 
5 m. long. How many times will it vibrate in an hour ? 

351. What must be the ratio of the lengths of two simple 
pendulums that one may make three vibrations while the other 
makes four? 

352. A seconds pendulum loses 8 sec. per day when carried 
to another station. Compare the values of g at the two places. 

353. A pendulum is carried upward with an acceleration 
equal to g. What will be the effect on its period ? 

What would be the effect if it moved downward 
with the same acceleration ? 

354. AC is a light rigid rod suspended at A. B 
and C are two small heavy spheres attached to the 

rod. 

AB = 30 cm., AC 80 cm. 

Mass of B, 20 g. ; mass of C, 50 g. 
(a) Find the periodic time of each if the other 
were absent. 

(6) Find the periodic time of the system. 

/V 2 

The expression r = 2 TT\ becomes, in this case, 

* MgR 



30 



- 

(20 + 50)980 -66 c 

Taking the numerator and dividing it by the total mass, we 
have" A" for this case. 

Hence if R had been given, the actual masses need not be known. 

82 



PENDULUMS. MOMENTS OF INERTIA 83 

355. In the system shown in Fig. 21, all lengths are measured 
from 5. 

Find (a) the Swr 2 ; 

(b) the distance from 5 to center of gravity. 

(c) the periodic time of the system. 
(Neglect weight of the rod.) 

356. Find the time of vibration of a compound pendulum 
consisting of a uniform cylindrical rod 2 m. long, radius 2 cm., 
knife edges 40 cm. from end. 

V^ fM ^2 
, what do you mean by R ? Name two 

o 

values of R which could not be used in finding g. 

358. Find the moment of inertia of a thin uniform rod : 

(a) When the axis is _L to end of rod. 

(b) When the axis is J_ to middle point. 



becomes 



ox^dx, 

O 



What is the relation between these two values and the center 
of gravity of the rod ? 

359. Find the moment of inertia of a thin rod whose density 
increases uniformly from one end to the other : 

(a) When axis is -L to light end. 

(b) When axis is _L to heavy end. 
(Note that p = p Q + kx.) 

360. What relation exists between the two values above and 
the energy which the rod would have with a given angular 
velocity in the two cases ? 

361. Find the moment of inertia of a rectangular area, axis 
through the center and in the plane of the figure parallel to 
one side. 

362. Find the moment of inertia of a thin circular plate, axis 
any diameter. 



8 4 



PROBLEMS IN PHYSICS 



363. Find the moment of inertia of a circular plate of uni- 
form density, axis through center and perpendicular to plane 
of the circle. 

364. Find the moment of inertia of a circular plate, axis 
perpendicular to plane of circle and through its center, when 
the density increases uniformly from the center outward. 

365. Find the moment of inertia of a right circular cylinder, 
axis through center and perpendicular to axis of the cylinder, 
length of cylinder /. 



Et* 

ff$\ r 
:-.-4--,-;HW4-- 

R/ 



V 




Fig. 41. 

By direct integration we may consider the volume element as having a 
base rdOdr, and a thickness dx. 



Then 



dm = prdrdOdx, 



It may be observed that this result is the sum of two parts, the first the 
same as Ex. 358 (), the second the same as Ex. 362. The energy of the 
rotating cylinder is, in fact, made up of two parts, one due to the motion of 
the center of gravity of each circular lamina, the other due to the rotation of 
these laminae about their diameter with the same angular velocity as the axis 
of the rod. 

In all cases of finding moment of inertia, we have to express ^mr 2 as an 
integral whose form and limits are determined by the problem in hand. It 
should be remembered by the student in physics that energy of rotation is the 
thing of real interest and importance rather than the particular mathematical 
machinery involved. 



ELASTICITY 

366. Define elasticity of solids ; of fluids. 

367. When is a body said to be highly elastic and when 
inelastic ? To which of these classes does rubber belong ? 
glass ? 

368. State what is meant by the term stress. What is the 
stress when 40 kg. rests on a horizontal surface 10 cm. 
square ? 

369. A vertical rod 4 sq. cm. cross-section sustains a weight 
of 100 kg. What is the 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 
iron wire. The ratio '"" is taken as * and 



- for f a PP lied - asKg. 42shows the result. What 
area of cross-section 

does the straight portion OB represent ? What does the slope 

85 



86 



PROBLEMS IN PHYSICS 



of that portion mean ? Estimate the safe load. What does 
the bend indicate ? 




Fig. 42. 

376. Define Young's Modulus. It was found that if the 
elastic limit would permit so great an extension, it would 
require a force of 17- IO 11 dynes per unit area of 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 - t -, ^, etc., from A. 
o 4 

" (a) If the wheel is turned 16, through what angle would each 
pointer turn ? 



OF Tin: 



ELASTICITY 



(b) If J/was 10 kg. in case (a), what would be the twist pro 
duced by 25 kg. ? 

(c) If M were as in case (a) and R were multiplied by 2-|, 
how would the distortion compare with that in b ? 





Fig. 43. 

(d) If the length were half as great, compare the moments 
required to turn the wheel through the same angle. 

(e) If the radius of the cylinder were reduced one-half, how 
would the angles mentioned in a be altered if the length and 
the moment of the applied force were unchanged ? 

B 





Fig. 44. 



381. If A and B, Fig. 44, are the 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) the mean displacement of these elementary areas. 

(c) the mean return forces per unit area. 

(d) the mean leverages for these return forces. 

(e) the total torques or moments tending to restore 

the cylinders to their former positions. 

382. How does the torque vary with the length of the 
cylinder ? 

383. By reference to 380 and 381, find the moment of torsion 
for a brass wire 3 m. long, 5 mm. radius, given the coefficient 
of rigidity for brass = 38 io 10 . 



Show that T= n , etc. 



384. The moment of torsion of a wire 240 cm. long, radius 
.7 mm. is 17.7. What force applied 2 cm. from its axis and 
perpendicular to a radius would twist one end of a meter length 
of this wire 360 ? 



LIQUIDS AND GASES 

385. Distinguish between a liquid and a gas. 

386. State fully the reasoning by which the following con- 
clusions are reached : 

(a) At any point in a liquid at rest the pressure is equal in all 
directions. 

(b) The pressure at any point on a submerged surface is 
normal to that surface. 

387. Show that the intensity of pressure in a homogeneous 
heavy liquid varies directly as the depth. 

388. Explain what is meant by a "head" of h feet of water, 
a pressure of h cm. of mercury. 

389. Express a pressure of 100 Ib. per square inch in kilo- 
grams per square meter. 

390. Is it essential that a barometer tube be of uniform bore ? 

391. A barometer tube inclined from the vertical by 5 reads 
765 mm. Find the correct reading. 

392. Compute the height of the " homogeneous atmosphere" 
when the barometer stands at 740 mm. 

393. Express in atmospheres the pressure existing at a depth 
of 20 m. in sea water. 

394. Find the pressure at a depth of 6 cm. in mercury sur- 
mounted by 4 cm. of water of unit density ; and this, again, by 
12 cm. of oil of density .9, atmospheric pressure not considered. 

395. Neglecting atmospheric pressure find the intensity of 
pressure due to a head of 10.37 m - (34 ft.) of water; (a) in 
grams weight, (b) in dynes. 

89 



90 PROBLEMS IN PHYSICS 

396. Find in centimetres of mercury the pressure at a depth 
of 20 m. in water of unit density, the barometer standing at 
76 cm. 

397. The pressure at the bottom of a lake is 3 times that at 
a depth of 2 meters, what is the depth of the lake ? 

398. At what depth in mercury will be found a pressure 
equal to that existing in sea water at a depth of i km. ? 

399. The air sustains a column of water 33 ft. (10.0 m.) high. 
To what internal pressure is the tube of a syphon subjected at 
a height of 30 ft. above the reservoir ? 

400. Explain the action of an ordinary suction pump. What 
is the maximum theoretical height to which water can be raised 
by such a pump ? 

401. A body of volume 24 cc. weighs in air at o and 760 
mm. 16.142 grams. Correct the reading for the weight of dis- 
placed air, neglecting the air displacement of the weights. 

402. Two liquids that do not mix are contained in a U tube, 
the difference of level is 4 cm., and the distance between the 
free surface of the heavier liquid and their common surface is 
6 cm. Compare their densities. 

403. A U tube 16 cm. high contains mercury to a height of 
4 cm. ; how many centimeters of chloroform can now be poured 
into one arm ? 

404. Alcohol is poured into one arm of a U tube containing 
mercury ; when equilibrium obtains it is found that the free 
surface of the alcohol is 17 times as high as that of the mercury 
above the common surface of the two liquids ; what is the den- 
sity of alcohol ? 

405. Find the pressure on the upper surface of a horizontal 
plane 12 cm. square when immersed to a depth of 30 cm. in a 
solution of density .12. 




HYDROSTATIC PRESSURE 91 

On every square centimeter of the plane the pressure is the weight of a 
column of the solution i sq. cm. in section and 30 cm. high plus the pressure 
of the atmosphere on i sq. cm. of the free surface. This 
gives as total pressure on one side of the plane, the ba- 
rometer reading 76 cm. 

144 [(30 x 1.2) + (76 x 13.6)] = 

The pressure on the under surface of the plane is equal and 
opposite to this. 

406. To what depth must the plane in the 
last problem be sunk in order that the pres- 

sure on its upper surface may be double the atmospheric 
pressure ? 

407. A square of area 1.24 sq. m. has its upper edge in the 
free surface of a body of water and its lower edge 80 cm. below 
the free surface. Find the liquid pressure upon one side of it. 

Note that here we have an intensity of pressure varying uniformly from 
zero at the surface of the liquid to a maximum at the lower edge of the area. 
We need to find the mean intensity of pressure. 

408. By what law would the pressure on the area mentioned 
in the last problem vary with its inclination to the free surface ? 

409. Sketch the form of a dish such that the hydrostatic 
pressure on its bottom shall be (a) greater than, (b) equal to, 
and (c) less than, the weight of the contained liquid. 

410. A hole 15 cm. square is punched in the hull of a sea- 
going vessel at a depth of 3.2 m. below the surface of the water. 
Compute the force necessary to hold a board over the opening. 

411. The water in a pond is confined by a dam of rectangular 
surface. After heavy rains the water rises by J its normal 
height, although still not overflowing the dam, the surface area 
of the pond increases at the same time twofold. How does the 
total pressure on the dam vary ? 

412. Find the total pressure on a rectangular sluice-gate 8 ft. 
wide and 6 ft. deep when the water stands at a height of 5 ft. 



92 PROBLEMS IN PHYSICS 

413- Find the center of pressure of a rectangle whose upper 
edge is in the free surface of the liquid. 

The resultant pressure does not pass through the geometrical center of the 
rectangle because the distribution of pressure is not uniform but varies as 
the depth. Let b the breadth of the rectangle. Im- _ 
agine the total fluid pressure on the right of the rectan- 
gle to be concentrated at a certain point distant x from 
the surface. Then if we imagine equilibrium to still 
exist, we must have the sum of the moments of the 
various pressures about the upper edge as an axis = o. . 
The pressure on a horizontal strip d/i wide and b long. ~ p. 
is h.bdh. Its moment about the upper edge is hbdh. 
Summing these moments, together with the moment of P. which is negative, 
we have 



p 



= Px 
bh* 

- =fx 

Remembering that P = (mean depth) x area, 
we have finally x\h. 

414. Find the center of pressure of a rectangle whose upper 
edge is horizontal but submerged to a depth of Ji v 

415. If the rectangle were inclined at an angle a to the sur- 
face of the liquid, would the center of pressure change ? 

416. A right cone, vertex upward is filled with water. Show 
that the resultant pressure on the curved surface is equal to 
twice the weight of water in the cone and acts vertically 
upward. 

The volume of the cone is equal to \ the volume of a right cylinder of the 
same base and altitude. If such a cylinder be placed over the cone, and the 
space between it and the conical surface filled with water and the water inside 
the cone removed, the pressure on the curved surface would remain unaltered. 
Using this fact the proposition is readily proved. 

417. The diameter of the small plunger of a hydrostatic press 
is 8 cm. That of the large plunger is i m. The pressure ap- 
plied to the small plunger is 260 kg. What load is sustained 
on the large plunger? 



HYDROSTATIC PRESSURE 93 

418. The diameters of the two plungers of a hydrostatic press 
are 4 in. and 3 ft., both being circular. The smaller plunger is 
worked by a lever whose arms are in the ratio 10: i. Find 
the total load that can be lifted by a man exerting a force of 
120 Ib. 



SPECIFIC GRAVITY AND PRINCIPLE OF 
ARCHIMEDES 

419. A man can just lift a cylindrical jar when filled with 
water. How many men would be required to lift the same jar 
filled with a liquid of sp. gr. 12 ? 

420. To what height could the jar be filled with mercury in 
order that one man could just lift it ? 

421. Why is it easier to swim in salt than in fresh water? 

422. Explain why a balloon filled with hot air rises. 

423. Four spheres of the same size are made of Pt, Pb, 
Ni, and Al respectively. Compare their weights. 

If of the same weight, compare their radii ; their volumes. 

424. A gold and a silver coin are exactly similar in form 
and of equal weight. What is the ratio of their volumes ? 

425. Explain why the actual intensity of gravity need not be 
known in finding specific gravity. 

426. If a place could be found where g is o, could specific 
gravity still be found, and if so, how ? 

427. Suppose the space V in a liquid (Fig. 47) to contain 
matter of steadily increasing density. At first 

one-tenth that of the liquid, and finally ten 
times as dense. Show how the resultant force 
should vary. Draw a curve using density as 



& 



x, and resultant force on Va.sy. Fig. 47. 

428. A bottle filled with water weighs 172 g. ; the bottle 
weighs 72 g. What will it weigh when filled with sulphuric 
acid? Mercury? Oil of turpentine? 

94 



SPECIFIC GRAVITY 95 

429. A cube of silver and one of gold are of equal size. 
Compare their weights. If of equal weight, compare their edges. 

430. A body in air weighs 40 g. ; immersed in water, it 
weighs 30 g. Find its specific gravity. 

431. A body weighing 80 g. and sp.gr. 4 is immersed in a 
liquid sp. g. 2. How much weight does the body lose ? 

432. A body of volume 8 c.c., sp. g. 6, is immersed in liquid 
of sp. gr. 4. What is its loss of weight ? 

433. What force would be required to hold a mass of 80 g., 
sp. gr. 5, under the surface of a liquid of sp. gr. 13.6? 

434. A body weighed in water loses 25 g. ; weighed in a 
liquid of unknown density it loses 50 g. Find density of the 
liquid. 

435. A body in air weighs 50 g. ; its sp. gr. is 8. When 
weighed in a liquid, it loses 10 g. What is the specific gravity 
of the liquid ? 

436. A body immersed in one liquid loses 20 per cent of its 
weight ; when immersed in a second liquid it loses 40 per cent 
of its weight. Find the ratio of the specific gravities of the 
liquids. 

437. A sinker in water weighs 40 g., a block of wood in 
air weighs 30 g. ; both in water weigh 20 g. Find specific 
gravity of the wood. Draw the force system when both are 
weighed in water. 

438. A cork in air weighs 8 g. ; a sinker in water weighs 
60 g. ; both in H 2 O weigh 28 g. Find the specific gravity of 
the cork. 

439. The specific gravity of a body is 4. What would be its 
acceleration due to gravity when in water, neglecting friction ? 

What if specific gravity were .4 ? 

440. A body floating in water is placed under the receiver of 
an air pump and the air is exhausted. Will the depth to which 
the body sinks be altered ? Explain your answer fully. 



96 PROBLEMS IN PHYSICS 

441. A sinker, volume 80 c.c., sp. gr. 8, is fastened to a piece 
of wood weighing 35 g. in air; both in water weigh 525 g. 
What is the specific gravity of the wood? 

442. Does specific gravity depend on the units of mass, etc., 
employed ? 

443. A cork, sp. gr. .6 and volume 15 c.c., is attached to a 
brass sinker, sp. gr. 8. What must be the volume of the brass 
in order that the combination may just sink in water? 

444. What must be the edge of a hollow brass cube I cm. 
thick that will just float in water? 

445. A sinker of lead, sp. gr. 11.3, is attached to a fish line 
weighing .005 g. per centimeter and sp. gr. .1. What must be 
the volume of the lead to pull 10 m. of the line under water? 

446. A uniform rod weighted at the bottom is immersed suc- 
cessively in several liquids whose densities increase uniformly. 
What will be the relation of the volumes immersed ? 

447. A block of lead in air weighs 330 g. When suspended 
in water it is found that the water and containing vessel gains 
30 g. in weight. What is the specific gravity of lead ? 

448. Eighty c.c. of lead, sp. gr. 11.3, 20 c.c. of cork, sp. gr. .2, 
and 10 c.c. iron, sp. gr. 7.8., are fastened together. What would 
they weigh in water ? 

449. Compute the specific gravity of glass from the following 
data : 

Weight of bottle 20 g. 

Weight of bottle and H 2 O 100 g. 

Weight of powdered glass 1 5 g. 

Weight of bottle containing glass and filled up with H 2 O . 1 10 g. 

450. A specific gravity bottle is counterbalanced ; it is then 
filled with water, and 19.66 g. more are needed to keep it bal- 
anced. When filled with alcohol only 15.46 g. are needed. 
What is the specific gravity of alcohol ? 



SPECIFIC GRAVITY 97 

451. A hydrometer weighing 100 g. sinks to a certain mark 
in water, and requires 20 g. additional to sink it to the same 
mark in another liquid. What is the specific gravity of the 
second ? 

452. The specific gravity of a block of wood is .9. What 
proportion of its volume will be under water when it floats ? 

453. A block of wood, sp. gr. .7, is to be loaded with lead, 
sp. gr. 11.4, so as to float with .9 of its volume immersed. What 
weight of lead is required if the wood weighs i kg. : (i) When 
the lead is on the top ? (2) When the lead is immersed ? 

454. Show how to compute the specific gravity of a mixture 
of two or more liquids when the volumes mixed and their specific 
gravities are known : 

(a) When new volume is the sum of the volumes of com- 
ponents. 

(b) When there is a decrease of volume. 

455. Two liquids which do not mix and of specific gravities 
2 and 5 are placed in a beaker. A body of unknown specific 
gravity is observed to sink until .3 of its volume was in the 
lower liquid. What was its specific gravity ? 

456. Eight parts by volume of a liquid whose sp. gr. is 6 are 
mixed with five parts of a liquid sp. gr. 3. Find the specific 
gravity of the mixture when there is no reduction of volume. 
Find it when the total volume is reduced 5 per cent. 

457. What is the difference between hydrometers of constant 
immersion and those of variable immersion ? 

458. Explain how each is used, giving an example. 

459. A Nicholson's hydrometer weighs 100 g. and sinks to a 
certain point in H 2 O when 40 g. are added. It sinks to the 
same point in another liquid when 20 g. are added. Find 
specific gravity of second liquid. 

460. A long test-tube with mercury in the bottom and of 
uniform cross-section is used to determine the specific gravity 

H 



9 8 



PROBLEMS IN PHYSICS 



of a number of liquids lighter than water. Show how to cali- 
brate when the point to which it sinks in two liquids of known 
specific gravity is given. 

461. A piece of lead, volume 20 c.c., sp. gr. 1 1.4, is suspended 
from one arm of a balance and is immersed in oil, sp. gr. .9. 
From the other end an irregular mass of gold, sp. gr. 19.3, is 
suspended in turpentine, sp. g. .8. What is the volume of the 
gold if the beam remains horizontal ? 

462. A brick, sp. gr. 2, is dropped into a vessel containing 
mercury and water. Find its position of equilibrium. 

463. Two equal cubes of oak and pine respectively are placed 
in water. The edge of each is 20 cm. What height of each 
will be above the surface? 

464. A cylindrical rod of wood and iron is to be made so as 
to just sink in water. Specific gravity of wood, .5 ; of iron, 7.5. 
The length of the iron rod is 75 cm. How long must the 
wood be ? 

465. According to Boyle's law pv = k at constant tempera- 
ture. Give two definitions of k from a consideration of the 
formula. Also show graphically the meaning of k. 

466. A cylinder 24 in. long contains 2 cu. ft. of air at a 
pressure of 15 Ib. per square inch. 

The cylinder is slowly pushed in. 
(a) Find the pressure at several points 
of the stroke and lay them off as 
ordinates, thus forming a pressure- 
volume curve with axis as shown. 
Discuss this curve. (b) What is 
the total pressure on the inner sur- 
face of the piston ? 

467. Show that it follows from Fig ' 48 ' 
Boyle's law that the pressure of a gas at constant tempera- 
ture must be proportional to its density. 




BOYLE'S LAW 99 

468. Forty c.c. of air are enclosed in an inverted tube over 
mercury. The difference of level is 50 cm. The tube is 
depressed until the difference of level becomes 30 cm. What 
is the volume of the enclosed air ? 

469. A glass tube 60 cm. long and closed at one end is sunk, 
open end down, to the bottom of the ocean ; when drawn up it 
is found that the water has wet the inside of the tube to a point 
5 cm. below the top ; what is the depth of the ocean ? 

470. An air bubble at the bottom of a pond 6 m. deep has a 
volume of o. I c.c. Find its volume just as it reaches the 
surface, the barometer showing 760 mm. 



HEAT 



TEMPERATURE 

471. Define temperature. Is the sense of touch a reliable 
measure of temperature ? Explain fully. 

472. Bodies at different temperatures are sometimes said to 
be at different thermal levels. What is meant ? Explain the 
difference between temperature and quantity of heat. 

473. What does a 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. If the coefficient of cubical expansion of the liquid in a 
thermometer is less than that of the envelope, what effect will 
be produced on heating the thermometer ? 

477. Reduce to Fahrenheit readings, the following Centi- 
grade temperatures: 45, 12, 20. 

478. Reduce to Centigrade readings the following Fahrenheit 
temperatures: 212, 72, 32, 30. 

479. Plot Centigrade temperatures as abscissas and corre- 
sponding Fahrenheit readings as ordinates, and discuss the 
locus. Also, take from the 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 

480. At what temperature will both Fahrenheit and Centi- 
grade thermometers give the same reading ? What happens to 
mercury at this temperature ? 

481. The temperature of a given liquid is taken by both 
Fahrenheit and Centigrade thermometers. The Fahrenheit 
reading is found to be double the Centigrade reading. What 
is the temperature of the liquid in degrees Centigrade ? 

482. Define the coefficient of linear expansion and establish 

the formula 

l t = / (i + A/), 

where l t is the length of a bar of given material at temperature 
t, / its length at zero, and A the mean coefficient of expansion 
for the material between o and t. 

If a bar of given material be heated, it lengthens. Every unit of the 
original length elongates for every degree rise of temperature an amount A. 
This is the coefficient of linear expansion. Between narrow limits of tem- 
perature the elongation may be taken as proportional to the temperature rise. 
The total elongation for a temperature rise of / degrees from zero must there- 
fore be / A/, which makes the new length 

/, = / Q + / A/ = / (i + A/). 

When t is large, / can no longer be taken as a linear function of the tem- 
perature, but is represented by 

/, = / (i + A/ + A7 2 + ..). 

483. Show that the true linear expansion coefficient at 
temperature / is given by 

idi 

~l,7t 

484. A platinum wire is 4 m. long at o ; find its length at 
100. 

We have / 100 = / (i + .000009*) 

= 4 x 1.0009 
= 4.0036 m. 

485. Show that the value of X is independent of the unit of 
length used, but depends upon the thermometric scale used. 



102 PROBLEMS IN PHYSICS 

486. A lead pipe has a length of 12.623 m. at 15 ; find its 
length at o. 

487. Why is platinum wire well adapted for use in the 
" leading in " wires of a glow lamp, or in any circumstances 
in which it needs to be fused into glass ? 

488. A certain induction coil has 20,000 turns of copper wire 
in its secondary coil. If climatic changes cause a rise of 40 in 
its temperature, express the resulting expansion in turns of 
mean length. 

489. The length of a brass wire at 3 is 12 m. ; find its length 
at 33. 

In this example we might first find the length of the wire at zero degrees, 
and then by resubstitution find the length at 33. A sufficiently accurate 
result, however, is obtained by an approximation. We have 



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

// / * + A/' 

It i t -- , 

i + A/ 

= /,[i+A(/'-/)], 
very approximately when A is small. [See V.] 

490. Assuming that 43 is the maximum temperature to 
which steel rails, 10 m. long at o, are ever subjected during 
the changing seasons, compute the space which should be left 
between them when laid at 15. 

491. Measurements are made at 25 upon a brass tube by a 
steel meter scale, correct at o. The result is 6.426 m. Find 
the length of the tube at o. 

One should here consider that the result of these measurements is a 
number which shows the ratio of the length of the tube to the length of the 
scale at the temperature at which the measurements are made. Since the 
length of the tube at zero is required, the number obtained is too large because 
of the expansion of the thing measured and too small because of the expansion 
"of the unit. The result sought will therefore be found by multiplying the 
number by the ratio of the expansion factor of steel to the expansion factor 
of brass. 



EXPANSION OF SOLIDS 



103 



d 



492. A brass rod is found to measure 100.019 cm. at 10 and 
100.19 cm. at 100. Find the mean coefficient of linear ex- 
pansion of brass between 10 and 100. 

The student should work this example first by the accurate method and 
then by use of the approximate formula (see V.) and compare the results. 

493. A platinum bar originally at 15 is placed in a glass- 
blower's furnace. The increase in length is .96 per cent. Find 
the temperature of the furnace. 

494. When it is desired that a point 

p shall remain at a constant distance d 
from a support, an arrangement built 

on the principle shown in the figure 
may be used. The rods a, a, and b are 
of one metal and the rods c, c, are of 
another. This principle is used in the 
"gridiron" clock pendulum. Derive 
the conditions for compensation. 

495. A lever at A controls a distant 
railway signal at B. If A and B are 
connected by a rod, changes in temper- 
ature may cause a movement of the signal independent of any 
motion of the lever. Devise a scheme by which this may be 
avoided, the same rod being retained. 

496. A clock which keeps correct time at 22 has a pendu- 
lum made of iron. If the temperature fall to 8, how many 
seconds per day will the clock gain ? 

NOTE. The time of vibration of a pendulum is proportional to the square 
root of its length. 

497. Show that if X be taken as the coefficient of linear ex- 
pansion of a given material, the coefficient of volume expansion 
of the same material is approximately 3 X. [See V.] 

498. A silver dish has a capacity of 1.026 1. at 75 ; at what 
temperature will its capacity be just one liter? 



P 

Fig. 49. 



104 PROBLEMS IN PHYSICS 

499. A steel boiler has a surface area of 9.2 sq. m. at 6 ; 
find the per cent increase in this area for a rise in temperature 
of 80. 

500. Find the mean coefficient of volume expansion of tin on 
the Fahrenheit scale. 

501. Explain how density varies with temperature, and show 
that when t is small 

& t = S,(i 
and further that 



NOTE. These results are obtained by approximate methods. [See V.] 

502. The density at o of a specimen of wrought iron is 7.3, 
and the density at o of a specimen of tin is 7.4 ; at what tem- 
perature will these two specimens have the same density ? 

503. Distinguish between real and apparent expansion of 
liquids. Show that the coefficient of real expansion of a liquid 
is sensibly equal to the coefficient of apparent expansion to- 
gether with the coefficient of cubical expansion of the envelope. 

504. The coefficient of apparent expansion of mercury in 
glass is erVo > tne coefficient of real expansion of mercury is 
-^Vo- Find the coefficient of volume expansion of glass. 

505. A graduated glass tube contains 40 c.c. of mercury at o. 
If the whole be heated to 32, what is the apparent volume of 
the mercury ? 

If glass and mercury had the same coefficient of expansion, the apparent 
volume would remain unaltered. But taking the expansion coefficient of 
mercury at 182 x io~ 6 and that of glass at 3 x 85 x io~ 7 , it is evident that 
the volume of the mercury increases more rapidly than the volume of the tube. 
This means that the apparent volume of the mercury will increase. 

506. A glass flask holds 842 g. of mercury at o. How much 
will overflow if the whole be heated to 100 ? 

-507. Taking the density of mercury at o at 13.6, calculate 
the density at 200. 



EXPANSION OF LIQUIDS 105 

508. Taking the density of mercury at 60 as 13.45, find the 
density at 100. 

509. It is desired to study the true expansion of water. If 
the proper amount of mercury be placed in a glass bulb, the 
expansion of the mercury, for any rise of temperature, will 
equal that of the bulb itself. The volume above the mercury 
will thus remain constant, and may be filled with water. Any 
observed increase in the volume of water must therefore be 
its true expansion. What fraction of the volume of the bulb 
at zero must be filled with mercury to secure this result ? 

510. Describe the manner in which water behaves between 
zero and 10. 

511. The surface of a pond of water is observed to be just 
freezing. Would you expect the water at the bottom of the 
pond to be at the same temperature and density as that at 
the top? 

512. Describe the weight thermometer. The bulb of a ther- 
mometer contains 2.4 kg. of mercury at o. The whole is heated 
to t t causing an overflow of 40 g. Required t. 

Let M total mass of mercury. 

m = overflow. 

8 = density of mercury. 

K = coefficient of expansion of glass. 

a = coefficient of expansion of mercury. 
Now the volume of the thermometer at o is 

M 

8' 

which becomes, at /, 



The mass of mercury filling the thermometer at / D is 

M m, 
its volume at o is 

M- m 



106 PROBLEMS IN PHYSICS 

and this volume expands at f to 

M-m, 
-y-(i+0. 

But the volume of the expanded mercury is the same as that of the expanded 
bulb, from which relation t is readily found. 

513. A weight thermometer containing i kg. of mercury at 
o is placed in an oil bath, and the mass of expelled mercury 
is found to be 26.4 g. Find the temperature of the bath, the 
coefficient of apparent expansion of mercury in glass being 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 at 273 ? What is this temperature 
called ? If temperatures be reckoned from this point, how is 
the expression for the law modified ? 

516. A mass of gas at 15 occupies 120 c.c. Find its volume 
at 87, the pressure remaining constant. 

We have according to Charles' law, 




~ 120 x -= 120 x 1.25 
288 

= 150 c.c. 

517. Take volumes as ordinates and temperatures as abscis- 
sas, and give a graphical representation of Charles' law. 

518. At what temperature will the volume of a given mass 
of gas be three times what it is at 17 ? 

.519. A volume of hydrogen at 11 measures 4 1. If the 
temperature be raised, at constant pressure to 82, what is 
the change in volume ? 



