Mechanics

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^^

il GENERAL FOREWORD

is not necessarily dull or fenced off by a barrier of
technical jargon. Of course, specialisation in this as in
other subjects is not for everyone, but the publication of
this series of books enables any man or woman to learn,
any child to be taught, to pass with imderstanding and
safety the *^ Thresholds of Science."

THSB8E0LD8 OF SCIENCE

MECHANICS

THRESHOLDS OF SCIENCE

Otber Volttinci In Preparation i
MATHEMATICS hj C A. Laiuuit
ZOOLOGY by E* Basdket.

BOTANY by E. Bcocker.

CHEMISTRY by Geofgei Dar^cna.

THRESHOLDS OF SCIENCE ' ' "

^7

MECHANICS

BY

C. E. GUILLAUME
I

ILLUSTRATED

4-

• if- *

f

Garden City New York

J)OUBLEDAY, PAGE & COMP^Ny

1915

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THE NEW YORK

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I

METEIC MEASUEES

1000 metres

100 m.

10 m.

1 metre

•1 m.

•01 m.

•001 m.

I

1000 grammes
100 g.
10 g.

1 gramme

I

•Ig.
•01 g.

•001 g.

LENGTH

1 kilometre = 1093*6 yards

= about g mile.
1 hectometre.
1 decametre.
39-37 ins. = 109 yard.
1 decimetre.
1 centimetre.
1 millimetre = abont ^ in.

MASS

1 kilogramme = about 2 J lbs.

1 hectogramme.

1 decagramme.

weight of 1 c.c! of wat^r at

4? C. = 16'43 grains =

•036 oz.
1 decigramme.
1 centigramme.
1 milligramme.

H

r

9

=r-CM

:zr-r-i

£ /&.

CONTENTS

PAGB

General Forewobd to the Sebies • • • i
Metbic Measures v

FIRST PART.

PEEPARATION FOR THE STUDY OP MECHANICS.

Chapter I.

HOW TO study nature.

1. Observation and Experiment

2. Approximation and Simplification

3. The Need for Simplification .

4. Limits of Experiment .

5. Illusion .....

6. Education of the Senses : Measurement

7. Induction and Deduction • •

Chapter II.

MOTION.

1
2

4
5
7
9
14

8. How Velocity is Defined 20

9. Relative and Absolute Velocity . • • • 22

10. lUusion in the Observation of Motion • • • 25

11. Distance, Time, Velocity, and Acceleration • • 29

12. Direction of Velocity. Composition and Resolution

of Motions ,«.,,,. 34

VIII

CONTENTS

SECOND PART.

EXPOSITION OF PRINCIPLES.

CnAPTER III.

FORCE, THE CAUSE OP ACCELERATION.

PAGE

13. Velocity, Accoloration, Force, and Inertia . . 38

14. Laws of Falling Bodies 42

16. Doiliiotion from the Laws of Falling. Symbols . 45

16. Exaggeration of the Disturbing Causes ; their Study

and Elimination 47

17. Gonoralisatiou of the Laws of Falling • . .51

Chapter IV.

THE WORK OF FORCES.

18. The Meaning of Work .

19. Storage and Restitution of Work
Mass .....

20

21. Kinetic Energy .

22. Power

23. Conservation of Work .

24. Action and Reaction. Momentum

53
54
57
62
65
67
70

Chapter V.

FORCES without MOTION.

26. Reactions of Matter ; Elasticity and Friction . 74

26. Forces in Equilibrium ; Composition and Resolution

of Forces :...... 78

27. The Law of Attraction ; Weight and Mass . . 85

28. Couples 92

29. The Lever 95

30. Centre of Gravity 98

31. Pressure •..»•••• 102

CONTENTS

ix

Chaptbb VI.

REST AND MOTIOK DUB TO FOftCES.

PAGE

32. Virtual and Beal Acceleration • • • • 108

33. The Price of Acceleration . . . • .110

34. Centripetal Acceleration and Centrifugal Force . 113

35. Central or Eccentric Percussion ; Reciprocal Centres 120

36. Moment of Inertia 122

Chapter VII.

OSOILLATORT MOTION.

37. Periodic Forces and Oscillations .

38. Representation of Oscillatory Motion

39. Laws of Oscillatory Motion .

40. The Pendulum and Weight •

41. Resonance . • • • •

125
126
127
133
136

THIRD PART.

DEVELOPMENT AND APPLICATION.

Chapter VIII.

CALCULATIONS IN MECHANICS.

42. Qualitative and Quantitative Relations •

43. Units in Mechanics • • • •

140
143

Chapteb IX.

impact.

44. Definition of Impact .

45. Representation of Elastic Impact

46. Real Impact

47. Inelastic Impact .

48. Forms of Energy. . •

147
148
151
154

156

X CONTENTS

Chaptxb X.

RESISTANCE OF MATEBIALS.

PAGE

40. Different Properties of Matter . . . .159
60. Elastic Limit, Breaking Strain, and Modnlna of

Elasticity 160

51. The Bending of Bars 162

62. Fragility, Plasticity, and Hardness . . . 166

53. Gradual Rupture and Sudden Rupture . . .168

64. Conservation of Momentum, and Deadening of Shock 171

65. Speed of Transmission in Sudden Rupture • . 174

Chapter XI.

