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EbUic T 1C2II, f^.^H-l

Darvarb CoUeae Xibrarj

FXOH

MaBBAchueetts. Ins.tltut*
of Technology

3 2044 102 874 203

f y* t St, Jamai^i Fl*
— Hj^ —

ii.

'3[he l^oBal School ^etieo.

ELEMENTARY MECHANICS ;

PIEST LESSONS IN NATUBAL PHILOSOPHY.

W. JEROME HARRISON, F.G.S.,

SECOND YEAR'S COURSE.

T. NELSON AND SONS, PATERNOSTER ROW. .

tiJtuc.T'a- o ? I f'S 4 . a H 1

^>UW<UcCi«^ IvaittLit ed T-'L'-k^Un^

CONTENTS.

-»♦-

I. FOBCE AND MATTEB,
n. CLASSIFICATION OF FORCES, ...
m. THE FOBOE OF GRAVITY,
IV. EFFECTS OF THE FORCE OF GRAVITY,
V. FALLING BODIES,
VI. FALLING BODIES,
Vn. THE FIRST LAW OF MOTION: INERTIA OF MATTER
Vm. INERTIA OF MATTER IN MOTION,
IX. FRICTION,
X. MASS AND MOMENTUM,

XL COMPOSITION AND RESOLUTION OF FORCES,
XIL THE SECOND LAW OF MOTION,
Xm. THE THIRD LAW OF MOTION, ...
XIV. WORK, AND HOW TO MEASURE IT,
XV. ENERGY,

XVI. POTENTIAL ENERGY, ...
XVn. KINETIC ENERGY,
XVm. INDESTRUCTIBILITY OF ENERGY,
XIX. THE NATURE OF HEAT,
XX. HEAT AS A FORM OF ENERGY,

AT REST,

102

107

113

118

127

134

APPENDIX.
QUESTIONS AND EXERCISES,

140

ELEMENTAEY MECHANICS

-♦♦-

I.— FORCE AND MATTER

1. Introduction— 2. Examples of Force— 3. Definition of Force— 4. Kinds of
Motion — 5. Molecular Motion — 6. Distinction between Matter and Force—
7. How Forces become known to us.

1. Introduction. — Let us consider what is meant
by the word force, and what are the nature and
character of the various forces at work around us,
of which we often speak collectively under the
name of the " forces of nature."

Day hy day we see these forces at work, acting
in various ways, and producing many and different
results ; but they very rarely excite our curiosity,
for we have grown accustomed to them from see-
ing them so often. A storm excites our wonder,
and we are curious to know more about that won-
derful and mighty force which causes the vivid
lightning and the deafening thunder. But how many
of us think, even for a moment, about the ever-
acting and far more important force which causes all
substances to fall towards the ground, and which
enables our bodies and all things around them to
maintain their place upon the surface of the Earth ?

8 FORCE AND MATTER.

This force of gravitation acts so constantly and so
unchangingly, always producing the same unvary-
ing results, that we do not recognize anything won-
derful in it ; and it is only by looking very closely
into the results which this force produces, and by
thinking long and deeply about them, that we are
at last enabled to recognize how wonderful and
beautiful even the commonest of such occurrences
really is. *

2. Examples of Force. — If we take a piece of paper,
set it upon edge against some object, as a book, and
then pull it by means of a piece of thread attached
to it, we notice that the paper falls. To pull the
paper down we had to bring into use a power — the
power of the muscles of the arm ; and this power,
carried along by the thread, caused the paper to
fall. Or we may give the paper a push, and again
it falls. This time we also use the power of our
muscles, but in a different way. But we can cause
the paper to fall by means of a very different power.
Let us take a piece of sealing-wax, and, having well
dried and warmed it, rub it with a piece of warm
dry flannel or fur, and bring it within an inch or
two of the paper. Now, although nothing touches
the paper, we see it move towards the sealing-wax
and fall. In the first case we pulled the paper over
by means of the power of our muscles, which was
transmitted by the string; in the next, the seal-
ing-wax must have exercised a power to cause the
paper to fall. Now, this power to set matter in
motion, whether exerted by the hand or by the wax,
is an instance of what we mean by the word force.

FORCE AND MATTER. 9

But power or force may be used for other purposes
than moving a piece of paper. When a cannon is
fired, the gunpowder with which it is loaded exerts
force to propel the cannon-ball ; and the target of
iron against which the cannon-ball strikes must
exert force to stop it. So also a boy playing
cricket exerts force in striking the cricket-ball with
his bat, causing it to fly swiftly across the field;
and the boy who catches the ball exerts force in
stopping it. Let^ us suppose, however, that the
cricket-ball were a very large and very heavy one :
the boy with the bat might try to move it, but fail ;
yet he would have exerted force in trying to move
it ; and were such a ball made to move, the other
boy would expend force in trying to stop it, although
he again might fail.

3. Definition of Force. — Thus we see that force is
that which moves or tries to move a body, or which
changes or tries to change the motion of a body.

4. Kinds of Motion. — We have seen that force
frequently produces motion, and always tends to do
so. We must now pause for a moment in order
that we may learn something about motion. When
a cannon is fired, we know that the gunpowder
exerts a force which compels the cannon-ball to fly
from the cannon, and to continue in motion imtil it
comes to rest again perhaps a mile from the cannon.
Here the cannon-ball as a whole moves a distance of
one mile. Again, a locomotive engine may start from
London, and at the end of about three hours may
have arrived at Birmingham. Here again we notice
that the whole body of the engine moves from one

10 FORCE AND MATTER.

place to another. Motion of this kind we speak
of as motion of the body as a whole.

5. Molecular Motion. — Let us now take the cannon*
ball of which we have already spoken, and place it
on a large fire. After a time it will become white
hot, and will glow with heat. The body as a whole
is perfectly still ; it remains where we placed it on
the fire. But is there any motion at all there ?
Yes, we think there is. . We suppose that all bodies
in the world, and the cannon-ball amongst them,
are made up of an immense number of tiny pieces,
too small to be seen, called molecules. When we
place the cannon-ball on the fire, we think that
these molecules are made to move by the heat — to
swing backwards and forwards, as it were ; and that
the more we heat the ball the faster we make the
molecules move, so that when the ball is white hot
each molecule moves backwards and forwards with
immense quickness. Thus, although the ball as a
whole is perfectly at rest, every little molecule in it
is swinging to and fro with almost inconceivable
rapidity. This kind of motion we call ^motion of
the molecvZes of a body. .Thus we have two kinds
of motion: — (1.) Motion of the body as a whole;
(2.) Motion of the molecules of a body, or molecular
motion. Now, whether the piece of matter be large
or small — ^whether it be a body or a molemle — it
will never move of itself. If it be moving, some-
thing must have acted on it to set it in motion, and
the power which produces that motion is called
a force.

6. Distinction between Matter and Force. — When

FORCE AND MATTER. 11

we look around us, we see an immense number of
different substances. Even within the four walls
of a room we can count, perhaps, twenty or thirty
distinct things, or even more. These substauces are
of different kinds. Some we can see, as wood, iron,
glass ; others we can only feel, as air ; others, again,
we can both see and smell, as oranges or musk ;
and others still we can taste. They all affect some
one or more of our senses. If these substances are
examined with the aid of a delicate balance, it will
be found in addition that they all have weight.
Now, all substances which can affect our senses, or
which have weight, are spoken of under the general
name of matter.