EXPANSION OF GASES IO/ 

520. The temperature of a constant volume of gas is raised 
from o to 91. Find the per cent increase in pressure. 

521. Show that for a given mass of gas the quantity *-= t or 
pressure x volume 



-, is invariable. 



absolute temperature' 

522. Find the dimensions of the product/^. 

523. Find the volume of 2 Ib. oxygen at a pressure of 3 
atmospheres and temperature 27, the volume of I Ib. oxygen 
at o and i atmosphere being 11.204 cu - ft- 

The volume at o and i atmosphere is 

z/ 2 x 11.204 cu. ft. 
If the gas is heated at constant pressure to 27, it expands by Charles' law to 

v r = fff x 2 x -11.204 cu. ft. 
Now if the pressure be increased three-fold at constant temperature, 

V" \ X f f f X 2 X 11.204 CU. ft. 
= 8.2 CU. ft. 

524. Find the numerical value of ^ for a mass of i g. 
of air. 

Now ^ = ^^o, where v is the volume of i g. at o and p Q is a pressure 

i 
of i atmosphere. 

A = J 3-596 x 76 
in grams' weight per square centimeter 

T = 273- 



.001293 



c.c. 



Therefore, ^ = '3-596 x 76 = 

r c 273 x .001293 

i f) f l) 

525. Compute the value of ~; for a gas s times heavier 

than air, of which m grams are taken. Show that the value 
of this constant depends on the quality and quantity of the 
gas used. 

526. The pressure on a given mass of gas is doubled, and 
at the same time the temperature is raised from o to 91. 
How is the volume affected ? 



108 PROBLEMS IN PHYSICS 

527. The pressure of a given mass of air is that due to 
1 20 cm. of mercury, its volume is 1000 cu. cm., and temperature 
15. If now the pressure be increased to 250 cm., the volume 
becomes 300 c.c. ; what is the temperature ? 

i)"V 

528. Find the value of *-=, where / is measured in pounds 
per square foot, v in cubic feet, and T in Fahrenheit degrees. 

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

530. Show that the final temperature resulting from mixing 
M grams of a substance of specific heat c and at a temperature 
/"with m grams of water at a temperature t is 

mt 



Me + m 

531. Solve the equation of 530 for the specific heat c, and ex- 
tend the problem to the case in which the thermal capacity of 
the calorimeter is considered. 

SUGGESTION. Some of the heat liberated by the hot body goes to 
warm the calorimeter, which is assumed to be carried through the same tem- 
perature range as the water. This amount of heat is therefore M c c' (9 /), 
where M e is the mass of the calorimeter, and c' the specific heat of the material 
of which it is made. 

532. How many minor calories are required to raise the tem- 
perature 3 kg. of copper from 16 to no ? 

533. Equal masses of iron and aluminum cool through the 
same range of temperature ; compare the amounts of heat lost. 

534. Assuming no loss of heat, how much heat must be 
imparted to 2 gal. of water, initially at 14, in order to raise it 
to the 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 ? 

538. Into 12 kg. of water at 30 are dropped, at the same 
instant, i kg. of copper at 100 arid 1.2 kg. of zinc at 60 ; find 
the resultant temperature. 

539. If a calorimeter be made of material of specific heat c' , 
and if it have a mass m' , the product m'c f is sometimes called 
the water equivalent of the calorimeter. What justifies the use 
of the term ? 

540. A copper calorimeter weighs 62 g. ; what is its water 
equivalent ? 

541. In determining the water equivalent of a calorimeter 
the following data are observed : 

Weight of calorimeter 52.66 g. 

Weight of calorimeter + cold water .... 302.71 

Initial temperature n 

Temperature of hot water ...... 80 

Final temperature ........ 14.8 

Total weight after addition of hot water . . . 317.61 

Compute the water equivalent. 

542. Compare the result obtained in the last problem with 
the computed value, assuming the calorimeter to be made 
entirely of copper. 

543. A silver dish weighing 50 g. contains 500 g. of water at 
1 6 ; a piece of silver weighing 65 g. is heated to 100 and then 
plunged into the water; the resulting temperature is 16.50; 
what is the specific heat of silver? 

544. A mass of 200 g. of copper is heated to 100 and 
placed in 100 g. of alcohol at 8 contained in a copper calorim- 
eter, whose mass is 25 g., and the temperature rises to 28.5. 
Find the specific heat of alcohol. 



110 PROBLEMS IN PHYSICS 

545. An iron ball is heated to 100 and then dropped in 
3 1. of water at 6, causing a rise of temperature of 2 ; what 
is the diameter of the ball ? 

5450. The specific heat of most substances is not a constant, 
but is a function of the temperature. If the quantity of heat 
necessary to raise one gram of a substance from o to / be 

given by 

Q, = at + bt* + a*, 

show that the specific heat at a temperature t is 

C=a + 2 bt + 3 cP, 

and that the mean specific heat between f and tf is 
C m = a + b (t + S) + c (/ 2 + tf + t") . 

546. One starts with 100 g. of water at 10, and to this one 
adds successive amounts of water from a reservoir maintained 
always at 100. Express the temperature of the mixture as a 
function of the amount of hot water added. Plot a curve 
between amounts of water added (abscissas) and final tempera- 
tures (ordinates). Note the limit beyond which the curve has 
no physical meaning. 

547. Show from the equation for the final temperature in the 
method of mixtures, that loci similar to that in the last problem 
are hyperbolas. Discuss fully. 

548. Define heat of fusion. What seemed to justify the term 

latent heat ? 

> 

549. Taking temperatures as ordinates and quantities of heat 
as abscissas, plot the relation between these quantities for the 
case in which ice at 10 is converted into water at 90. 

550. How many calories must be supplied to 15 kg. of ice at 
o to completely melt it ? 

551. How many grams of ice at o must be added to 1000 g. 
of water at 30 to produce a final temperature of 5 ? 



CHANGE OF STATE III 

552. In a determination of the heat of fusion of ice, the fol- 
lowing data are observed : 

Weight of calorimeter 71.5 g. 

Water equivalent . . . . . . . . 8.5 g. 

Weight of calorimeter and water 156 g. 

Temperature of water . . .... 54 

Temperature after ice is melted 32 

Weight after addition of ice ...... 174.5 g. 

Compute the heat of fusion of ice. 

553. Required, the amount of heat necessary to raise 3 kg. 
of lead at 10 to the 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 is meant by regelation? In what substances 
should we look for the phenomenon ? 

565. Explain the making of snowballs, the formation of ice on 
pavements, and the flow of glaciers, as phenomena of regelation. 

566. Why is iron an excellent metal for casting ? Why are 
coins stamped instead of being cast ? 

567. Punched rifle bullets pursue a straighter course than do 
cast bullets. What reason can be given for this ? 

568. What property of wrought iron enables it to be readily 
welded ? How does sealing-wax behave when heated ? 

569. What amount of heat must be supplied to 10 kg. of 
water at 100 to convert it into steam at the same temperature ? 

To convert I g. of water at 100 into steam at the same temperature requires 
536 calories (heat of vaporization of water) . In this case we must have 
H 536 x io 4 = 5360 calories. 

570. Find the numerical value of the heat of vaporization of 
water in terms (a) of pound and degree Centigrade units, (b) in 
terms of pound and degree Fahrenheit units. 

571. Explain why evaporation cools. If a few drops of ether 
be placed on the bulb of a thermometer, an immediate lowering 
of the mercury is observed ; but when the thermometer is dipped 
in a bottle of the ether, no lowering is observed. Explain. 



CHANGE OF STATE 113 

572. A kettle contains 2 kg. of water at 40. How much 
heat must be supplied in order to boil the water away? 

573. A calorimeter contains 316 g. of water at 40. Steam 
at 1 00 is passed into the water until the mass of water becomes 
336 g. What is the temperature ? 

The mass of steam condensed is 

336- 316 = 20 g., 
which yields the heat of vaporization, 

20 x 536 calories. 

Further, the 20 g. of condensed steam in cooling to the final temperature 6 
yields 

20 (100 0) calories. 

The 316 g. of water originally in the calorimeter is raised from 40 to 0, 
which means a gain of heat of 

316 (0 40) calories. 

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

574. In a determination of the heat of vaporization of water 
by passing steam into a calorimeter containing cold water, the 
following data are obtained : 

Weight of calorimeter . . . . 71.5 g. 

Water equivalent of calorimeter . . . . 8.5 g. 

Weight of calorimeter and water . . . 173 g. 

Temperature of cold water 17 

After passage of steam : 

Weight of calorimeter and water . . . 181 g. 
Temperature 41 

Compute the heat of vaporization. 

575. What is meant by the total heat of steam ? 

576. What amount of steam at 100 must be passed into 
1 6 kg. of water at o in which 4 kg. of ice are floating, in order 
to raise the whole to 30 ? 

577. Calculate the heat necessary to raise to the boiling- 
point, and to completely vaporize 120 g. of alcohol at 12. 



114 PROBLEMS IN PHYSICS 

578. What is meant by a saturated vapor ? Upon what does 
the pressure of a saturated vapor depend ? 

579. Some values from Regnault's determination of the max- 
imum pressure of water vapor are given below. Plot them. 

Temperature Pressure 

(abscissas). (ordinates). 

o 0.46 cm. 

10 0.91 cm. 

20 1.74 cm. 

30 3.15 cm. 

40 . . . . . . 5.49 cm. 

50 9.20 cm. 

60 14.90 cm. 

70 23.30 cm. 

80 35-5 cm. 

90 ... ... 52.50 cm. 

100 76.00 cm. 

580. Into a barometer tube in which the mercury stands at 
760 mm. a few drops of water are introduced. (a) Explain 
what happens. (&) If the temperature be 30, and there still 
remain a little water on top of the mercury, what will be the 
reading of the barometer ? (The height of the layer of water 
is neglected.) (c) What are the effects of raising and of lower- 
ing the barometer tube, supposing the cistern to be deep 
enough to admit of this ? 

581. In a closed chamber saturated water vapor in contact 
with its liquid exists at a pressure of 23.3 cm. What is the 
temperature ? If means are provided for pumping out the 
vapor, what happens ? 

582. How define the 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 Quito, Ecuador, the mean barometer reading is 
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? Draw a curve showing 
the distribution of energy in the visible and non-visible spectra. 

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

606. Explain how the specific heat of a substance may be 
determined by the method of cooling. 

607. Equal masses of water and alcohol cool successively 
through the same range of temperature in the same dish in 
times whose ratio is J^-. Compute the specific heat of alcohol 
for the range of temperature used in the experiment. 

608. What is meant by the radiation constant of a calorim- 
eter ? How is it determined experimentally, and how is it used 
in a specific heat determination by the method of mixtures. 

609. What is meant by the term mechanical equivalent of 
heat? Describe any method by which it has been determined. 

610. Express 20 calories in ergs. 

From Introduction, we take as the value of J, 4.2 x io 7 ergs. Hence 
20 calories = 8.4 x io 8 ergs. 

611. Show that the numerical value of J in gravitational 

units varies as unit of temperature. 
unit of length 

612. To raise I gr. of water i C. requires 4.2 x io 7 ergs. 
Find the number of 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 ice at o p be fired against an 
impenetrable wall in order to be completely melted, assuming 
that no heat is lost ? 



n8 



PROBLEMS IN PHYSICS 



615. Why does the specific heat of a gas at constant pres- 
sure differ from the specific heat at constant volume ? 

616. Describe an experiment to show that air is not cooled 
by expansion if no external work is done. Is this result true 
of all gases ? 

617. A cubic meter of air at o 
and 76 cm. pressure is contained in a 
cylinder whose piston moves without 
friction. If the air be heated to 100, 
what is the external work done ? 




Fig. 49 (a). 



By the conditions of the problem, external 
work is done against the pressure of the atmos- 
phere. This pressure is 

p = 76 x 13.6 grams' weight per square centimeter. 
Since the gas expands at constant pressure, the increase in volume is 

100 



Whence the work is 



= x io 6 c.c. 



100 



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

618. Compute the heat supplied to cause this expansion. 
This is readily done by finding the mass of the air in the 

cylinder and using the specific heat at constant pressure. 

619. Compute the heat required to raise the temperature of 
this mass of air at constant volume. 

620. One liter of air at o is confined by a weightless piston 
in a cylinder whose sectional area is i sq. dm. The pressure 
of the atmosphere is 76 cm. The temperature of the gas is 
raised to 273, thus increasing the volume to 2 1. Compute 
the mechanical equivalent of heat. [Ratio of specific heat at 
constant pressure to specific heat at constant volume =1.41.] 

621. What is an isothermal line? an adiabatic line? Why 
is the adiabatic line through any point of the 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 port of the cylinder is 
closed, when the expansion is assumed to take place in accord- 
ance with Boyle's law, the back pressure being zero. 




----- -X 

j 



Fig. 50. 

Let the positions of the piston at different times be laid off along OX and 
the corresponding pressures along OY. At E, when the piston has proceeded 
a distance a, 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 ( l pdx. 

Ja 

But the condition that pv = constant gives 

p'Aa = pAx, 



so that 



= Ap'a log, -. 



a 
Note that Ap'a is the work done on the piston during admission 



I2O PROBLEMS IN PHYSICS 

624. Find an expression for the entire effective work of the 
forward stroke of an engine working under the conditions above 
named except that there is a constant back pressure (condenser 
pressure) p c . 

Note that the pressure of admission is constant, as is also the back pressure. 
The work due to these pressures is readily calculated. 

625. (a) Apply the results of the last problem to finding the 
work per forward stroke when the numerical data are : 

Area of piston = 100 sq. in. 

Length of stroke = 14 in. 

Boiler pressure = 60 Ib. per square inch. 

Back pressure = 2.5 Ib. per square inch (actual). 

Cut-off at T 3 T stroke. 

If an ordinary steam gauge shows 60 Ib., the actual pressure is 60 -f 14.7 Ib. 
per square inch. 

(b) The engine is 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 light uncharged 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 ? 

647. In using an electrophorus we may divide the process 
into four parts : (i) the approach of the metallic plate to the 
charged one; (2) "grounding" the upper plate; (3) separating 
the two ; (4) the discharge of metallic plate. 

648. Draw diagrams showing the distribution of charge in 
each case of Example 647. 

649. Draw the lines of force in each case of Example 647. 

650. Two equal light insulated conducting spheres are sus- 
pended so as to hang near together. One is charged positively. 
Will it attract the other ? The second is grounded. Will the 
force action be altered, and how ? 

. 651. If instead of grounding the second they had been 
brought in contact and then separated, what change in the 
force action would be observed ? 



POTENTIAL 123 

652. Give numerical values to the charge and distance 
between the centers of the spheres in the latter case, and find 
the force action before and after contact. 

653. Define electrical potential at a point. In what units is 
it measured? 

654. An isolated charge causes a potential of 25 at a point 
near it. What would the potential be if the charge were in- 
creased fourfold ? if a charge opposite in sign, and twice as 
large, were combined with the first ? 

655. Show that for a single charge q the potential, at a dis- 

Q 

tance r, is - 
r 



656. Find the potential at a point midway between A and B 
in Fig. 5 1 ; between B and C. 

Q=160 q'--80 

AB = i m., BC ' = 20 cm. @- ~B 

Fig. 51. 

657. How much work would be required to move a charge of 
2 + units from a point on AB 10 cm. from A to one 10 cm. 
from B} (In Fig. 51.) 

658. A small sphere is charged with 40 + units. Draw the 
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 its ends would be 40 
and 10 respectively, if the conductor were absent. How would 
the potential of these points be altered by the introduction of 
the conductor ? 

660. What takes place on the conductor when it is moved 
across the equipotential surfaces of the field ? 



124 PROBLEMS IN PHYSICS 

66 1. Two spheres of equal radii are suspended by silk 
threads, and each is grounded. After the "ground" is broken 
charged bodies are brought in the neighborhood, such that 
the potentials at the points occupied by the center of the 
spheres would be at potentials 10 and 10 respectively. What 
changes would occur if the spheres were connected by a wire ? 

662. A sphere of radius 10 is charged so that the surface 

density is IOQ . What quantity is required ? What is the 

4?r 
potential of a point just outside the sphere? What is the 

electric force at that point ? Would any of these quantities be 
altered if the sphere were immersed in turpentine ? Explain. 

663. What work is done in moving a charge of + 30 from a 
point where V 40 to one where V= 100? 

664. To move a charge of + 4 from V = 10 to V = + 10 
will require how much work ? 

665. A small sphere has a charge of 84- units. Draw six 
equipotential surfaces; three having V< I, one V= i, two 



666. Indicate, briefly, the change in these surfaces if a 
charge of 4 were brought to a point 9 cm. from the first 
sphere. 

667. A charged sphere A is brought near to an insulated 
conductor B. 

Describe the electrical state of B (charge C B ^) 



and potential) : Fig. 52. 

(a) When A is placed near B. 

(b) After grounding B. 

(c) When B is again insulated and A removed. 

. (d) When B is again insulated and A brought nearer 
than before. 

'668. Draw the lines of force in each case of Ex. 667. 
669. Draw the equipotential surfaces of Ex. 667. 



POTENTIAL 125 

670. Two equal charges are 80 cm. apart. If each charge is 
+ 40, what is the potential 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 of force and equipotential surfaces of a uniform field. 
What is the meaning of the ratio of the difference in potential 
between two points on the line to the distance between the 
points ? 

675. What is the meaning of the above ratio when the field 
is not uniform? when the field is variable, but the distance 
between the points is very small ? 

676. Assuming that the charge on an isolated sphere acts on 
a small charge just outside the sphere as though the entire 
charge were placed at the center, show that the electric force 
just outside is 4717) (p = surface density). Since independent of 
radius of the sphere, what follows in regard to an infinite 
charged plane ? 

677. Can two equipotential surfaces intersect ? Can an 
equipotential surface intersect itself ? Explain your answers. 



126 PROBLEMS IN PHYSICS 

678. Explain how an insulated conductor in the presence of 
charged bodies remains an equipotential region. 

679. A charged sphere is placed between two very large 
conducting plates. Draw the lines of force and equipotential 
surfaces. 

680. What peculiarity of the distribution of the lines of force 
indicates a strong field ? of the equipotential surfaces ? 

681. Draw a curve showing the relation between the charge 
and potential of an isolated conductor, using Q as x and V as y. 
What does the slope of the line mean ? What does the area of 
the curve mean ? 

682. After Q has reached a certain value, a grounded con- 
ductor is placed near the first, and Q is again increased. What 
changes in the Q V line would indicate this ? 

683. When Q is stationary, and the second conductor is near, 
they are both surrounded by paraffine and Q is again increased. 
Show how the Q V line would differ from the preceding. 

684. A conducting sphere A is charged and placed on an 
insulating support at a great distance from all other conductors. 
Another conductor, B (uncharged), is brought near A. Will 
the charge on A be altered ? the distribution ? the potential 
of A ? the force at neighboring points ? If the distribution of 
force is altered, where would it be increased and where dimin- 
ished ? Answer the same questions if B were "grounded." 

685. A straight line is drawn in any direction in a uniform 
field. If the potential at each point of the line be taken as y, 
and distances from a fixed point on the line as x, what kind of a 
curve will be found ? What will the slope mean? What will 
the slope be when the given line is drawn perpendicular to the 
lines of force ? When will the slope be a maximum ? 

686. Explain fully the difference between the electric force 
at a point, and the electric potential at that point. What rela- 
tion is there between them ? 



CAPACITY 



127 



687. Is potential a directed quantity or vector? Find the 
dimensions of electric potential. 

688. The difference of potential between two points is 500 ; 
the distance between them is 40 cm. What is the average field 
strength between them ? 

689. The average field strength between two points is 50; 
they are 2 m. apart. What is the difference of potential ? 

690. Find the term C, V, or Q, omitted in the following table, 
where C = capacity of a con- 
ductor ; V= the potential to 

which it is raised ; Q = charge 
required to give a potential V. 

691. Find the energy in each 
case of Example 690. 

692. What is meant by the 
term capacity as applied to a 
conductor or condenser ? 



Q 


V 


C 


80 




20 


20 




80 


80 


20 




20 


80 






80 


20 




2O 


80 



693. A charge of 400 raises the potential of a sphere from 
o to 100. What is its radius ? 

694. Three spheres, capacities 4, 8, 12, respectively, are 
charged to potentials 24, 16, and 8. What is the quantity on 
each ? The spheres are connected by a wire of negligible 
capacity. What will be the common potential ? 

695. What energy is required to charge a sphere of radius 
10 to a potential of 100 ? of radius 100 to a potential of 10 ? 
to charge a sphere of radius 10 with a charge of 100? 

696. The radius and charge on a sphere are each increased 
threefold. How is the potential affected ? the energy ? 

697. (a) Upon what does the electrical capacity of a con- 
ductor depend ? Explain why the capacity of a body is altered 
by bringing a grounded conductor near, (b) If a body whose 
capacity is 200 C.G.S. is charged to a potential of 4 (C.G.S.), 



128 PROBLEMS IN PHYSICS 

what is the quantity of electricity ? How much work is done 
in charging the body ? (If formulas are needed derive them.) 
(Winter, '96.) 

698. A and B are two spheres, radius of each i cm. What 
is the capacity of each* ? 

699. A is given a charge of -f- 80. B is given a charge of 
40. The distance between their centers is 50 cm. Locate a 
point on the line joining their centers where V= o ; -f 2 ; 3. 

700. B is brought in contact with A and then replaced. 
How would the charges be altered ? What change in potential 
would occur at each of the points mentioned above ? 

701. What do you mean by a condenser? Upon what does 
the capacity of a condenser depend ? 

NOTE. Unless otherwise stated, it will be assumed that one coating of a 
condenser is grounded, i.e. V o. 

702. State the analogy between electric condensers and 
water reservoirs. 

703. A condenser of capacity 1000 is charged with 500 units. 
Half of this charge escapes. What proportion of the energy 
has been lost ? 

704. A quantity Q charges a condenser to a potential V. 
What energy is stored ? 

705. The area of the plates, the thickness of the dielectric 
and its specific inductive capacity are each doubled. How will 
its capacity be changed ? 

706. Define specific inductive capacity. 

707. A certain condenser when air is used as the dielectric 
has a capacity of 400 ; when glass is substituted, the capacity is 
found to be 2600. What is the specific inductive capacity of 
the glass ? 

708. The force action between two charged plates is found to 
be 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 r 2 , specific inductive capacity 
of dielectric k. 

710. Derive the expression for the energy required to charge 
a condenser in terms of Q and V\ in terms of Q and C; in 
terms of C and V. 

NOTE. dW = VdQ. But V is a function of Q, V- Q. 



711. Compare the energy required to charge two spherical 
condensers to the same potential when the radii of the shells 
of one are 20 cm. and 20.1 cm., sp. ind. cap. of dielectric 2, while 
for the other these quantities are 40, 40.2, and 6. 

712. A condenser of capacity 50 and charge 400 is connected 
by a poor conductor to earth until its energy is reduced to one- 
sixteenth of its initial energy. What charge escapes ? How 
much is the potential decreased ? 

713. It is observed that the energies of discharge of two 
jars charged from the same source to earth are as I to 9. Find 
the ratio of their capacities. 

714. A and B are two reservoirs of the same volume, but of 
unequal height. P is a pump powerful enough to force water 
to the top of A. 

(a) Which would possess 
the more potential energy 
when filled ? 

(b) Which would exert the 
greater pressure when full ? 

(c) The 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 is I sq. m., the distance apart is .1 cm., 
the specific inductive capacity of the dielectric being 4 ? 

721. How much energy is required to charge such a con- 
denser to a potential of 100 ? 

722. In the discharge of a condenser what becomes of the 
energy ? What experiments confirm your statement ? 

723. How would you proceed in order to charge a Leyden 
jar? 

724. Find the energy of discharge of a condenser when the 
plates are of potentials V l and V^ and the capacity is C. 

725. There are three Leyden jars, A, B, and C, equal in 
capacity, having their outer coatings connected to earth. A is 



CONDENSERS 131 

first charged. Its knob is then connected with the knob of B. 
It is then disconnected from B and connected with C. Finally 
the knobs of A, B, and C are connected. Find the energies 
of the several discharges. (Larden.) 

726. When are two or more condensers said to be connected 
in " series " ? When in parallel or multiple ? 

727. The inner plates of four similar condensers are joined, 
and each outer plate is grounded. What is the ratio of the 
capacity of the set to that of a single one ? 

Compare : (a) The potential to which a given charge would 
raise the system with that to which it would raise one. (b) The 
energy required to raise the system to a given V with that 
required for one ? 

728. Four similar condensers are joined in series ; the outer 
plate of the last is grounded, the inner plate of the first is 
charged to a potential V. The capacity of each condenser is 
C. What is the potential of each jar? What is the total 
charge ? What is the entire energy stored ? 

729. Two spheres, A and B, radii 5 and 2 respectively, and 
charges +40 and 10 are joined by a wire of negligible 
capacity. Find the capacity of the system ; the quantity on 
each sphere ; the amount of electricity which has passed along 
the wire ; the initial energy and the final energy. 



CURRENT ELECTRICITY 

730. State Ohm's law. For what kind of conductors and 
under what conditions is it true ? 

The units used in measuring current, electromotive force or potential 
difference, and resistance are named the ampere, volt, and ohm respectively. 
The relation of these to the C.G.S. system will be illustrated later (see p. 187). 
Ohm's Law is not dependent on the units employed. Hence in any system 

potential difference . . 

/ = . In the practical system, current in amfieres 

resistance 

_ potential difference in volts 
resistance in ohms 

731. When potential difference = 80 volts, resistance = 40 
ohms, what current will flow ? What quantity will pass each 
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. V A is + , Vg is . 
They are connected by a poor conductor. What changes of 
potential will take place ? 

132 



OHM'S LAW 



133 



737. In the above case, if the charge on A is reduced 80 + 
units and the charge on B is reduced 80 units, what is the 
total quantity which has passed along the connecting wire? 

738. If this transfer takes place in 5 sec., what is the aver- 
age current strength ? 

739. Two bodies of different potential are joined by a moist 
thread. It is observed that the change of potential is slow and 
the current is small. Explain. 

740. What do you mean by the resistance of a conductor ? 
What effect does the resistance of a conductor joining two 
points of constant difference of potential have ? 

741. Find the terms omitted, /, potential difference, or R, in 
the following table : 



Potential 
Difference. 


j? 


' 


120 


5 




5 


200 




500 


250 




25 




5 


I! 5 




20 


340 




17 




35 


7 




400 


50 




2OOO 


.0005 



742. The terminals of a wire of resistance 60 ohms are kept 
at potentials of 100 and 10 for 5 min. ; the terminal of lower 
potential is then " grounded " and the potential of the other 
reduced to 90; current flows again for 10 min. Compare the 
quantities transferred. 

743. If in the equation V = / R, we take each quantity in 
turn as constant and the others as x and y, what loci would 
be obtained ? 

744. A uniform wire AB is kept at a uniform temperature, 
and its ends at a constant difference of potential. Draw a 



134 PROBLEMS IN PHYSICS 

diagram showing the relation between the fall of potential and 
length of the wire. 

745. If in Example 744 V A V B = 100 volts, and V A = 200 
volts, what will be the potential midway between A and B ? 
at 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 relation between current, potential difference, and resistance through- 
out a circuit may often be best understood by a properly constructed diagram. 
We may choose either of two ways, according to the end in view. We may 
assume any potential we please as our arbitrary, o, since we are concerned 
only with differences of potential. Then K may be plotted as y and R as x, 
or we may use V as y and distances measured from an arbitrary point in cir- 
cuit as x. In case the circuit contains sources of electromotive force, we may 
usually consider the rise of potential through them as sudden, and the line 
becomes a broken one. If, however, the source of electromotive force is dis- 
tributed like the armature of a dynamo, the line in such places would be 
curved. Potential-resistance curves are of considerable importance, and the 
student is advised to study carefully the simpler cases explained below before 
drawing those of more complicated circuits. 



POTENTIAL DIAGRAMS 



135 



Take the case of a single cell, electromotive force 3 volts, internal resist- 
ance 6 ohms, external resistance 10 ohms. Starting at any point as /?, and 





Fig. 55. 



assuming MB as representing the potential at B, Ohm's law states that along 

/?/?' 

BC the potential falls uniformly, so that = /. 

B' C 

At C we may suppose an abrupt rise of potential taking place at the bound- 
ing surface of liquid and plate, then another uniform fall due to the resistance 
of the cell, another rise at D 1 falling again 
along DB to the value MB. Note that the 
lines of fall are all parallel, which is equiva- 
lent to the statement that the current is the 
same throughout the circuit. Suppose now 
that the external resistance were increased, / 
must decrease, and all of the sloping lines 
would become more nearly parallel with OX. 
But the vertical lines CC V and DD l are con- 
stant in length and independent of R ; it follows then, in order that C^D may 
remain parallel to BC and D^B, either CC l must fall or DD^ rise, or both. 
This is the same as saying that the difference of potential of the terminals of 
a battery depends upon the external resistance, and approaches the electro- 
motive force of the cell as this resistance is increased. 

When potential and distance from a fixed point are used as 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 H w 
connected to a tank 7", from which leads a straight pipe A } A 2 -~S; A^H y 
A^Hfr etc., a series of vertical pipes opening from the main whereby the 



136 



PROBLEMS IN PHYSICS 



pressure at each point can be measured ; 6" a stop-cock whereby the flow in 
the main can be checked. When S is open and the pump working, so that 



H 




L H 1 














HI 














"""--..^ 


H, 
















H. 








T 






"""--^.^ 


H 4 


H 


g 












^223 




, 


*i i 


^ 2 > 


\3 ^ 


^4 / 


^5 S 








Fig. 5 


6. 







the current is steady, the pump will be unable to keep T full up to // , and it 
will be found that the tops of the water columns will be in a straight line 



751. What is the electrical analogue of : 

(a) The friction of the pipe ? 

(b) The friction of the pump ? 

(c) The pressure at A 1 ? 

(d) The difference between A^H^ and A^H^t 

(e) The ratio, 

pump pressure _ difference of pressure between A l and A 2 ? 
total friction friction between A 1 and A z 

(/) The height of line H^H^ vertically above 5 ? 

(g) Current and quantity ? 

(h) The changes which occur when 5 is slowly closed ? 