ARTILLERY.

56. (reneral Conditions in the Firing of a Projectile . 176

67. Actual Firing 177

68. Around the Projectile ...... 181

69. Within the Projectile 184

60. The Journey from the Earth to the Moon . • .186

Conclusion 189

Notes and Problems 191

Index 196

xii LIST OF ILLUSTRATIONS

FTO. PAGE

23. A shot sent with sufficient speed would travel round

the earth 88

24. A ton weighs five kilogrammes more at the pole

than at the equator 90

25. A carriage wheel turns because it is under the action

of a couple 92

26. The equilibrium of the bar can be brought about in

many ways ; but always when equal couples are
opposed at the point of suspension ... 94

27. In order to balance a couple, a couple of the same

moment is required . . . . . .96

28. The lever allows movements to be amplified . . 98

29. A body supported at its centre of gravity has no

tendency to rotate 99

30. The centre of gravity can often be determined geo-

metrically 100

31 AND 32. The position of the centre of gravity is some-
times surprising 101

33. The pressure on the horizontal bottom of a vessel

is independent of the shape of the vessel . .104

34. How centrifugal force is produced . . . .114

35. Equilibrium of a ball on the surface of a paraboloid

turning at a suitable speed 117

36. The projection of rotation gives, the motion of

oscillation 126

37. A mass under the action of a force restoring it to its

position describes an oscillation . • . .128

38. The most complicated oscillations can be repre-

sented by the motion of a tooth of a gear-
wheel . .132

39. The force that brings the pendulum to its position

of equilibrium increases more slowly than the

distance moved 133

40 AND 41. A spring forces apart two cars running
towards each other. The diagram of the forces
acting explains the variations of velocity . .148

42. In an inelastic impact, the forces during separation

are less than during the approach . . .154

43. The shaded areas represent the proportion of work

recovered in an inelastic impact . • • .155

LIST OF ILLUSTRATIONS xiii

PIG. PAGE

44. A stax made of bread rebounds without being

deformed . • 159

45. The mechanical properties of metals can be studied

by measuring the elongation undergone by A wire
loaded with a weight 160

46. Extension and compression 163

47. In rupture by a falling weight, the wire first absorbs

the kinetic energy of the fall . . .169

48. A slow pull breaks the upper wire, a jerk the lower.

In the first case, the ball acts by its weight ; in

the second, by its mass 173

49. The throwing effect and shattering effect depend

on the development of the pressure in a gun . 179

50. At the start of a shell the fuse is primed, and on its

arrival the striker fires the charge . . . 185

2 MECHANICS

different routes, because each one is acted upon by forces
peculiar to itself.

If we were gifted with a penetration a hundred times
keener than that of Sherlock Holmes, we might, by dint
of perseverance, determine the complex forces that each
stone obeys. But we are naturally dull and slow of
apprehension ; we must therefore simpUfy our task by
persuading nature to disclose its secrets.

For this purpose we must arrange a place where these
actions can happen in an unusually simple manner. We
must replace direct observation by a kindred process —
eayperiment.

An experiment is a question addressed to nature, which
is always prepared to answer correctly, if the question is
correctly put. There will be no false evidence to mislead
us, no lawyers seeking to prejudice the inquiry. Nature
discloses itself in all its sincerity, and our art will consist
merely in asking it a simple question to which it can
return only a simple answer.

Replace the mountain-side by a smooth plank, the
stone by a ball made by a skilful turner, and let the ball
run down the incUned plank. After our experiment, we
shall be able to describe very closely not only the path
that all balls will follow on planks having the same
incUnation, but other details of their descent ; if our
observation has been complete, we shall even know the
moment at which the ball will pass any point on the
plank, as well as the time occupied in travelling the whole
length.

II.— Apppoximation and Simplification.

Can we describe accurately every feature of the descent
of the ball ? No ; because the plank is not perfectly
straight nor the ball perfectly round ; in any other
experiment both ball and plank would have been different
in these respects. Yet the plank is much straighter than

4 MECHANICS

What we have done is what all men of science do;
they work as accurately as they can, and interpret the
result by eUminating the minor interferences that attend it.

A well-conducted experiment will not then attempt
an impossible perfection ; it will approach as closely as
possible ; it will render a single action dominant in such
a way that the others almost disappear, and subsequent
reasoning will reject them entirely.

The details rejected in a first experiment can often be
examined separately ; thus, in the descent of the ball,
the undulations are of great interest, for they reveal the
vibration of the house shaken by movements from the
street. If these movements of the ground have not, as
is the case in towns, an artificial origin, they are called
seismic, and are of extreme importance ; but that is
another problem, to be approached by other methods.

It has always been thus. The early naturalists de-
scribed great animals in their general forms, then others
examined their anatomy ; next came histologists, who
examined. their tissues under the microscope ; last of all,
the study of microbes, of which they are the hosts, has
concentrated upon the infinitely small the largest amount
of interest. These are separate problems whose import-
ance appears successively. The subject is first approached
by simpHfying, and retaining only the main facts ; then
little by Uttle, the neglected detail is taken into account.

III.— The Need for Simplification.