If an iron nail or a needle be placed on a piece
of cork, and made to float on the surface of water
in a basin, and a magnet is then brought near, the
iron will be drawn towards the magnet. To produce
this result the magnet must have exerted force.
But the force cannot be seen, nor can it be detected
by the aid of any of our senses. Since matter is
said to be anything that affects our senses, and since
force does not affect our senses, force cannot be a
kind of matter. Or let an iron ball be carefully
weighed, and afterwards raised to a red heat in a
fire ; then let it be weighed a second time, and it will
be found to weigh exactly as much as at first. Since
the iron weighs no more when hot than when cold,
even when tested by the most delicate balance, it is
clear that heat has no weight, and therefore cannot
be matter. Thus we say that force is not mattery
hit is that which a^ta upon matter.

12 FORCE AND MATTER.

7. How Forces become known to ns. — It is clear
that force itself cannot be perceived by the aid of
our senses ; but force acting upon matter produces
certain effects, and it is these effects that are recog-
nized. Then from our observations conclusions are
drawn as to the nature of the force which has pro-
duced these effects. Thus we cannot see, or feel, or
weigh the force called magnetism ; but we can see
the magnet draw the iron nail towards it. We
therefore feel sure that there must be some power
or force existing in the magnet, and to this force
we give the name of Magnetism.

IL— CLASSIFICATION OF FORCES.

8. The Forces of Nature— 9. The Physical Forces— 10. The Force of Gravity—
11. The Force of Cohesion— 12. The Force of Sound— 13. The Force of
Light— 14. The Force of Heat— 16. The Force of Magnetism— 16. The
Force of Electricity— 17. Chemical Force— 18. Muscular Force.

8. The Forces of Nature. — By the "forces of
nature " we mean all those powers that are at work
in the world, and which, by their action on mat-
ter, have made the world what it is. They may
be divided into two classes: I. Physical Forces;
II. Chemical Force.

9. The Physical Forces. — The physical forces are
usually considered to be seven in number — ^namely,
gravity, cohesion, sound, light, heat, magnetism, and
electricity. We shall consider them in the above
order.

10. The Force of Oravity. — If a ball is placed on
a table and made to roU towards the edge, when
it reaches the side of the table it will fall to the
floor. Why should it fall ? If no force acted
on the ball, it would remain suspended in the air ;
but as it moves toward the ground, we conclude
that some force is pulling it downward, and to that
force the name of Gravity is given. If the ball
were taken to twice the height, or to any still

14 CLASSIFICATION OF FORCES.

greater elevation, we know that it would fall in a
similar way. Thus it is clear that the force of
gravity can act at a distance. If the ball were
examined after its faU, no cha^e would be found
to have taken place in the matter of which it was
composed ; it would retain in all probability its old
shape, and if it were made of india-rubber, for
instance, this substance would be found to have
undergone no change. The force of gravity, there-
fore, not only acts at a distance, but also produces
no change in the substance on which it acts. These
two facts will be found to be characteristic of all
the physical forces ; and if a new force were to be
discovered to-morrow which had these two proper-
ties, it would be classed at once among the physical
forces.

11. The Force of Cohesion. — All the substances
or bodies which form the world are believed to be
each of them composed of an infinite number of very
minute pieces called molecules. These molecules are
too small to be seen, even with a microscope, and
cannot be broken or divided by any physical means.
The molecules of solid bodies are not free to move,
but are held together, each in its own place, by a
force to which the name of Cohesion has been given.
If the molecules be separated from one another by
too great a distance, as when the body is cut with
a knife, cohesion cannot act, and the parts of the
body fall asunder. In this respect cohesion differs
somewhat from the other physical forces, since it
can act only over the very small distances which
separate molecules. But cohesion produces no change

CLASSIFICATION OF FORCES. 15

in the matter of the body; for whether the matter
of solid water (ice), in which cohesion is strong, or
liquid water, in which cohesion is weak, or gaseous
water (steam), in which cohesion is absent, be exam-
ined, it will be found to be always alike in its
composition, always made of the two elements oxy-
gen and hydrogen — always water.

12. The Force of Sound. — When a body is struck, a
sound is generally produced. If such a body as a
bell be struck and then carefully watched, it will
be seen to vibrate — that is, its parts will move to
and fro; and if the hand be placed upon it, the
vibrations may be felt. These vibrations are taken
up by the air and carried away by it on all sides, in
a manner very similar to that in which waves spread
away on all sides of a stone dropped into water.
When these vibrations of the air (or sound-waves)
reach our ears they produce the sensation we call
sound.

13. The Force of Light. — Light, like sound, is caused
by a vibratory or to-and-fro movement of the mole-
cules of the body which is producing the light.
When a body emits rays of light, it is supposed that
its molecules are vibrating with enormous rapidity,
and that these vibrations are carried away in all
directions by a very thin gaseous substance or fluid
called the ether. This ether is supposed to be
extremely thin, or rare, and to be spread through
all space and all matter. It is believed to fill the
enormous space which exists between the Earth and
the sun, moon, and other heavenly bodies, thus
enabling their light to travel over the millions of

16 CLASSIFICATION OF FORCES.

miles which lie between them and us. We can-
not prove the existence of this ether, but it is very
convenient to suppose that such a substance exists.

14. The Force of Heat. — ^Very closely connected
with the force of light is another force termed Heat.
Heat is produced by a motion of the molecules of a
body very similar to that which produces light.
The molecules of all bodies are supposed to be in a
state of motion. When this motion is slow, the
body is said to be cold ; and it is supposed that as
the motion of the molecules becomes quicker so the
body becomes hotter. Heat may be produced in
many ways. If a button be rubbed smartly on a
piece of wood, its molecules are set in motion, and
it soon becomes too hot to be touched by the naked
hand. Heat is also produced in all cases of burn-
ing ; but whether produced in this or in any other
way, heat always consists of a motion of the mole-
cules of the heated body.

15. The Force of Magnetism. — When a common
horse -shoe magnet is brought near a small piece
of iron — an iron key, for instance — the iron moves
toward the magnet; and if they are allowed to
touch, the iron may be lifted up by the magnet.
In order that the key may be so lifted, the magnet
must exert a certain power or force, and this power
which enables the magnet to lift the iron is called
Magnetism. Even before the magnet touches the
iron, the iron may be seen to move under the influ-
ence of the magnet. If the iron be afterwards ex-
amined, it will be found to be quite unchanged by
the action of the magnetic force. This clearly in-

(766)

CLASSIFICATION OF FORCES.

dicates that magnetism, in common with the other
forces we have named, is a physical force, since it
can act at a distance and without changing the
properties of the matter it acts upon.

16. The Force of Electricity. — By eleetricity we
mean a power by which, for example, certain sub-
stances can attract, or draw towards themselves,
light bodies. This electric power or force may be
produced by various means, as by friction or rubbing,
by chemical action, and in several other ways. If
a piece of well-dried
glass be rubbed with
dry silk and brought
near some small pieces
of paper, they will be
seen to jump up to-
wards the glass. The
same effect may be
obtained by the aid
of a piece of sealing-wax which has been rubbed
with a hot dry flannel. The force of electricity is
known to be very closely related to the force of
magnetism.

17. Chemical Force. — Having examined some of the
physical forces, it is now necessary to consider the
second division of the forces of nature — namely, the
Chemical Force. As an example of this force, let
us take an ounce of loaf-sugar and grind it to a
powder; mix with it two ounces of a white substance
called chlorate of potash, similarly powdered. If the
mixture is left alone, it will remain unchanged for
almost any length of time. But let the end of a

(765) 2

Fio. 1.
Rubbed Glass Rod and Bits of Paper.