752. Would the analogy hold if the pipe were bent ? if it 
were enlarged at some point ? 

753. State a case in flow of water analogous to cells in series ; 
in multiple. Explain fully. 

754. In the circuit shown (Fig. 55), a point in the external 
resistance is "grounded." Draw the potential-resistance curve. 
What change in your diagram would indicate a change in the 
position of the ground ? 

755. Determine what external resistance is required in the 



v^ ^-* v 

f OF THB 

f rjNrVEHi 



OOF THB 
'NTVKP' 



POTENTIAL DIAGRAMS 



137 



circuit of Fig. 57 in order that the potential difference of A and 
B may be I volt? i^- volts ? \ volt ? 




Fig. 57. 

756. If the resistances of the cells in Fig. 58 are very small, 
draw the 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 



c 1 ID E 



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 1 A, AB, etc.) are so large that their 
resistance can be neglected. A is connected to the earth. 




Fig. 60. 

(a) Draw diagram to show the variation of potential along 
A' CDS', (b} Compute the difference of potential between 
C and D. 

763. Name four things upon which the resistance of a wire 
depends. 

764. Two copper wires are of the same 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 is Ohm's law applied to find how multiple resist- 
ance depends on the resistance of the separate branches ? 

767. The length of a wire is increased fourfold. How much 
must its radius be changed that its resistance may be the same 

as before ? 

*- . ' 

768. An iron wire of a certain length and 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 

-^-- = --\ = , or A R B = =3 ohms. 

A X B 4 12 12' 4 



140 



PROBLEMS IN PHYSICS 



774. A copper wire of length / is divided in the ratio of 
3 to 5, and the pieces joined in multiple. What lengtJi of the 
same wire might have been taken to get the same resistance ? 

In dealing with a complex circuit it is well to compute each multiple resist- 
ance first, and then deal with 
the set in series. 5 



775. Find the total 
resistance of the circuit, 
Fig. 62. In this sys- 
tem we may compute 
the resistance from A 
to B, then from C to D, 
finally add together all 
the resistances in se- 
ries. 

776. Find the total 
resistance of the cir- 
cuit, Fig. 63. (Com- 
pute each multiple re- 
sistance first.) 



B 




Fig. 62. 




Fig. 63. 



777* Find the resistance of the system shown in Fig. 64. 

778. AC and BD (Fig. 65) are two metal plates of o resist- 
ance. A and B are joined by a wire of 
10000 ohms resistance. Find x^ so that 
when placed in multiple with the first the 
combined resistance is 1000. Then x^ so 
that multiple resistance of the three is 100, 
etc. 3 





Fig. 65. 



MULTIPLE RESISTANCE 



141 



779. Prove that the resistance of two wires in multiple is 
always less than that of either. 

780. Prove the following construction for computing multiple 
resistance. 

Lay off on O Y a length r^. 

Lay off any line || to O Y a length r v 




Fig. 66. 

Join the upper end of each line with the lower of the other. 
The ordinate of the intersection of these lines is the resistance 
required. 

For three or more resistances we may extend the construction as for r s . 
By using cross-section paper the results may be quickly obtained. 




Fig. 67. 

781. Prove the following construction for resistances in 
multiple. Take ;r=r 1 , / = r 2 ; join their extremities. Then 
the resistance of r^ and r 2 in multiple is given by the 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 
cells in multiple and in series we must be careful to consider both the electro- 
motive force and the resistance of the combination. It is assumed that the 
cells are exactly alike, both in resistance and electromotive force, unless 
otherwise stated. 

The electromotive force of any number (?/) of cells in series 

sum of electromotive forces. 

The electromotive force of any number (;/) of cells in multiple 

= electromotive force of one. 
The resistance of any number () of cells in series 

sum of resistances. 

The resistance of any number (n) of cells in multiple is computed just 
as any other multiple resistance. 

782. Six cells, resistance of each 12 ohms, electromotive 
force of each 2 volts are connected in series. Find combined 
electromotive force and resistance. Find them when in mul- 
tiple. 




Fig. 68. 

783. A system of ten cells, electromotive force 3 volts, r 
6 ohms, are connected as shown in Fig. 68. Find the electro- 
motive force and resistance. 

784. A system of fifty cells, electromotive force I volt, r .4 
ohms, are placed "ten in a row" (series), and the five rows 
in multiple. What is the internal resistance of the battery? 
the electromotive force ? 

785. Find the current strength when each circuit (Examples 
783 and 784) is closed by an external resistance of 200 ohms. 



DIVIDED CIRCUITS 



143 



786. Given twenty-four cells, electromotive force 2 volts, r 4 
ohms, external resistance 5 ohms. Separate 24 into its various 
factors (as 2, 12; 3, 8 ; etc.); choose each factor in turn as the 
number of cells in a row, and the other as the number of rows. 
Compute the current strength in each case. 

787. Do the same when external resistance = I ohm ; 
200 ohms. 

788. When two or more wires are joined in multiple, at each 
junction they have a common potential. Hence by Ohm's law 
the current through any wire will be the common potential 
difference between A and B divided by the R of that wire. 

789. Three wires in multiple (Fig. 69); potential difference 
between A and B = 24 volts ; resist- 
ances as shown. What current flows 

in each branch ? What is the total 
current ? 

790. The currents in two branches 
of a divided circuit are as 4 to 12. 
What is the ratio of their resistances? 

791. In the circuit shown in Fig. 70, find 

(a) The total electromotive force. 

(b) The total resistance. 

(c) The total current. 

(d) The fall of potential between K and G. 

40 15 




Fig. 69. 




HH 

Fig. 70. 

(e) The fall of potential between A and B ; C and D ; 

E and F. 

(/) The current in each branch between A and B. 
State your reason for each step in the numerical work. 



144 PROBLEMS IN PHYSICS 

792. Twenty 5ovolt lamps, each requiring 1.2 amperes, are 
connected as shown. The resistance of BB' and CO is nearly 
o, that of AB + DC is i ohm. 

A B R 



D 

Fig. 71. 

Find (a) The resistance between B and C. 

(b) The total current. 

(c) Difference of potential between B and C. 

(d) Difference of potential between A and D. 

(e) The heat developed per minute in the lamps. 

(/) What change takes place when five pairs of lamps 

are turned off ? 
(g) What objection would there be to 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 I R x l I x . 

But /* = 9 I R . Fig. 73. 

.-. 100 I R = x^I R . 

x l = - l -^ = 1 1 i ohms. 
(Compare Example 778.) 

797. Prove that when a shunt of resistance s is placed around 

a wire of resistance r the current is r = total current. 

s + r 

Extend this to three or more resistances in multiple. 

f 

In general, 7 ri = \ **''*"'' 



798. A galvanometer of 1980 ohms resistance is "shunted"" 
by a wire r = 20. What proportion of the total current passes 
through the galvanometer ? 

799. The difference of potential between A and B, Fig. 63, 
is to be measured by placing an instrument (voltmeter) in shunt 
with the resistance between A and B. What change in this 
difference of potential is caused by the insertion of the instru- 
ment ? 

800. In the circuit of Fig. 70, what is the smallest resist- 
ance a voltmeter could have that when placed in shunt with 
AB the difference of potential between A and B may be changed 
only 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 A is 50 volts, and at B is 40 volts, what 
range of potentials may be found along APB1 along AQB ? 

805. If P and Q are two points of the same potential and 
the key k is closed, would the distribution of the potential 
be altered ? 

806. When is it certain that if any point P is chosen on the 
upper branch, a point of the same potential can be found on the 
lower one ? Explain fully. 

807. If a source of electromotive force were in any part of the 
circuit between A and B, would it always be possible to find for 
any potential along APB a corresponding point in AQB1 

808. Find the relation between the resistances AP, PB, etc., 
when V P = V fi , in case of no electromotive force between A 
and B. (Wheatstone's bridge.) 

809. Show that the best arrangement of a given number 
of cells is that which makes the external and internal resist- 
ances as nearly equal as possible. 

nE [" = electromotive force of one cell. 



/ = 



nr [r = resistance of one cell. 

m [/? = external resistance. 

mnE [n = No. cells in a " row." 

nr + mR \m No. of rows. 



ARRANGEMENT OF CELLS 147 

Since mn = number of cells, the numerator is constant. 

.-. /is a maximum when nr + mR is a minimum ; 

i.e. nr -f- mR is a minimum by variation of n and m. 

.. rdn + Rdm o. 
But mn constant. 

.. mdn -\-ndm = o. 

Whence = or = R. 

m n m 

It does not follow that the two simultaneous equations mn = JV and 
~ R have integer roots ; and as fractional parts of a cell are meaningless. 
we must choose the two factors of A 7 " which make as nearly^? as possible. 

810. Deduce from the statement of how to group for maxi- 
mum current a rule when the external resistance is very great ; 
very small. 

811. How would you group twenty-four cells, each r = 6, 
E=3, R=i6, for a maximum current? ^=36? ^=9? 

R = 25 ? 

n-6 
-m~ = l6 > 

mn = 24. 
Multiply these equations, 

2 -6 = 16- 24, 

n 2 16 . 4 or n = 8. 
.-. m = 3. [8 in a row, 3 rows. 

812. Apply Kirchhoff s laws to the circuit shown in Fig. 75, 
where electromotive force of the cell is E and the resistance of 
cell and connecting wires is r. 

These laws are often stated as fol- 
lows : 

(1) If any number of conductors 
meet in a point S/ = o ; or there is no 
accumulation of electrification at the 
point. 

(2) In any complete circuit 




In applying the first law, if we consider the current flowing toward A as 
+ ,we must consider those from A as . While in the use of the second 



148 PROBLEMS IN PHYSICS 

law, if we start from A toward B, i.e. 'with the current, and call S l r l +, we 
must, when returning along r 2 , take 7 2 r 2 as . 

By (i) 7=7 1 + 7 2 +7 3 . 

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

I\ r \ - 7 3 r 3 = O, 

7 2 r 2 - 7 3 r 3 = o, 
7r + 7^ = E. [Where E = electromotive force of cell.] 

From the first and second of (2) we may express 7 2 and 7 3 in terms of I v 
r v r 2 , and r. y 

Substituting in (i), 



.. 



. for / and / 



If R = equivalent resistance of r v r 2 , and r 3 , 

7^? + Ir = , 
7^ + fr = E, 
7^ + Sr = , 
7 3 r 3 + 7r - . 

Add last three and equate to three times the first. Solve for R, using 7j 



above. 



813. Find the distribution of current in a set of five un- 
equal resistances joined in multiple. 




Fig. 76. 

814. In the circuit of Fig. 76, show that 



( _ r i r s ^ - f r * - r * r * r < 

where / = total current. 



/ I--P-J =/ 4 1+1* + ^ - 2l , 
V ri + r?) 4 V r 2 r s r, + rj a ' 



KIRCHHOFPS LAWS 149 

815. The resistance of ADB is 10, of ACS is 40. Find the 
current in AB. 

Assuming direction of currents as indicated by the arrows, 
7 L = 7 2 + / 3 , 

7 3 40 / 2 20 = 10, 
7 L 10 + / 2 20 = 2. 

Eliminating / x and 7 3 , we have / 2 = 7 V amperes. What does the nega- 
tive sign mean ? Solve when the arrow from A to B is reversed. 

816. Find 7 2 when one cell is re- . _ 
versed (Fig. 77). 

817. What electromotive force 
must be inserted in branch (i) 
(Fig. 77), that no current shall pass 
through (2) ? 

Put / 2 = o. and the third equation = e. 
Whence E = 2% volts. 

8 1 8. Three cells, electromotive force E v E^ E& internal 
resistances r lt r^ r 3 , are joined in multiple and the external 
resistance is R. Find the total current. Test your answer by 
reference to the case when the cells are alike. 




819. Assume, in Example 8 1 8, E l = 2, E^ 4, E z = 6, r = 3, 
r 2 = 6, r 8 = 12, ^ = 40. Find the current in amperes. 

820. A and B are two points in a circuit which is carrying a 
current of 10 amperes. A R B = 100 ohms. What work is done 
in this portion of the circuit per minute? What becomes of 
this energy ? 

821. How much heat is developed per second in a portion of 
:a circuit, potential difference of the ends 50 volts, and the 
current 50 amperes ? 

Current in amperes x potential difference in volts = energy in watts. 
Heat per second in calories = wa s = watts x .24. 
Or/f=/. V .24=72^ . .24. '" 



150 PROBLEMS IN PHYSICS 

822. The resistance of a conductor is doubled and the cur- 
rent halved. How is the heat developed affected ? 

823. The current in a wire is multiplied by three. How 
much must the resistance of the conductor be altered that the 
loss by heat shall be unchanged ? 

824. A current of 10 amperes develops 144. io 4 calories per 
minute. What was the resistance ? What quantity passes per 
minute ? What potential difference is required to maintain the 
current ? 

825. State clearly the meaning of the terms watt and joule. 
Watts x time = ? 

826. A current of 40 amperes flowing in a coil causes a 
difference of potential of 20 volts between its terminals ; 

(a) How much energy is consumed in i hour ? 

(b] How much heat is developed ? 

827. Four wires of equal length and diameter, but of differ- 
ent specific resistances, are joined in series. For example, soft 
steel, copper, platinum, and silver are used. Find the ratios of 
the heat developed in the wires. 

828. Given mn similar cells, each E.M.F. = e> resistance of 
each = r ; external resistance R. How must they be arranged 
to secure the greatest heating effect ? 

829. A wire of resistance 1000 ohms is found to develop 
heat enough in io sec. to raise 24 kg. of water 10. What 
current does the wire carry ? What difference of potential was 
required to maintain it ? 

830. If work is done by the current in addition to overcoming 
resistance, would IE and I 2 R have the same value ? Explain. 

831 Find the distribution of heat in the circuit shown in 
Fig;. 72, when there is no back electromotive force. 

832. When a given set of generators are connected so as to 
give a maximum current through a given external resistance, 
show that one-half the total heat is developed in the generator. 



TRANSMISSION OF ENERGY 151 

833. Three copper wires of equal length, diameters .1 mm., 
.3 mm., .5 mm., respectively, are joined in multiple. The elec- 
tromotive force of the junctions is kept constant. Find the 
ratio of the heats developed in the wires. 

834. Why are large conductors usually used to transmit 
electrical energy ? Why is copper used in many cases rather 
than iron ? What determines which shall be used ? 

835. Why is it desirable to transmit electrical energy at high 
potential ? 

836. Why is it desirable to transform a small current at high 
potential to a larger current at lower potential at the point where 
it is used ? 

837. A current of 40 amperes is sent over a line of 10 ohms 
resistance. What is the fall of potential in the line? If the 
end of higher potential is at V= 1000, what energy per second 
is delivered at the end of lower potential ? What is the heat 
loss per second ? Answer the last two questions if V at the 
higher end were 2000 volts. 

838. The voltage at which a certain amount of power is 
supplied to a line is doubled. What is the effect on the heat 
losses ? How much could the length of the line be increased 
and still have no more loss in the line than at the lower voltage ? 
How might the 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. 
The statement that the specific resistance of copper is ly-io- 7 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 = 5 = .17 ohm. 

io~ 2 

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. 


Length. 


Radius. 


Specific Resistance. 


Hard steel 
Soft steel 


io m. 
io m. 


.5 mm. 
.5 mm. 


3i4-io~ 7 
I C7-IO" 7 


Copper 


I km. 


.2 mm. 


I7-IO" 7 


Platinum 


100 m. 


2 mm 


I 3 C I O~ 7 


Silver 


100 m. 


2 mm 


I C-IO" 7 


German silver .... 
Carbon 


100 m. 
i m. 


.2 mm. 
.1 mm. 


*3 !< 

236- io~ 7 

59350-10-7 



843. From the table of specific resistances above, compute 
the resistances of wires i m. long and I sq. mm. cross-section 
in each case. 

844. A wire is drawn out into an extremely long circular 
cone. If its radius at each point is a times the distance from 
the end, and the specific resistance of the metal is 35 icr 7 , find 
the resistance of the wire. 

Form the expression for the resistance of a length dl and integrate. 

As a first approximation, and between certain limits of temperature, the 
change of resistance of a wire with temperature may be expressed as a certain 
percentage of the resistance at o times the temperature above o. The state- 
ment that the temperature coefficient of copper is .00388 means that for each 
degree a copper wire is heated above o, its resistance is increased the .oo388th 
part of its resistance at o. 



TEMPERATURE COEFFICIENTS 153 

845. The resistance of a coil of copper wire at o is 1785 
ohms. What will it be at 40 ? 

The increase is .00388 -40 1785. 

^4o =I 785[i + .1552], etc. 

846. The resistance of an iron wire at 20 C. is 1010.6 ohms. 
The temperature coefficient is .0053. What is its resistance at 
o ? 40 ? 80 ? 

847. Taking the specific resistance of copper as 17- io~ 7 , and 
temperature coefficient as 39- io~ 4 , &n& ^assuming this coefficient 
as constant, at what temperature would copper have no re- 
sistance ? 

848. The temperature coefficient of a certain iron wire is 
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 is the range of temperature through which it may 
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. 

852. The electrochemical equivalent of H is 1038- io~ 8 (for I 
coulomb). The atomic weight of sodium is 23, its valence I. 
Find the electrochemical equivalent of sodium. 

853. A current of 2 amperes passes through a copper sul- 
phate solution for i hour. If the anode is a copper wire, how 
much copper will be deposited on the cathode ? 



154 PROBLEMS IN PHYSICS 

854. Compute the following electrochemical equivalents : 



Substance. 


Atomic Weight. 


Valence. 


Electrochemical 
Equivalent. 


Hydrogen 


I 




104. io~ 7 


Potassium 
Gold 


39-i 

Io6 2 






Copperic salts .... 
Copperous salts . . . 
Lead 


63.18 
63.18 

2O6.4 


2 
I 
2 













855. A deposit of 8.856 g. of copper is made by a current in 
ij hours in a Cu-CuSO 4 -Cu voltameter. What was the cur- 
rent strength ? 

856. A copper and a silver voltameter are placed in series. 
Find the ratio of the deposits formed. 

857. Explain how you would arrange your apparatus in order 
to "plate" an article with silver. 

858. A magnetic needle free to turn is placed in a uniform 
magnetic field. A new field at right angles to the first is then 
developed. Show by diagram what position the needle will 
assume. Does it depend on the pole strength or length of 
the needle ? What would be the effect of reversing either field ? 
both fields ? 

859. A wire carrying current is stretched north and south. 
The current flows from south to north. What position will a 
compass needle take when held over the wire ? How will its 
position alter as it is brought nearer the wire ? What position 
would it take if placed under the wire ? if placed midway 
between two such wires carrying equal currents in the same 
direction ? if in opposite directions ? when between, but nearer 
to one than to the other ? 

860. A piece of wire i cm. in length is bent into a circu- 
lar arc of I cm. radius. A current of I ampere flows in the 
conductor. What force would act on a + unit pole at the 



GALVANOMETERS 1 5 5 

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

(a) I=i ampere, one complete turn ? 

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

(c) I=\ ampere, n turns, radius = r? 
Note that i ampere = ^ C.G.S. unit of current. 

861. A circular coil of wire is placed in a north and south 
plane with its axis horizontal. A current is sent through the 
coil, flowing north on the upper side. What effect would the 
current have on a freely suspended magnetic needle when 
placed directly above the coil ? directly below ? in the same 
plane and just north? south? at the center? 

862. What would be the strength of the magnetic field at 
the center of a coil of n turns, mean radius R, I = one ampere ? 
From this derive the law of a tangent galvanometer, consisting 
of one large coil and a short (?) needle at the center. 

863. What do you mean by the term constant of a galvanome- 
ter? What is a tangent galvanometer? a sine galvanometer? 
Is a galvanometer of necessity one or the other ? 

864. Compute the current in each of the following cases, 
where 7 = galvanometer constant, B = deflection in degrees : 

Tangent galvanometer, / = 4.5, 8 = 25. 



/ = 42.icr 6 , S = 20. 

^- 

What would the currents be if the galvanometer were a 
" sine " galvanometer ? 

865. When is a galvanometer said to be sensitive ? 

866. Explain how a sensitive galvanometer is constructed. 

867. Explain how a given galvanometer may be made more 
sensitive. 



156 PROBLEMS IN PHYSICS 

868. If 7=10 tan S, H=.i$, 72=10, 8 = 25, what 

r 

must be the radius of the coil if / = 2 amperes ? How would 8 
be changed if H were reduced 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 it have on the galvanometer constant : 
(a) when placed in the same plane as the needle, and just north 
or south of it ? (b) when in the same plane east or west ? 
(c) when placed just below? 

871. How would the action of the soft iron in Example 870 
differ from that of a magnet ? 

872. The 7 of a certain tangent galvanometer is 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 H = .2. What current would 
give the same deflection where H is . I ? 

874. The needle of a tangent galvanometer is observed to 
make 40 complete vibrations in one minute. 7 at that point is 
34*io~ 6 . When moved to another place it is found to make 25 
complete vibrations in one minute. Find the constant in the 
new position. 

875. (a) Give a diagram showing the construction of a simple 
type of tangent galvanometer. Explain in what position it must 
be placed in measuring current, and derive formula, (b) State 
the distinction between magnetic and diamagnetic substances. 
Describe an experiment by which the behavior of each, when 
placed in a magnetic field, can be shown. 



BALLISTIC GALVANOMETER 157 

876. A tangent galvanometer is connected in series with a 
generator of constant electromotive force and a known resist- 

o 

ance which can be varied. A series of resistances are inserted, 
and corresponding deflections are observed. If these resistances 
are taken as x, and tangents of deflection as y, what sort of 
a curve will result ? Does the entire curve have a physical 
meaning ? 

877. How is the quantity of electricity measured when it 
passes as an intense and variable current for a very short time. 
(Examples, condenser discharges and induced currents.) 

878. What is a ballistic galvanometer ? What is meant by 
the term constant of a ballistic galvanometer? 

From the theory of the ballistic galvanometer we find that 

Q= I0 . r.^ sm i0 = smi0, 

7T G 

while from the magnetic pendulum 



where H - horizontal component of earth's field. 

T '= periodic time of magnet in that field. 
G = "true" constant of the galvanometer (tan.). 
M pole strength x distance between poles = ml. 
K a moment of inertia. 
= angle of maximum deflection. 



T 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 H is nine times 
as great, if the needle is remagnetized and its magnetic moment 
increased fourfold ? 

882. The constant of a ballistic galvanometer at a point 
where T is 4 sec. is .045. What will the constant be where 
T = 2 sec. ? 

883. A coil of 100 turns, mean radius 40 cm., is turned 180 
about a diameter which is perpendicular to the lines of force of 
a field of strength 10. The coil is connected with a ballistic 
galvanometer, and a deflection of 20 is observed. Resistance 
of the circuit 1 5 ohms. Find Q Q . 




Fig. 78. 

If ds is a current element so short that it may be regarded as straight, the 
laws concerning the magnetic force due to ds at any point A may be stated as 
follows : 

(1) The force is _L to the plane APQ. 

(2) The force is proportional to the length of ds. 
(3) The force is inversely proportional to AP . 

(4) The force is proportional to the " broadside " projection of ds, i.e. to 

PR = PQ sin PQR = ds sin (9. 

Summing up the last three of these four laws, we may say that 
T? _ ^ ds sin 

** K 9 

r 2 
[k depends on current strength and units used.] 



FIELDS DUE TO CURRENTS 159 

The field at the point A is then found by integrating this expression. In 
order to perform the integration a relation between the variables must be 
given, i.e. the shape and position, with reference to A, of the circuit must 
be specified. 

In the case of a very long straight wire, we have, if p = perpendicular 
distance from A to the wire, 

ds 2.k 



For a wire of finite length 2 /, A in the plane perpendicular to its middle point, 
the limits would be / and -f /. 

884. Find how long the wire must be in order that when p 
is 5 cm. the field is within one per cent of that due to an 
infinite wire with the same current. 

885. The horizontal component of the earth's field at a 
certain point in the Cornell Physical Laboratory is .145. At 
what distance from a long straight wire carrying 10 C.G.S. 
units of current would the field due to the current have the 
same intensity? (Here ki.} 

886. What current must flow in an infinite straight wire that 
the magnetic field 10 cm. from the wire may exert a force on 
unit pole equal to the weight of I g. ? 

887. Find the field strength at the center of a square, if a 
current passes around it. 

888. Find the force exerted on a + unit pole placed at the 
intersection of the diagonals of a rectangle, sides a and b, and 
carrying a current /. 

889. Apply the formula FA = k ds ' * in 6 to the case of a 

circular wire of radius R, when A is taken in the line perpendicular 
to the plane of the circle, and through its center. (Axis of coil.) 

Show by diagram the direction of the force for each element, 
and for the complete circle. 

What is the force component along the axis ? Where is this 
a maximum ? 



MAGNETISM 

890. State the law of attraction or repulsion between magnet 
poles. Where do similar laws occur in physics ? Show how a 
definition of unit magnet poles follows directly from the law. 

891. Find the force in dynes between two unlike magnet 
poles of strength 8 and 12 units respectively when the distance 
between them is .04 m. 

The force varies according to the law ^i. 

a 2 " 

8 T2 

Expressing d in centimeters F = 6 dynes. 

16 

892. Two like magnet poles, of strengths 10 and 27 units 
respectively, are separated by a distance of 30 mm. Find the 
force in milligram's weight between them. 

893. When two magnet poles are placed a distance apart of 
I cm. the force between them is 12 dynes. How must the 
distance be varied in order that the force may increase to 48 
dynes ? 

894. What is a magnetic field of force ? a magnetic line of 
force ? 

895. (a) Map the field of force around an ordinary bar mag- 
net. (&) Map the field around two magnets placed with their 
like poles (supposed of equal strength) near each other and 
their axes at right angles. 

896. A bar magnet is laid on a horizontal plane with its 
axis north and south, and its 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 of H is .2026. Find the total strength of field. 



162 PROBLEMS IN PHYSICS 

908. The angle of dip at New York is 70 6', and the total 
strength of field at that point is .61. Find the horizontal and 
vertical components. 

909. Why is the earth's field simply directive in its action on 
a suspended magnet ? 

910. Why does not an ordinary compass needle dip or tend 
to dip ? 

911. Define magnetic moment. Find the dimensions of mag- 
netic moment, and compute the moment of a magnet .13 m. 
long, and of pole strength 42, the magnetization being assumed 
uniform throughout the. length of the magnet. 

912. A magnet having a moment M is broken into n equal 
pieces of the same 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 of an ordinary pendulum, and draw conclusions as to the character of the 
motion produced in each case. See 741, 742. 



MAGNETOMETERS 163 

915. Show that the magnetic moment of a uniformly magne- 
tized bar is proportional to the volume of the bar. Whence 
define intensity of magnetization. 

916. A bar magnet has a 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 is magnetic potential measured ? Find 
the potential of a point distant .6 m. from a magnet pole of 
strength 72. 

919. Find the work done in carrying a pole of strength 4 
units from a point distant 5 cm. from a magnet pole of strength 
100 units to a point distant 2 cm. from this pole. 

920. Find the potential at a point 6 cm. distant from the 
north pole of, and in line with the axis of, a bar magnet 10 cm. 
long and of pole strength 80. 

921. A point P is distant OP from the center of a small 
magnet whose magnetic moment is M. Show that the potential 

at P is 2 > where is the inclination of OP to the axis of 

the magnet. 

N 

t 

-it 

+m[^^F 

-m +ra 



m 



i 
H i 

I Fig. 79. 

S 

922. When the left hand magnet, Fig. 79, is so short com- 
pared with d that the lines joining their poles may be considered 



164 



PROBLEMS IN PHYSICS 



as parallel with that joining their centers, what is the torque 
exerted by the large magnet on the small one ? 

Treat force action of each pair of poles separately. Then take moments 
and add. 

923. What torque is exerted by the earth's field ? 

924. By means of the last two examples show that when 
small magnet is in equilibrium 

H 2d 

[Where l=\ distance between poles of large magnet. 

925. Explain why pole strength of small magnet need not be 
known. Why could it not be reduced to zero and yet have 
-equation of Example 924 true ? 

926. Show that when d is very great in comparison with /, 



2 ml 



tan 



-pm 



927. If H= .24, d \ m., / = 10 cm., 

4> = 25. 

What is the pole strength of the magnet ? 

928. Prove that when magnets are placed 
as in Fig. 80 [the length of the small mag- 
net being small compared with d~\ 

2ml 



d 



H 



= |X 2 4-/ 2 ] 



929. When / may be neglected, show that 
2 ml 



H 



= d tan 



+m 




S w 



rn 



930. How do the results of Examples 926 
and 929 compare. Explain why such a dif- 
ference should be expected. 

931. If the large magnet were reversed, what change of 
position would the small one experience ? 



Fig. 80. 



MAGNETIZATION 165 

932. If the magnets were exactly alike, and each were sus- 
pended so as to be free to move, would each turn through the 
same angle in Fig. 79 ? in Fig. 80 ? 

933. Taking axes parallel and normal to the axis of a magnet, 
plot curves showing (a) the variation of potential and (b) the 
variation of magnetic force with distance along the axis. Dis- 
cuss the relation existing between these curves. (Only one 
pole of the magnet is to be considered.) 

934. Define magnetic induction (B), permeability (/j), and 
susceptibility (K). Imagine a piece of soft iron placed in a 
weak field. Further, imagine the field to gradually increase 
in strength. Show by means of a curve the changes which 
take place in the induction in the iron with the increase in the 
field strength. 

Such a curve is called a magnetization curve, and is of great practical value. 
It is usually plotted with the induction B and the field strength H as co- 
ordinates. Obviously the ratio of any ordinate B to the corresponding 
abscissa H is the permeability /x of the iron. 

935 Which is the more easily magnetized, soft iron or steel? 
Which retains the greater amount of magnetism when the mag- 
netizing force is removed ? Explain answers fully in accordance 
with the molecular theory of magnetism. 

936. Why is magnetism removed by heating ? Why are iron 
rods subjected to tapping or jarring liable to become magnetized ? 