Whether it is from natural indolence or* from need of
clearness, the mind prefers what is understood without
much difficulty and, accordingly, it constantly resorts to
simplification. When, we say the number of inhabitants of
France is 39 millions, we knowingly make a mis-statement.
The truth is that we have not remembered the number of
inhabitants. By saying 38,893,654 we should perhaps be
nearer the truth, but that is not certain ; and, given two

6 MECHANICS

of otir senses ; they vary with the means adopted fof
examination.

An echo is an image of sound as a reflection in a mirror
is the echo of a face. If the reflection is good it must be
formed on a surface from which every trace of unevenness
or irregularity has been removed. An echo, on the
contrary, is returned by a stretched sheet, by a wall, and
by a screen of trees — not very distinctly in the case of
the last, but very clearly in the case of the house, which
is therefore a very suitable mirror for sound, as it returns
the image without appreciable distortion.

Why this difference ? The structure of light is much
finer than that of sound ; the latter, like the labourer
with the plank, declares the wall perfectly smooth, while
the light penetrates and discloses all its irregularities, as
the laccmaker separates and classifies threads.

This difference in the nature of hght and sound is
discernible in all their manifestations ; it enables us to
hear the words of a person through a wall, while, in order
to see him, we must be able to look directly at him.

In studying nature, we must be perfectly aware of the
degree of dehcacy of our process. Sometimes it has to be
exact, sometimes it is preferred less exact. Look at a
good engraving through a lens ; it is easy to distinguish,
lighter and darker, and separated \>y more or less black,
all the separate points that constitute the picture ; but
the picture, though now not so pleasant to look at, will
not have completely disappeared. Let us now examine
it under the microscope ; only the points remain, and we
cannot unite them. Just as trees prevent us from seeing
the forest, so the points prevent us from seeing the picture.
Too close an examination exaggerates the detail and leaves
the whole indistinct.

The roughness of a process of examination brings about
that simplification which otherwise our minds would be
forced to effect. At first glance, the line on the plank
appeared straight, but close examination proved it not so ;

\

8 MECHANICS

The inyentor of this inusion desired to deceive our
sight, and he has completely succeeded ; our eyes in
turn deceive us, without intending to do so.

Illusions are found everywhere. ^Vhen, in the six-
teenth century, the thermometer enabled temperature to
be determined otherwise than by touching a bodj- with the
hand, men were astonished to discover that springs are
colder in winter than in summer, since the evidence of the
senses had always taught the contrary. The cause of this
illusion we shall recognise if we plunge the right hand
into cold water, the left into warm water, and then both
into tepid water. The water will appear cold to the left
and warm to the right.

Bodies of equal weight seem heavier to us in inverse
proportion to their size. Tlie following experiment is
always successful. Put upon the hands of a friend a
cardboard box and a lead ball of equal weight. Ask him
the question, " How much heavier is the ball than the
box?*' The reply is generally, "Three or four times."
We never trust, then, the evidence of our hands if we
have to buy anything by weight.

Occupation enables us to forget the passing of time ;
but when a day has been very full of events, although it
may have appeared short at the time, the morning seems
very far off when evening comes. On a journey we often
confuse yesterday with the day before.

Waiting makes time seem endless ; the popular expres-
sion *' I have waited an age for you," is a significant proof.

The loss of illusions is a cause for regret, and in ordinary
life we arc happier if we retain some of them. Painters,
engravers, and all those who present images for our
enjoyment appeal to our capacity for illusion ; and the
better they succeed, the more grateful we are.

When, instead of believing, we wish to know, we must
rid ourselves of illusions, and try to see things as they
really are, within the range of power and penetration of
Qur senses,

HOW TO STUDY NATURE 19

By using in turn each method of investigation, we are
going to form an idea of the laws of Mechanics. The
experiments we shall make, and the reasoning to which
we shall submit the causes, would strictly lead to the
discovery of the facts of the science, if they were not.
already known. Our sole advantage in working together
will be like that of a party of tourists, exploring, by the
aid of a map, a country covered with good roads and with
a sign-post at every crossing. Having explored this, to
us, new country, we shall feel a great admiration for the
pioneers who opened it up and for the engineers who
mapped it and made the roads. If, further, we succeed
in knowing it well, we shall have gained, in addition to
the enjo5'Tnent of the scenery, a facility in finding our
way in future exploration in a wilder country, where
the paths are aimless and the heights hidden in mist.

c^

28 MECHANICS

propagation of sound from A to B. We should therefore
hear the projectile at D before hearing it at A. It is also
obvious that the sound from C would reach us after that
from D. There is accordingly a certain point in the
trajectory in respect of which sound is first heard ; sounds
proceeding from any other points of the flight are heard
subsequent to this. In other words, the projectile seems
to start its flight from a certain point and to be imme-
diately divided into two projectiles, which niove, one
towards the gun, the other towards the target.*

This example is an indication of the caution ^vith which
the subject of motion must be approached. We certainly
shall not err in trusting the evidence of our eyes so lon<][
as we are dealing with terrestrial objects, but a like
affirmation cannot be ventured with regard to celestial
objects. We believe that we see a star at a definite
point in the sky ; yet the light from some stars reaches
us only after many years — hundreds or thousands of
years. While the light has been travelUng, the earth
has rotated thousands of times and has travelled thousands
of millions of miles. The star is then far from the point
where we judged it to be.