18 CLASSIFICATION OP FORCES.

glass rod, moistened with strong sulphuric acid (oil of
vitriol), be now brought near a portion of the mix-
ture. As long as the acid does not touch the mixture
no change takes place. It may be brought within
the smallest distance short of actual contact and
nothing happens. But let the acid touck the pow-
der, and the latter at once bursts into a brilliant
violet flame, giving pff a dense white smoke, and in
a moment or so nothing remains but a few black-
ened cinders. Here, then, is an action totally dif-
ferent from any we have hitherto mentioned. It
was necessary, before any effect was produced, that
the substances should be actually touching ; and the
result of the action was a substance totally different
from those we had before. The force which pro-
duced this great change is called the Chemical
Force.

As another illustration of chemical force, let a
small quantity of bichloride of mercury be dissolved
in water in a glass, and let a little iodide of potassium
be treated in the same way in another glass. The
two substances may be held as close to each other
as possible, but no change will be visible. Let the
one liquid, however, be poured into the other, and
instantly a beautiful salmon-coloured substance will
be formed, quite different in every respect from the
two colourless liquids from which it was derived.
This second instance of the action of chemical force
clearly points out the facts, that this force, unlike the
physical forces, cannot act at a distance ; and that
when it does act, it completely changes the nature
of the substances upon which it acts.

CLASSIFICATION OF FORCES. 19

18. MuBcnlar Force. — There is one force which is
perhaps the most familiar to us of all the forces.
This is the force possessed by the masses of red flesh
called roascies, by
means of which we
move our bones,
lift weights, etc.

The way in which '

a muscle exerts
force is by con-
tracting. When, .
for example, we
bend our arm, it
is easy to feel
the muscle in the

upper part of the pw. s.

arm becoming
short and thick.
We must notice, however, that the muscles can only
exert force when the animal to which they belong
is alive. For this reason the muscular force is also
called the vital force, ot the force possessed by
living beings.

III.— THE FORCE OF GRAVITY.

19. Division of the Study of Forces— 20. Why Bodies fall towards the Earth—
21. The Force of Gravity acts at ail Distances— 22. The first Law of Gravita-
tion— 23. Experimental Proof of the first Law of Gravitation— 24. Second
Law of Gravitation— 25. Illustrations of the second Law of Gravitation—
26. Why Rain fails.

19. Division of the Study of Forces. — It would not
be possible to include, in the study of Mechanics, an
account of all the forces of nature. The study of
the modes of action of the chemical force constitutes
the science of chemistry. Sound, light and heat,
electricity and magnetism, are also considered as
distinct sciences. The way in which the muscular
force of our bodies is produced and maintained
belongs to the science of animal physiology.
Cohesion we spoke of in the first part of this book.
There remains therefore only the force of gravity,
which we must now examine more closely and
endeavour to understand.

20. Why Bodies fall towards the Earth. — If any
portion of matter, as a stone, be not supported —
that Ls, be not held up in any way — it will be seen
to move towards the Earth. This is a fact which
comes under our notice every day, and in what-
ever part of the world we may be it remains true.
We have learned that that which produces motion

THB rORCB 07 6RAVITT. 31

13 a force, therefore some force must have acted
upon the stone to cause it to move
toward the Earth ; and as in all
t parts of the world things fall in
the same direction — that is, to-
' ward the Earth — it is only
natural that we should conclude
that the force which causes the
stone to fall resides in or is pos-
sessed by the Ekrth, To this
Fio. a. force the name of Gravity has

OM°°iiSt^hu^d*iS ^6^ given; a word derived from
wirdiiihownbjthscHidiB. t^jg LatJQ ^ord gravis, which

Onma being eitlngalshed. ...

means heavy; and gravity is said
to be "that force which causes

all bodies to fall to the Earth
when not supported." This
is plainly the case with sol-
ids and liquids. We have
only to take away their sup-
port, and they immediately
fall. So also do heavy gases,
such as carbonic acid gas,
which may be poured out of
any vessel containing it just
like so much water (Fig. 3).
But light gases, such as hy-
drogen, do not seem to obey
this law. If a small balloon p„ ,

be filled with hydrogen, it Balloon flUod wKh Hydrogen Oas
. > , 1 J i« , I rlilDff through tha .\lr.

nses to the top or the room.

But this is only because it is pushed upward by

22 THE FORCE OF GRAVITY.

the heavier air; just as a cork is forced up to
the surface of any water in which it may be im-
mersed, by the water (which is heavier than the
cork) getting imdemeath it. If the balloon fiUed
with hydrogen is placed where there is nothing to
press it up or to support it, as in a vacuum or
completely empty space, then the hydrogen will fall
towards the Earth just as the heavier carbonic acid
gas did in the open air.

21. The Force of Gravity acts at all Distances. — ^We
can set no bounds to the action of gravity. If a
body be taken up in a balloon miles above the Earth,
we know that gravity is still acting on it, and we
have only to set the body free to see it fall back to
the Earth. Gravity, however, acts over distances far
greater than any that can be reached by man. By
this force the planets are made to circle round the
Sun, and the Moon to revolve round the Earth.
When used in this universal sense, meaning the
attraction of the heavenly bodies for one another,
the term gravitation is generally used ; while we
speak of the force of gravity when we mean only the
attraction of the Earth for bodies near its surface.

22. The first Law of Gravitation. — We owe the
greater part of our knowledge concerning the laws
which regulate the force of gravity to Sir Isaac New-
ton (bom 1642; died 1727). It is said of this great
man that being one day in an orchard he saw an apple
fall from a tree, and immediately asked himself why
and how the apple fell. Led by this apparently small
circumstance, he began a series of experiments which
ended in his discovery of the laws according to which

THE FORCE OF GRAVITY. 23

gravity acts. These laws are two in number. The
first law states that " Every body attracts every other
body with a force proportionate to its Tnass" From
this we learn that the attraction of gravitation is
not only an attraction of one heavenly body for an-
other, or of the Earth for bodies near it, but that every
body in the universe attracts every other body. From
this it is clear that when two balls are suspended a
few inches apart, each attracts the other, and tends
to approach it. So, also, when the Earth attracts a
ball and causes it to fall, the ball also pulls the
Earth and makes it rise. If this be so, how is it
that we only see the ball fall, and do not see the
Earth rise ? In order to understand this clearly,
we must know what is meant by the word mass as
used in the law just stated. By " mass " is meant
the quantity of matter which any body contains.
Thus, if we take two ivory balls of the same size,
weight, and kind, they will each contain the same
quantity of matter, or have the same mass, and the
two together will have twice the mass of one of
them. These balls, if hung up at a distance of one
foot apart, will each attract the other with a certain
force. Let one of them be made twice as heavy as
the other, and the power of attraction of this ball
will be doubled. Let it be made four times as
heavy, and its power of attraction will be fourfold
that of the lighter ball. Now imagine one ball to
be increased to the size of a mountain, and its
power to attract will be increased in the same pro-
portion. The attractive force of a little ball is too
small to be measured, but the force of gravity pos-

THE FORCE OP GRAVITY.

sessed by a mountain may be made clearly per-
ceptible.

23. Experimental Proof of the first Law of Gravita-
tion. — In the year 1774, Dr. Maskelyne suspended
a small ball near the mountain Schiehallion, in Perth-
shire, and he found that the ball was plainly attracted

towards the
mountain.