937. An iron tube is driven into the earth in the Northern 
Hemisphere. What would be its magnetic condition ? 

938. What kind of iron would you choose for the construction 
of permanent magnets ? of telegraph instruments ? 

939. Show that B, H, and / are quantities of the same kind 
or dimensions. What must therefore be true of //, and K? 

940. Explain the principle of magnetic screening, as when 
a galvanometer needle is protected by an iron screen. 



166 PROBLEMS IN PHYSICS 

941. A sample of transformer iron gives the following data. 
Plot and discuss the magnetization curve. 

H B 

1.32 1324 

2.0 3650 

4.64 8800 

/.I lOQSo 

IO./3 12450 

14.65 13320 

19.42 13920 

37.0 15032 

49.8 15465 

942. Compute the data requisite to plot a permeability curve, 
using H and /* as 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 H along the middle of the solenoid, (a) when 
no iron is present, (b) when iron giving the data of Problem 941 
is present, the current in both cases being 5 amperes. 

955. What is hysteresis ? What is represented by the area 
of a hysteresis loop ? 

956. A transformer core contains 3840 cu. cm. of iron. The 
hysteresis loss is 16300 ergs per cycle per cubic centimeter. 
If this transformer be supplied with an alternating current of 
frequency 120 periods per second, what is the power (in watts) 
lost in hysteresis ? 

957. How does the energy spent in hysteresis appear ? What 
is the effect of jarring on hysteresis ? 

958. State clearly the meaning of the symbols in the formula 



/ K 
for the magnetic pendulum, T= 2 TT \ * . 



168 PROBLEMS IN PHYSICS 

959. Explain how the magnetic pendulum differs from the 
gravitational pendulum. Would there be any objection to using 
a magnetic pendulum for a clock ? 

960. What must be the pole strength of a magnet, moment 
of inertia 1800, distance between the poles 10 cm., that it may 
make 20 complete vibrations in 4 min., where H = .145 ? 

961. A large block of soft iron is placed beneath a horizontal 
vibrating magnet. What will be the effect on T? 

962. A magnet is set in vibration where H is .16, and T is 
found to be 3 sec. When taken to another place, T' is found to 
be 3.2 sec. Find H' . 

963. Derive the equation T 2 TT "\ ^^r, explaining any 
approximations or assumptions made. 

964. If a magnet is struck several blows, what will be the prob- 
able effect on its time of vibration as a magnetic pendulum ? 

965. A strip of lead is bound to a magnetic pendulum. 
What is the effect on 7\ ? 

In the study of the magnetic forces due to currents, of tendencies of con- 
ductors carrying current to move in a magnetic field, and of the direction of 
induced currents, it will be found that the concept of lines of force is one of 
great utility. Remembering that two magnets placed parallel, with their like 
poles contiguous, will tend to separate, we see that if this action is to be 
ascribed to a property of lines of magnetic force we should say that lines of 
force parallel and in the same direction repel. 

It will be found convenient to suppose that the characteristics of lines of 
magnetic force are in part as follows : ^.~_--^--r-^-_^-^ 

(a) Magnetic lines of force parallel and in same f,^' ~^<\ 
direction repel each other. l ;i 

() Magnetic lines of force parallel and in oppo- fill 

site directions attract. vj^ 

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

elastic threads which first bend, and then ^---3^-^--. 

break when a conductor moves across i'^^ ~"A"~' 

them. dV '*' 

(d) These lines tend to shorten and also to pass /L^ 



through iron rather than air. '^^ g 

These, together with the fact that when a cur- r -^== 

rent flows lines of magnetic force tend to form pig. 81. 



FIELDS DUE TO CURRENTS 



169 



circles around it, are very useful in indicating the relations of currents to 
varying fields, etc. 

The direction of current and the positive direction of the lines of force due 
to it are related to each other in the same way as are the direction of transla- 
tion of a 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 
of the clock or "clockwise. 11 If current pass down perpendicular to the paper 
at A, the entire plane has lines directed as shown. 

For convenience in diagram, we shall indicate that a line of force is coming 
up through the paper by a , going down by a x . Thus, if current flows in 
the line MN in the plane of the page, the magnetic lines are vertical circles 
encircling MN clockwise, looking from M to N. 

This is not suggested as the only way in which these relations may be 
remembered, but as one found of considerable convenience in practice. 

A few diagrams are added to show the application of these statements. 



(1) 

X X X X X 



(3) 



X X X X X X X 



XXX XXXXX 

xxxxxxxxx 



(2) 



Fig. 83. 

(1) Two parallel currents in the same direction attract. 

(2) Two parallel currents in opposite directions repel. Likewise for con- 
ductors inclined to each other. 

(3) Two rectilinear currents perpendicular to each other. AB free to turn 
about A. B moves to the left. Similarly, if CD is a circle and AB a radial 
current. 

(4) Current down perpendicular to plane of magnet. At A conductor and 
magnet tend to approach ; at B to separate. (See Fig. 81.) 

The property of magnetic lines of force assumed in (c} may be conveniently 
used in determining the direction of induced currents. We might look at 



PROBLEMS IN PHYSICS 



the matter of relative motion of a conductor and lines of magnetic force some- 
what, as indicated by Fig. 84. 

Let A be the intersection of a conductor with the plane of the paper, and 
let the lines of force be parallel to this plane. When A is moving to the 
right or the field moves to the left, we may consider the lines of force from 




c d 



MOTION 



Fig. 84. 



a to c as crowded together and stretched, d is stretched so far that lateral 
compression is forcing it to encircle A, e has gone through the phases b, c, d, 
and the points corresponding to P and Q of d have united as at s, leaving e r 
encircling the wire. Current tends then to flow down, just as current would 
flow to set up like lines or to 
oppose the motion. 

966. The case of an 
east and west wire in 
the earth's field is a 
good example. If MN 
and OP (Fig. 85) repre- 
sent two lines of the 
earth's field, AB an east 
and west wire, then, if 




INDUCED CURRENTS 



I/I 



AB is moved up, the lines tend to encircle it as shown. Which 
way does current tend to flow ? Does the current help or 
oppose the motion ? 

967. Draw the diagram when the wire is falling. 

968. A telegraph wire is stretched east and west. The 
direction of the earth's field is 75 with the horizontal. Show 
by diagram the direction of the induced currents 

(a) When it falls vertically downward. 

(b) When it is raised vertically. 

Show also in what direction to move it in order to get a 
maximum current ; a minimum current. 

969. Two parallel wires are placed as in Fig. 86. When 
the key k is closed, what takes 

place in the other wire ? If the 
wires moved apart with a velocity 
equal to that of light, would the 
same effect be observed ? 

We may consider circular lines of mag- 
netic force as springing out from the first 
wire. Their radii increasing at what rate ? Fi - 86 ' 

970. The north pole of a magnet passes through the bottom 
of a cup C. Mercury covers the bottom, and a wire suspended 





\ 


1 


\ 


1 


\ 


t 


\ 


\ / 



\ 



Fig. 87. Fig. 87 (a). 

vertically above N dips below the surface of the mercury. If 



1/2 PROBLEMS IN PHYSICS 

current flows from A to B, show that B will move away from 
and rotate around N. 

Consider the projection of the lines of force due to the magnet on the 
surface of the mercury. (See Fig. 87.) 

971. Extend to the case of a flexible conductor. 

The student should apply this method to cases of action of magnetic fields 
described in 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 
electromagnet. It is observed that it is harder to maintain its 
motion when current is passing through the coils of the magnet. 
Explain this. What becomes of the energy used in turning 
the plate ? Does the magnet tend to move ? 

977. Show in what direction a magnet may move with refer- 
ence to a fixed wire in order that no electromotive force may 
be set up in the wire. 

978. In the figure of Example 1000, in what direction must 
the coil turn that current may flow from A to D ? 

979. A solenoid is wound so that it looks like a 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. 

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

of energy in the case of an x~7<r?OC7C7\7C" > v 

induced current produced ( [ )[ J[ J f ]( ]( } / COIL WITH 

by motion in a magnetic 

field ? (Winter, '96.) 

983. Find the force act- 
ing on a pole of 60 units' 



CURRENT 
+ 



o 



strength at a distance of Fig - 88 - 

5 cm. from an infinitely long straight conductor carrying a cur- 
rent of 5 amperes. 

984. To reduce the force in the foregoing case by 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 io 9 magnetic lines of force in 2 sec. What is the average 
electromotive force induced ? 

The E.M.F. induced is proportional to the rate of cutting. To reduce the 
result to practical units (volts), divide by io 8 . 



OF TftK 

UNIVERSITY 



174 PROBLEMS IN PHYSICS 

987. A wire 30 cm. long is moved through a field of strength 
6000 lines per sq. cm. at the rate of 10 m. per second. Find 
the induced electromotive force in volts. 

988. A centimeter length of a straight wire is placed at right 
angles to the lines of force of a uniform magnetic field, i C.G.S. 
unit of current flows through the wire. The strength of the 
magnetic field is 1000. What force acts on the wire ? If the 
current is ten times as great, the field 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 ? 

991. Would it make any difference in the quantity if the 
loop were turned slowly or quickly ? 

992. How can a straight wire be moved in a magnetic field, 
and yet have no electromotive force developed in it ? 

993. If a closed loop of wire be moved without change of 
plane through a magnetic field of uniform strength, will any 
current flow in it ? Will any electromotive force be developed 
in it ? 

994. A wire 2 m. long, and lying horizontally east and west, 
is allowed to fall freely, (a) Find the value of the induced 
electromotive force at the end of 3 sec. (b) Find the mean 
value of the induced electromotive force during a fall of 5 sec. 
(c) Find the time elapsing before the electromotive force shall 
be just i volt. 

995. AA' and BB' are a pair of copper rails, so large that their 
resistance may be neglected in comparison with that of the rest 



INDUCTION 



175 



of the circuit. 5 is a wire of resistance I ohm, sliding without 
friction over the rails, and at right angles to them. Resistance 
of galvanometer circuit, 3 ohms. If the rails are in a field of 
3000 lines per sq. cm., the direction of the field being upward, 




[ ,1,1 




B' 



'B 



Fig. 89. 



normal to the plane of the rails, and the distance between the 
rails be 40 cm., find : 

(a) The velocity required to develop an electromotive force 
in S of i volt. 

(b) The direction of this electromotive force when the motion 
is in the direction indicated. 

(c) The current in the circuit when k is closed. 

(d) The work done in the circuit. 

(e) The force necessary to propel 5 at this velocity. 

996. Show that the quantity of electricity set in motion by 
any displacement of the slider is independent of the velocity 
with which that displacement takes place. 

997. If the velocity of the slider were doubled, what would 
be true of the work done in the circuit ? 

998. If the galvanometer were replaced by a cell developing an 
electromotive force of I volt, and having a resistance of 3 ohms, 
in what direction and with what velocity would the slider move ? 

999. How can the slider and rails of Problem 995 be used to 
show that the dimensions of resistance in the electromagnetic 
system are those of a velocity ? 

1000. A rectangular loop of wire .1 m. wide and .2 m. long 
rotates uniformly at a speed of 1200 revolutions per minute in a 



1 7 6 



PROBLEMS IN PHYSICS 



field of 4000 lines per square centimeter. Find the average 

value of the electromotive force 

induced. 

Since all that is desired is the aver- 
age value of the induced electromotive 
force, we have only to find the total 
change in the number of lines thread- 
ing the loop per revolution, and divide 
this by the time of one revolution. 

1001. With the direction of 
field and of rotation as indicated, 
what is the direction of the in- 
duced electromotive force ? 

1002. When such a coil ro- 
tates in a uniform field, to what Fig. 90. 

are the instantaneous values of electromotive force propor- 
tional ? 

1003. If a loop of wire rotating in a magnetic field form part 
of a closed circuit, the resulting current is an alternating one. 
Sketch and describe a device by which the current may be 
caused to flow always in the same direction in the external 
circuit. 




W 




Fig. 91. 

1004. A wire w is caused to rotate around the north pole of 
a magnet by means of a cord on a pulley. Contact is made in 



INDUCTION 177 

the mercury cups a, a', the closed circuit being aa'g. The 
strength of pole is 72. The wire is caused to rotate with a 
speed of 600 revolutions per minute. The resistance of the 
circuit is .01 ohm. What is the current in amperes ? 

Would current flow if the wire extended the entire length of 
the magnet ? 

1005. If the wire were fixed and the magnet were placed on 
a pivot so as to be free to turn about its axis, what would happen 
when current is passed through the wire ? 

1006. A Faraday disc has a radius of 15 cm. It rotates with 
a speed of 2400 revolutions per minute in a field normal to the 
disc of average density 2000 lines per square centimeter. Com- 
pute the electromotive force of the machine. 




Fig. 92. 

1007. What essential differences are found in the following 
types of dynamos : (a) magneto, (b) series, (c) shunt, (d) com- 
pound ? 

1008. What type of dynamo is best adapted to incandescent 
lighting ? 

1009. Which would suffer most from a short circuit, a shunt 
or a series dynamo ? 

1010. What is meant by residual magnetism ? What impor- 
tant part does it play in the operation of dynamos ? 

ion. A certain 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. It is required to find the number of ampere turns 
needed on the field limbs to set up in the air gap a magnetic 
density of 6000 lines per square centimeter. Concerning this 
machine the following data are known : 

Diameter of armature core 25 cm. 

Length of armature core 36 cm. 

Mean length of magnetic circuit in field (i.e. 

dotted line abed} 145 cm. 



DYNAMO FIELD 



179 



Permeability of armature iron for a magnetic 

density of 6000 1120 

Coefficient of magnetic leakage for this type of 

circuit 1.5 

Permeability of field iron for a magnetic density 

of 1.5 x 6000 2250 

Depth of double air gap 0.72 cm. 

The work done in carrying a + unit magnet pole around the path indicated 
by the dotted line is 



10 



where S is the number of turns of wire on the 
field, and i the current in them. Considering 
the magnetic circuit as made up of three sepa- 
rate parts, in each of which the value of H is 
assumed to be constant, we have 



rm 



rrn 



10 . J 

the subscripts a, g, and/ referring to the arma- 
ture, air gap, and field, respectively. 

Taking the computations in the order indi- 
cated, we have 

/? a 6000 
rl a = = , 

Pa I I 20' 

6000 




Fig. 93. 



and 



-25 = 134- 



I 1 20 

For air, /x = i, 

hence H g l g = 6000 x 0.72 4320. 

Now in every dynamo there is a certain amount of stray field, or waste 
magnetic flux, which forms closed loops by various paths outside the air gap. 
The amount of stray field is readily found for different types of machines by 

total magnetic flux . 
experiment. The ratio useful ma * netic flux is called the coefficient of mag- 

netic leakage. The induction to be provided for in the field is, therefore, 
kB a = 1.5 x 6000 = 9000, 



and we have 



HJ f - 



145 = 580, 
2250 

= 134 4 4320 + 580 
= 5034- 



l8o PROBLEiMS IN PHYSICS 

The requisite number of ampere turns is therefore 



St = = 4000 nearly. 
i. 26 

The student should note that in the foregoing method certain assumptions 
are made which are not rigorously true. The method, however, gives results 
which meet all the requirements of practical dynamo design. 

1021. The armature of this dynamo has upon its surface 184 
conductors, and it makes 1200 revolutions per minute. Com- 
pute the electromotive force. 

Since the pole pieces are not likely to cover more than 80 per cent of the 
armature, the magnetic density may be taken, as in the preceding case, as the 
same in air gap and armature. The 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 is on open circuit. To what two causes is 
this waste of power due? How may it be diminished ? 

1023. What is meant by a characteristic curve? A series 
machine gives the following data. Plot it, using current on 
the 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 : 

~ Jv- . total electrical energy developed 

Gross efficiency = = = : = ^ -^- 

total mechanical energy supplied 



Net efficiency 



_ useful electrical energy developed 
~ total mechanical energy supplied 



, . . .-.. useful electrical energy 

Electrical efficiency = = = : -& 

total electrical energy 

Since every machine has some internal resistance, the electrical efficiency 
can never reach 100 per cent. 

1031. A certain dynamo develops electric power to the 
amount of 10 kilowatts. If the gross efficiency of the machine 
is 85 per cent, how many 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 is 70 per cent ; 
the gross efficiency is 84 per cent. What is the electrical 
efficiency of the machine ? 

1034. A. certain dynamo requires 8 kilowatts when driven at 
full capacity. The net efficiency being 82 per cent under these 
conditions, and the pressure at the terminals being 105 volts, 
what is the maximum current output ? 

1035. A shunt dynamo has a field resistance of 70 ohms, and 
an armature resistance of .022 ohm. When running at full 
load the machine develops 80 amperes at an available potential 
difference of no volts. What is the electrical efficiency of the 
machine ? 

1036. A house is to be lighted with 40 glow lamps, each re- 
quiring. 5 ampere and no volts. Allowing for a loss of 4 per 
cent in the mains, a net efficiency in the dynamo of 84 per cent, 
and a reserve power in the engine of 15 per cent more than 
that actually required to run the lamps, what should be the 
horse-power of the engine installed ? 

1037. What determines the practical limit of long-distance 
transmission of power ? 

1038. When current is supplied to a direct-current dynamo it 
runs as a motor. Explain by reference to Problem 995. 

1039. An ammeter is introduced into a motor circuit. The 
current is found to be stronger when the armature is held still 
than when it is allowed to run. Explain. 

1040. If the wheels of a street car were securely locked, the 
controller could not safely be turned so as to let maximum cur- 
rent flow. Why ? 

1041. A wire i m. long, carrying a current of 20 amperes, is 
held in a uniform field of 6000 lines per square centimeter. 
Find the restraining 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' 2 r. 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 that it follows from the foregoing that the effi- 
ciency of a motor doing maximum useful work is but 50 per cent. 

1045. Under what conditions will a motor run at maximum 
efficiency ? 

1046. A 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 
it be possible to use one as motor and the other as a dynamo ? 
What would such an arrangement be called, and what uses might 
it have ? 

1049. ( a ) What is meant by the period of an alternating cur- 
rent ? (b) A small 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\ sin ado. 
Jo 



da 

which is readily integrated. 

The mean value of an harmonic current is similarly found from the expres- 

sion * = /since. 

NOTE. In the treatment of alternating currents it is usually justifiable to 
consider them as harmonic even though they depart somewhat from the sine 
law. In the following problems the current is assumed to be a simple sine 
function of the time. 

1051. The maximum value of an alternating current is 120 
amperes. What is the mean value ? 

1052. What is the maximum value of an alternating current 
that will cause the same quantity to flow across any 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 the mean square of a current whose maximum value is /. 

The " square root of the mean square " of an alternating current is called 
its virtual value, and is of great importance. 

1054. The virtual value of an alternating current is 35.3 
amperes. What is its maximum value ? its mean value ? 



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. If a current of 2.1 amperes flowing in a coil of 100 
turns set up through that coil a magnetic flux of .084 x io 8 
lines, what is the coefficient of 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 8 lines. The change in the current is 2.1 
amperes. Therefore the inductance of the circuit is 

100 X .084 X 10" = h 
2.1 X IO 8 

1059. An harmonic current of 20 amperes (virtual value) is 
flowing in a given circuit. If the frequency be 120 periods per 
second and L .06 henry, what is the electromotive force of 
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. If at full load, the primary power is 4000 

watts and the primary current 2 amperes. What are the values 
of the secondary E.M.F. and current, the efficiency of the 
transformer being 90 per cent ? 

1070. What is necessary that an ordinary alternator may run 
as a motor ? 

1071. What is meant by a rotating magnetic field? How 
may it be produced ? 

1072. A magnetic field whose instantaneous strength is 
given by the equation 

b = 6000 sin wt 

is combined at right angles with another of strength 

' = 5000 sin (wt - -\ 
Find the magnitude of the resultant field. 



MAGNETIC AND ELECTRICAL UNITS 187 

1073. What are the important differences between synchron- 
ous motors and induction motors ? 

Magnetic and Electrical Units. We have seen how from 
the arbitrarily chosen units of mass, length, and time a con- 
venient and consistent system of mechanical units is built up. 
From the same fundamentals, and in a similar way, the units 
necessary for electrical and magnetic measurements may be 
derived. In every case the definition of the unit is based on a 
physical law or a deduction from a physical law. It is evident 
that more than one unit might easily be chosen according as 
different physical phenomena were made the basis of the selec- 
tion. Thus two distinct C.G.S. systems of electrical units 
have arisen. One, the electrostatic system, is based on the 
definition of unit quantity of electrification as defined from the 
experimentally proved relation between the magnitudes of 
electric charges and the force, in air, between them. This 
relation is 



Now since unit length is a fundamental, and unit force has 
been already chosen, it is consistent to say that unit quantity is 
such a quantity that acting on an equal quantity at unit dis- 
tance will repel it with a force of one dyne. Unit quantity is 
thus made to depend directly upon the units of force and dis- 
tance. To ascertain the way in which the fundamentals are 
involved in any measurements of quantity we must pass to 

dimensions ; thus, 

O 2 
unit force = ML T~ 2 = J^ 

whence Q = M*L*T~\ 

Unit current is said to flow in a circuit when unit quantity is 
conveyed in unit time. This makes the dimensions of current 






1 88 PROBLEMS IN PHYSICS 

PROBLEM. Suppose that the unit of time were increased 
threefold, and the unit of length were doubled. How would 
the C.G.S. electrostatic unit of current be affected? 

Making these changes in the fundamentals, we have for the 
new unit of current 



That is, the new unit is smaller than the old, the ratio being 

TWO' 

Hence a given current would appear to be -$$- times as 
great. 

The other system is called the C.G.S. 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, 

7 2 Trrm 
r = A - 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 is meant by a vibratory motion? Does the bob 
of a pendulum have such motion ? Does the balance wheel of 
a watch have such motion ? State any examples of vibration 
which occur to you. 

1081. In what ways do the motions of different particles along 
a clock pendulum differ? In what respects are their motions 
alike ? 

1082. What kind of motion does the end of the minute hand 
of a clock have ? How does its motion differ from that of the 
hour hand ? the second hand ? 

1083. Compare the angular velocities of the hour, minute, and 
second hands of a clock. 

1084. An elastic ball is dropped and allowed to bound and 
rebound from the floor until it comes to rest. Is the motion 
vibratory ? Draw the time and height curve. Draw the time 
and velocity curve approximately. Explain any peculiarities of 
these curves. (See falling bodies.) 

1085. C anc l E are tw balls in circular and elliptic grooves 
on a horizontal table. OP is a rod turning about the common 
center of the ellipse and circle with a uniform angular velocity, 
and pushing the balls around. Compare the linear velocities of 
the two balls at AA', BB' , etc. Compare the average linear 
velocity of E with the velocity of C. The periodic time of C 
is 40 sec. What is that of E ? Is the motion of the balls 
vibratory ? (See Fig. 94.) 

191 



1 9 2 



PROBLEMS IN PHYSICS 



1086. If OA', Fig. 94, is very 
small, what kind of motion will the 
ball moving in the ellipse approach ? 

1087. How does the motion of 
the piston of an engine differ from 
that of a point in the 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 
Where will he walk the fastest ? The first goes clear 
in 20 min. What is his angular velocity ? What 
periodic time of the second man ? Fig. 95. 
B 




have ? 
around 
is the 





Fig. 96. 

1089. If P 1 P 2 = P 2 P& does M 1 M 2 = M 2 M 3 ? The time re- 
quired for the first to move from P 1 to P z is the same as from 
P 2 to P s , and equals that for the second to go from M 1 to M z 
or M 2 to M 3 . How has the motion of the second man changed 
in going from M l to M z ? Fig. 96. 

If P moves uniformly in a circle of radius #, and M is the 
foot of the perpendicular dropped from P on a diameter OA, 



SIMPLE HARMONIC MOTION 193 

we have from trigonometry OM= a cos <. Making all measure- 
ments from OA, and calling CD the angle turned through in i 
sec., we have < = wt. 

Then displacement of M from center is 
OM = x a cos &)/. 

The period is the same as that of P ; 

^ 2 TT 2 TT 

i.e. T= or ft) = . 

27T . 

. '. x = a cos - /. 

1090. When, i.e., for what values of / is x a maximum ? a 
minimum ? How does the velocity of M vary ? 

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 is dropped vertically into a pond of still water. 
It is observed that when ten circular crests have started outward, 
the outer one has a radius of 6 m. What is the wave length ? 
If 40 sec. are required for the outer crest to acquire a radius 
of 5 m., what is the period ? 

1099. If a vertical section is made through the center of the 
wave system described above, draw the curve of intersection 
with the surface approximately. Would this curve change in 
form from instant to instant ? Would it change in position ? 

noo. A system of water waves X = i m., v 4m., is moving 
across a lake parallel to a row of fine wires 25 cm. apart. These 
wires, starting at a certain point, are numbered o, I, 2, 3, 4, 5. 
etc. At a given instant a crest is observed at the wire marked o. 

State (i) At which other wires crests would be found. 

(2) At which other wires hollows or troughs would be 

found. 

(3) At which other wires the water is at its natural 

level. 

(4) At which other wires the water is at its natural 

level, but falling. 
194 



WAVES 195 

1 10 1. When crests are observed at two wires 4 m. apart, how 
many crests would there be between them ? How many troughs ? 

1 1 02. Suppose that each individual particle moves in a circle, 
how many times would a particle go around its circle while a 
crest was traveling 20 m. ? 

1103. A system of water waves is moving across a lake. The 
wave length is 5 m. The velocity of propagation is 6 m. per 
second. A crest is observed at a stake at a given instant. 
Where will that crest be in 10 sec. ? Where was it 20 sec. 
before? At the instant when the crest is at the stake men- 
tioned, what was the condition at a stake 10 m. back? 15 m. 
back ? \6\ m. back? 17-^ m. back ? i8| m. back ? 

1104. Two exactly similar wave systems are moving in oppo- 
site directions. Show by diagram how "nodes" and "loops" 
will be formed. 

NOTE. The student can easily trace or copy a sine curve on a card, and 
then cut out a pattern so as to readily draw two like curves. Then compound 
them by the ordinary method. Now move one ^ A to the right and the other 
the same distance to the left, and again compound them. Move each again, 
etc. It will be found that certain points will be permanently at rest and 
others vibrate with greater or less amplitude. 

1105. Distinguish clearly between a progressive and a station- 
ary wave system. Show how a stationary system may be pro- 
duced. 

1106. A system of progressive waves is moving in a straight 
line. The wave length and velocity of propagation is known 
and the complete history of the motion of one particle is given. 
What can be inferred from this ? 




Fig. 97. 

1107. A wave motion of simple harmonic type is propagated 
along OX (Fig. 97). The wave length is X, the velocity of 



196 PROBLEMS IN PHYSICS 

propagation is v. The circle at the left is called the circle 
of reference, which means that as P moves around the circle 
with uniform angular velocity the line PM, varying harmoni- 
cally, is a representative of the actual motion of every disturbed 
particle of the medium. How far will the wave travel through 
the medium while P goes once around the circle ? 

1108. Show that T= = -, where T is the common pen- 

to v 
odic time. 

1 109. What relation is there between the angle turned through 
by p and the distance traversed by every portion of the wave 
disturbance in that time ? 

i no. Use this relation to modify y = a sin wt so as to express 
a progressive wave disturbance of simple harmonic type. 

mi. Show that y a sin (a>t + otf ') 



= a sin [vt -f- x\. \x = vt 1 . 

A- 

1 1 12. Show that if the displacement at 5 is 

y = a sin (vt + x), 
\ 

it is identical with that which was at the origin - sec. before. 

1113. The displacement at 5 is now given by 

y=a sin (vt + x}. 

A, 

What will represent it when it reaches R, a distance / beyond ? 
What was it represented by when at' a point /units back of 5? 

1114. If y 4 sin [10 / + 5 x\ is the expression for a progres- 



PROPAGATION OF WAVES 197 

sive wave, what is the periodic time ? the wave length ? the 
velocity of propagation ? 

1115. Waves of length 2 m. pass a certain point. It is ob- 
served that four pass per second. Write the expression for 
their motion. 

1 1 16. From the equation y = a sin (vt + x), we see that as 

X ^ 

/ increases so that t' t= T= , y takes all values between 

v 

+ a and a. While if t is constant, that is, at any instant of 
time, all possible values for y may be found by varying x from 
x to x H- X. What fact does this express ? 

1117. How does the energy distribution of a progressive wave 
system differ from that of a stationary system ? 

1118. Two progressive wave systems, wave lengths 2 : 3, are 
compounded. Sketch approximately the resultant in various 
phase relations. 

1119. What do you mean by the terms like phase, opposite 
phase, retardation of (2 n + i ) ? 

1 1 20. Two wave systems of equal frequency are compounded. 
Sketch approximately the resultant wave form in the following 
cases : 

(a) When the phases are alike and amplitudes equal. 

(b) When the phases are alike and amplitudes are as I : 2. 

(c) When the phase difference is 45, and amplitudes are 

as 1:2. 

(d) When the phase difference is 90, and amplitudes equal. 

(e) When the phase difference is 180, and amplitudes equal. 
(/) When the phase difference is 180, and amplitudes i : 2. 
1 1 21. The displacement of a point is given by y l + y%> where 



j/ 2 = A 2 cos (tot 

Find the resultant displacement, and discuss the expression 
obtained. 



SOUND 

1 122. If a sounding body were in the air, and at a considerable 
distance from the earth, what would be the form of the wave 
front if the temperature were uniform ? What would be the 
direction of motion of those air particles in the same vertical 
line as the source of sound ? the same horizontal line ? in a 
line at an angle of 30 with the vertical ? 

1123. If the velocity of sound in air is different in different 
directions, how would the wave form be altered ? 




Fig. 98- 

Suppose the air in an indefinitely long tube disturbed by the motion of the 
piston, connected as shown in Fig. 98. Let the wheel be imagined to make 
one revolution at a uniform angular velocity in the 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 P when piston makes just one 
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 travel when crank pin moves 
through an angle of 30? 60 ? 90? 180? 270? What part 
of a wave length in each case ? 

1129. Consider two points in the tube a distance x apart, the 
velocity and displacement of the first given at a time t. How 
long before the second will acquire that velocity and displace- 
ment ? Through what angle will crank pin move in that time ? 

1130. The velocity of sound at o C. = 33240 cm. per second. 
Find the velocity when temperature is 25 C. 

1131. Show that if V, = V Vi -f .003665 t, velocity increases 
nearly 60 cm. per second for i rise in temperature. 

1132. The report of a cannon is heard 10 sec. after the flash 
is seen. The temperature of the air is 20 C. How far was 
the observer from the gun ? 

1133. How much is the wave length of the air wave sent out 
by a 256 fork altered by a rise of temperature from o to 20 ? 