In addition, a star sometimes appears, shines for
some time, and then disappears. Thus, on the 21st of
February, 1901, a splendid and hitherto unknown star
was observed in the constellation Perseus. It rapidly
increased in brilliance, then darkened and disappeared.
The examination that could be made of it during the
short time it existed was sufficient to allow its distance
from the earth to be calculated and the time its light
had required to reach us to be determined It was
consequently found that, at an immense distance, a fearful
conflagration had taken place in the middle of the
sixteenth century.

* M. Durand-Greville has observed that the explosion
frequently attributed to aerolites is simply due to an aualogoas
illusion.

86 MECHANICS

Suppose the balloon, instead of moving in a straight line,
had described a curve. Each part of this line could be
replaced by its two components, or by its two projections,
as they are called ; and the motion of the balloon would
be known by describing the motion of two points simul-
taneously displaced along the straight lines A'C and
A'B , so that by drawing from these points the perpen-
diculars to the two axes the position of the balloon would
be at their intersection.

It is somewhat after this manner that the position of a
warship forcing an entrance into a channel is determined,
when a mine is to be fired beneath it. Two observers
on the shore follow the course of the vessel with telescopes ;
each ascertains one of the projections of its movement,
and, by prearranged signals, they communicate their
observations. One of them, in charge of the mines, which
are exactly marked on a map, traces on the map the course
of the ship, and when it passes near a mine he causes an
explosion by means of an electric current.

Let us watch, from a distance, a wheel turning in the
horizontal plane and carrying a light. We can see only
a motion from right to left, and from left to right, rapid in
the middle of its course, but much slower at the ends;
that is all our observation reveals, and we are reduced to
conjecturing the real motion of the light. If another
observer follows the motion from a point in the plane of
the wheel, but at right angles to our Une of sight, he will
see exactly the same movement, except that, when the
light is at the end of its course for us, it is at the middle
for him. He will be as incapable of defining its motion
as we are, for he will have observed only one of its elements.
Yet, if wc know his observation at every instant, or if he
knows ours, the position of the luminous point can be
determined, and hence the circumference that it is
describing can be constructed.

We can also measure the distance travelled by the
luminous point and calculate iU vdooity or acceleri^tiont

MOTION 87

In addition, we may consider the motion of the straight
line joining it to the centre of the circle and describing
angles. This motion will be known if we can indicate the
angular velocity, expressed in revolutions per second, or,
generally, by the quotient of any unit angle divided by
any unit of time.

One peculiarity of the observation is noteworthy ;
while the light moves in a circle with a constant velocity,
it appears to come and go, to stop and start. For each
observer the apparent motion is subject to an acceleration,
sometimes positive, sometimes negative. The uniform
motion, of which the projection possesses variable velocity,
is affected by an acceleration of direction, as it is called.
The importance of this will soon be understood.

The analogy between an oscillation and a circular motion
is not as distant as it appears ; it is very close, and we shall
see later (§ XXXVIII.) that the consideration of oscillation
as the projection of a uniform circular motion is full of
consequences.

SECOND PART

Exposition of Principles

CHAPTER III

FORCE, THE CAUSE OF ACCELERATION

XIIL— Velocity, Acceleration, Force, and Inertia.

Experts in the interesting game of bowls always
choose with the very greatest care the piece of ground on
which they are to play. They unhesitatingly reject
ploughed land, and have no liking for gravel. They look
for a place with- a flat, smooth surface — either sand or
rolled turf. Ice likewise would not suit them, for, if they
disregarded the danger of slipping and falling, they would
experience little pleasure in chasing bowls which a slight
effort would send rolling undesirabl j distance j over the
smooth surface.

Let us follow the movements of the bowl from the
moment it is taken in hand until it comes to a dead stop at
the end of its course. The player takes hold of it, throws
it forward with a swing of his arm, thereby imparting to
it a certain velocity, and releases it ; it does not fall
right down at his feet, for it has acquired a new property
by virtue of which it falls obliquely and then rolls away
along the surface of the ground. Upon prepared ground
it rolls a fairly moderate distance ; on ploughed, soft,
soil, however, it would very quickly come to rest, perhaps
at its very first contact therewith, or perhaps after re-
bounding once or twice ; on gravel, the motion would be

44

MECHANICS

The lengths of the successive lines are greater and
greater, which shows that in equal times the vertical
distances travelled by the ring in its fall have kept on
increasing — in other words, its velocity has been increasing.

If we measure the distances fallen and divide them by
the distance corresponding to the point 1, we shall find
that they are represented very nearly by the figures
4, 9, 16, i.e., the distances fallen are in the ratio of the
squares of the natural numbers.

We shall hereafter see that certain causes have con-
tributed to falsify slightly the result of our experiment.
Let us disregard them for the moment, and assume that
the differences between the numbers found and the squares
of the natural numbers are due to the effect of these
interfering forces which we have agreed to disregard.
Let us apply the principle of simpUfication already
established, and let us accept as a natural law that a body
falling from rest traverses distances which are as the
squares of the times the body has been faUing.

Now set down these numbers.

Times.

Heights.

First differences
(average velocities.)