Now we
can under-
stand why
the Earth
cannot be
seen to move

Pig. 5.— The SchiehaUlon Experimeiit. tOWards a

The baU, b, is suspended by the fine thread, a b. If the falling ball,

mountain, m, were removed, the ball would hang vertically, rni "C^orf 1i*q

in the direction a h'. But the matter of which the moun- -'- "® XLiartin b

tain is composed attracts the ball, and draws it a little to- ][xiaSS is SO
wards the mountain. The same thing occurs on whichever

side of the mountain the experiment is performed ; the sus- many mlll-

pended ball is always drawn a little towards the mountain. . . .

ion times
greater than the mass of the ball, that the Earth
moves towards the ball through a distance too
small to be measured or seen. So, also, it is clear
why the Earth circles round the Sun, and not the
Sim round the Earth : it is because the mass of the
Sun is much greater than the mass of the Earth.
For the same reason the Moon goes round the Earth,
and not the Earth round the Moon. The mass of
the Sun is no less than three hundred thousand
times greater than the mass of the Earth ; and the
mass of the Earth is forty-nine times greater than
the mass of the Moon.

THE FORCE OF GRAVITT. 25

24. Second Law of Qravitation. — Not only does the
force of gravitation vary with the mass of bodies,
but it also changes according to the distance by
which they are separated from one another. If the
two ivory balls before mentioned be placed near each
other, they will each attract the other with a certain
force ; if the distance is increased the attraction will
become less ; while if the distance is lessened the
attraction of the balls for one another will increase.
Newton's second law of gravitation gives the rela-
tion between the distance and the attracting force.
It says : " All bodies attract each other inversely
as the square of the distance between them" We
will first explain the meaning of the two words
inversely and square. To invert a thing is to place
it upside down. The number 3 will be unaltered
if we write it thus, f , but f inversely becomes J
(one-third) ; so 4 or ^ inversely is t ; and 9 or f in-
versely is J. The number 1, being represented thus, |,
will be unchanged if taken inversely.

The word square, as used in this law, is easily
explained. A number is squared when it is multi-
plied by itself once. Thus 3 squared, or the square
of 3, is 3 X 3 = 9 ; and 9 squared will be 9 x 9 — that
is, 81 ; and so on.

25. Illustrations of the second Law of Gravitation. —
Let two balls be suspended one foot apart. They
will attract each other with a certain force which
we will represent by the number 1. Now let the
balls be removed to a distance of two feet apart.
How great then is the attraction between them ? It
will be only one-fourth as great as at first. To

26 THE FORCE OP GRAVITY.

prove this, let the respective distances be represented
by the numbers 1 and 2. Now the attraction does
not vary as the distance, but as the square of the
distance; so that the numbers 1 and 2 must be
squared, giving 1x1 = 1, and 2x2 = 4. But the
attraction is not merely as the square of the dis-
tance, but inversely as the square of the distance
between the two balls; therefore the numbers 1 and
4 must be inverted, becoming 1 and J. If, then,
two bodies attract each other with a certain force
when they are one foot apart, at double that dis-
tance the attraction is only one-fourth as great, at
three times the distance one-ninth as great, at a
distance of four feet the attraction is only one-six-
teenth as great as at one foot ; and so on. Since a
body on the Earth's surface is 4,000 miles from the
centre of the Earth, it is clear that if we could raise
the body to a height of 4,000 miles above the sur-
face, it would be attracted only one -quarter as
strongly, at a height of 8,000 miles only one-ninth
as strongly as on the surface ; and so on. If the
body weighed one pound at the surface, it would
weigh only one-quarter pound at 4,000 miles high,
and one-ninth poimd at 8,000 miles; the weight
being tested by a spring balance.

In order to obtain a perfectly clear idea of what
is meant by this law of inverse squares, it will be
necessary to work out numerous examples in the
manner above indicated, and to endeavour to express
correctly the answers in words.

Suppose the only pieces of matter in the whole
universe were two raindrops, at a distance, say, of

THE FORCE OF GRAVITY. 27

one million miles apart. Since each drop possesses the
force of gravity, they would attract each other and
would begin to move one towards the other. At
first, as the distance between them is so great, the
attraction would be very small, and they would
move very slowly ; but as that distance decreased,
the attraction would increase, and they would move
faster and faster, till at last they would meet mid-
way, each having travelled exactly half a million
miles.

26. Why Bain falls. — In exactly the same way a
raindrop formed in the clouds attracts our Earth
and is attracted by it. Each moves towards the
other — 'the raindrop tdwards the Earth and the Earth
towards the raindrop ; but while the little ball —
the raindrop — moves downwards, say, one mile, the
great ball — the Earth — moves upwards only the
smallest imaginable fraction of an inch. The con-
sequence is that we can see the raindrop fall, but
we cannot see the Earth rise to meet it.

IV.— EFFECTS OF THE FORCE OF GRAVITY.

27. Meaning of the Words "up" and "down"— 28. The Force of Gravity is the
Cause of Weight — 29. Centre of Gravity— 30. Centre of Gravity of regular
Bodies— 81. Centre of Gravity of irregular Bodies— 32. The Centre of
Gravity of any Body seeks to place itself in the lowest possible Position—
83. Stable and unstable Equilibrium— 34. Neutral Equilibrium— 35. Illustra-
tions of Equilibrium.

27. Meaning of the Words "up" and "down." — When
a body is dropped we say that it falls down ; and in
whatever part of the world we may, be living we
make use of the same expression. As our Earth is
shaped like a ball, it is clear that in different places
on its surface bodies falling towards the centre of
the Earth must fall in different directions. Thus
a ball dropped in England will move in a direc-
tion exactly opposite to that taken by a ball dropped
in New Zealand, and a ball dropped at the equator
will move in a direction nearly at right angles to
both.

Let A B c D (Fig. 6) be four points above the
surface of the Earth, from which stones are let fall,
and let the circle represent the surface of the Earth ;
let e m 8 be the points on the surface of the Earth
on which the stones will fall when allowed to move.
The four stones will move in the directions shown by
the four arrows. The stone A moves in an opposite

EFFECTS OF THE FORCE OF GRAVITY. 29

direction to the stone c ; the stone B will move from
right to left, while the stone D will move from left
to right. The four movements are thus in four dif-
ferent directions ; but if these directions be continued
as indicated by the dotted lines, they will each pass
through the centre of the Earth, which is repre-
sented by the letter X. Thus, when we make use
of the word down we mean toviard the centre of
the Earth ; and by the
word up we mean in
the opposite direction
— namely, away from
the centre of the Earth.

A straight line
drawn from any point
toward the centre of
the Earth is called a
vertical line, and a line
drawn at right angles
to this is called a hori- fio. a.

zontal line. All bodies, »»*«'»" t™^ "^^ ™'" '" «" ^^■
then, in all parts of the world, fall vertically. From
this we see that the force of gravity acts as if it
were seated at the Earth's centre, which is therefore
called the centre of gravity of the Earth. Instead
of saying that bodies fall toward the centre of the
Earth, it will mean the same thing if we say that
they fall toward the centre of gravity of the Earth,
for this is situated at the centre of the Earth.

28. The Force of Oravity is the Cause of Weight. —
When a body is supported, as when it is lying on
a table, it appears at first sight as if gravity had

30 EFFECTS OF THE FORCE OP OEAVITY.

no effect upon it. But if the body be placed on
the band, a certain pressure will be felt. This Is
caused by the force of gravity attracting the body,
and endeavouring to make it move downward ; in
so doing it causes the body to press on the hand.
This pressure is called the w.eight of the body.
Weight, then, may be defined to be " the dovmward
pressure of any body ca/iiaed by the attraction of
gravity r We can now better understand why
the attraction of the Earth bears the name of gravity;
for the Latin word gravis, from which it is derived,
means heavy, and the heaviness or weight of any
body is due to the force of gravity.