1134. A whistle giving 1000 vibrations per second is 156.20 m. 
distant. How many complete waves between it and the obser- 
ver ? Temperature o C. 

1135. The flash of a gun is seen, and 20 sec. later the report 
is heard. The distance is known to be 6932 m. What was the 
temperature ? 

1136. Show that *y has the same dimensions as a velocity. 



200 PROBLEMS IN PHYSICS 

1137. Apply the formula to the case of iron, taking the value 
of Young's modulus as 18- 10" ; density 7.67. 

1138. Find the ratio of the velocity of sound in brass to that 
in iron. 

1139. A string makes 256 complete vibrations per second. 
When the velocity of sound is 34600 cm. per second, what is 
the wave length of the sound ? 

1140. If the temperature of the air were increased, what quan- 
tities referred to in Example 1139 would be altered? 

1141. A 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 ; 



{-A 



F 



area 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. 

1149. A string making 400 vibrations per second has its 
length and stretching force each divided by 4, and its mass per 
unit length multiplied by 4. What effect on the pitch if the 
string is made no less flexible ? 

1150. A wire, I m. of which weighs I g. and is 80 cm. long, 
is made to vibrate in unison with fork n = 128. What force is 
used to stretch it ? 

201 



OF THB 

TJNIVERSITY 



202 PROBLEMS IN PHYSICS 

1151. Why is the base string of a guitar wound with fine 
wire ? If the wire makes each centimeter of the string four 
times as heavy, how will the number of vibrations be altered ? 
What objection is there to lowering the pitch by increasing the 
radius of the string ? 

1152. Explain why it is often more desirable to shorten all 
the strings on a banjo by means of a clamp in order to raise 
the pitch rather than to increase the tension of the strings. 

1153. Draw a diagram to scale, showing the relative positions 
of the frets on a 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 _ I J9 ' 98 ' I0 * 

"512^ 98.10-* 

= J 
512 

1156. Two steel wires, mass of I m., respectively .98 and .45, 
are stretched side by side. The length of the larger is observed 
to be 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 it is found that triangular portion retains its 
shape and moves along the cord at a uniform velocity. Draw 
diagrams showing what happens at B. 

1163. A uniform stretched wire is distorted as shown, A and 
B being rigidly fixed. The distorted portion retains its form 
and moves along the cord at a uniform velocity. Draw diagrams 
showing reflection at D. 




1164. Two like distortions are moving in opposite directions, 
and with the same velocity along a string as shown. Draw a 
series of diagrams showing their positions at several successive 
short intervals of time. Explain why the point (P) midway 
between 3 and 4 remains at rest (Fig. 101). 




Fig. 101. 

1165. Show by diagram how a string may vibrate in various 
modes at the same time. 



STRINGS GENERAL 



It is shown in books on acoustics that the equation of motion for an elastic 
string executing small free vibrations about a position of equilibrium is 



where 



W^r&i (Fig. 102) 

m = mass per unit length, 
F stretching force, 

y = displacement of a point x distant from the origin, 
at a time / 



Fig. 102. 

(1) Show that the equation is of consistent dimensions. 

(2) Writing the equation in the form 



m 



m 



show by substituting that a possible relation between y, a, x, 
and / is 

y = A s\i\px cospat. [A independent of x, y, t. 

(3) If the string is fastened at the point x = o and also at the 
point x = / (i.e. at those points y o for all values of /), find the 
least value of /. 

SUGGESTION. Sin// = o. Hence what set of values may pi have. 

204 



VIBRATION OF STRINGS 205 

(4) Any part of the string between x = o and x = /, in other 
words, any point of the string free to move, will have what kind 
of motion ? 

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

(6) What other frequencies may occur ? What are the tones 
due to these called? Is "A" the same for all of these fre- 
quencies ? 

(7) Does the solution given correspond to a displacement 
when / = o, or to an initial velocity ? 

(8) Show that 

IB sin px sin pat 
C cos/;trcos/tf/ 
D cos/^r sin pat^ 

each satisfy the original equation, and that the sum of any 
number of such terms is also a solution. 

(9) Would the last two be consistent with a fixed point 
at x = o ? 

(10) If y B s'mflx sin pat is a consistent solution, and the 

point x = were touched lightly, what would happen ? 

1166. Draw diagrams showing places of maximum and of 
minimum pressure changes in an open pipe : (a) when vibrating 
in its fundamental mode ; (b) for the first overtone ; (c) for the 
third overtone. 

1 167. Do the same for maximum and minimum displacements. 

1168. Draw similar diagrams for a closed tube. 

1169. An open pipe is vibrating in its fundamental mode; a 
hole in its side large enough to allow considerable air to pass in 
or out is suddenly opened. If the hole is at the middle of the 
tube, what effect will be produced ? 

1170. If the end of the pipe in Example 1 169 is closed and the 
hole left open, what differences will be observed ? 



206 PROBLEMS IN PHYSICS 

1171. Distinguish between "flue" and "reed" pipes, and 
name instruments of each class. 

1172. A closed organ pipe is 60 cm. long. What is the wave 
length of its fundamental ? 

1173. What is the wave length of its first overtone ? 

1174. What is the wave length of the fourth overtone ? 

1175. When the velocity of sound in air is 34800 cm., what is 
the number of vibrations per second in each of the above cases ? 

1176. Would increase of temperature change the pitch of an 
organ pipe ? 

1177. An open tube is 100 cm. long. Find the wave length 
and frequency when the velocity of sound is 34000 cm. per 
second. 

1178. What is the wave length and frequency of its first three 
overtones ? 

1179. A fork making 332 vibrations per second is fixed in 
front of a cylindrical tube, and the length adjusted to resonance 
when temperature is o. How much must the length be 
altered to resound at 25 ? 

1180. A closed pipe is made just long enough to reinforce a 
fork at its mouth, frequency of the fork 64. What must be the 
frequencies of the next four forks of higher pitch which it will 
also reinforce ? 

1181. What would they be if tube were open ? 

1182. A whistle making 4000 vibrations per second is moved 
slowly away from a wall. What is the first position of reinforce- 
ment ? the second ? 

1183. How far will the whistle be from the wall when there 
are four nodes between it and the wall, and the sound is re- 
inforced ? 

1184. How many beats per second will be heard when two 
forks make 250 and 255 vibrations per second respectively ? 



INTERFERENCE 2O/ 

1185. How could you determine, if 6 beats per second were 
heard, which fork was the higher in pitch ? 

1186. Show by diagram how the wave giving beats is made 
up of two differing slightly in frequency and wave length. 

1187. Explain the fluctuations in the intensity of sound from 
a 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. If the scale in Konig's apparatus for the determination 
of the velocity of sound in air is 40 cm., what would be the 
lowest pitch which could be used as a source ? For what pitch 
would there be found just three points where the flame was 
stationary ? 

1190. A tuning-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 the same fundamental pitch ? If so, explain 
the cause. 

1192. Three shortest possible tubes containing respectively 
air, oxygen, and hydrogen, velocities of sound, 33200, 31700, 
126900, resound to a fork giving 1000 vibrations per second. 
What are their lengths ? 

1193. A locomotive whistle makes 1000 vibrations per second. 
When moving 50 km. per hour, what will be the alteration in 
pitch when approaching the observer? when receding? Tem- 
perature of air o C. 

1194. A locomotive whistle makes 3000 vibrations per second. 
Find the apparent number of vibrations : 

(a) When approaching the station at the rate of 100 km. per 
hour. 



208 PROBLEMS IN PHYSICS 

(b) When at rest and the observer is approaching the train at 
the same rate. 

(c) When they are moving away from each other each at the 
rate of 100 km. per hour. 

1195. Draw a diagram showing the effect of motion of the 
source relative to the air upon the wave length in air. 

1196. Indicate clearly the difference between motion of the 
source when observer is at rest and motion of observer when 
source is at rest. 



LIGHT 



REFLECTION 

1197. State the laws of reflection of light. 

1198. Show how reflection is explained on the wave theory. 

1199. If a mirror were perfect, could it be seen ? 

1200. Indicate how the form of a reflected wave front may be 
found when the form of the incident wave and of the reflecting 
surface is known. 

1201. An object is placed in front of a plane mirror. Show 
by diagram the path of the rays by which the image is seen. 
What relation is there between the size of the object and the 
size of the image ? 

1202. A plane mirror is used to reflect a beam of parallel light. 
The mirror is turned 10. Through what angle is the reflected 
beam turned ? Give diagram. 

1203. Show that the image formed by a plane mirror appears 
to be as far back of the mirror as the object is in front. 

1204. Show how spherical waves reflected at a plane surface 
have their curvature reversed. 

1205. Two mirrors are placed at an angle of 90, with a candle 
between them. How many images will be seen ? Locate them. 

1206. If a wave after reflection is to converge to a point, 
what must be the wave form ? 

1207. Two mirrors are inclined at any angle, and a luminous 
point is placed between them. Show that all the images are on 

p 209 



210 



PROBLEMS IN PHYSICS 



a circle, and determine its radius and center. Show how to 
find the angular position of each image. 

1208. Two plane mirrors are placed parallel to each other, and 
50 cm. apart. An object is placed 20 cm. from one of them. 
Show how the images will be spaced. Draw the path of the 
rays by which the fourth image on one side is seen. 

1209. Explain why it is difficult to read the image of a printed 
page in a plane mirror. 

1210. A printed sheet is laid on a table between two parallel, 
vertical, plane mirrors. Which of the images are easily read ? 

1 21 1. A train of mirrors are placed vertical, and inclined to 
each other. Given the angle of incidence on the first, and the 
angle between the planes of each of the mirrors, find the devia- 
tion after successive reflection from each. 

1212. The walls of a rectangular room are plane mirrors. A 
candle is placed at any point in the room, and a person standing 
at a given point, with his eye 

in the same horizontal plane 
as the candle, wishes to ob- 
serve it by rays reflected in 
succession from each of the 
walls. Find the point at which 
he must look. Find the ap- 
parent distance of the image 
seen. 




Fig. 103. 



Notation 


used i 


(Fig. 103). 


C . . 




MN . 




A . . 




P . . 




Q 

CA =R 




P 1 . . 




F . . 



in problems relating to spherical mirrors 

center of curvature. 

aperture of mirror. 

vertex of mirror. . 

luminous point. 

point of incidence. 

radius of curvature. 

intersection of reflected ray and PA. 

principal focus. 



CURVED MIRRORS 211 

Lengths to the right from A are taken + . 
AP' = p 1 = image distance = P' Q approximately. 
AP p = object distance = PQ approximately. 
AF =f= principal focal distance. 

1213. Derive the formula showing the relation between/,/', 
and R. 

1214. What is meant by the term principal focus f 

1215. The radius of a concave spherical mirror is 20 cm. The 
sun's rays fall normally on a small portion of its surface. How 
far from the mirror will the image of the sun be formed ? 

1216. If R = 20 cm., find /' when / = 40 cm. ; 35 ; 25 ; 20 ; 
15; 12; 10; 8; 5. 

For which values of/ above will a real image be formed ? 

1217. If the object is an arrow 5 cm. high, find the size of the 
image in each of the cases of Example 1206. (Size refers to 
linear dimensions.) 

1218. Construct the image as formed by a concave mirror 
when / > R, /</ < R, / </ When is it real ? when virtual ? 
when larger than the object ? when smaller ? 

1219. Show by diagram that if the aperture of a concave 
mirror is large the image formed will be distorted. 

1220. With a given concave mirror where must an object be 
placed so that the image may be real and twice as large as the 
object ? virtual and three times as large as the object ? 

1221. What must be the radius of a concave spherical mirror 
that an image of an object 20 ft. from a screen may be projected 
on the screen and be magnified three times, the object being 
placed between the mirror and the screen ? 

1222. Show how to find the position and size of the image 
formed by a convex mirror : (i) geometrically, (2) analytically. 

1223. Derive the formula for a convex mirror, stating clearly 
the approximations made. 



212 PROBLEMS IN PHYSICS 

1224. A convex mirror R = 80 cm. is placed 30 cm. from a 
candle flame. Where will the image appear to be ? Construct 
it. Find its size if the flame is I in. high. 

1225. An object is moved from a point very near a convex 
mirror to a great distance away from it. How far does the 
image move ? How would its size change ? 

1226. The radius of curvature of a concave mirror is 9 cm. ; 
an object is 10 cm. in front of it. If the mirror is flattened out, 
i.e. if r increases to oo , trace the changes in size and position 
of the image, neglecting the decrease of/. 

1227. The radius of curvature = 100 cm. The object is 90 
cm. from the mirror and is moving outward with a velocity of 
10 cm. per second. How fast is the image moving and in which 
direction ? 

1228. A luminous point is placed at the focus of a parabolic 
mirror. Find the path of the reflected rays. Find the form of 
the wave front. 

1229. Can a very small element of any wave surface be con- 
sidered as spherical? If so, what would the center of the sphere 
mean ? What surface would the center of the sphere trace as 
the surface element moved over the surface of the wave ? 

1230. State the laws of refraction. Show by diagram what 
you mean by the terms used in stating the law. 

1231. Derive the "sine law" from consideration of the velo- 
city of propagation of waves in the two media. 

1232. If the velocity of light is altered in passing from one 
medium to another, does the frequency change ? Does the wave 
length change ? 

1233. Does the index of refraction vary with the wave 
length ? 

1234. Show by diagram the path of a ray when passing from 
water to air at angles of incidence less than the critical angle ; 
just at this angle. 



REFRACTION 213 

1235. What is the critical angle for glass to air, index 

a u = - 3 - ? 

r'ff 2 * 

1236. If the angle of incidence is observed to be 20 and of 
refraction 15, find the index of refraction from each substance 
to the other. 

1237. If the angle of incidence is 40 and the index is J, find 
the angle of refraction. 

1238. A beam of light falls on the surface of still water at 
an angle of 15 with the vertical. Find its direction in the 
water, index > w = . Illustrate by a diagram drawn to scale. 

1239. If the angle of incidence is 45 ; 60 ; 75 ; find the 
direction in the water. 

1240. If the angle of incidence is 45 in passing from water 
to air, what is the direction in air ? 

1241. Light is incident at an angle of 50 in water and passes 
into air. Find path of ray. 

1242. If the direction of a ray is reversed so that it passes 
from water to air, what will be the index? 

1243. A ray passes from water to air, angle of incidence 15. 
Find direction in air. 

1244. Does the critical angle depend on wave length? If 
so, which wave lengths would you expect to have the greater 
critical angle ? 

1245. The velocity of light in air is approximately 3.jo 10 cm. 
per second. What is its velocity in water, //, = ^ ? What in 
glass, (J, = f ? in CS 2 , p = 1.63 ? 

1246. How much longer would it take light to reach the 
earth from the sun if the space were filled with water, neglect- 
ing the difference in velocity in air and vacuo ? Mean distance 
earth to sun, 148. io 6 km. 

1247. A plate of glass is immersed in water with its surface 
horizontal. Light is incident at an angle of 60 on the surface 
of the water. Find its direction in the glass, a /* w = , a p ff = f . 



214 PROBLEMS IN PHYSICS 

1248. The index from air to glass is 1.5. The index from 
air to CS 2 is 1.6. Find the index from glass to CS 2 . 

1249. A beam of monochromatic light is divided ; one part is 
sent through i m. of water, the other part through an air path, 
so that there may be no relative retardation. What is the air 
path required ? 

1250. Light is incident at an angle of 30 on a parallel plate 
of glass 3 cm. thick. Draw the path of the ray. How much is 
the beam displaced in passing through the plate, JJL = | ? 

1251. An observer estimates the depth of a pond, looking 
vertically downward, as 30 ft. What is the depth ? 

1252. If he looked from water at an object 30 ft. above the 
surface, how far above the surface would it appear to be ? 

1253. A fish is 8 ft. below the surface of the water. A 
man shoots at the place where the fish appears to be, holding 
his gun at an angle of 45 with the surface of the water. Does 
the bullet pass above or below the fish ? (Neglect any change 
of direction of bullet.) 

1254. Show by diagram how a straight stick held partly in 
water at an angle of 60 appears to a person in the air. How 
would it appear if the eye were under water ? 

1255. Under what circumstances may light be propagated in 
curved rather than straight lines ? 

1256. Explain how the sun may be seen after it has passed 
below the horizon. 

1257. Prove that if A is the refracting angle of a prism, //. 
the index of refraction, S the angle of minimum deviation, 

sin k(A + &) 
^ = sin 1 A 

1258. IfA= 60, B = 53, find p. 

1259. When A = 60, /* = |, find 8. 
When A = 30, /* = , find 8. 



REFRACTION 215 

1260. Compare the minimum deviation produced by a 30 
water prism and that of a similar 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 l apparent source to an eye is second medium, 

PQ p = PA, Z.PQC =2, 



( DENSE 



3-*^ 



n ^>- TP; , 


^__p 


C. Pi-R 5 




\ P-R 



Fig. 104. 

The A />Cg and P^CQ have a common angle C. 
sin / ft R 



sin 6" 
sin C 


P 
P\ 


Law of 
P-R 


sinr 
sin / 


P\- R 

; LL -^ 1 



sinr 





Pi-R' P 



i.e. 
or 



R P, P 

(A) may be used to derive the formula for a lens if care is taken to 
observe : 

(1) The index from first medium to the second is the reciprocal of the 
index from second to first. 

(2) Distances to right are + , to left . 

216 



LM JL V 



LENSES 



217 



(3) The thickness of the lens may be neglected. 

(4) p l should be eliminated between the expressions for refraction in 
and out. 

For example, the biconvex lens, radii R v R 2 (A) becomes 



- i /, i . 



_ /A i f Since / 1 is the virtual 

- 7T - - TI -- 1~ OUt. 

R. 2 p p l source. 

. / _ !\r_L _i__L I _ JL _ 1. ["Multiply second by /A and 
) \-R l R. 2 \~ ' p' p L add the equations. 

If p' is negative, we have a real image or the light converges, and, 
changing the signs, 



1269. A convex lens is placed between a source of light and 
a screen so as to give an image of the source on the screen. 
How many such positions for the lens may be found ? Compare 
the sizes of the image and object in each case. 

1270. A double convex lens, the ratio of whose radii is 6 to i, 
is used as a condenser for a magic lantern. When the light is 
at a distance of 2 in., the emerging rays are parallel. What 
are the radii, the material of the lens being crown glass ? ('78. ) 

1271. A candle is / cm. from a wall. A converging lens 
forms an image on the wall; when moved a distance d it also 

/2 _ ,J% 

forms an image. Prove that f = -- 

4/ 

1272. In a lens where SB construct the image of an 

/ / / 
object placed between lens and F; when placed beyond F. 

1273. Write a rule for the construction of images in case of 
spherical lenses and mirrors. 

1274. The focal length of a converging lens is 3 m. Find 
the distance from the lens (assumed thin) to the image in each 
of the following positions of the object : 4 m. ; 5 m. ; 8 m. ; 
10 m. ; 20 m. ; i km. ; 3 m. ; 2 m. ; i m. ; 5 cm. 



218 PROBLEMS IN PHYSICS 

1275. Show by construction the position and size of the 
image when /= i m. ; / = 3m.; /= 2m.; / = .5 m. 

1276. In the derivation of the formulae for lenses, what 
assumptions are made which are only approximately correct ? 

1277. What do you mean by a converging lens? by a diverg- 
ing lens ? 

1278. Assuming that a biconvex lens gives a real image, 
construct it, and assuming that the lens is thin, prove that 

- H - = - by use of similar triangles. 

* P . size of image /' 
Show also that -: . . . & = 

size of object / 

1279. By means of the formula A, 

Find the formula for a biconcave lens. 
Find the formula for a 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 , *\ = r 2 = 30 cm. 

1281. A lens of focal length 25 in air, >, = f. What will 
be the focal length in water, > w = |. 

1282. A 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 lt //, / 2 , / 2 ', etc., are joined 
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 1 are taken as 
abscissas and ordinates, what kind of a curve will be found ? 

1293. To what does the other branch of the curve correspond ? 

1294. A small object is placed slightly beyond the principal 
focus of a biconvex lens. The image formed is viewed through 
a biconvex lens placed nearer to the image than the principal 
focal distance. What is such an arrangement called ? Draw a 
diagram showing formation of the image seen, and find the 
ratio of its height to that of the object. 

1295. Draw diagrams showing what is meant by "short" 
sight or myopia. What form of lens is needed to correct 
myopic vision ? 

1296. What is meant by "long" sight, and how may it be 
corrected ? 

1297. A person is unable to see clearly objects 30 cm. from 
the eye. Give two possible explanations of this. 

1298. Indicate by diagram how inability to decrease the 
radius of curvature of the crystalline lens would affect vision. 
What kind of glasses would be needed ? 



INTERFERENCE 

1299. What must be the relation between the elements of 
two light waves in order that interference may be possible ? 

1300. Explain three general methods by which interference 
may be obtained. 

1301. Find the effective retardation of a ray of light reflected 
from B over one reflected from C. Fig. 105. 



E 




Fig. 105. 

Consider parallel rays incident at A and C such that the ray refracted at A, 
reflected at B, and refracted at C proceeds along the same path CE as the ray 
reflected at C. When 2 strikes the surface, the phase is the same as at D in i . 
Draw CB' perpendicular to AB. Then i travels from D to C, while 2 travels 
from A to B' . 

Apparent retardation is B' B + BC. 

Extend AB to C', making BC' = BC. 

Then BB' + CB = 8, 

CC' = 26. 

.. 8 = 2 e cos r. 

.. 8 = 2 fie cos r. 



[Retardation due to glass path. 
[Equivalent retardation in air. 



But one reflection is with change of phase. 
.. effective retardation, 

8 = 2 fie cos r + -. 

It follows that if white light is reflected as shown in the figure, light of wave 
length A. will be a minimum when 2 fie cos r = n\. (n any integer.) 

220 



LENSES 221 

1302. What is the least thickness of crown glass, index -|, 
which will give interference for sodium light when r = 45 ? 

1303. What thickness of a film, index -|, would retard light 
of wave length 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. 

1307. Explain the production of color in the soap-bubble. 
How can the wave length of light be measured ? Derive the 
formula. Give diagram of apparatus used in projecting these 
colors on a screen. ('88.) 

1308. Derive the formula for "Newton's rings." 

p = "Y & sec r> (2n+i)- for bright ring. 



p = ^/R sec r- n\ for dark ring. 

1309. If red light X = 76- io~ 6 is used and R = 9 cm., r = 45, 
find the radii of the first four bright rings. 

1310. What would be the ratio of the radii of rings of the 
same order for X = 76- io~ 6 and X = 52 io~ 6 ? 

1311. Find the general expression for the width of the rings 
for a given wave length. Do they increase or decrease in width 
as r is increased ? 



DIFFRACTION 



1312. Explain why the shadow of a twig cast by an arc light 
on a frosty pane of glass is often fringed with color. 

1313. A slit in a piece of cardboard is held close to the eye 
and parallel to the filament of an incandescent lamp. Explain 
the colored fringes observed. Are the colors pure spectral 
colors ? 

1314. White light diverging from a narrow slit falls on two 
parallel narrow slits very close together. Show how the ap- 
pearance on a screen beyond the apertures depends on the 
wave length considered and on the distance between the two 
parallel slits. 

1315. Light from a small source is divided and passes by 
two paths of slightly different length to a screen. Explain 
briefly the difference in the phenomena observed when the 
light is white and when it is monochromatic. 

1316. Parallel rays 
of white light fall nor- 
mally on a transmis- 
sion grating and the 
diffracted rays are 
brought to a focus by 
a lens. Show by dia- 
gram how spectra are 
formed and derive the 
formula (Fig. 106). 

1317. Two gratings 
are placed one above 

222 




Fig. 106 



DIFFRACTION 223 

the other in a horizontal beam of white light from a vertical 
slit. If one has twice as many lines per centimeter as the 
other, how will the spectra differ ? 

1318. If d= icr 3 , x=59.icr 6 , find 6 1 - # 2 ; 3 . 

1319. For a certain wave length and grating, # 3 = 6 ; for a 
different wave length, # 2 = 6. Find the ratio of the two wave 
lengths and explain overlapping spectra. 

1320. Show from the expression = sin # n how the length 
of the spectrum will change with d. 

1321. Sunlight passing through a narrow slit falls normally 
on a transmission grating 800 lines per centimeter. The spectra 
are focused on a screen 10 m. from the grating. Find the 
position and length of the first spectrum. 

1322. Light of wave length 589- io~ 7 passes through the slit 
and falls on a grating G, Fig. 107. An eye placed just back of 
the grating observes a series of images of the slit, as S v 5 2> 
5 3 , etc. Explain how these images are formed. 

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

t-s" 

-S' 
s 



S 2 
Fig. 107. 

1323. How do the spectra formed by diffraction differ from 
those formed by refraction ? 

1324. What assumptions are made in the derivation of the 
formula for a grating which are only approximately true ? 

1325. Derive the formula for a reflection grating if the angle 
of incidence = i and the grating space = d. 

1326. Show by diagram the formation of the first spectrum 
by a reflection grating. 



TABLES 



[In these tables the admirable arrangement made use of 
in Bottomley's Four-Figure Mathematical Tables has been 
followed.] 



226 



LOGARITHMS 





O 


1 


2 


3 


4 


5 


6 


7 


8 


9 


123 


456 


789 


10 


oooo 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 0374 


4 8 12 


I 7 21 25 


29 33 37 


11 

12 
13 


0414 
0792 
1139 


0453 
0828 

"73 


0492 
0864 
1206 


0531 
0899 
1239 


0569 

0934 
1271 


0607 
0969 
1303 


0645 
1004 
1335 


0682 
1038 
1367 


0719 
1072 
1399 


755 
1106 

H30 


4811 
3 7 10 
3 6 10 


15 J 9 23 

14 17 21 
13 I 6 19 


26 30 34 
24 28 31 
23 26 29 


14 
15 
16 


1461 
1761 
2041 


1492 
1790 
2068 


1523 
1818 
2095 


1553 

1847 

2122 


1584 

1875 
2148 


1614 
1903 
2175 


1644 
I93 1 

22OI 


1673 
1959 
2227 


1703 
1987 

2253 


1732 
2014 
2279 


369 
368 

3 5 8 


12 15 18 

II 14 17 

ii 13 16 

IO 12 15 

9 12 14 
9 ii 13 


21 24 27 

20 22 25 

18 21 24 


17 
18 
19 


2304 

2553 
2788 


2330 

2577 
2810 


2355 
2601 

2833 


2380 
2625 
2856 


2405 
2648 
2878 


2430 
26 7 2 
2900 


2455 
2695 

2923 


2480 
2718 
2945 


2504 
2742 
2967 


2529 
2765 
2989 


2 57 
2 5 7 
247 


17 20 22 

16 19 21 
16 18 20 


20 


3010 


3032 


3054 


3075 


3096 


3 Il8 


3139 


3160 


3181 


3201 


246 


8 ii 13 


15 17 19 


21 
22 
23 


3222 
3424 
3617 


3243 
3444 
3636 


3263 
3464 
3655 


3284 
3483 
3674 


3304 
3502 
3692 


3324 

3522 

37" 