Second differences
(average accelera-
tions.)

1

2
3
4
5

1

4

9

16

25

1

3
6

7
9

2
2
2
2

The differences between the distances traversed in
successive equal periods of time are proportional to the
average velocities during these periods. These velocities
are represented by the series of odd numbers.

The differences between these average velocities are aU
represented by the number 2. Thus we can assert that
the average acceleration is constant.

46 MECHANICS

of each of the velocities and the time during which it
lasted. In the present instance, it cannot properly be
said that any single velocity lasts any time, because it is
continually changing. But the time can be conceived as
divided infinitesimally, and then the velocities can be
multiplied by the periods of time. The product will be

the area of the triangle OCt, which is equal to — , and
as » = flrf, this area {i.e., the distance traversed) is given
by -0-. The course of the body can be represented by

the curve OP, which is that of a parabola.

In the special case of falling bodies, it has been cus-
tomary to denote acceleration by the letter g.

The three laws applicable to freely falling bodies can
be written do\vn thus : —

Constant acceleration . . . , . . = g

Velocity varying as time . . . . v = gt

Distance traversed var3dng as the ^

square of the time . . . . . • ^ = ^

In order to discover the laws associated with falling
bodies, we have had recourse to a series of symbols. We
have represented time graphically in such a way as to
convey a meaning to our mind. Such a mode of repre-
sentation is not unfamiliar. The hand of a clock indicates
time by the distance it has travelled ; the angle which it
describes measures the time, the rotation of the earth
furnishing the unit.

We can equally well employ a symbol to represent an
acceleration, a velocity, or a distance. In fact, we find
that in the third diagram a distance has been represented
by another distance ; this is quite permissible. We have
only represented a relatively long line by a short line.
This is the method adopted in drawing maps.

We could have given a more striking example of the
parabola; it is met with continually in innumerable

52 MECHANICS

figure, though a very small addition to the weights in the
pan had visibly stretched the spring. We conclude that,
within the Umits of our movements and measurements,
which are at least as exact as those in the experiment
on the fall of the ring, the force exerted by the weight on
its support is the same ; the ring therefore possessed the
same weight throughout the whole extent of its fall, i.e,,
it exerted upon itself a constant force. We can then give
our result in the following more general form : —

A body acted upon by a constant force possesses a constant
acceleration and a velocity proportional to the duration of
application of the force ; it describes a distance which,
calculated from the position of rest, is proportional to the
square of the time during which the force acts.

This law must be remembered, for it is fundamental
and is encountered everywhere.

It was useful to make by means of the spring a pre-
liminary survey of the field of force in which we operate.
If the scale-pan had carried a piece of iron into the neigh-
bourhood of a powerful magnet, we should have found
that neither the force nor the acceleration was constant.
The magnet would have been one of the accidental com-
plications mentioned in our first experiment, and would
have had to be eliminated. Fortunately for the discovery
of mechanical laws, powerful magnets are rare and of
modern invention ; moreover, all bodies do not possess
the properties of iron. If magnets had long been of
common use, and if many substances were magnetic, the
fundamental laws would perhaps be unknown even now.

70 MECHANICS

a person weigh when he jumps ? " " It depends," we
shall reply. Yes, it depends on many causes. The work
of his fall must be equalised by work done at the
moment he touches the ground. If the ground is soft, he
can alight as he pleases ; the work will be met by the
force between his feet and the ground ; but, if he lands
on hard ground, he must use up the work on his muscles
by relaxing his knees and all other joints. If he
omits to do so, he is subjected to forces that operate over
such a very short distance, that they may be great enough
to break has bones. This explains why a man on skis,
relying too much on th 3 ela ticity of his fall, may be left
on the frozen snow with a broken leg.

Here, again, the skill of the packer is great. Though
he exerts a great force on the head of a nail, he must,
on the other hand, protect fragile objects from the effects
of force while on their journey, no great care being taken,
at stations and wharves, even if the cases are marked
" fragile." When a case falls heavily, its Contents acquire
work and velocity. They lose the latter by giving up
their work ; to avoid violent forces, the work must
be done over a sufficiently great distance. Hence the
need for the elasticity of the shavings, etc., used in
packing.

In order to understand thoroughly what happens in
consequence of a shock, certain properties of matter must
be known. Before undertaking the study of this subject,
we have first to work through a wide field.

XXIV.— Action and Reaction. Momentum.

Watch a bird fly from the end of a flexible twig ; it
bends its legs under it, extends them suddenly, and takes
to the wing. At this instant pay attention to the twig ;
it will be seen to move quickly backwards, and, after
some oscillations, to return to its normal position.

Why does this twig do this ? Obviously because the

TS MECHANICS

These equal impnbes can be compaied with another
product.

We have seen that the forec/is the prodinrt of mass m
and acceleration a, Le, —

/= win.

Elsewhere, we saw that vdocity rs tiie piodact of
acceleration and time —

at = v^

Multiplying these two equations^ we have

ft = mr.

The product of the mass and velocity of the diot ought
then to be the same as for the gun, no other force having
acted. The action of the gases preponderates to such a
degree, during the very short time ctf the firing of the gun
(about the one hundredth of a second), that the state-
ment is very nearly correct for the instant the shot leaves
the gun. We are not concerned with what happens after
the projectile begins to travel in the air.