In. the fact that different bodies have different
weights we have a good illustration of the law that
gravity attracts bodies in proportion to their mass.
Thus, if a solid piece of iron of a certain size
be taken, it
will be pulled
down with a
certain force ;
suppose the
force to be
equal to one
pound. If a
second piece
of iron of ex-
actly the same

Fio 7 -Volume Kiii Mmj kind, but of

twice the size,
be taken, it will be pulled downward with twice
the force ; that is, with a force equal to two

EFFECTS OF THE FORCE OF GRAVITY. 31

pounds. A body with three times the mass will
be attracted three times as strongly ; and so on. A
piece of platinum, whose volume is, say, one cubic
inch, weighs three times as much as a piece of iron
of the same volume, because it is attracted three
times as strongly by the Earth ; hence we say that
the mass of the platinum is three times as great as
the mass of the iron, and we believe that there is
three times as much matter in it. In Fig. 7 we
have represented the relative volume, or bulk, of
equal masses of platinum, water, air, and hydrogen.

29. Centre of Gravity. — Bodies are composed of an
immense number of tiny pieces or molecules. The
force of gravity acts on each of these molecules, draw-
ing it downwards, and not only on those on the out-
side of the body. Thus, if a ball of clay be taken,
every molecule of that ball will be acted on by
gravity; and the weight — that is, the force with
which gravity pulls the piece of clay — will be un-
altered, in whatever shape it may be moulded, so
long as the number of molecules remains the same.
Thus the clay may be flattened into a sheet, or
moulded into a brick, or rolled into a rod, and yet
its weight remains unaltered.

Now there is one point within all bodies at which
their weight may be considered to be concentrated.
This point is called " the centre of gravity" of the
body ; and if this point be supported, the whole body
will be supported. Thus if a ruler be marked
exactly in the middle and then suspended at that
point, it will usually be f oimd to balance. The centre
of gravity of the ruler must therefore be situated at

32 EFFECTS OF THE FORCE OF GRAVITY.

the middle of the ruler. We can understand this if
we consider that on each side of the middle point the
ruler is made up of an equal number of molecules
which balance one another ; and we might add equal
weights to each end of the ruler without at all dis-
turbing its balance, since they would neutralize each
other.

30. Centre of Gravity of regular Bodies. — In the
case of a body of regular shape it is a compara-
tively easy matter to find the position of its
centre of gravity. It will only be necessary to
find a point situated so as to have an equal quantity
of matter on all sides, or, in other words, to find the
middle point of the figure, and this will be the
centre of gravity. In a ball or sphere, the centre
of gravity is the centre of the sphere. In a cube,
the centre of gravity is at the point which we may
call the centre of the figure — namely, the point
where straight lines joining opposite corners would
cross. The same rule gives the centre of gravity
of a body shaped like a brick, which may be
called oblong bodies. The centre of gravity of
a cylinder is midway between the centres of
the circular ends. The centre of gravity of a
circular piece of card-board or of a ring is the centre
of the circle. In a piece of card-board of triangular
shape the centre of gravity may be found by draw-
ing lines from the middle points of two sides to the
opposite angles : the point in which these lines cut
one another is the centre of gravity of the triangle.
The centre of gravity of a cone or pyramid is found
by the following rule : — ^Join the point or apex of

EFFECTS OF THE FORCE OF GRAVITY.

the cone or pyramid with the centre of the base, and
measure oflf three-quarters of the length of this
straight line from the apex. The point so obtained
is the centre of gravity of the cone or pyramid.

31. Centre of Gravity of irregular Bodies. — In the
case of bodies v^rhich have not a regular shape we
cannot so easily determine, by calculation, the posi-
tion of the centre of gravity. It can, however, be
always determined by experiment, as follows : — Take
any irregularly-shaped body, as a piece of card-board
(a B c D, Fig. 8) ; make a hole in it with a brad-awl
at A, and hang it up from
this point by means of a
piece of string. The string
will of course hang verti-
cally. Draw a line in this
vertical direction across the
card, as A c. Now suspend
the card in a similar way
from any other point in it,
as B, and, as before, draw

a vertical line, B D. The '''''' 8.-Centre of Gravity.

point E, where these lines cut, will be the centre
of gravity of the body ; and if the card be sus-
pended from any other point, it will be found that
the line drawn vertically downward from that
point will always pass through the centre of
gravity, E. If the end of the finger be placed under
the centre of gravity, the card will be found to
balance about that point ; but if the finger be moved
only an inch to the right or the left, the card will
no longer balance. From this we see that if a body

(766) 3

34 EFFECTS OF THE FORGE OF GRAVITY.

is to remain at rest it must have its centre of gravity
supported; for if this be not done, the body must
move in some direction or other until it finds a sup-
port for its centre of gravity.

32. The Centre of Qravity of any Body seeks to place
itself in the lowest possible Position. — As gravity pulls
all bodies toward the centre of the Earth, it is clear
that they will come to rest with their centres of
gravity as near to the centre of the Earth as possible.
Thus if a weight be suspended by a string, it will
remain at rest only when it is as low as it can get ;
and this will be when the string hangs straight up
and down, or vertically. If the weight be now moved
ever so short a distance to the one side or the other,
its centre of gravity will be slightly raised; and

when the body
is set free it will
return to its old
position.

Fig. 9 shows
a roller shaped
like a double

Fig. 9.— Double Cone rolling (apparently) up-hill. , .

cone resting on
two inclined pieces of wood, which touch at A,
but are wide apart at B and c. When this roller
is set free at A, it begins at once to roll towards B c,
although to do this it apparently rolls up-hill. If,
however, the vertical height above the table of the
centre of gravity of the roller be measured, first at
A and then at B c, it will be found to be nearer the
table at b c than at a ; the reason being, that the
conical shape of the roller allows its centre of gravity

EFFEC?rS OP THE FORCE OF GRAVITY.

to move down more than the inclined pieces of wood
raise it up. In the same way we may explain the
action of the toy shown in Figs. 10 and 11. This
toy, we might think, ought to remain lying on its
side when we place it so (Fig. 10), but it does not;

/.<-->. \

/
/

Fig. 10.

Fig. U.

it always gets up again (Fig. 11). Now we may be
perfectly sure tiiat there is some arrangement by
which the centre of gravity is in its lowest position
when the image is standing upright. This is effected
by putting a piece of lead inside the lower part of
the image, as at a. Fig. 11.

33. Stable and unstable Eqitilibzium. — ^We have seen
that if a body is to remain at rest in any given posi-
tion, its centre of
gravity must be
supported. When
the centre of grav-
ity of a body is
supported in such

a way that the Three kinds ofiqumbrium.

body remains at

rest, it is said to be in equilibrium. It is usually pos-

36 EFFECTS OF THE FORCE OF GRAVITY.

sible to support a body in more than one way. Thus
a cone may be made to rest upon its base (Fig. 12, a).
or upon its apex (Fig. 1 2,B),or upon its side (Fig. 1 2,c).
In the first case (a) the centre of gravity is as low as
it can be placed, and if the cone be slightly inclined
to either side it will return again to its old position.
In this position the cone is said to be in stable
equilibrium. In the second case (b) not only would
there be considerable trouble in getting the cone to
balance on its point, but the slightest disturbance
afterwards would cause it to fall. A body in this
state is said to be in unstable equilibrium.

34. Neutral Eauilibrium. — In the third case, how-
ever (c), when the cone rests upon its side, it
may be moved, and will perhaps roll some distance,
but will at last come to rest, still lying on its side.
This condition is said to be one of neutral equilibrium.
A ball placed upon a table is also in neutral
equilibrium, for it will rest indifferently on any
part of its surface ; but if a hole be cut in one
side of the ball, and a heavy substance, such as a
piece of lead, be placed in the hole, then the ball
will come to rest only when that point rests upon
the table : the ball will then be in stable equili-
brium, for the centre of gravity will be in its lowest
possible position.