3345 
3729 


3365 
3560 

3747 


3385 
3579 
3766 


3404 
3598 
3784 


246 
246 
24 6 


8 IO 12 
8 10 12 

7 9 ii 


14 16 18 
14 15 17 
13 15 17 


24 
25 
26 


3802 

3979 
415 


3820 

3997 
4166 


3838 
4014 

4183 


3856 
4031 
4200 


3874 
4048 
4216 


3892 
4065 
4232 


3909 
4082 
4249 


3927 
4099 
4265 


3945 
4116 
4281 


3962 

4U3 
4298 


2 4 5 
2 3 5 
2 3 5 


7 9 ii 
7 9 10 
7 8 10 


12 14 16 

12 14 15 


27 
28 
29 


43H 
4472 
4624 


433 
4639 


4346 
4502 

4654 


4362 

45 l8 
4669 


4378 
4533 
4683 


4393 
4548 
4698 


4409 
45 6 4 
4713 


4425 
4579 
4728 


4440 

4594 
4742 


445 6 
4609 

4757 


2 3 5 
2 3 5 
i 3 4 


689 
689 
679 


II 13 I 4 

II 12 14 
IO 12 11 


30 
33 


477i 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


i 3 4 


679 


10 ii 13 


4914 
5i85 


4928 
5065 
5198 


4942 

5079 
5211 


4955 
5092 
5224 


4969 
5105 
5237 


4983 
5"9 
525 


4997 
5132 
5263 


5011 

5H5 
5276 


5024 

5 ! 59 
5289 


5038 
5172 
5302 


3 4 
3 4 
3 4 


678 

I I I 


10 II 12 
9 II 12 
9 IO 12 


34 
35 
36 


5315 
5441 
5563 


5328 
5453 

5575 


5340 
5465 
5587 


5353 
5478 
5599 


5366 
5490 
5611 


5378 
55 02 
5623 


539i 
55H 
5635 


5403 
5527 
5647 


5539 
5658 


5428 
5551 
5670 


3 4 
2 4 

2 4 


568 
5 6 7 
5 6 7 


9 10 ii 
9 10 ii 
8 10 ii 


37 
38 
39 

~40~ 


5682 
5798 
59H 


5 6 94 
5809 
5922 


5705 
5821 

5933 


5717 
5832 
5944 


5729 
5843 
5955 


5740 

5855 
5966 


5752 
5866 

5977 


5763 
5877 
5988 


5999 


5786 

5899 
6010 


2 3 
2 3 
2 3 


5 6 7 
5 6 7 

4 5 7 


8 9 10 
8 9 10 
8 9 10 


6021 


6031 


6042 


6o53 


6064 


6075 


6085 


6096 


6107 


6117 


2 3 


4 5 6 


8 9 10 


41 
42 
43 


6128 
6232 
6335 


6138 
6243 
6345 


6149 
6253 
6355 


6160 
6263 
6365 


6170 
6274 
6 375 


6180 
6284 
6385 


6191 
6294 
^395 

6493 
6590 
6684 


6201 
6304 
6405 


6212 
6314 
6415 


6222 
6325 
6425 


2 3 
2 3 
2 3 


4 5 6 
4 5 6 
4 5 6 


7 8 9 
7 8 9 
7 8 9 


44 
45 
46 

48 

49 i 


6435 
6532 
6628 


6444 
6542 
6637 


6 454 
6$ 


6464 
6561 
6656 


6474 

6571 
6665 


6484 
6580 
6675 


6503 
6599 
6693 


6513 
6609 
6702 


6522 
6618 
6712 


2 3 
2 3 
2 3 


4 5 6 
456 

4 5 6 


7 8 9 
7 8 9 
7 7 8 


6721 
6812 
6902 


6730 
6821 
6911 


6739 
6830 
692O 


6749 
6839 
6928 


6758 
6848 
6937 


6767 
6857 
6946 


6776 
6866 
6955 


6785 
6875 
6964 


6794 
6884 
6972 


6803 
6893 
6981 


2 3 
2 3 

I 2 3 


4 5 5 

4 4 5 
4 4 5 


678 
678 
678 


50 

~5T 
52 
53 


6990 


6998 


7007 


7016 


7024 


733 


7042 


7050 


7059 


7067 


i 2 3 


3 4 5 


678 


7076 
7160 
7243 


7084 
7168 
7251 


7093 

7177 
7259 


7101 

7185 
7267 


7110 
7!93 
7275 


7118 
7202 
7284 


7126 
7210 
7292 


7135 
7218 
7300 


7H3 
7226 
7308 


7152 
7235 


I 2 3 
I 2 2 
I 2 2 


3 4 5 
3 4 5 
3 4 5 


678 
677 
667 


54 


7324 


7332 


734 


7348 


735 6 


7364 


7372 


738o 


7388 


7396 


I 2 2 


3 4 5 


667 



LOGARITHMS 



227 





O 

7404 


1 


2 


3 


4 


5 


6 


7 


8 

7466 


9 


123 


456 


789 

5 6 7 


55 


7412 


74i9 


7427 


7435 


7443 


745i 


7459 


7474 


I 2 2 


345 


56 
57 
58 


7482 
7559 
7 6 34 


7490 
7566 
7642 


7497 
7574 
7649 


755 
7582 

7657 


7513 
7589 
7664 


7520 

7597 
7672 


7528 
760^ 
7679 


7536 
7612 
7686 


7543 
7619 
7694 


755i 
7627 
7701 


2 2 
2 2 


345 

3 4 5 
344 


5 6 7 
5 6 7 

5 6 7 

5 6 7 
5 6 6 
5 6 6 


59 
60 
61 


7709 
7782 
7*53 


7716 
7789 
7860 


7723 
7796 
7868 


773i 

7803 

7875 


7738 
7810 
7882 


7745 
7818 
7889 


7752 
7825 
7896 


7760 
7832 
7903 


7767 

7839 
7910 


7774 
7846 
7917 


I 2 
I 2 


344 
3 44 
344 


62 
63 
64 


7924 

7993 
8062 


793i 
8000 
8069 


7938 
8007 

8075 


7945 
8014 
8082 


7952 
8021 
8089 


7959 
8028 
8096 


7966 
8035 
8102 

8169 


7973 
8041 
8109 


7980 
8048 
8116 


7987 
8055 
8122 


I 2 
I 2 
I 2 


334 
334 
334 


5 6 6 
5 5 6 
5 5 6 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8176 


8182 


8189 


I I 2 


334 


5 5 6 


66 
67 
68 


8195 
8261 
8325 
8388 

8451 
8513 


8202 
8267 
8331 


8209 

8274 
8338 


8215 
8280 
8344 


8222 
8287 
8351 


8228 
8293 
8357 


8235 
8299 
8363 


8241 
8306 
8370 


8248 
8312 
8376 


8254 
8319 
8382 


I 2 
I 2 
I 2 


334 
334 
334 


5 5 6 
5 5 6 
4 5 6 

4 5 6 
4 5 6 

4 5 5 

4 5 5 
455 
4 5 5 


69 
70 
71 

^T2~ 
73 

74 


8395 
8457 
8519 


8401 
8463 
8525 


8407 
8470 
8531 


8414 
8476 
8537 


8420 
8482 
8543 


8426 

8488 
8549 


8432 
8494 

8555 


8439 
8500 
8561 


8445 
8506 
8567 


I 2 

[ 2 
I 2 


234 
234 
234 


8573 
8633 
8692 


8579 
8639 
8698 


8585 
8645 
8704 


859i 
8651 
8710 


8597 
8657 
8716 


8603 
8663 
8722 


8609 
8669 
8727 


8615 
8675 
8733 


8621 
8681 
8739 


8627 
8686 

8745 


I 2 
I 2 
I 2 


234 
234 
234 


75 

^6~ 

77 
78 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


I 2 


2 33 

233 
233 
233 


4 5 5 

4 5 5 
445 
445 


8808 
8865 
8921 


8814 
8871 
8927 


8820 
8876 
8932 


8825 
8882 
8938 


8831 
8887 
8943 


8837 
8893 
8949 


8842 
8899 
8954 


8848 
8904 
8960 


8854 
8910 
8965 


8859 

8915 
8971 


I 2 
I 2 
I 2 


79 
80 
81 


18976 
9031 
9085 


8982 
9036 
9090 


8987 
9042 
9096 


8993 
9047 
9101 


8998 

9053 
9106 


9004 
9058 
9112 


9009 
9063 
9117 


9015 
9069 
9122 


9020 
9074 
9128 


9025 
9079 
9133 


I 2 
I 2 
I 2 


233 
2 3 3 
233 


445 
445 
4 4 5 


82 
83 
84 


9138 
9191 

9243 


9H3 
9196 
9248 


9149 
9201 
9253 


9154 
9206 
9258 


9159 
9212 
9263 


9165 
9217 
9269 


9170 
9222 
9274 


9175 
9227 
9279 


9180 
9232 
9284 


9186 
9238 
9289 


I 2 
I 2 
I 2 


233 
233 
233 


445 
4 4 5 
445 


85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 


I 2 


233 


4 4 5 

445 
344 
344 


86 
87 
88 


9345 
9395 
9445 


935 
9400 

945 


9355 
9405 
9455 


9360 
9410 
9460 


9365 
9415 
9465 


937 
9420 
9469 


9375 
9425 
9474 


938o 
943 
9479 


9385 
9435 
9484 


9390 
9440 
9489 


I 2 

o 
o 


233 
223 
223 


89 
90 
91 


9494 
9542 
9590 


9499 
9547 
9595 


954 

955 2 
9600 


959 
9557 
9605 


9513 
9562 
9609 


95i8 
9566 
9614 


9523 
957i 
9619 


9528 

957 6 
9624 


9533 
958i 
9628 


9538 
9586 

9633 


o 
o 


223 
2 2 3 
223 


344 
344 
344 


92 
93 
94 


9638 
9685 
9731 


9643 
9689 

9736 


9647 
9694 
9741 


9652 
9699 
9745 


9657 
973 
975 


9661 
9708 
9754 


9666 
97*3 
9759 


9671 
9717 
9763 


9675 
9722! 
9768 


9680 

9727 
9773 




223 
223 
223 


344 
344 
344 


95 

~96~ 
97 
98 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 


I 


223 


344 

344 
344 
344 


9823 
9868 
9912 


9827 
9872 
9917 


9832 
9877 
9921 


9836 
9881 
9926 


9841 
9886 
9930 


9845 
9890 

9934 


9850 
9894 
9939 


9854 
9899 
9943 


9859 
9903 
9948 


9863 
9908 
995 2 


I 
I 
D I 


223 
223 
223 


99 


995 6 


9961 


9965 


9969 


9974 


9978 


9983 


9987 


9991 


9996 


D I I 


223 


334 



228 



NATURAL SINES 





O' 


6' 


12' 18' 


24' 


3O' 


36' 


42' 48' | 54' 


1 2 3| 4 5 




~~F 

2 
3 

~4~ 
5 
6 

~T~ 

8 
9 


oooo 


0017 


0035 0052 


0070 


0087 


0105 


0122 


0140 


OI 57 


369 


12 I 5 


0175 
0349 
0523 


0192 
0366 
0541 


0209 0227 
0384 0401 
0558 i 0576 


0244 
0419 
0593 


0262 
0436 
0610 


0279 

0454 
0628 


0297 
0471 
0645 


0314 

0488 
o663 


0332 
0506 
0680 


369 
369 
369 


12 I 5 
12 I 5 
12 I 5 


0698 
0872 
1045 


7 J 5 
0889 
1063 


0732 
0906 
1080 


0750 
0924 
1097 


0767 
0941 
i"5 


0785 
0958 
1132 


0802 
0976 
1149 


0819 

0993 
1167 


0837 

IOII 

1184 


0854 
1028 

I2OI 


369 
369 
369 


12 I 5 

12 14 
12 14 


1219 

1392 
1564 


1236 
1409 
1582 


1253 
1426 

1599 


1271 

1444 
1616 


1288 
1461 
1633 


1305 

1478 
1650 


1323 
H95 
1668 


1340 

1513 

1685 


1357 

153 

1702 


1374 

1547 
1719 


369 
369 
369 


12 14 
12 14 
12 14 


10 

"IT 
12 
13 


1736 


1754 


1771 


1788 


1805 


1822 


1840 


1857 


1874 


1891 


369 


12 14 


1908 
2079 

2250 


1925 
2096 
2267 


1942 
2113 
2284 


1959 
2130 
2300 


1977 
2147 
2317 


1994 
2164 
2334 


2OII 

2181 

2351 


2028 
2198 
2368 


2045 
2215 
2385 


2062 
2232 
2402 


3 6 9 
3 6 9 
3 6 8 


II I 4 
II 14 
II 14 


14 
15 
16 


2419 
2588 
2756 


2436 
2605 
2773 


2453 
2622 
2790 


2470 
2639 
2807 


2487 
2656 
2823 


2504 
2672 
2840 


2521 
2689 
2857 


2538 
2706 
2874 


2554 
2723 
2890 


257i 
2740 
2907 


3 6 8 
368 
368 


II 14 
II 14 
II 14 


17 
18 
19 


2924 
3090 
3256 


2940 

3*07 
3272 


2957 
3123 
3289 


2974 
3140- 

3305 


2990 
3156 
3322 


3007 
3i73 
3338 


3024 
3190 

3355 


3040 
3206 

337 1 


3057 
3223 
3387 


3074 
3239 
3404 


368 
368 
3 5 8 


II 14 
II 14 
II 14 


20 


3420 


3437 


3453 


3469 


3486 | 3502 


35i8 


3535 


355i 


35 6 7 


3 5 8 


II 14 


21 
22 
23 

~24~ 
25 
26 


3584 
3746 
3907 


3600 
3762 
3923 


3616 

3778 
3939 


3633 
3795 
3955 


3 6 49 
3811 

397 1 


3665 
3827 
3987 


3681 

3843 
4003 


3697 
3859 
4019 


37*4 

3875 
4035 


3730 
3891 

405 i 


3 5 8 
3 5 8 
3 5 8 


II 14 
II 14 
II 14 


4067 

4226 

4384 


4083 
4242 
4399 


4099 
4258 
4415 


4"5 

4274 

443i 


4131 
4289 
4446 


4H7 
435 
4462 


4163 
432i 

4478 


4179 
4337 
4493 


4195 
4352 
459 


4210 
4368 
4524 


3 5 8 
3 5 8 
3 5 8 


II I 3 
II I 3 

10 13 


27 

28 
29 


4540 

4695 
4848 


4555 
4710 
4863 


457 1 
4726 
4879 


4586 

474i 
4894 


4602 

4756 
4909 


4617 
4772 
4924 


4633 
4787 
4939 


4648 
4802 
4955 


4664 
4818 
4970 


4679 
4833 
4985 


3 5 8 
3 5 8 
3 5 8 


10 13 

10 13 
10 13 


CO CO CO CO 
CO tO H* O 


5000 


5015 


5030 


5045 


5060 


5075 


5090 


5105 


5120 


5135 


3 5 8 


10 13 


5 ! 5 
5 2 99 
5446 


5165 
53H 
546i 


5180 
5329 
5476 


5195 
5344 
5490 


5210 
5358 
555 


5225 
5373 
5519 


5240 
5388 
5534 


5255 
5402 

5548 


5270 
5417 
5563 


5284 
5432 
5577 


2 5 7 
257 
2 5 7 


IO 12 
IO 12 
10 12 


34 
35 
36 


5592 
5736 
5878 


5606 

5750 
5892 


5621 

5764 
5906 


5635 
5779 
5920 


5650 
5793 
5934 


5664 
5807 
5948 


5678 
5821 
5962 


5 6 93 
5835 
5976 


577 
5850 
5990 


572i 
5864 
6004 


2 57 
2 5 7 
257 


IO 12 
10 12 

9 12 


37 
38 
39 

lib" 


6018 

! 6l 57 
6293 


6032 
6170 
6307 


6046 
6184 
6320 


6060 
6198 
6334 


6074 
6211 
6347 


6088 
6225 
6361 


6101 
6239 
6 374 


6115 

6252 
6388 


6129 6143 
6266 6280 
6401 6414 


2 5 7 
2 5 7 
247 


9 12 
9 ii 
9 ii 


6428 


6441 


6455 


6468 


6481 


6494 


6508 


6521 


6534 6547 


247 


9 ii 


41 
42 
43 


6561 
6691 
6820 


6574 
6704 

6833 


6587 
6717 
6845 


6600 
6730 
6858 


6613 

6743 
6871 


6626 
6756 
6884 


6639 
6769 
6896 


6652 
6782 
6909 


6665 
6794 
6921 


6678 
6807 
6934 


247 
2 4 6 
246 


9 ii 

9 ii 
8 ii 


44 


6947 


6959 


6972 


6984 


6997 


7009 


7022 


7034 7046 


759 


246 


8 10 



NATURAL SINES 



229 





0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


123 


4 5 


45 


7071 


7083 


7096 


7108 


7120 


7i33 


7*45 


7i57 


7169 


7181 


246 


8 10 


46 
47 
48 


7'93 
73H 
743i 


7206 
7325 
7443 


7218 
7337 
7455 


7230 

7349 
7466 


7242 
736i 
7478 


7254 
7373 
7490 


7266 
7385 
750i 


7278 
7396 
75i3 


7290 
7408 
7524 


7302 
7420 
7536 


246 
2 4 6 
246 


8 10 
8 10 
8 10 


49 
50 
51 


7547 
7660 
7771 


7558 
7672 
7782 


757 
7683 
7793 


7581 
7694 
7804 


7593 
7705 
7815 


7604 
7716 
7826 


7 6l 5 
7727 

7837 


7627 
7738 
7848 


7638 
7749 
7859 


7649 
7760 
7869 


2 4 6 
2 4 6 
2 4 5 


8 9 
7 9 
7 9 


52 
53 
54 


7880 
7986 
8090 


7891 

7997 
8100 


7902 
8007 
8111 


7912 
8018 
8121 


7923 
8028 
8131 


7934 
8039 
8141 


7944 
8049 
8151 


7955 
8059 
8161 


7965 
8070 
8171 


7976 
8080 
8181 


2 4 5 
2 3 5 
235 


7 9 
7 9 
7 8 


55 


8192 


8202 


8211 


8221 


8231 


8241 


8251 


8261 


8271 


8281 


2 3 5 


7 8 


56 
57 
58 


8290 

8387 
8480 


8300 
8396 
8490 


8310 
8406 
8499 


8320 

8415 
8508 


8329 
8425 
8517 


8339 
8434 
8526 


8348 
8443 
8536 


8358 
8453 
8545 


8368 
8462 
8554 


8377 
8471 

8563 


2 3 5 
2 3 5 
235 


6 8 
6 8 
6 8 


59 
60 
61 


8572 
8660 
8746 


8581 
8669 
8755 


8590 
8678 
8763 


8599 
8686 
8771 


8607 
8695 
8780 


8616 
8704 
8788 


8625 
8712 
8796 


8634 
8721 
8805 


8643 

8729 
8813 


8652 
8738 
8821 


3 4 
3 4 
3 4 


6 7 

a ? 


62 

63 
64 


8829 
8910 

8988 


8838 
8918 
8996 


8846 
8926 
9003 


8854 
8934 
9011 


8862 
8942 
9018 


8870 
8949 
9026 


8878 
8957 
9033 


8886 
8965 
9041 


8894 

8973 
9048 


8902 
8980 
9056 


3 4 
3 4 
3 4 


1 i 

5 6 


65 


9063 


9070 


9078 


9085 


9092 


9100 


9107 


9114 


9121 


9128 


2 4 


5 6 


66 
67 
68 


9135 
9205 
9272 


9H3 
9212 
9278 


915 
9219 
9285 


9157 
9225 
9291 


9164 
9232 
9298 


9171 
9239 
9304 


9178 
9245 
93" 


9184 
9252 
9317 


9191 
9259 
9323 


9198 
9265 
9330 


2 3 
2 3 
2 3 


5 6 

4 6 
4 5 


69 
70 
71 


9336 
9397 
9455 


9342 
9403 
9461 


9348 
9409 
9466 


9354 

9415 
9472 


9361 
9421 
9478 


9367 
9426 

9483 


9373 
9432 
9489 


9379 
9438 
9494 


9385 
9444 
9500 


939i 
9449 
955 


2 3 

2 3 
2 3 


4 5 
4 5 
4 5 


72 
73 

74 


95 11 
9563 
9613 


95 l6 
9568 
9617 


9521 
9573 
9622 


95 2 7 
9578 
9627 


9532 
9583 
9632 


9537 
9588 
9636 


9542 

9593 
9641 


9548 
9598 
9646 


9553 
9603 
9650 


9558 
9608 

9655 


2 3 

2 2 
2 2 


4 4 
3 4 
3 4 


75 


9659 


9664 


9668 


9673 


9677 


9681 


9686 


9690 


9694 


9699 


I 2 


3 4 


76 

77 
78 


973 
9744 
978i 


9707 
9748 
9785 


9711 

9751 
9789 


9715 
9755 
9792 


9720 

9759 
9796 


9724 
9763 
9799 


9728 
9767 
9803 


9732 

977 
9806 


9736 
9774 
9810 


9740 
9778 
9813 


2 
2 
2 


3 3 
3 3 
2 3 


79 
80 
81 


9816 
9848 
9877 


9820 

9851 
9880 


9823 
9854 
9882 


9826 

9857 
9885 


9829 
9860 
9888 


9833 
9863 
9890 


9836 
9866 
9893 


9839 
9869 

9895 


9842 
9871 
9898 


9845 
9874 
9900 


I 2 




2 3 

2 2 
2 2 


82 
83 
84 


9903 
9925 
9945 


9905 
9928 

9947 


9907 
993 
9949 


9910 
9932 

995 * 


9912 
9934 
995 2 


9914 
9936 
9954 


9917 
9938 
995 6 


9919 
9940 
9957 


9921 
9942 
9959 


9923 
9943 
9960 





o 


2 2 
I 2 
I I 


85 


9962 


9963 


9965 


9966 


9968 


9969 


9971 


9972 


9973 


9974 


001 


I I 


86 
87 
88 


9976 
9986 
9994 


9977 
9987 

9995 


9978 
9988 

9995 


9979 
9989 
9996 


9980 
9990 
9996 


9981 
9990 
9997 


9982 
9991 
9997 


9983 
9992 
9997 


9984 
9993 
9998 


9985 
9993 
9998 


I 
000 

o o o 


I I 
I I 

O O 


89 


9998 


9999 


9999 


9999 


9999 


I -000 

nearly. 


I 'OCX) 

nearly. 


I'OOO 

nearly. 


rooo 

nearly. 


I'OOO 
nearly. 


o o o 






230 



NATURAL COSINES 





O' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


123 


4 5 





rooo 


I'OOO 

nearly. 


rooo 

nearly. 


rooo 

nearly. 


rooo 

nearly. 


9999 


9999 


9999 


9999 


9999 


000 





1 

2 
3 


9998 

9994 
9986 


9998 

9993 
9985 


9998 

9993 
9984 


9997 
9992 

9983 


9997 
9991 
9982 


9997 
9990 
9981 


9996 
9990 
9980 


9996 
9989 
9979 


9995 
9988 

9978 


9995 
9987 

9977 


O O O 
O O O 
I 


O O 

I I 
I I 


4 
5 
6 


9976 
9962 
9945 


9974 
9960 

9943 


9973 
9959 
9942 


9972 

9957 
9940 


997 i 
995 6 
9938 


9969 
9954 
9936 


9968 
9952 
9934 


9966 

995 i 
9932 


9965 
9949 
9930 


9963 
9947 
9928 


O O 
I 
O I 


I 2 
I 2 


7 
8 
9 


9925 
9903- 
9877 


9923 
9900 

9874 


9921 
9898 
9871 


9919 

9895 
9869 


9917 

9893 
9866 


9914 
9890 
9863 


9912 
9888 
9860 


9910 
9885 
9857 


9907 
9882 
9854 


9905 
9880 

9851 


I 
O I 
O I 


2 2 
2 2 
2 2 


10 


9848 


9845 


9842 


9839 


9836 


9833 


9829 


9826 


9823 


9820 


I I 2 


2 3 


11 
12 
13 


9816 
9781 
9744 


9813 
9778 
9740 


9810 
9774 
9736 


9806 
9770 
9732 


9803 
9767 
9728 


9799 
9763 
9724 


9796 

9759 
9720 


9792 
9755 
97'5 


9789 
975 1 
9711 


9785 
9748 
9707 


112 
I I 2 
112 


2 3 
3 3 

3 3 


14 
15 
16 


973 
9659 
9613 


9699 

9655 
9608 


9694 
9650 
9603 


9690 
9646 
9598 


9686 
9641 
9593 


9681 
9636 
9588 


9677 
9632 
9583 


9673 
9627 

9578 


9668 
9622 
9573 


9664 
9617 
9568 


I I 2 
I 2 2 
122 


3 4 
3 4 
3 4 


17 
18 
19 


9563 
95 11 
9455 


9558 
9505 
9449 


9553 
9500 

9444 


9548 
9494 
9438 


9542 
9489 
9432 


9537 
9483 
9426 


9532 
9478 
9421 


9527 
9472 

9415 


952i 
9466 
9409 


95i6 
9461 
9403 


I 2 3 
I 2 3 
I 2 3 


4 4 
4 5 
4 5 


20 


9397 


9391 


9385 


9379 


9373 


9367 


936i 


9354 


9348 


9342 


I 2 3 


4 5 


21 
22 
23 


9336 
9272 
9205 


9330 
9265 
9198 


9323 
9259 
9191 


93 ! 7 
9252 
9184 


93" 
9245 
9178 


934 
9239 
9171 


9298 
9232 
9164 


9291 
9225 
9157 


9285 
9219 
9150 


9278 
9212 
9M3 


I 2 3 
I 2 3 
I 2 3 


4 5 
4 6 
5 6 


24 
25 
26 


9135 
9063 


9128 
9056 
8980 


9121 
9048 
8973 


9114 
9041 
8965 


9107 
9033 
8957 


9100 
9026 
8949 


9092 
9018 
8942 


9085 
9011 
8934 


9078 
9003 
8926 


9070 
8996 
8918 


I 2 4 

i 3 4 
i 3 4 


5 6 

\ I 


27 

28 
29 


8910 
8829 
8746 


8902 
8821 
8738 


8894 
8813 
8729 


8886 
8805 
8721 


8878 
8796 
8712 


8870 
8788 
8704 


8862 
8780 
8695 


8854 
8771 
8686 


8846 

8763 
8678 


8838 

8755 
8669 


i 3 4 
i 3 4 
i 3 4 


I I 

6 7 


30 


8660 


8652 


8643 


8634 


8625 


8616 


8607 


8599 


8590 


8581 


i 3 4 


6 7 


31 
32 
33 


8572 
8480 

8387 


8563 
8471 
8377 


8462 
8368 


8545 
8453 
8358 


8536 
8443 
8348 


8526 
8434 
8339 


8517 
8425 
8329 


8508 

8415 
8320 


8499 
8406 
8310 


8490 
8396 
8300 


2 3 5 
235 
2 3 5 


6 8 
6 8 
6 8 


34 
35 
36 


8290 
8192 
8090 


8281 
8181 
8080 


8271 
8171 
8070 


8261 
8161 
8059 


8251 
8151 
8049 


8241 
8141 
8039 


8231 
8131 
8028 


8221 
8121 
8018 


8211 
8m 

8007 


8202 
8100 
7997 


2 3 5 
2 3 5 
235 


7 8 
7 8 
7 9 


37 
38 
39 


7986 
7880 
7771 


7976 
7869 
7760 


7965 
7859 
7749 


7955 
7848 

7738 


7944 
7837 
7727 


7934 
7826 
7716 


7923 
7815 
7705 


7912 
7804 
7694 


7902 
7793 
7683 


7891 
7782 
7672 


245 

2 4 I 
246 


7 9 
7 9 
7 9 


40 


7660 


7649 


7638 


7627 


7 6l 5 


7604 


7593 


758i 


757 


7559 


246 


8 9 


41 
42 
43 


7547 
743i 
73H 


7536 
7420 
7302 


7524 
7408 
7290 


7513 
7396 
7278 


75 01 
7385 
7266 


7490 
7373 
7254 


7478 
736i 
7242 


7466 

7349 
7230 


7455 
7337 
7218 


7443 
7325 
7206 


246 
2 4 6 
246 


8 10 
8 10 
8 10 


44 


7193 


7181 


7169 


7157 


7H5 


7133 


7120 


7108 


7096 


7083 


246 


8 10 



N.B. Numbers in difference-columns to be subtracted, not added. 



NATURAL COSINES 



2 3 I 





0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


123 


4 5 


45 


7071 


759 


7046 


7034 


7022 


7009 


6997 


6984 


6972 


6959 


246 


8 10 


46 
47 
48 

^49" 
50 
51 


6947 
6820 
6691 


6934 
6807 
6678 


6921 
6794 
6605 


6909 
6782 
6652 


6896 
6769 
6639 


6884 
6756 
6626 


6871 

6743 
6613 


6858 
6730 
6600 


6845 
6717 
6587 


6833 
6704 

6574 


246 
246 
2 4 7 


8 ii 
9 ii 
9 ii 


6561 
6428 
6293 


6547 
6414 
6280 


6534 
6401 
6266 


6521 
6388 
6252 


6508 

6374 
6239 


6494 
6361 
6225 


6481 

6347 
6211 


6468 

6334 
6198 


6455 
6320 
6184 


6441 
6307 
6170 


2 4 7 
2 4 7 
2 5 7 


9 ii 
9 ii 
9 ii 


52 
53 
54 


6018 

5878 


6i43 
6004 

5864 


6129 
5990 
5850 


6115 

5976 
5835 


6101 
5962 

5821 


6088 
5948 
5807 


6074 
5934 
5793 


6060 
5920 
5779 


6046 
5906 
5764 


6032 
5892 
575 


2 5 7 
2 5 7 
2 5 7 


9 12 
9 12 
9 12 


55 


5736 


572i 


5707 


5 6 93 


5678 


5664 


565 


5635 


5621 


5606 


2 5 7 


10 12 


56 
57 
58 


5592 
5446 
5299 


5577 
5432 
5284 


5563 
5417 
5270 


5548 
5402 

5255 


5534 
5388 
5240 


5373 
5225 


555 
5358 
5210 


5490 
5344 


5476 
5329 
5180 


53H 
5165 


2 5 7 
2 5 7 

2 57 


IO 12 
IO 12 
IO 12 


59 
60 
61 


5150 

5000 

4848 


5135 
4985 
4833 


5120 
4970 
4818 


5105 

4955 
4802 


5090 
4939 
4787 


575 
4924 
477 2 


5060 
4909 
4756 


5045 
4894 


4879 
4726 


5oi5 
4863 
4710 


3 5 8 
3 5 8 
3 5 8 


10 13 
10 13 
10 13 


62 
63 
64 


4695 
4540 
4384 


4679 
4524 
4368 


4664 
459 
4352 


4648 
4493 
4337 


4633 
4478 
4321 


4617 
4462 
435 


4602 
4446 
4289 


4586 

443i 
4274 


457i 
4415 
4258 


4555 
4399 
4242 


3 5 8 
3 5 8 
3 5 8 


10 13 
10 13 

II 13 


65 


4226 


4210 


4195 


4179 


4163 


4147 


4131 


4"5 


4099 


4083 


3 5 8 


II 13 


66 
67 
68 


4067 

3907 
3746 


4051 
3891 
3730 


4035 
3875 
37H 


4019 
3859 
3697 


4003 

3843 
3681 


3987 
3827 
3665 


397i 
3811 

3649 


3955 
3795 
3633 


3939 
3778 
3616 


3923 
3762 
3600 


3 5 8 
3 5 8 
3 5 8 


II 14 

II 14 
II 14 


69 
70 

71 


3584 

3420 
3256 


35 6 7 
3404 
3239 


3387 
3223 


3535 
337i 
3206 


3355 
3190 


35 2 
3338 
3173 


3486 
3322 
3156 


3469 
3305 


3453 
3289 
3123 


3437 
3272 
3107 


3 5 8 
3 5 8 
368 


II 14 
II 14 
II 14 


72 
73 

74 


3090 
2924 
2756 


3074 
2907 
2740 


3057 
2890 

2723 


3040 
2874 
2706 


3024 

2857 
2689 


3007 
2840 
2672 


2990 
2823 
2656 


2974 
2807 
2639 


2957 
2790 
2622 


2940 

2773 
2605 


368 
368 
368 


II 14 
II 14 
II 14 


2588 


2571 


2554 


2538 


2521 


2504 


2487 


2470 


2453 


2436 


3 6 8 


II 14 


76 

77 
78 

80 
81 


2419 
2250 

2079 


2402 
2233 
2062 


2385 
2215 

2045 


2368 
2198 
2028 


235 1 
2181 

201 I 


2334 
2164 

1994 


2317 
2147 
1977 


2300 
2130 
1959 


2284 
2113 
1942 


2267 
2096 
1925 


368 
369 
369 


II 14 

II 14 
II 14 


1908 
1736 


1891 
1719 
1547 


1874 
1702 

1530 


i857 
1685 

1513 


1840 
1668 
1495 


1822 
1650 
H78 


1805 

1633 
1461 


1788 
1616 
1444 


1771 

1599 
1426 


1754 
1582 
1409 


369 
369 
369 


12 14 
12 14 
12 14 


82 
83 
84 


1392 

1219 

1045 


1374 

1201 
1028 


1357 
1184 
ion 


1340 
1167 
0993 


1323 

"49 
0976 


1305 
1132 
0958 


1288 
0941 


1271 
1097 
0924 


1253 
1080 
0906 


1236 
1063 
0889 


369 
369 
369 


12 14 
12 14 
12 14 


85 


0872 


0854 


0837 


0819 


0802 


0785 


0767 


0750 


0732 


0715 


369 


12 I 5 


CO t> 00 O 

oo oo oo oo 


0698 
0349 


0680 
0506 
0332 


o663 
0488 
0314 


0645 
0471 
0297 


0628 

454 
0279 


0610 
0436 
0262 


593 
0419 
0244 


0576 
0401 
0227 


0558 
0384 
0209 


0366 
0192 


369 

3 6 9 
369 


12 15 
12 15 
12 I 5 


0175 


I57 


140 


OI22 


0105 


0087 


0070 


0052 


0035 


0017 


369 


12 I 5 



N.B, - Numbers in difference-columns to be subtracted, not added. 