The velocities are in opposite directions; and, if we
agree to give a positive sign to one directi<Mi, the other
is consequently negative, and the algebraic sum of the
products is zero.

Now imagine the gun placed in the forward turret of
an ironclad, which is firing directly forward. To an
observer in the turret, the conditions are the same as in
the preceding experiment. But an observer of the
motions from the shore would have to add the velocity
of the vessel to each velocity measured by the observer in
the turret. As the result of calculation, it would be
found that after, as before firing, the sum of the products
of the masses and their various velocities would be equal
to the sum of the masses multiplied by the velocity of
the vessel.

Descartes called the product of mass and its velocity
qtumliiy of motion^ i.e., momentum. Our experiment, as
well as any other we may do, proves that the momentum of
a number of masses acting upon each other^ xoithout inter^

THE WORK OF FORCES 78

ference from any external force^ is constant. This is the
principle of conservation of momentum.*

Thus a body cannot be displaced by internal force,
alone. A man seated in a boat or a carriage cannot put
it in motion without an external point to which to apply
his force. He can make a light boat move alternately
backward and forward by moving to and fro. When he
has a velocity, the boat has ; when he stops, it stops ;
when he sits down, it resumes its original position. To
displace it, some external force must act — pressure on the
water, the pull of a rope, or the action of the wind. The
same conditions apply to a leap from a boat to the bank.
If any one' tries it, he will see that the boat is indeed
driven back, because, having lost its load, it is changed ;
so is he, for he will have fallen into the water.

We shall meet applications of the principle just
explained ; but we must now try to discover how a state
of rest is brought about, since forces seem always to
produce motion.

* " Although motion is only a condition of matter which is
moved, there is nevertheless a fixed quantity of it that never
increases or diminishes, even if there is sometimes more and
sometimes less in certain parts." — ^Descabtes 2 " Principles
of PhHosophy," 1644.

CHAPTER V

FORCES WITHOUT MOTION

XXV,— Reactions of Matter ; Elasticity and Friction.

Forces do not always produce motion. Innumerable
objects around us are under the action of forces and yet
they do not move. Books on the shelf, an inkstand on
the table, and the table resting on the floor are all sub-
jected to a force — their weight ; but at the point where
each object rests is developed an antagonistic force
(reaction) which exactly counteracts the action of the
weight. The principle of equality of action and reaction,
already recognised in the case of bodies in motion, is true
for bodies at rest. We should have readily said that the
principle would be applicable, a fortiori^ if one of them
could be more true than the other. In reaUty, they are
equally true, but the reaction of a support on a body at
rest. is not so apparent as the reaction of an object free
to move under a force exerted by another object.

Any one going out on a frosty day and observing the
difficulty of keeping his feet must have been forcibly
reminded of that property of the ground whereby he
is enabled to maintain an upright position without great
effort. The same idea may have come to him when his
bicycle has slipped on wet paving or when he has seen a
cab-horse fall on an asphalt road. He may have seen,
too, some kind-hearted person throw a shovelful of cinders
under the animal's feet, with the result that its hoofs
gripped the ground and it could rise.

In consequence of these examples, it is easy to realise
bow much we owe to the beneficent action of friction.

8g MECHANICS

in the conditions ; so long as the trajectory was entirely
within a space in which the verticals could be considered
parallel, the curve was a parabola. Then its shape changed
as the lines of action of its weight included greater and
greater angles ; finally, when the velocity was such that
the variation in the direction of the trajectory was
equivalent to the variation in the direction of the weight,
the trajectory cut at right angles the perpendiculars
from the places passed, i.e., it was horizontal at every

point. By an exact adjustment of acceleration and
tendency to travel away in a straight line, a shot sent
horizontally could maintain a horizontal trajectory, and
continue round the earth indefinitely. A speed of a little
over 5 miles per second would be necessary — seven
times greater than that of the fastest projectiles— but
much less than that of certain meteorites that, crossing
the orbit of the earth, deviate from their way and continue
their course through space. Those of the meteorites that
eDt«i the atmosphere lose velocity, become heated, and

92 MECHANICS

like its colour ; a definition of mass in terms of weight
would seem as artificial as that of a crab in an old
dictionary of the French Academy : — " Small red fish that
walks backwards."

The idea that mass is the quotient of weight by accelera-
tion leads to strange errors.

Some years ago an engineer, who had remembered
only the formulae of Mechanics, developed the following
theory : — A balloon in equilibrium in the air weighs
nothing ; its weight being nothing, so is its mass ; if it

has no mass, --— = 0, whatever its speed ; therefore a

dirigible balloon can have no kinetic energy ; it cannot
maintain its motion, even for an infinitely short time,
and, if it is to be kept moving, it must be constantly
propelled.

True, it could be given a velocity without any effort
at all. But the engineer had not drawn this last con-
clusion ; he had gone no further than to condemn the
balloon for the former reason. Apparently he did not
know that, from time to time, a balloon knocks down a
chimney or breaks a branch off a tree.

XXVIII.— Couples.