35. Illustrations of Equilibrium. — Carts loaded with
heavy weights, as iron, can travel along an uneven
road without being upset, while those with a load of
equal weight but of lighter material, such as hay, are
likely to be overturned on the same road. In the first
case the centre of gravity of the load and cart is low;

EFFECTS OF THE FORCE OF GRAVITY. 37

but in the second case, owing to the bulky nature of

the load, the centre of gravity is higher up, and the

cart is in a state of imstable equilibrium (Fig. 13). So

long as a vertical line drawn from the

centreof gravityfalls between the wheels,

the weight is supported by the road; but

if the road slopes so much as to cause

this line to fall outside either wheel,

the weight is unsupported, and the cart

must topple over sideways. So, too, the j.^^ ^3

centre of gravity ig much higher when Hay- cart in a

the crew of a small boat are on their SStbie^equmbrium

feet, than when they are seated, and S,u "^^p^^^,"^***-

' •/ ' Tbecentxeof grav-

it requires very little force to upset ity(ata)is so situ-

.,,,.,,- rm J. • ated that the verti-

the boat in the former case. Ihat is cai une, a 6, fails
why we so often read of accidents inewLeL^lfthe
caused by persons standing up in small eaxt were loaded
boats. In all cases the larger the base ubriom would be
of any body and the lower its centre rlts^'^ntr:;
of gravity, the more stable will the «^^^y "^^^^^ ^

, , , lower, as at e, and

equilibrium be; the smaller the base the vertical iine« or
and the higher the centre of gravity, 7™n '*?hr^two
the more unstable is the equilibrium, ^i»e«i«-
and the more easily will the body be overturned.

There is a famous tower at Pisa, in Italy (see
Fig. 14), which leans considerably to one side; but
it is perfectly safe, for the vertical line connecting
the centre of gravity of the tower with the centre
of gravity of the Earth falls within the base of the
tower.

,. ^ '«^-

v.— FALLING BODIES.

86. The Force of Orayitj causes all Bodies to fall with the same Velocity—
37. Besistance of the Air to falling Bodies— 38. Experimental Illnstrations—
89. Bodies falling in a Vacunm— 40. Unifonp Velocity— 41. Variable Ve-
locity.

36. The Force of Gravity causes all Bodies to fall with
the same Velocity. — When a body is supported in
any way — as, for example, by the hand — the effect
of the action of gravity on it is shown by the
weight of the body ; if the body is not supported,
however, the effect produced by this force is per-
haps clearer, for the body falls toward the centre
of the Earth in a direction which we have learned
to call vertical. Very wrong or uncertain ideas
about falling bodies were held for a long time ;
and it was reserved for an Italian philosopher
named Galileo to find out by observation and
experiment, about the year 1590,. the laws which
govern the motion of falling bodies. His experi-
ments consisted in watching the fall of various
bodies dropped from the top of the Leaning Tower
of Pisa in Italy (Fig. 14). One of the most
important facts he learned was, that all bodies,
when their motion is not interfered with in any
way, fall equally fast.

FALLING BODIES. 39

Let US conaider the case of two leaden balls, of
the same size and weight, dropped side by side
from a height of, say, twenty feet. Clearly they
will move at the same rate and reach the ground
at the same time. The balls may be brought so

Fio. 14,— LuDlDg Tovsr at Pl»t.

near together that they will touch each other ; still
they will fall at the same pace. Let them be joined
into one ball, and still no change will take place in
their rate of motion. If instead of two balls we have

40 FALLING BODIES.

a hundred, which are allowed to fall at the same
time from the same height, these also will all reach
the ground in exactly the same time ; and if they
were all rolled into one, this would produce no
change whatever in their rate of motion. From
this it is clear that all bodies, light and heavy, fall
with the same velocity. Thus a two-pound weight
and a one-pound weight let fall from the same
height would reach the ground at the same time.
This can easily be proved by experiment.

This may also be rendered clear in another way.
We have seen that gravity attracts all bodies in
proportion to their mass. Thus a ball of a certain
mass will be attracted twice as strongly as another
ball whose mass is only one-half as great. If
these two balls be placed on a table, it will take
just twice as much force to move the heavy one
horizontally as it will to move the light one ; and
if we let a force of two pounds act on the large
ball, while a force of one pound acts on the small
ball, they will both roll at the same speed along
the table. Now this is exactly the way in which
gravity acts: the larger the mass of a body the
greater is the force of gravity acting on it : hence
all bodies, heavy and light, fall towards the centre
of the Earth at the same rate, if they are dropped
from the same place.

37. Resistance of the Air to falling Bodies. — But it
may be said that if a gold coin and a piece of gold
leaf be dropped at the same time, the coin will reach
the ground long before the gold leaf. That is per-
fectly true ; but the cause of the difference is the

FALLING BODIES. 41

resistance which the air offers to the fall of all
bodies. When a body falls it has to push aside
the air lying between it and the Earth, just in
the same way as a ball falling into a heap of sand
has to push aside the particles of sand in order to
move downward. Now, the coin being much the
heavier, pushes aside the particles of air more easily
than the very thin and light piece of gold leaf
can, and therefore the coin falls the more rapidly of
the two bodies.

38. Experimental Illiistrations. — Take a piece of
paper about two inches square, and a leaden bullet ;
let them fall together, and the bullet will quickly
out-distance the paper, as it more easily overcomes
the resistance of the air. Now roll the paper up
into a ball, and let both the bodies fall a second
time. The paper now, having much less surface,
will not be so much hindered by the air, and both
paper and bullet will reach the table at the same
time. Again: cut out a circular piece of thin paper,
slightly smaller than a penny, and drop the penny
and the paper side by side : the paper will lag be-
hind. Now place the paper on the top of the
penny, and let them fall together. The penny,
being underneath, prevents the paper from feeling
the resistance of the air, and they both reach the
table at the same time.

39. Bodies faUing in a Vacnum. — But perhaps the
most convincing proof is this : — If it be the air which
resists the falling of bodies, then when we cause very
heavy and very light bodies to fall in a place entirely
empty of air, they should all fall with the same speed.

FALLIHO BODIE&

We can take all the air out of a vessel by means
of an air-pump, and thus obtain what is called a
vacuum. Place in the long glass tube (Fig. 15) a
piece of gold leaf, a feather, a gold coin, and a bullet,
and draw out all the air by an air-pump. Now

quickly turn the tube upside down, and
all the bodies, light and heavy, will be
seen falling together to the bottom of
the tube. As all bodies, light and
heavy, fall with the same speed in a
vacuum, it must be the resistance of the
air which causes the diflference in their
rates of motion when they fall, some
slowly and some rapidly, as we usually
see them do in the air around us.

40. Uniform Velocity.— It is now neces-
sary that we should try to understand
exactly the manner in which bodies fall.
To do this, it will be necessary to use
a few words, the meaning of which we
will try to make clear beforehand.

The word velocity, as generally used,
indicates great speed or quickness ; but
in mechanics the word simply means the
speed or rate at which a body moves,
whether that speed be great or small.
Thus we may speak about a velocity
of one mile per day or a velocity of
sixty miles per minute with equal ac-
curacy. There is, however, another point con-
nected with this statement which it would be
well to notice. When we say that a railway train

Fia. 15.
Falling Bodies.