232 



NATURAL TANGENTS 





0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


123 


4 5 





oooo 


0017 


0035 


0052 


0070 


0087 


0105 


OI22 


0140 


OI 57 


369 


12 14 


1 

2 
3 


0175 
0349 
0524 


0192 
0367 
0542 


0209 
0384 
0559 


0227 
0402 
0577 


0244 
0419 
0594 


0262 

0437 
0612 


0279 

0454 
0629 


0297 
0472 
0647 


03H 
0489 
0664 


0332 
0507 
0682 


369 
369 
369 


12 I 5 
12 I 5 
12 I 5 


4 
5 
6 


0699 
0875 

1051 


0717 
0892 
1069 


0734 
0910 
1086 


0752 
0928 
1104 


0769 
0945 

1122 


0787 
0963 
"39 


0805 
0981 
H57 


0822 
99 8 
"75 


0840 
1016 
1192 


0857 
io33 

I2IO 


369 
369 
369 


12 I 5 
12 I 5 
12 I 5 


7 
8 
9 


1228 

1405 
1584 


1246 

1423 
1602 


1263 
1441 
1620 


1281 

H59 
1638 


1299 

H77 
1655 


1317 

H95 
1673 


1334 
1512 
1691 


1352 
1530 
1709 


1370 
1548 
1727 


1388 
1566 
1745 


369 

369 
369 


12 I 5 
12 I 5 
12 I 5 


10 


1763 


1781 


1799 


1817 


1835 


1853 


1871 


1890 


1908 


1926 


369 


12 I 5 


11 
12 
13 


1944 

2126 

2309 


1962 
2144 
2327 


1980 
2162 
2345 


1998 
2180 
2364 


2016 
2199 

2382 


2035 
2217 
2401 


2053 

2235 
2419 


2071 

2254 
2438 


2089 
2272 
2456 


2IO7 
2290 
2475 


369 
369 
369 


12 15 
12 I 5 
12 I 5 


14 
15 
16 


2493 
2679 
2867 


2512 

2698 
2886 


2530 
2717 
2905 


2549 
2736 

2924 


2568 
2754 
2943 


2586 

2773 
2962 


2605 
2792 
2981 


2623 
28ll 
3OOO 


2642 
2830 
3019 


2661 
28 4 9 
3038 


369 
369 
369 


12 16 

13 16 
13 16 


17 
18 
19 


3057 
3249 

'3443 


3076 
3269 
3463 


3096 
3288 
3482 


3"5 

3307 
35 2 


3134 

3327 
3522 


3i53 
3346 
354i 


3172 
3365 
35 61 


3191 

3385 
3581 


3211 

3404 
3600 


3230 
3424 
3620 


3 6 10 
3 6 10 
3 6 10 


13 16 
13 16 
13 17 


20 


3640 


3659 


3679 


3699 


37'9 


3739 


3759 


3779 


3799 


3819 


3 7 I0 


13 17 


21 
22 
23 


3839 
4040 

!'4245 


3859 
4061 
4265 


3879 
4081 
4286 


3899 
4101 

4307 


3919 
4122 

4327 


3939 
4142 

4348 


3959 
4163 

4369 


3979 
4183 
4390 


4000 
4204 
4411 


4O2O 
4224 
4431 


3 7 I0 
3 7 10 
3 7 I0 


13 17 

14 17 
14 17 


24 
25 
26 


i'4452 
4663 

4877 


4473 
4684 
4899 


4494 
4706 
4921 


45'5 
4727 
4942 


4536 
4748 
4964 


4557 
477 
4986 


4578 
479i 
5008 


4599 
4813 
5029 


4621 
4834 
5Q5 1 


4642 
4856 

573 


4 7 10 
4 7 ii 
4 7 ii 


14 18 
14 18 
15 18 


27 
28 
29 


'595 
5317 
'5543 


5"7 
5340 
5566 


5U9 

5362 
5589 


5161 

5384 
5612 


5184 
5407 
5635 


5206 
5430 
5658 


5228 
5452 
5681 


5250 
5475 
574 


5272 
5498 
5727 


5295 
5520 

575 


4 7 ii 
4 8 ii 
4 8 12 


15 18 
15 J 9 
15 19 


30 


'5774 


5797 


5820 


5844 


5867 


5890 


59H 


5938 


596i 


5985 


4 8 12 


16 20 


31 
32 
33 


6009 
6249 
6494 


6032 
6273 
6519 


6056 
6297 
6544 


6080 
6322 
6569 


6104 
6346 
6594 


6128 

6371 
6619 


6152 

6395 
6644 


6176 
6420 
6669 


6200 

6445 
6694 


6224 
6469 
6720 


4 8 12 
4 8 12 
4 8 13 


1 6 20 

16 20 

17 21 


34 
35 
36 


6745 
7002 
7265 


6771 
7028 
7292 


6796 
754 
73 J 9 


6822 
7080 
7346 


6847 
7107 
7373 


6873 
7133 
7400 


6899 

7159 
7427 


6924 
7186 
7454 


6950 
7212 
7481 


6976 

7239 
7508 


4 9 13 
4 9 13 
5 9 H 


17 21 
18 22 

18 23 


37 
38 
39 


7536 

:gi 


7563 
7841 
8127 


7590 
7869 
8156 


7618 
7898 
8185 


7646 
7926 
8214 


7673 
7954 
8243 


7701 
7983 
8273 


7729 
8012 
8302 


7757 
8040 

8332 


7785 
8069 
8361 


5 9 H 
5 10 14 

5 I0 J 5 


18 23 

19 24 

20 24 


40 


8391 


8421 


8451 


8481 


8511 


8541 


857i 


8601 


8632 


8662 


5 I0 *5 


20 25 


41 
42 
43 


86 93 
9004 
9325 


8724 
9036 
9358 


8754 
9067 

939i 


8785 
9099 
9424 


8816 
9131 
9457 


8847 
9163 
9490 


8878 
9195 
9523 


8910 
9228 
9556 


8941 
9260 
9590 


8972 
9293 
9623 


5 IO *6 
5 ii 16 
6 ii 17 


21 26 

21 27 

22 28 


44 


9657 


9691 


97 2 5 


9759 


9793 


9827 


9861 


9896 


9930 


9965 


6 ii 17 


23 2 9 



NATURAL TANGENTS 



233 



45| 


O' 


6' 12' 


18' 


24' 30' 


36' 


42' 


48' 


54' 


123 


4 5 


I'OOOO 


0035 0070 0105 


0141 0176 


O2 1 2 


0247 


0283 


0319 


6 12 18 


24 30 


46 

47 
48 


1-0355 
1-0724 

1-1106 


0392 
0761 
"45 


0428' 0464 
0799 0837 
1184 1224 


0501 

0875 
1263 


0538 
0913 

1303 


0575 
095 l 
1343 


0612 
0990 
1383 


0649 
1028 
1423 


0686 
1067 
1463 


6 12 18 
6 13 19 
7 13 20 


25 3i 

25 32 

26 33 


49 
50 
51 

~52~ 
53 
54 


1-1504 
1-1918 
1-2349 


1544 
1960 

2393 


1585 

2OO 2 
2437 


1626 
2045 
2482 


1667 
2088 
2527 


1708 
2131 

2572 


175 
2174 
2617 


1792 
2218 
2662 


1833 
2261 
2708 


1875 
2305 
2 753 


7 H 21 
7 14 22 
8 15 23 


28 34 
29 36 
30 38 


1-2799 
1-3270 
i'3764 


2846 
3319 
38H 


2892 
3367 
3865 


2938 
34i6 
3916 


2985 

3465 
3968 


3032 
35*4 
4019 


3079 
35 6 4 
4071 


3127 

3613 
4124 


3i75 
3663 
4176 


3222 

37 J 3 
4229 


8 16 23 
8 16 25 
9 17 26 


3i 39 
33 4i 
34 43 


55 ! 


1-4281 


4335 


4388 


4442 


4496 


455 


4605 


4659 


4715 


477 


9 18 27 


36 45 


56 
57 
58 


1-4826 

1-5399 
1-6003 


4882 
5458 
6066 


4938 
5517 
6128 


4994 
5577 
6191 


5051 
5637 
6255 


5108 

5697 
6319 


5 l66 

5757 
6383 


5224 
5818 
6447 


5282 
5880 
6512 


5340 
594i 
6577 


10 19 29 
10 20 30 

II 21 32 


38 48 
40 5 
43 53 


59 
60 
61 

~62~ 
63 
64 


1-6643 
1-7321 
1-8040 


6709 6775 
7391 7461 
8115 8190 


6842 
7532 
8265 


6909 6977 
7603! 7675 
8341! 8418 


745 
7747 
8495 


7"3 

7820 

8572 


7182 

7893 
8650 


7251 

7966 
8728 


ii 23 34 

12 24 36 

13 26 38 


45 5 6 
48 60 
5 1 64 


1-8807 
1-9626 
2-0503 


8887 8967 
9711 9797 
0594 0686 


9047 
9883 
0778 


9128 
9970 
0872 


9210 
0057 
0965 


9292 
0145 
1060 


9375 
0233 
"55 


9458 

0323 
1251 


9542 

0413 
1348 


14 27 41 

15 29 44 
16 31 47 


55 68 
58 73 
63 78 


65 


2-1445 


1543 1642 


1742 


1842 


1943 


2 45 


2148 


2251 


2355 


'7 34 5 1 


68 85 


66 
67 
68 


2-2460 
2-3559 
2-475 i 


2566 2673 
3673 3789 
4876 5002 


2781 
3906 
5 I2 9 


2889 
4023 

5257 


2998 
4142 
5386 


3109 
4262 

5517 


3220 
4383 
5649 


3332 
454 
5782 


3445 
4627 
59i6 


18 37 55 

20 40 60 

22 43 65 


74 92 
79 99 
87 1 08 


69 i 

70 1 
71 

~72~ 
73 

74 


2-6051 
2 '7475 
2-9042 


6187 6325 
7625 7776 
9208! 9375 


6464 
7929 

9544 


6605 
8083 
9714 


6746 
8239 
9887 


6889 
8397 
0061 


734 
8556 
0237 


7179 
8716 

0415 


7326 
8878 

595 


24 47 71 
26 52 78 
29 58 87 


95 II8 
104 130 

"5 M4 


3-0777 
3-2709 

3-4874 


0961 
2914 
5105 


1146 
3122 

5339 


1334 
3332 
5576 


1524 
3544 
5816 


1716 

3759 
6059 


1910 

3977 
6305 


2106 
4197 

6554 


2305 
4420 
6806 


2506 
4646 
7062 


32 64 96 
36 72 108 

41 82 122 


129 161 
144 180 
162 203 


75 


3-732I 


7583 


7848 


8118 


8391 


8667 


8947 


9232 


9520 


9812 


46 94 139 


i 86 232 


76 

77 
78 


4-0108 
4-33I5 
4-7046 


0408 
3662 

7453 


0713 
4015 
7867 


1022 

4374 
8288 


1335 
4737 
8716 


'653 
5 I0 7 
9152 

3955 
9758 
6912 


1976 
5483 
9594 
4486 
0405 
7920 


2303 
5864 

0045 


2635 
6252 

0504 


2972 
6646 
0970 


53 107 i 60 
62 124 186 

73 146 219 


214 267 
248 310 
292 365 


79 
80 
81 


5^446 
5'67i3 
6-3138 


1929 
7297 
3859 


2422 
7894 
4596 


2924 
8502 
5350 


3435 
9124 
6122 


5026 
Fo66 
8548 


5578 
1742 

9395 


6140 
2432 
0264 


87 175 262 


35 437 


Difference-columns 
cease to be useful, owing 
to the rapidity with 
which the value of the 
tangent changes. 


82 
83 
84 

~85~ 


7-II54 

8*1443 
9'5 144 


2066 
2636 
9-677 


3002 
3863 
9-845 


3962 
5126 

10-02 


4947 
6427 
10-20 


5958 
7769 
10-39 


6996 

9152 
10-58 


8062 

0579 
10-78 


9158 
2052 
10-99 


0285 
3572 

IT20 


"43 


11-66 


U'9 1 


I2-I6 


12-43 


12-71 


13-00 


13-30 


13-62 


I3-95 


86 
87 
88 


14-30 
19-08 
28-64 


14-67 
19-74 
30-14 


15-06 

20-45 
31-82 


I5- 4 6 
2 1 -2O 
33-69 


15-89 
22-02 
35-8o 


16-35 
22-90 
38-19 


16-83 
23-86 
40-92 


17-34 
24-90 
44-07 


17-89 
26-03 
4774 


18-46 
27-27 
52-08 


89 


57-29 


63-66 


71-62 


81-85 


95'49 


114-6 


143-2 


191-0 


286-5 


573-o 



234 



NATURAL COTANGENTS 





O' 


& 


12' 


18' 


24' 3O' ! 36' 


42' 


48' 54' 


Difference-columns 
not useful here, owing 
to the rapidity with 
which the value of the 
cotangent changes. 





Inf. 


573-0 


286-5 


191-0 


143-2 1 14-6195-49 


81-85 


71-62 63-66 
31-8230-14 
20-45 1974 
15-06 14-67 


1 

2 
3 


57-29 
28-64 
19-08 


52-08 
27-27 
18-46 


47-74 
26-03 
17-89 


44-07 
24-90 

I7-34 


40-9238-1935-80 

23-86 22-90 22-02 
16-83 16-35 15-89 


33-69 

2 1 -2O 
15-46 


4 
5 
6 


14-30 
"'43 
9-5I44 


I3-95 

IT2O 

3572 


13-62 
10-99 
2052 


13-30 
10-78 

579 


I3-OO 1271 112-4;: 
10-58 10-39 I0'20 
9152! 7769 6427 


I2-I6 
IO'O2 
5126 


11-91 n-66 
9-845 9-677 
3863 2636 


7 
8 
9 


8-1443 
r"54 
6-3138 


0285 
0264 
2432 


9158 

9395 
1742 


8062 
8548 
1066 


6996 5958 
7920 6912 
0405J 9758 


4947 
6122 
9124 


3962 
5350 
8502 


3002 2066 
4596| 3859 
7894 7297 


10 


5-6713 


6140 


5578 


5026 


4486 


3955 


3435 


2924 


2422 1929 


123 


4 5 


11 
12 
13 


5^446 
47046 

4-33I5 


0970 
6646 
2972 


0504 
6252 
2635 


0045 
5864 
2303 


9594 
5483 
1976 


9152 
5107 
1653 


8716 
4737 
1335 


8288 

4374 

1022 


78671 7453 
4015; 3662 
0713 0408 


74 148 222 

63 125 188 
53 107 160 


296 370 
252 314 

214 267 


14 
15 
16 


4-0108 
37321 

3^874 


98l2 
7062 
4646 


9520 
6806 
4420 


9232 

6 554 
4197 


8947 
6305 

3977 


8667 
6059 
3759 


8391 
5816 

3544 


8118 
5576 
3332 


7848 

5339 
3122 


7583 

5 I0 5 
2914 


46 93 139 

41 82 122 

36 72 108 


i 86 232 
163 204 
144 i 80 


17 
18 
19 


3-2709 

3*0777 
2-9042 


2506 

595 
8878 


2305 

0415 
8716 


2106 

0237 
8556 


1910 
0061 
8397 


1716 
9887 
8239 


5 2 4 

97H 
8083 


1334 

9544 
7929 


1146; 0961 

9375 ' 9208 
7776 7625 


32 64 96 

29 58 87 
26 52 78 


129 161 

"5 !44 

104 130 


20 


2 '7475 


7326 


7179 


734 


6889 6746 


6605 


6464 


6325) 6187 


24 47 7 1 


95 II8 


21 
22 
23 


2-6051 

2-4751 
2-3559 


5916 
4627 

3445 


5782 
454 
3332 


5649 
4383 
3220 


5517 
4262 
3109 


5386 
4142 
2998 


5257 
4023 
2889 


5 I2 9 
3906 
2781 


50021 4876 

3789! 3673 
2673; 2566 


22 43 65 
20 40 60 
18 37 55 


87 108 
79 99 
74 92 


24 
25 
26 


2-2460 
2-1445 
2-0503 


2355 
1348 

0413 


2251 
1251 

0323 


2148 
"55 
0233 


2045 j 1943 
1060 0965 
0145 0057 


1842 
0872 

997 


1742 
0778 

9883 


16421 1543 
0686' 0594 
9797! 9711 


17 34 5 1 
16 31 47 
15 29 44 


68 85 
63 78 
58 73 


27 
28 
29 


9626 
8807 
8040 


9542 
8728 
7966 


945 s 
8650 

7893 


9375 
8572 
7820 


9292 
8495 
7747 


9210 

8418 
7 6 75 


9128 
8341 
7603 


9047 
8265 
7532 


8967 
8190 
7461 


8887 
8115 
739i 


14 27 41 
i3 26 38 

12 24 36 


55 68 
51 64 
48 60 


30 


7321 


7251 


7182 


7"3 


7045 


6977 


6909 


6842 


6775 


6709 


" 23 34 


45 S 6 


31 
32 
33 


6643 
6003 
'5399 


6 577 
594i 
5340 


6512 

5880 
5282 


6447 
5818 
5224 


6383 
5757 
5166 


6319 

5697 
5108 


6255 
5637 
5051 


6191 

5577 
4994 


6128 
4938 


6066 

ml 


II 21 3 2 

10 20 30 
10 19 29 


43 53 
40 5 
38 48 


34 
35 
36 


4826 
4281 
3764 


477 
4229 
3713 


4715 
4176 

3663 


4659 
4124 

3613 


4605 
4071 
3564 


4550 
4019 

35 '4 


4496 
3968 
3465 


4442 
39i6 
34i6 


4388 
3865 
3367 


4335 
3814 
3319 


9 18 27 
9 17 26 
8 16 25 


36 45 
34 43 
33 4i 


37 
38 
39 


3270 
2799 
2349 


3222 

2753 
2305 


3175 
2708 
2261 


3127 
2662 
2218 


3079 
2617 
2174 


3032 
2572 
2131 


2985 
2527 
2088 


2938 
2482 
2045 


2892 
2437 

2OO2 


2846 

2393 
1960 


8 16 23 
8 15 23 

7 14 22 


3i 39 

30 38 
29 36 


40 


1918 


1875 


1833 


1792 


1750 


1708 


1667 


1626 


1585 


1544 


7 H 21 


28 34 


41 
42 
43 


i54 
1106 
1-0724 


1463 
1067 
0686 


H23 
1028 
0649 


1383 
0990 
0612 


1343 
095 i 
0575 


1303 
0913 

0538 


1263 
0875 
0501 


1224 

0837 
0464 


1184 
0799 
0428 


"45 
0761 
0392 


7 13 20 
6 13 19 

6 12 .18 


26 33 
25 32 
25 3i 


44 


i'0355 


0319 


0283 


0247 


O2I2 


0176 


0141 


0105 


0070 


0035 


6 12 18 


24 30 



N.B. Numbers in difference-columns to be subtracted, not added. 



NATURAL COTANGENTS 



235 





0' 


6' 


12 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


123 


4 5 


45 


ro 


0-9965 


0-99300-9896 


0-9861 


0-9827 


0-9793 


0-9759 


0-9725 


0-9691 


6 ii 17 


23 29 


46 

47 
48 


9657 
9325 
9004 


9623 
9293 
8972 


9590 
9260 
8941 


9556 
9228 
8910 


9523 
9195 

8878 

^57^ 
8273 
7983 


9490 
9163 
8847 


9457 
9UI 
8816 


9424 
9099 

8785 


939i 
9067 

8754 


9358 
9036 

8724 


6 ii 17 
5 ii 16 
5 10 16 


22 28 
21 27 
21 26 


49 
50 
51 


8693 
8391 
8098 


8662 
8361 
8069 


8632 
8332 
8040 


8601 
8302 
8012 


8541 
8243 

7954 


8511 
8214 
7926 


8481 
8185 
7898 


8451 
8156 
7869 


8421 
8127 
7841 


5 i J 5 
5 10 15 
5 I0 M 


20 25 
20 24 
19 24 


52 
53 
54 


78i3 
7536 
7265 


7785 
7508 

7239 


7757 
748i 
7212 


7729 
7454 
7186 


7701 
7427 
7 1 S9 


7 6 73 
7400 

7!33 


7646 

7373 
7107 


7618 
7346 
7080 


7590 
73i9 
754 


75 6 3 
7292 
7028 


5 9 H 
5 9 H 
4 9 13 


18 23 
18 23 
18 22 


55 


7002 


6976 


6950 


6924 


6899 


6873 


6847 


6822 


6796 


6771 


4 9 13 


17 21 


56 
57 
58 


6745 
6494 
6249 


6720 
6469 
6224 


6694 

6445 
6200 


6669 
6420 
6176 


6644 

6395 
6152 


6619 

637 1 
6128 


6594 
6346 
6104 


6569 
6322 
6080 


6544 
6297 
6056 


6519 
6273 
6032 


4 8 13 
4 8 12 
4 8 12 


17 21 

16 20 
16 20 


59 
60 
61 


6009 
'5774 
'5543 


5985 
575 
5520 


596i 

5727 
5498 


5938 
574 

5475 


59H 
5681 
5452 


5890 
5658 
5430 


5867 
5635 
5407 


5 8 44 
5612 

5384 


5820 
5589 
S3 62 


5797 
5566 

5340 


4 8 12 
4 8 12 
4 8 ii 


16 20 
15 19 
15 *9 


62 
63 
64 


5317 
5095 
4877 


5295 
573 
4856 


5272 

505 1 
4834 


5250 
5029 

4813 


5228 
5008 
479i 


5206 
4986 
477 


5184 
4964 
4748 


5161 

4942 
4727 


5'39 
492i 
4706 


5"7 

4899 
4684 


4 7 ii 
4 7 ii 
4 7 ii 


15 18 
15 18 
14 18 


65 


4663 


4642 


4621 


4599 


4578 


4557 


4536 


45'5 


4494 


4473 


4 7 10 


14 18 


66 
67 
68 


*445 2 
4245 
4040 


443i 
4224 
4020 


4411 

4204 
4000 


4390 
4183 
3979 


4369 
4163 

3959 


4348 
4142 

3939 


4327 
4122 

3919 


4307 
4101 

3899 


4286 
4081 
3879 


4265 
4061 
3859 


3 7 10 
3 7 I0 
3 7 I0 


14 17 
14 17 
13 17 


69 
70 

71 


3839 
3640 

'3443 


3819 
3620 

3424 


3799 
3600 

3404 


3779 
358i 
3385 


3759 
356i 
3365 


3739 
354i 
3346 


3719 
3522 
3327 


3699 
35 2 
3307 


3679 
3482 
3288 


3659 
3463 
3269 


3 7 10 
3 6 10 
3 6 10 


13 17 
13 17 
13 16 


72 
73 
74 


3249 
3057 
2867 


323 
3038 
2849 


3211 
3019 
2830 


3 J 9i 
3000 
2811 


3172 
2981 
2792 


3 : 53 
2962 

2773 


3134 
2943 
2754 


3H5 

2924 
2736 


3096 
2905 
2717 


3076 
2886 
2698 


3 6 10 
369 
369 


13 16 
13 16 
13 16 


75 


2679 


2661 


2642 


2623 


2605 


2586 


2568 


2549 


2530 


2512 


369 


12 16 


76 

77 
78 


2493 
2309 
2126 


2475 
2290 
2107 


2456 
2272 
2089 


2438 

2254 
2071 


2419 

2235 
2053 


2401 
2217 
2035 


2382 
2199 
2016 


2364 
2180 
1998 


2345 
2162 
1980 


2327 

2144 
1962 


369 
369 
369 


12 I 5 
12 15 
12 I 5 


79 
80 
81 


1944 
1763 
1584 


1926 

1745 
1566 


1908 

1727 
1548 


1890 
1709 
1530 


1871 
1691 
1512 


1853 
1673 
H95 


1835 
1655 
H77 


1817 
1638 
M59 


1799 
1620 
1441 


1781 
1602 
1423 


3 6 9 
369 
369 


12 15 
12 I 5 
12 I 5 


82 
83 
84 


1405 
1228 
1051 


1388 

1210 

I0 33 


137 
1192 
1016 


1352 
"75 
0998 


1334 
H57 

0981 


1317 

H39 
0963 


1299 

1122 
0945 


1281 
1104 
0928 


1263 
1086 
0910 


1246 
1069 
0892 


3 6 9 
369 

3 6 9 


12 15 
12 I 5 
12 15 


85 


0875 


0857 


0840 


0822 


0805 


0787 


0769 


0752 


0734 


0717 


369 


12 15 


86 
87 
88 

~89~ 


0699 
0524 
0349 


0682 
0507 
0332 


0664 
0489 
3H 


0647 
0472 
0297 


0629 

0454 
0279 


0612 

0437 
0262 


0594 
0419 
0244 


0577 
0402 
0227 


0559 
0384 
0209 


0542 
0367 
0192 


3 6 9 
369 
369 


12 15 
12 15 
12 15 


0175 


OI 57 


0140 


0122 


0105 


0087 


0070 


0052 


0035 


0017 


3 6 9 


12 14 



N.B. Numbers in difference-columns to be subtracted, not added. 



ANSWERS 





66. 


Last part, 35280. 


5. V/ 2 + P -f /fc 2 , dir. cosines l\b\h. 


6. S ,0 = tan-if. (90 o 


69. 


First, 1470; second, 22050 cm. 


i or - i. (180.) 


70. 


49kg. 


7. (o.) 


72. 


122.5 m -5 2 4'5 m - P er sec - 


7. o. 


73. 


5.87 sec. 


12. 5,8.66. 


74. 


4427+. 


23. O) 1936; (^)"35-405; 


75. 


36.3. 


(0 983-5- 


76. 


44.1 m. 


24. 45+ mi. per hr. 


77. 


90.4 m. 


26. 40 mi. per hr. 


81. 


10.4 m. up; 9.2 m. down. 


27. 96.56. 


83. 


0.5 sec., nearly. 


29. Area 2. x io, \_x = instantaneous 


85. 


20.4 m. 


length of side. 


86. 


2.04 sec.; 4.08 sec. 


Volume 3 x^ io, [x = instantaneous 


87. 


416?. 


length of side. 


94. 


485 cm. per sec.; 0.5 sec. 


30. 1162 m. If t = oC. 


96. 


5-83- 


33. See Introduction I, "Dimensions." 


98. 


655'- 


34. 75 cm. per sec. 2 


99. 


913.8 cm. per sec. 


35. 4015.7 km. per hr. 2 


102. 


264 ft. per min. 


36. 1 20 cm. per sec. 


103. 


66f ft. per sec. 


39. i, 3, 5, ... (2 - i). 


104. 


V2:l. 


40. 234 cm. 


105. 


8.54 mi. per hr.; 5725i' E. of S. 


41. O) 1152 cm. ; () 270 cm. 


106. 


7.071 mi. per hr. 


42. (a) 1264 cm. ; () 284 cm. 


107. 


36.56 km. per hr. 


44. -^ km. per hr. 2 


108* 


5 1. 96m. per min.; 30 m. per min.; o. 


45. 8th sec. 


109. 


26.4 ft. per min. 


46. 48 km. 


110. 


17.39; 12.30; 4.658; -2. 


47. 27.5 hrs. 


111. 


30.53+, 71 with "a," nearly. 


48. V = o; a = 2. 


112. 


47.1; 3.219. 


49. 3 sec. 


113. 


19.05 ; o ; 22. 


50. () 8.66 sec. ; () 3.54 sec. 


114. 


8.659 sec. 


56. 1600 dynes. 


115. 


7.14 sec. 


60. 500 sec. 


120. 


326.53 m.; 653.1 m. 


64. 196 io 4 dynes. 


121. 


(I) 2.49 sec.; (2) 498.4 ft.; (3) 


65. 623 io 5 dynes. 




215 ft.; (4) 2i5o', nearly. 



237 



238 



PROBLEMS IN PHYSICS 



[Exs. 124-272 



124. () .1000 radians per min. 


1 L r 9 , on P0 r 

i * I-XT+XIXZ+XI JH [-*2 + -ri] 




fK\ 2 


125. Angular velocity alike; linear as 




I : 2. 


0) \h. 


127. 4 TT radians per sec. 


209. 2000 ergs. 


129. 523.6 mi. per hr.(\vhen r =4000 mi.). 


210. 2i6.io. 


130. 33:8. 


211. 98- io 6 ; 294. io 6 . 


131. 25 m.; 39f m. ; o. 


213. 2352 io 7 . 


132. 4.1 grams. 


214. i6m. 


134. 2.5 ft. per sec. 


215. loo m. 


135. 131+ Ib. 


216. 34640. 


136. 10.35 k - w t- ; 4-35 kg. wt. 


2Yf 2OOOO 


137 T Mm . a AT-m 


COS IO 


* M+ m" M + m 


218. 5-io 3 . 


139. Uniform motion; 7^ = gM. 


219. 2-I0 5 . 


140. \g. 


220. 32 - io 5 gr. cm. 


142. 130! io 5 dynes. 


221. 96 io 5 gr. cm. 


144. 5.625,4.375. 


224. 72 io 3 kg. m. 


145 a M & T mM S 


226. 2 io 6 ergs. 


M + m ' M + m 


228. 49-10"; 24.5- io". 


146. M=-. 


230. W=mal. 


m 3 


231. as large. .-. Numeric 4 times as 


150. 53 io 4 dynes. 


great. 


151. 15; 3; 14.5; 13.9; 10.82; 7.93; 


234. 4 - IO IG ergs. 


4.84. 