Have you ever wondered why the wheels of a carriage
turn round ? The horses draw on the traces (automobiles

are quite different), and the force
is transmitted to the axles, which
draw the wheels forward ; but
these are on the ground and, as
we have seen, are kept in place
by friction. The two forces AB,
CD (Fig. 25) are not directly
i<w^ opposite ; one acts on the hub.
^ . the other on the rim. Whenever

'^""s recauJ^tf^^der ^^^^^ ^^ ^he case, the object on which
the action of a couple, the forces act begins to rotate.

\%

MVvllVW^i

^ X \ \\

1

::?n of the

: : r>? in

:_■-.: even

_: ~.}.y be

_ r ■. .J

'l'

r > • I.

•w - r^-

104 MECHANICS

and continued as a thin tube. If the liquid rises to the
same height as in the cyUndrical vessel, that part of the
bottom vertically below the tube will support the same
force as before, for the tube can be supposed produced
to the bottom so as to isolate a small column of liquid.
In reality, the surrounding liquid forms the walls of this
extension of the tube ; and in the neighbourhood of the
bottom, the hquid is pressed as if at the bottom. If it
is not to escape, the surrounding Uquid must support it
with the same force, which is communicated to the sides
and the whole of the bottom. The vessel might be made

very flat and surmounted by a very
high tube. The bottom would then be
under a great force, produced by a
small amount of hquid. This invari-
[^ able condition is so incredible that it
has been given the name of the
hydrostatic paradox.

If the bottom of the vessel supports

Pjq 33 ^he pres- such a force, ought it not to be very

sure on the hori- heavy, and would not a balance show
zontal bottom of a a weight greater than that of the vessel

denTof \he sTJe ^^^ ^^^^^ together ? The difficulty is

of the vessel. easily solved ; the Uquid, under pres-

sure in all directions, is held with equal
force against the dome of the vessel, and exerts as much
pressure on it upwards as it exerts on the bottom down-
wards, the small surface of the tube excepted. If the
dome is pierced, the water escapes in a strong jet.

This fact can be expressed by saying that the pressure
at any point whatever in a hquid depends only on the
height of the column above it, and not on the section.

However small the fissures in rock, water succeeds
in penetrating, and the further it descends the more
easily it makes a way, for it is under pressure from the
whole column. This is why the cutting of tunnels such
as the Simplon is so difficult. When the tunnel is under

114

MECHANICS

acceleraiian^ and will exercise upon the string an outward

force the stronger as the ball is turned
^ the more rapidly. This is the force
called centrifugal force.

We have seen (§ XXVII.) a similar
rotation, on a vast scale, in the move-
ment of stars in a planetary system ; by
resuming this problem, we shall lean
how to calculate the value of centri-
fugal force.

Consider, for example (Fig. 84), the
rotation of the moon round the earth.
E If the moon were free, it would travel
Pig. 34.— How cen- away in a straight line MA ; but, in

SSSd ^""^"^ ^ ""^^^^^ '^ trajectory is inclined

towards B. For a very short time,
during which the straight lines joining the centre of the
moon and the earth may be taken as parallel, the laws of
falling may be applied to the motion of the moon ; if its
acceleration is a, the distance it will travel towards tie

earth will be — , after a time t. The mass of the moon

being m, the force to produce the supposed accderation
is ma.

Now, the theorem of Pythagoras enables us to write

EM2 + Mm2 = Em2 = (EM + nmy^
or, substituting r for EM and v for the velocity of the

moon,

r2 + (i;^^ = {r +^y.

Suppressing r\ this reduces to v-i^ = raP + ??r.

Since t is extremely short, f* is extremely small; B
can therefore be neglected, giving

v-v = raP, or v^ = ra.

2 —

188 MECHANICS

to produce real resonance. It is \viser to break up the
impulses whenever possible ; for this reason the order is
always given to troops passing over a suspension bridge
to break step and to walk like a flock of sheep. It is a
wise precaution, suggested by the catastrophe at Angers.

The effects of resonance would be disastrous more
often, if a closely associated action did not always interfere
and restrict it ; this is the limiting action of friction and
all the other causes that consume kinetic energy. The
consumption increases with the amplitude of the motion,
and finally equals the energy of the impulse ; at this stage
the oscillation has reached its maximum value. The
difficulty is to determine whether this value is below that
of the dangerous amplitude ; if not, the limiting action
must be increased, or the period of oscillation either of
the moving body or of the exciting cause must be altered.
This process is well known to engineers, who frequently
make use of it.

The tenor Tamburini used to cause a cut glass vessel
to break in pieces, by giving loudly the note corresponding
with its natural vibration. The vibration, stimulated by
the sound waves, gradually increased and exceeded the
hmit of deformation that cut glass can bear. It would
have been enough to touch the rim with a feather to
prevent the vibration from reaching the dangerous limit,
and even to restrict it to a very small amplitude.

Thus, in turn, we utilise or avoid resonance. The
commonest example of sustained resonance is furnished
by the regulator of clocks and watches ; the shock induced
by the oscillating part itself restores to it the energy that
it has expended in overcoming the resistance of the air
and in moving the wheels.