FALLING BODIES. 43

moves with a velocity of sixty miles per hour, we
understand that the train, moving along always at
the same speed, travels sixty miles in an hour. So,
too, if we say that a boy is walking at the rate of
four miles per hour, we mean that if he were to
continue moving along steadily for one hour at that
rate, he would cover four miles. Motion of this kind
is called uniform motion, and the velocity is called
uniform velocity, because it is constant and un-
changing. This kind of motion, too, is easily
measured. Thus sound travels with the uniform
velocity of 1,120 feet per second. If it be required
to know how far a sound-wave will move in one
minute, it is only necessary to multiply 1,120 by 60
(the number of seconds in one minute), and the
answer is found to be 67,200 feet. Or, if a train has
a uniform velocity of 30 miles per hour, we may want
to know how far it will travel in one second. In 30
miles there are 158,400 feet; and in an hour there
are 3,600 seconds. Divide 158,400 by 3,600. The
answer is 44 ; so that we may say that the railway
train moves with a vielocity of 44 feet per second.

41. Variable Velocity. — ^Uniform motion is the
simplest kind of motion, but we soon find that it
is not the only kind. Thus, if an arrow be shot
straight upward, the eye can easily see that as the
arrow rises higher and higher it also gets slower and
slower ; and if a cricket ball be driven a long way
over the ground by a stroke from a bat, it moves
more and more slowly until it comes to rest. This is
called variable motion ; and bodies having this kind
of motion are said to move with variable velocity.

VL—FALLING BODIES.

(Continued.)

42. The Velocity of a falling Body is variable— 43. Falling Bodies move with
accelerated Velocity— 44. Breturded Velocity — 45. Velocity of falling Bodies
— 46. Amount of Acceleration of falling Bodies— 47. Space passed over by
a falling Body— 48. Problems on falling Bodies.

42. The Velocity of a falling Body is variable. — It is
much harder to catch a cricket ball that has been
thrown to a great height than one that has been
merely tossed up a few feet. Again, a boy may
jump from a chair to the ground without incon-
venience, but if he try tp jump from the housetop
he will probably break his legs. These facts show
that the further a body falls the faster it falls.
But why is it that falling bodies move with this
changing or variable velocity ? Let us suppose
that a ball is dropped from any point. At the end
of one second it will be moving with a certain
velocity ; this velocity has been given to it during
that second by the attraction of gravity. During
the next second, if it continues falling, it will gain
as much velocity as it did during the first second,
and this it will have in addition to what it had
before ; therefore, at the end of the second second
it will have twice the velocity that it had at the
end of one second. So also if it continues to fall

FALLING BODIES. 45

during a third second, it will gain during this
second as much velocity as it did during the first,
in addition to what it had at the end of the second
second ; the falling ball will therefore have three
times the velocity at the end of the third second
that it had at the end of the first second.

43. FaUing Bodies move with accelerated Velocity. —
Any force which makes a body move faster and
faster is called an accelerating (or hastening) force ;
and if it increases the velocity by the same amount
ervery second, it is called a uniformly accelerating
force. Gravity, then, as it makes bodies fall faster
the further they fall, and as it always gives to
them the same amount of extra velocity every
second, is said to be a uniformly accelerating force.

44. Retarded Velocity. — But when any body, as
a stone, is thrown upwards into the air, gravity,
instead of adding to its velocity, takes away from
it; and every second makes it move upward with
less and less speed, until at length the stone is
brought to rest in the air. If there were no force
of gravity, the stone would go upward for ever ;
but gravity, constantly attracting it, makes the stone
move more and more slowly, till at length it stops
for an instant in the air, and is then compelled to
return to the Earth. When the velocity decreases
in this way, the motion of the body is said to be
retarded,

45. Velocity of flailing Bodies. — When any body,
such as a ball, has been falling freely under the
action of gravity for one second, it will be found
to be moving at the rate of thirty-two feet per

46 FALLING BODIES.

second. Of course the velocity has been gradually
increasing, from the moment when the ball was
dropped to the end of the second ; and were the ball
allow-ed to continue falling, the velocity would still
increase up to the moment when the ball was
stopped. As the velocity is continually changing,
how can we speak of the velocity at any particular
instant ? When we say that at the end of the first
second the ball moves at the rate of thirty-two feet
per second, we mean that if the motion were to con-
tinue unchanged during the next second, the ball
would move through a distance of thirty-two feet.
Similarly we may see a train fly past a railway
station, and we say that it is going sixty miles per
hour. We may not see the train for more than
half a minute, but what we mean is clear enough.
It is not that the train will really travel sixty miles
in the next hour, but that, if it kept on at the same
pace for an hour, it would travel sixty miles.

46. Amoimt of Acceleration of falling Bodies. — If
the falling ball be watched at the end of the second
second, it will be found to be travelling more rapidly
than at the end of the first. Indeed, it will now
be going twice as fast — namely, at the rate of sixty-
four feet per second ; at the end of the third
second its velocity will be three times as great, —
namely, ninety-six feet per second ; and so on.
Thus we see that the velocity increases just as the
time increases ; and if we want to find the velocity
at the end of any given time, we have only to mul-
tiply by thirty-two the number of seconds during
which the ball has been falling. Thus, —

FALLING BODIES. 47

At the end of 1 second, the velocity is 32 x 1 = 32 feet per second.
At the end of 2 seconds, the velocity is 32 x 2 = 64 feet per second.
At the end of 3 seconds, the velocity is 32 x 3 = 96 feet per second.
At the end of 4 seconds, the velocity is 32 x 4 = 128 feet per second*
At the end of 5 seconds, the velocity is 32 x 5 = 160 feet per second.

And so on. For example, to find the velocity of a
falling body at the end of 7 seconds, multiply 7 by
32 ; this gives 224, At the end of 7 seconds the
body will be moving at the rate of 224 feet per
second. This uniform acceleration of 32 feet per
second, produced by the action of the Earth on fall-
ing bodies, is often represented in books on Mechan-
ics by the letter g,

47. Space passed over by a falling Body. — After a
body has been falling for one second, it is found
by experiment to have descended 16 feet. This is
nearly true ; but the exact distance fallen through
in one second depends to a certain extent on the
place where the body is dropped. Thus a ball will fall
a rather greater distance in one second at the north
pole than at the equator. The reason lies in the
fact that the force of gravity increases the nearer
we get to the Earth's centre. Now, as we are about
thirteen miles nearer the centre of the Earth at the
north pole than we are at the equator, we find that
the force of gravity is a little stronger at the former
than at the latter place. Thirteen miles is not much
out of nearly 4,000 miles, still it makes a slight
difference.

At the end of two seconds the body will have
fallen, not 32 feet, as we might think, but 64 feet;
at the end of three seconds, 144 feet; and so on.
These numbers (16, 64, 144) are in the proportion

48 FALLING BODIES.

of the squares of the numbers 1, 2, 3, indicating
the number of seconds. Thus, 64=16x2x2;
that is, 16 multiplied by 2 squared : and 144 equals
16 multiplied by 3 squared ; 16x3x3 =144. The
rule for finding the distance a body falls from rest
in any given time will be : Multiply the square of
the number of seconds by 16. Thus, —

In 1 second a body falls 16 x 1 x 1 = 16 feet.
In 2 seconds a body falls 16 x 2 x 2 = 64 feet.
_^ In 3 seconds a body falls 16 x 3 x 3 = 144 feet.

In 4 seconds a body falls 16 x 4 x 4 = 256 feet.
In 5 seconds a body falls 16 x 5 x 5 = 400 feet.

And so on.