235. 588 - io 10 ergs; 6 io 4 kg. m. 


152. 0.7265. 


236. 588 - io 6 ergs. 


153. 12.2; 37.4. 


237. MI = Y 1 ^ MZ', kinetic energy will 


154. 2. 


be acquired by the system. 


155. 60. 


239. 41552- io 6 ergs. 


156. 120. 


240. 6272 - io 6 . 


157. o. 


241. 197392 io 4 ; 49348 io 4 . 


162. 60. 


242. [11267 io 5 total energy]; 6.47cm. 


163. 4 kg. wt. 


243. 591 io" ergs approx. 


164. 7921.4 dynes; 15843 dynes. 


244. 4000 ergs. 


166. /'sec. 10 Ib. wt. 


245. H25-I0 7 . 


167. 11.5; 27.7.. 


246. 27- io 3 ; 51 . io 3 . 


168. 20; 20; 21.22. 


248. 64 io" dynes. 


169. 45 inch 


249. 48 - io 5 ergs. 


Algebraic sum = 282.8 gr. wt. 


250. 24 . io 4 . 


175. 911+ cm. 


251. O) 25 io 7 -m.; (b} 625 io 6 m.; 


176. 20000 Ib. wt. 


(tr) o; (d} 625 io 6 m. 


180. io cm. 


252. 980 . m. ergs. 


183. \ ap + f P ; l ap + f P. 


260. 45 io 9 ergs. 


186. 3600. 


261. 5- io 8 ergs; 5.1 kg. m. 


187. 50.9 [kg. cm.]. 


262. Vzgh. 


197. O) L- t 


271. 0.199. 
272. 49 - io 8 ergs. 



Exs. 281-452] 



ANSWERS 



239 



281. 


10053 kg- wt. 


351. 


iV 


285. 


IOOO: I. 


352. 


Ratio 1.000046. 


286. 


6| kg. wt. 


353. 


7 i - T 


287. 


1:24. 




V~2 


289. 


1 60 kg. 


354. 


(a} .875; 1.43; (<*) 1.253- 


302. 


98 io 6 . 


355. 




309. 


48 - 1012. 


356. 


1.718 sec. 


314. 


- 320; 1600. 


358. 


\Ml' 2 ', ^Ml-. 


318. 


^ 2 v _ 


359. 


(a) |po/ 8 + \kl*\ 




dx 2 " 




(fr} x o / 3 4- * kfi 


332. 


.02. 




Jbfvl 


333. 


142+. 


362. 


one-fourth mass X square of 


334. 


44.8 tons. 




4 
radius. 


335. 
336. 


l6 4 2'. 

28.62. 


363. 


^- one-half mass X square of 

2 


337. 


(I) .2. 




radius. 


339. 


Equate resultant force to (J/+Z), 


368. 


392 - io 3 . 




and solve for a. 


369. 


245 - io 5 , increased fourfold. 


341. 


g [sin 60 /A cos 60]. 


371. 


4 io~ 4 . 


342. 


2656 io*. 


372. 


625 io~ 6 . 


349. 


.8 sec. approx. 1= 16; 


376. 


17- io". 




1.14 sec. approx. / 32. 


377. 


.26 cm. 


350. 


802+. 


384. 


II3-54- 



LIQUIDS AND GASES 



393. 
395. 
396. 
398. 
402. 



About 3 A. 

96.4 gr. wt. 
123 approx. 

93.5 meters. 
5:3. 



417. 40560 kg. 

418. 97200 Ibs. 

419. 12. 

420. of its height. 
13-6 

423. 21.5:11.3:8.9:2.6. 
46.5:88.5:112:383. 
3.6:4.45:4.96:726. 

424. V,:V S = . 5 3 5 . 
428. 2 1426; 159. 

19.3; 2.66:2.15. 



429. 
430. 
431. 



257; 
10.5: 
4. 
40. 



432. 32. 

433. 137.6 gr. wt. 

434. 2. 

435. 1.6. 

436. i : 2. 

437. .6. 

438. .2. 

439. 735; 1470. 
441. .5. 

443. f. 

445. 4.37. 

446. Inversely as the densities. 

447. ii. 

448. 876. 

449. 3. 

450. .79. 

451. 1.2. 

452. .9. 



240 



PROBLEMS IN PHYSICS 



[Exs. 453-729 



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


459. f. 


455. 2.9 [Note that 5 = ~|. 


461. 1 1.3 c.c. 
462. 1 8% approx. in Hg. 


456. 4.84; 5.09. 


464. 975 cm. 


HEAT 


477. 113; 53-6; - 4- 


535. As 3 : 55 nearly. 


478. 100; 22.2; o; -344. 


536. 27.6. 


480. -40. 


537. 12.7 and 42.3 liters. 


481. 160. 


540. 5.78 grams. 


486. 12.618 m. 


541. 5.6 grams. 


488. The increase in length is equiva- 


542. 4.91 grams. 


lent to 13.6 added terms. 


543. .06. 


492. 189 x io~ 7 . 


544. .62. 


493. 1129. 


545. 3.29 cm. 


504. 3 T Ho- 


613. 4.9 grams. 


505. 40.197 c.c. 


614. 81363 cm. per sec. 


507. 13.11. 


618. 30618.75 calories. 


508. 13.35. 


619. 21851.7 calories. 


509. T %V 


620. 4.189 x i o 7 ergs. 


513. 176.25. 
532. 26226. 


624. W = Ap'a [i + log ^ J - ApJ. 


533. As n: 21. 


625. (a) 4386.3 ft.-lbs. () 47.85 H.P. 


534. 781052. 


626. 4-2 H.P. 


ELECTRICITY AND MAGNETISM 


632. F= .01 dynes repulsive. 


693. 4 cm. 


633. F= .64 dynes attractive. 


695. (a) 50000; () 5000; (<:) 500. 


634. /'=4/; r' = zr. 


697. 1600 ergs. 


635. q = 25.6. 


703. Loss |. 


636. r' = 2r. 


707. JP = 6.5. 


640. Surface density = 

47T 


711. Energy = i : 6. 
712. V and Q reduced initial values. 


643. 8000 dynes. 


719. energy remains. 


654. O) V = 4 V; V = -V. 


100000 


657. -42. 


7T 


662. Q = loooo; V = looo; force = 


721. 15.9- io 7 ergs. 


100. 


725. \ W. each jar. 


663. Work = 1800 ergs. 


729. Cap. = 7; change in large sphere 


664. 80 ergs. 


= 21.43; small sphere 8.57; en- 


670. V = 2; /=o. 


ergy over wire 18.57 units; initial 


688. /= 12.5. 


energy = 185 ; final energy = 


689. V = loooo. 


64.3; final potential = 4.29. 






Exs. 731-99] 



ANSWERS 



241 



731. 
732. 
733. 
734. 
735. 
742. 
746. 

755. 

758. ' 

759. 

762. 

767. 

768. 

770. 

771. 

772. 
773. 

774. 
775. 
776. 

777. 
778. 

781. 

782. 

783. 
784. 
789. 
790. 
791. 



792. 

795. 
798. 
820. 
824. 

826. 
829. 
837. 

840. 



7=2 amperes; 600 coulombs. 

8 amperes. 

A" = 3*. 

5 amperes. 

4800 coulombs. 

(a) 450; (J) 900. 

(a) .0377 amperes; () .377, .754, 

1.131, i. 808; (V) 3.77 volts. 
8 ohms. 
Ri = 25 ohms. 
2.58 volts. 
T 4 g volts. 
Radius doubled. 
i oooo ohms. 
1.66 ohms. 
(a) 1.19; (*) 14? (07-795 W 

140. 

A 2 = 35.26 ohms. 
3; if; 4^ ohms. 
Length = \l. 
2531 ohms. 



27 ohms. 

x= mi. ii ohms. 

Take intersection of line -- 1- 
r\ r% 

= i with x = y. 
7i = i2; n 72 series; = 2; 

r\ = 2 multiple. 
E = 12 volts; A* = io| ohms. 
.8 ohms. 
6; 3; 4; 13- 
3:1. 
(0) 30 volts; () 59 ohms; (c) 

.508+ amperes; (af) 16.3 volts; 

0) 5.54 (^ to 7?). 
(#) 8.332 ohms; () 12 amperes; 

(V) 100 volts; 0) 111.96 volts. 
.028 amperes in branch 10. 
i%. 

6 x io 5 joules. 
1008 ohms; 10080 volts; 600 cou- 

lombs passing per min. 
28.8- io 5 joules; 6.9. io 5 calories. 
1= 10.04 amperes; 10040 volts. 
7V = 64000 ergs; I*R = 16000 

ergs. 
45.11 ohms. 



841. 


.126 L. 


846. 


A 40 = 1021.2; A'so = 1042.4. 


847. 


256 C. 


848. 


2187.2 ohms. 


850. 


7.7 ohms at o. 


853. 


2.362 g. of copper. 


855. 


5 amperes. 


864. 


1.9017 amperes; .026 amperes; 




14.36 io~ 6 amperes. 


868. 


(#) radius =157 cm.; () 5 = 470. 


869. 


Total current = .0838 amperes. 


872. 


7 ' = 2.81 io- 3 . 


873. 


.1 amperes. 


874. 


7o' = 13.3 io~ 6 . 


881. 


7 = 138. 


882. 


7 = .0225. 


883. 


7 = 38.6 io 5 . 


884. 


/= ip. 


885. 


138 cm. 


886. 


490 amperes. 


887. 


k 
Force = 4\/2 




a 


899. 


Force || to bar equals .29 dynes. 


900. 


3912 dynes. 


901. 


M = 4.66 C.G.S. units. 


904. 


77=8. 


907, 


77= .208; V = .534. 


911. 


M 546. 


913. 


M ' 6000. 


914. 


(a) 102; () 72.114. 


916. 


140. 


918. 


1.2. 


919. 


1 20 ergs. 


920. 


V - 8.33 


944. 


k= 30.65; = 15438; 




fj. = 386. 


947. 


B = 3508. 


949. 


2524. 


950. 


I34-3. 


954. 


(#) 77 = 125.7 P er sc l' cm 


956. 


751.1 watts. 


960. 


M =. 214.765. 


962. 


.141. 


983. 


6 dynes. 


987. 


1.8 volts. 


988. 


1000 dynes; 100000 dynes. 


989. 


5 io 5 dynes _L to field. 


990. 


13.76 io 7 coulombs. 



2 4 2 



PROBLEMS IN PHYSICS 



[Exs. 994-1244 



994. O) 8.5-IO- 4 volts; (b\ 
volts; (r) 297.7 sec - 
1000. .32 volts. 

1031. 15.77 H.P. 

1032. Electrical eff. 92.6%. 

1033. Electrical eff. 83.3 %. 

1034. 62.5 amperes. 

1035. 96.5%. 

1036. 4.5 H.P. 
1041. 12 io 5 dynes. 
1046. Net eff. 91.2%. 



1094. 
1095. 
1101. 
1102. 
1114. 
1115. 
1124. 
1128. 

1129. 

1130. 
1132. 
1133. 
1134. 
1135. 
1139. 
1140. 

1141. 
1147. 
1148. 



1150. 
1156. 

or 



.419; 6; 77.4. 

8; 160; 1600; .314. 

2 crests, 3 troughs. 

20. 

(I) .6283; (2) 1.25 

y = a sin TT [8 1 -f x~\ . 

332m. 

(i) 27.7m.; (2) 55. 

x 

V 

34740 cm. 

3444m. 

X 20 = X 1.036. 

500 waves. 

23. 7 C. 

135.2 cm. 

Velocity and wave 
creased. 

328m. 

128; 362.1; 181. 

(i) Make string f c 
length; (2) incr< 
by the factor 1.26+. 

42 io 5 dynes. 

= 1-4. 0098] 6g . 

Fz l_9 .0045 J 

* n^-i 

F* |_9 -0045 J 



1157. 2s= 

/2 

'1158. 5 = 



) 8. 38- io- 4 


1047. 77%. 




1051. 76.44 amperes. 




1052. 100 amperes. 




1054. Max. 7=49.93 amperes; mean 




value = 31.3 amperes. 




1059. 843; 904.3. 




1061. Imp. = 454. 




1062. = 8 7 3 4 '. 




1063. 3300 watts. 




1066. Z = .024 henrys. 




1069. loo volts; 36 amperes. 


SOUND AND LIGHT 




1172. 240 cm. 


4- 


1173. 80 cm. 


T 






1174. 26.6cm. 




1175. 145; 435; 1305. 


; (3) 2. 


1177. 120 cm.; 170. 




1179. At o length 50 cm.; at 25 




length 52.2 cm. 


U (3) 83- 


1180. 192; 320; 448; 576. 




1181. 128; 192; 256; 320. 




1182. 2.1 cm.; 6.2 cm. from wall. 




1183. 18.7 cm. 




1184. 5 beats. 




1189. n 208; n = 1040. 




1190. 2.8; 8.5; 70.8 cm. 




1192. 8.3 cm.; 7.9 cm.; 31.9 cm. 




1215. io cm. 


length in- 


1216. 13.3 cm.; 14; 16.7 etc. 




1217. 1.7 cm.; 2.0 cm.; 3.3 cm. etc. 




1220. f /?; \R. 




1221. JP=i 5 ft. 


' its former 


r> 

1225. ; from natural size to zero. 


ase tension 


2 


. 


1227. 15.6 cm. per. sec. toward. 




1235. 4i49'- 


l 


1236. 1.3214; .7567. 


>. 


1237. 7 437'- 




1238. Angle of refraction = III2 ; . 





1239. 322'; 403o'; 46^25 '. 




1240. 7032'. 




1243. 20 11'. 




1244. Yes; critical angle increases with 


= 1.225* 


increase of wave length. 



Exs. 1245-1322] 



ANSWERS 



543 



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

1246. 165 sec. 

1247. Angle of refraction in glass 

= 2 5 4 0'. 

1248. 1.07 

1249. 1.33 rn. 

1250. .582 cm. 

1251. 40 ft. 

1252. 40 ft. 

1253. Above. 

1258. v. = 1.668. 

1259. 2338'; io22'. 

1260. For yellow light taking index of 

crown glass as 1.530, 



1270. 

1274. 
1280. 
1281. 
1282. 
1285. 
1286. 
1302. 
1303. 
1309. 
1310. 
1318. 
1319. 
1322. 



Taking 1.530 as index, 1.24 in. 

and 7.44 in. 

12 m.; 7.5 m.; 4.8 m. etc. 
30 cm. 
100 cm. 
.9 cm. 
/*=i.5. 
/= radius. 
3i2-io- 7 cm. 
76 . io~ 6 sec. 

22- io~ 3 ; 38 io- 3 -.-etc. 
1.21 : i. 

323'; 647'; io!2'. 
2:3. 
1059. 




INDEX 



ACCELERATION, 38, 39, 40, 41. 
Approximations, 33. 
Archimedes' principle, 94. 
Atmospheric pressure, 89. 
Averages, 31. 

BAROMETER, 89. 

Batteries, 137, 138, 139, 142, 143. 
best arrangement of, 146, 147. 
Boiling points, table of, 16. 
Boyle's law, 98. 

CALORIE, 108. 

Calorimeters, 109, no, in, 112, 113. 

Capacity, electrical, 124. 

specific inductive, 18, 128. 

thermal, 108, 109. 
Cells, best arrangement of, 147. 

grouping of, 142, 143. 
Center of inertia, 58, 61. 

of mass, 58, 61. 

of gravity, 58, 61. 
Coefficient, of expansion, 101. 

cubical, 103, 104, 105. 

of gases, 106, 107, 108. 
Coefficients of expansion, 1 6. 
Condensers, 128, 129. 
Conductivities, thermal, 17. 
Critical angle, 213. 
Current alternating, 183, 184, 1 86. 
Current electricity, 132. 

DENSITIES, tables of, 14. 
Diffraction grating, 222, 223. 
Dimensions, 5, 187-190. 
Dimensional equations, 5. 
Doppler's principle, 207. 



Dynamo, 179-183. 

characteristic curve, 180, 181. 
efficiency, 181, 182. 

ENERGY, of charge, 127. 

of discharge, 130, 131. 

kinetic, 67. 

of rotation, 67, 74. 

transformation to potential, 75-78. 
Elastic limit, 85. 
Elasticity, 85. 
Electric force, 123. 
Electrochemical equivalent, 153, 154. 
Electromagnetic units, 18, 187-190. 

attraction, repulsion, 169-172. 

induction, 169-182. 
Electromotive force, 142. 

of induction, 173-178. 
Expansion coefficients, 16. 

FALL, of potential in a wire, 135. 

of potential and electromotive force, 

135- 

Farad, 18. 
Faraday's disc, 177. 
Fields offeree, electric, 124, 125. 

magnetic, 161, 169, 173. 
Force, 40-43. 

systems, 53-57. 
Friction, 79. 

angle of, 79, 80. 

coefficient of, 80. 

GALVANOMETER, 155, 156. 

Ballistic, 157. 
Gases, 89 et seq. 
Graphic methods, 27, 30. 



245 



246 



INDEX 



HEAT, 100. 

specific, 108. 

specific variation of, no. 

of fusion, no- 1 1 2. 

of vaporization, 112, 113. 

in electric circuit, 149-151, 183. 
Heats, of liquefaction, table of, 15. 

of vaporization, table of, 1 6. 
Hydrometers, 97. 
Hydrostatic pressure, 90-92. 

press, 92, 93. 
Hysteresis, 167. 

INDICATOR diagram, 119. 
Indices of Refraction, 20. 

KILOGRAM, 8, 9. 
Kirchhoff s law, 147-149. 

LENSES, 216. 

images by, 217, 218. 

curvature of, 217-219. 
Light, reflection of, 209. 

velocity of, 213, 214. 

refraction of, 212-215. 

interference of, 220-223. 

diffraction of, 222, 223. 
Lines offeree, 122. 

magnetic, action of, 168-170. 
Liquids, and gases, 89. 

pressure, 89-91. 

MAGNETIC, field, due to currents, 159. 

induction, 165. 
Magnetism, 161. 
Magnetization curve, 165, 166. 
Magnetometer, 163, 164. 
Mass and weight, 7. 
Measurement, I. 

Mechanical equivalent of heat, 14. 
Melting points, table of, 15. 
Mirrors, 209. 

plane, 209, 210. 

concave, 210, 211. 

convex, 211, 212. 
Moment of inertia, 82-84. 
Motor, 183. 



Multiple resistance, 139-141. 
graphic methods, 141. 

NEWTON'S rings, 221. 

OHMS, various, 18. 
Ohm's law, 132 et seq. 
Overtones, 205, 206. 

PENDULUM, gravity, 82. 

magnetic, 167, 168. 
Potential, diagrams, 135-138. 

gravitational, 75-78. 
Prism, 214, 215. 
Pressure, of atmosphere, 89. 

of gases, 98, 99, 106-108. 

of liquids, 89, 92. 

center of, of liquids, 92. 
Projectiles, 49. 
Pulleys, systems of, 73, 74. 

REFRACTION, index of, 212-215. 

indices of, 20. 

law of, 212. 
Resistance, multiple, series, 138, 149. 

specific table of, 17. 

temperature coefficients of, 17. 

units of, 1 8. 

SELF-INDUCTION, 185. 

Shunts, 143-145. 

Simple harmonic motion, 191-193. 

Sound, 198. 

musical, 200. 

velocity of, 19, 199. 
Specific gravity, 94-97. 

gravity bottle, 96. 

heats, 14, 15. 

inductive capacity, table of, 18. 

resistance, 152. 
Spectra, 215, 222, 223. 
Static electricity, 121. 
Strain, 85, 86. 
Stress, 85, 86. 

TEMPERATURE, 100. 

Thermometer, 100. 

scales, 101. 



INDEX 



247 



Thermometer weight, 105, 106. 
Thin plates, 220, 221. 
Torsion, 87, 88. 

moment of, 88. 
Transformer, 186. 
Transmission of energy, electric, 151. 

UNITS, i. 

C. G. S. and practical, 190. 

electrical, magnetic, 187. 

fundamental and derived, 2. 

of area, 12. 

of force, 13. 

of heat, 1 6. 

of length, 12. 

of mass, 13. 

of power, 13. 

of resistance, 18. 

of stress, 13. 

of volume, 12. 

of work, 13. 

practical, in C.G.S., 18. 

transformation of, 190. 

VAPOR, pressure, 114, 118, 119. 
volume, 114, 1 1 8, 119. 



Vectors, 21. 

addition of, 21, 22. 

examples on, 23, 25. 
Velocity, of light, 19. 

of sound, 19, 199. 

of sound, temperature, effect on, 199. 
Vibration, 191. 

columns of air, 205-207. 

elliptic, 192. 

strings, 201-204. 

WAVE length of sound, 200. 

of light, 221, 223. 
Wave lengths of light, table of, 19. 
Waves, 194, 195. 

phase, 197. 

progressive, 196. 

retardation of, 197. 

sound, 198. 

Wheel and a^cle, 74, 75. 
Work, by torque, 65. 

constant force, 62. 

general expression for, 69, 70. 

principle of, applied to machines, 71. 

variable force, 63, 69. 

YOUNG'S modulus, 86. 



WITH NUMEROUS ILLUSTRATIONS. 

THE ELEMENTS OF PHYSICS. 

BY 

EDWARD L. NICHOLS, B.S., Ph.D., 

Professor of Physics in Cornell University, 
AND 

WILLIAM S. FRANKLIN, M.S., 

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

{Vol. I. Mechanics and Heat. 
II. Electricity and Magnetism. 
III. Sound and Light. 

Volumes I. and II. now ready. Price $1.50 net, each. 
Volume III. In Press. 



It has been written with a view to providing a text-book which shall correspond with 
the increasing strength of the mathematical teaching in our university classes. In most of 
the existing text-books it appears to have been assumed that the student possesses so 
scanty a mathematical knowledge that he cannot understand the natural language of 
physics, i.e., the language of the calculus. Some authors, on the other hand, have assumed 
a degree of mathematical training such that their work is unreadable for nearly all under- 
graduates. 

The present writers having had occasion to teach large classes, the members of which 
were acquainted with the elementary principles of the calculus, have sorely felt the need of 
a text-book adapted to their students. The present work is an attempt on their part to 
supply this want. It is believed that in very many institutions a similar condition of affairs 
exists, and that there is a demand for a work of a grade intermediate between that of the 
existing elementary texts and the advanced manuals of physics. 

No attempt has been made in this work to produce a complete manual or compendium 
of experimental physics. The book is planned to be used in connection with illustrated 
lectures, in the course of which the phenomena are demonstrated and described. The 
authors have accordingly confined themselves to a statement of principles, leaving the 
lecturer to bring to notice the phenomena based upon them. In stating these principles, 
free use has been made of the calculus, but no demand has been made upon the student 
beyond that supplied by the ordinary elementary college courses on this subject. 

Certain parts of physics contain real and unavoidable difficulties. These have not been 
slurred over, nor have those portions of the subject which contain them been omitted. It 
has been thought more serviceable to the student and to the teacher who may have occa- 
sion to use the book to face such difficulties frankly, reducing the statements involving 
them to the simplest form which is compatible with accuracy. 

In a word, the Elements of Physics is a book which has been written for use in such 
institutions as give their undergraduates a reasonably good mathematical training. It is 
intended for teachers who desire to treat their subject as an exact science, and who are 
prepared to supplement the brief subject-matter of the text by demonstration, illustration, 
and discussion drawn from the fund of their own knowledge. 



THE MACMILLAN COMPANY. 

NEW YORK: CHICAGO: 

66 FIFTH AVENUE. ROOM 23, AUDITORIUM. 



A LABORATORY MANUAL 



OF 



PHYSICS AND APPLIED ELECTRICITY. 

ARRANGED AND EDITED BY 

EDWARD L. NICHOLS, 

Professor of Physics in Cornell University. 



IN TWO VOLUMES. 



VOL. I. Cloth. $3.00. 

JUNIOR COURSE IN GENERAL PHYSICS. 

BY 

ERNEST MERRITT and FREDERICK J. ROGERS. 
VOL. II. Cloth, pp. 444. $3.25. 

SENIOR COURSES AND OUTLINE OF ADVANCED WORK, 

BY 

GEORGE S. MOLER, FREDERICK BEDELL, HOMER J. HOTCHKISS, 
CHARLES P. MATTHEWS, and THE EDITOR. 



The first volume, intended for beginners, affords explicit directions adapted to a 
modern laboratory, together with demonstrations and elementary statements of prin- 
ciples. It is assumed that the student possesses some knowledge of analytical 
geometry and of the calculus. In the second volume more is left to the individual 
effort and to the maturer intelligence of the practicant. 

A large proportion of the students for whom primarily this Manual is intended, 
are preparing to become engineers, and especial attention has been devoted to the 
needs of that class of readers. In Vol. II., especially, a considerable amount of 
work in applied electricity, in photometry, and in heat has been introduced. 



THE MACMILLAN COMPANY. 

NEW YORK: CHICAGO: 

66 FIFTH AVENUE. ROOM 23, AUDITORIUM. 



A LABORATORY MANUAL 

OF 

PHYSICS AND APPLIED ELECTRICITY. 

ARRANGED AND EDITED BY 

EDWARD L. NICHOLS. 



COMMENTS. 

The work as a whole cannot be too highly commended. Its brief outlines of the 
various experiments are very satisfactory, its descriptions of apparatus are excellent ; 
its numerous suggestions are calculated to develop the thinking and reasoning powers 
of the student. The diagrams are carefully prepared, and its frequent citations of 
original sources of information are of the greatest value. Street Railway Journal. 

The work is clearly and concisely written, the fact that it is edited by Professor 
Nichols being a sufficient guarantee of merit. Electrical Engineering. 

It will be a great aid to students. The notes of experiments and problems reveal 
much original work, and the book will be sure to commend itself to instructors. 

S. F. Chronicle. 



Immediately upon its publication, NICHOLS' LABORATORY MANUAL OF 
PHYSICS AND APPLIED ELECTRICITY became the required text-book in 
the following colleges, among others : Cornell University ; Princeton 
College ; University of Wisconsin ; University of Illinois ; Tulane 
University ; Union University, Schenectady, N.Y. ; Alabama Poly- 
technic Institute; Pennsylvania State College; Vanderbilt Uni- 
versity ; University of Nebraska ; Brooklyn Polytechnic Institute ; 
Maine State College ; Hamilton College, Clinton, N.Y. ; Wellesley 
College ; Mt. Holyoke College ; etc., etc. 

It is used as a reference manual in many other colleges where the 
arrangement of the courses in Physics does not permit its formal 
introduction. 



THE MACMILLAN COMPANY. 

NEW YORK: CHICAGO: 

66 FIFTH AVENUE. ROOM 23, AUDITORIUM. 



WORKS ON PHYSICS. 



A TEXT-BOOK OF THE PRINCIPLES OF PHYSICS. 

By ALFRED DANIELL, F.R.S.E., 

Late Lecturer on Physics in the School of Medicine, Edinburgh. 
Third Edition. Illustrated. 8vo. Cloth. Price, $4.00. 

" I have carefully examined the book and am greatly pleased with it. It seemed to me a particu- 
larly valuable reference-book for teachers of elementary physics, as they would find in it many sugges- 
tions and explanations that would enable them to present clearly to their students the fundamental 
principles of the science. I consider the book in its recent form a distinct advance in the text-books 
bearing on the subject, and shall be pleased to recommend it to the students in our laboratory as a 
reference-book." Prof. HERBERT T. WADE, Department of Physics, Columbia College. 

A LABORATORY MANUAL OF EXPERIMENTAL PHYSICS. 

By W. J. LOUDON AND J. C. flcLENNAN, 

Demonstrators in Physics, University of Toronto. 
Cloth. 8vo. pp. 302. Price, $1.90, net. 

The book contains a series of elementary experiments specially adapted for students who have had 
but little acquaintance with higher mathematical methods; these are arranged, as far as possible, in 
order of difficulty. There is also an advanced course of experimental work in Acoustics, Heat, and 
Electricity and Magnetism, which is intended for those who have taken the elementary course. 

WORKS BY R. T. GLAZEBROOK, M.A. 

MECHANICS AND HYDROSTATICS. Containing: I. Dynamics. II. Statics. III. Hydro- 
statics. One volume, pp. 628. Price, $2.25, net. 

STATICS, pp. 1 80. Price, 90 cents, net. DYNAMICS, pp. 256. Price, $1.25, net. 

HEAT. pp. 224. Price, $1.00, net. LIGHT, pp. 213. Price, $1.00, net. 

HEAT AND LIGHT, pp. 440. Price, $1.40, net. 

BY S. L. LONEY, 

Author of "Plane Trigonometry," "Co-ordinate Geometry," etc. 

ELEMENTS OF STATICS AND DYNAMICS. 
Complete in One Volume. Ex. Fcap. 8vo. Cloth. Price, $1.90. 
The Two Parts may be had, bound singly, as follows: 

PART I. ELEMENTS OF STATICS. Price, $1.25, net. 
PART II. ELEMENTS OF DYNAMICS. Price, $1.00, net. 

" The two volumes together form one of the best treatises that I know on the subject of elementary 
Mechanics, and are most admirably adapted to the needs of the student whose mathematical course has 
not included the Calculus and who yet desires to obtain a good idea of the groundwork of Mechanics." 
BENJAMIN W. SNOW, Professor of Physics, Indiana University. 

MECHANICS AND HYDROSTATICS FOR BEGINNERS. 
Ex. Fcap. 8vo. Second Edition, Revised. Price, $1.25. 

A TREATISE ON ELEMENTARY DYNAMICS. 
Second Edition, Reprinted. Cr. 8vo. Cloth. Price, $1.90. 



THE MACMILLAN COMPANY. 

NEW YORK: CHICAGO: 

66 FIFTH AVENUE. ROOM 23, AUDITORIUM. 



THIS BOOK IS DUE ON THE LAST DATE 
STAMPED BELOW 



AN INITIAL FINE OF 25 CENTS 

WILL BE ASSESSED FOR FAILURE TO RETURN 
THIS BOOK ON THE DATE DUE. THE PENALTY 
WILL INCREASE TO SO CENTS ON THE FOURTH 
DAY AND TO $1.OO ON THE SEVENTH DAY 
OVERDUE. 



JUL 



1948 
6 194 



WAR 3 1 1950 
'"' 11950 



, 



LD 21-20m-6,'82 




TC 32655 



Engineering 
Library