THIRD PART

Development and Application

Although, in our rapid survey of the principles of
Mechanics, we have always had every-day occurrences
to impress upon us the reaUty of these principles, we have
regarded them so far in a somewhat limited way, in
consequence of our having selected for special attention
only certain aspects of the facts under examination. We
have now to undertake more complex problems in which
various principles contribute to produce the effect we are
studying. Thus we shall grasp more fully their co-
ordination and relation, and after solving some problems
we shall be better prepared to apply the knowledge we
have gained. These problems will no longer be of a
general kind ; they will have reference to particular cases
and will give numerical results. We must first, therefore,
become acquainted with the units in whicli all the
quantities that come within the domain of Mechanics
are expressed.

146

MECHANICS

century ago, Laplace proposed to use as the unit of pressure
the mean pressure of the atmosphere at sea-level ; this
is called an atmosphere^ and is defined by a column of
mercury 76 centimetres high. The distribution of 1
kilogramme-force on 1 square centimetre gives a third
unit of pressure which is much used in industry.

The numerical relations of the dynamic units are
summarised below.

Force:
Work:

Power :

1 kilogramme-force
kjlog]
joule

1 poncelet

( 1 kjlogrammetre

( 1 jor

^— - [ \ ssr '^"*/^""

0*981 megadyne.
9*81 joules.

1 megadyne-decimetre.
100 lalogrammetres per

second.
0-981 kilowatt.
0*981 megabar.
1*013 megabar.

150 MECHANICS

The result, though surprising, is correct ; the velocities
are exchanged.

The theory of impact seemed so incredible to those
who established it, that the celebrated Huyghens, in order
to realise this exchange of velocities, utilised the very
elementary experiment that we have just performed.

We can arrive at this result by considering the work
done by the spring on the two cars.

The spring is compressed until the moment when the
two velocities are equal, i.^., until the cars are at relative
rest. In accordance with the principle of conservation
of momentum, the common velocity will be half the
initial velocity of the left-hand car. The momentum
before the impact was

m,2v + m.O = 2mv,
At the middle of the impact it is

2m X common velocity = 2nvo.

Therefore the common velocity is = ».

The work of the spring having given the right-hand
car a velocity u, the same amount of work done in the
opposite direction (areas APP' and BQQ') will destroy
the velocity of the left car.

We could have given the two cars any velocity what-
ever ; the two methods we have used would have led to
this final result : When two equal mousses meet in an elastic
impacty they exchange their velocities.

If the masses are unequal, it is evident that there can
be no exchange. A shell striking an armour-plate does
not, fortunately, communicate to it its own velocity. We
could easily discover formulae suitable for that case ; but
the time so spent could be more usefully employed ; it
will be enough to indicate the method. We should first
say that the momentum of the two masses together
remains constant ; then we should imagine we were on
one of the moving bodies against which the other is to
strike, such as a rubber ball against a wail. The velocity

170 MECHANICS

It is interesting to notice that the static rupture of the
rubber was brought about by one seventy-fifth of the
weight required for the steel ; but the energy necessary
for sudden rupture was seven times as much; according to
our method of estimating the relative merits of steel
and rubber, we can express them by numbers whose
ratios will vary from about 1 to 500. This example
makes it plain that the method of testing materials ought
to be perfectly definite, if any significance whatever is
to be attached to the numbers that indicate their qualities.

We shall understand, too, without its being emphasised,
that the steel would have resisted the shock much better
if it had been fastened to its support by a piece of rubber.
In order to withstand the static weight, this would have
had to be of large section ; but it would have served to
nullify the energy of the falUng weight in the second
experiment, and would have prevented rupture.

For a static weight, the breaking stress is obviously
almost independent of the length of the wire. It would
be absolutely independent if the wire were weightless, and
if it were quite homogeneous ; the longer it is, the greater
probability there is of a weak part whose breaking strain is
a little less than that of the average of the other sections.
The wire withstands a falling weight all the better if it is
long, for the quantity of kinetic energy that it can reduce
to the potential form is proportional to its length.

Let us now take an extreme case. A very short wire
ought to be broken by the energy of a very slight shock.
If we reduce the wire to a length of 1 millimetre, it ought
to be enough to allow a weight of 100 grammes to fall
from a height of about '8 milUmetre. But an experiment
would not confirm this apparently accurate result. The
reason is that the deformation of the clamps in contact
with the wire could no longer be neglected ; all these
deformations transform kinetic energy into potential
energy, and assist the wire to bear the stress.

Our young pupils will draw from these remarks all the

mCEGyHES

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INDEX

199

Time, 29

of pendulum - swing,

134, 135
measurement of, 11, 12
unit of, 143

Units, 143
Upward direction, 86

Velocity, 29, 38, 45
absolute, 22
angular, 37, 124
average, 21, 29
definition of, 20
direction of, 34
instantaneous, 21^

29
relative, 22
unit of, 144

Vertical, illusion of the, 86,

108, 116
Vibration, 125, 137
Virtual acceleration, 108, 112
von Guericke, Otto, 50

Wab of Secession, 167, 186

Watt, 145

Weight, 51, 111, 133, 135

and mass, 61, 64, 91,
173

Work, 53, 54, 63, 106

capacity for, 57, 61
conservation of, 67
definition of, 54
restitution of, 54
storage of, 54
unit of» 145

The Country Life Press, Gardkn City. N. Y.

THB COUNTRY UFE
GARDEN aTY, N.T.

•*

Op^