We may understand the way in which a body
falls, under the action of gravity, if we remember
that after having fallen 1 6 feet during the^rs^ second,
the body begins the next with a velocity of 32 feet
per second. This velocity alone would carry it over
32 feet during this second ; but, in addition, gravity
makes it fall as far as it did during the first second,
— ^namely, 16 feet. Thus the total distance fallen
in the second second will be32+16=48 feet. This,
added to the 16 feet it has fallen during the first
second, gives the 64 feet traversed during the first
two seconds. Again : the ball begins the third second
with a velocity which would carry it over 64 feet
during that second; add to this the 16 feet which
gravity alone would cause it to fall, and we have
80 feet traversed during the third second; this,
added to the 64 feet described in the two pre-
ceding seconds, gives the total distance of 144 feet
passed over in three seconds, by a body falling freely
from a state of rest.

FALLING BODIES.

Velocity«32 feet

VeIocltyB64 feet. ^

Velocity»96 feet.

• • •

©•

Space traversed daiiagjirst second^id feet.

Space traversed during second secondB48 feet

Space traversed during tMird second»8o feet.

■ Space traversed duxixtg/intrth secondBiia feet

Velodty-«za8 feet.

Fio. 16.— Diagram showing the relative Spaces passed over hj a Body falling freely

from rest, daring the first four seconds of its fall. (Scale, i inch to 16 feet)

Total distance traversed =16 + 48 + 80 + 112 =256 ft

48. Problems on falling Bodies. — By the aid of
this knowledge several interesting problems may be
worked. Suppose, for instance, that a stone is

(766) 4

50 FALLING BODIES.

dropped into a well, and that in two seconds it is
heard to strike the water. Since in two seconds a
body falls through sixty-four feet, we may take that
distance for the depth of the well. This is really a
trifle too great, because it takes some time, though
very little, for the sound of the splash to reach the
ear ; and thus the real time of the motion of the
stone is somewhat less than two seconds. This
makes so little difference, however, that in practice
we need not notice it.

Again : it is sometimes required to know how far
a body falls in some one given second, as, for ex-
ample, the fourth. Proceeding as above, we find
that in four seconds a body falls 16x4x4=256
feet, and that in three seconds it falls 16 x3 x 3=144
feet; therefore during the fourth second it falls
256-144=112 feet.

VII— THE FIRST LAW OF MOTION: INERTIA

OF MATTER AT REST.

49. Absolute and relative Motion — 50. The First Law of Motion — 61. Inertia
of Matter at Best — 52. Illustrations of Inertia— 63. Practical Applications
of Inertia.

49. Absolute and relative Motion. — ^By absolute
motion we mean the true motion of a body through
space, independently of any other motion. When the
motion of one body is compared with the motion
of another, it is called relative motion. We know
from the science of astronomy that our Earth is not
standing still, but that it possesses at least two dif-
ferent motions. (1.) I* moves in a circle around the
Sun once in a year; and (2.) It turns on its axis once
in twenty-four hours. Speaking roughly, we may
say that in consequence of the motion around the
Sun, the Earth moves through somewhat more thati
one and a half million of miles in a day. In con-
sequence of the Earth turning round on its axis, a
place on the equator also describes in one day a
circle, of which the circumference is about twenty-
five thousand miles. Now, all people and all things
on the Earth have these two motions ; and hence,
when in common language we say that an object on
the Earth is at rest, we mean that it is at rest as far

52 THE FIRST LAW OF MOTION.

as the Earth is coTicemed, and not that it is really
or absolutely at rest, for that is impossible. Thus
we see that no body on the Earth can be absolutely
at rest, since all must partake of the Earth's motions.
Bodies may be at rest so far as the Earth is con-
cerned, or as compared with other bodies on the
Earth ; but this is not an absolute state of rest.

In like manner, when bodies fall, the motion we
observe is not their whole motion, but only their
motion relatively to the Earth; for these falling
bodies are partaking at the same time of the Earth's
motions.

Again : if a box or a parcel be placed on the seat
beside a person in a moving train, the box is rela-
tively at rest as far as the person and the othei
things in the carriage are concerned— it does not
change its position with respect to them ; but it is
in motion compared with the trees and hedges be-
side the linQ of railway, and it has in addition the
motions of the Earth.

It is true that we do not notice the movements
of the Earth, but the fact that we may be in motion
and unconscious of it, is established by common ob-
servation. For example, let there be two railway
trains side by side in a station, and let one of them
begin to move ; the passengers in both trains are
often at a loss to know which of the trains is moving.
They see that there is relative motion, but, until
they look at some object which they know to be at
rest, they are uncertain whether their own train or
the other is at rest with respect to the station.

50. The First Law of Motion. — Sir Isaac Newton,

THE FIRST LAW OF MOTION. 53

to whom we are indebted for a very large part of
our knowledge concerning moving bodies, discov-
ered what are called " the laws of motion." These
laws are so important, and so many occurrences are
explained by their aid, that it is necessary to con-
sider them one by one. We shall now consider
the " First Law of Motion." This law states that
" W?ien a body is not acted on by any force, if it
be at rest it will remain at rest ; and if it be in
motion it will continue to move in a straight line
with a uniform, velocity!' This is a law which we
cannot absolutely prove, for w^e cannot place a body
on the earth so that it will not be acted on by any
force. Gravity, in the absence of any other force,
will always affect the body. In a case of this sort
we must take the law as it stands, and try it in as
many ways as we can. If it yields satisfactory
answers to all our questions, then we are justified
in regarding the law as true.

51. Inertia of Matter at Best. — Let us consider the
first part of Newton's law — namely, that " When a
body is not acted on by any force, if it be at rest
it will remain at rest."

Place a ball on a table. The ball is at rest rela-
tively to the other things in the room. How long
will it remain at rest ? For ever, if it be not acted
upon by some force. We clearly recognize the fact
that the ball cannot move of itself. If it does move,
we know that some force must have caused it to do
so. This property we call the inactivity or inertia
of matter. The word " inert " is often used to mean
dead or lifeless, and matter is inert in the sense that

64 THE FIRST LAW OP MOTION.

it cannot put itself in motion. Force is always re-
quired to produce motion in matter. Hence matter
is said to possess the property of inertia ; and the
first law of motion, which states this fact, is often
spoken of as the " law of inertia^ Let it be clearly
understood, however, that inertia does not indicate
any unwillingness, as it were, on the part of matter
to be moved : it will offer no active resistance to
any force acting on it. The law of inertia, when
clearly understood, simply means that there is no
power residing in matter by which a body can either
move itself or bring itself to rest if it be set in
motion. A stone can neither start itself nor stop
itself; it requires force to set a stone in motion,
and it also requires force to stop a moving stone.

52. Illustrations of Inertia. — (1.) If a small weight,
as a stone or a piece of lead, be placed on a sheet of
paper, when the paper is slowly drawn along the
table the stone or the lead will move with it. Gravity
is pulling the weight downward, and causing it to
press on the paper ; and as neither the paper nor
the weight is perfectly smooth, there will be a cer-
tain amount of rubbing or friction, as it is called,
between the paper and the weight. A portion of
the force exerted by the hand will therefore be
transmitted by the paper to the weight, and will
cause it to move along with the paper. But if the
paper be pulled with a jerk, it will be found that
the weight will be left behind. The reason is, that
the friction did not in this case last long enough to
pass on to the weight suflBlcient force to cause it to
move along with the paper; and as the weight

THE FIRST LAW OF MOTION. 55

could not move itself, it was left behind. Instead
of saying that the weight " could not move itself,"
we may say that it has the property of inertia.

(2.) Place a number of small wooden draughts-
men one upon the other, so as to form a small perpen-
dicular column. If the lowest man is pushed gently
along, the whole column will move forward, the
friction between the men being sufficient to com-
municate the motion from each man to the one next
above it. But if the lowest draught is pushed some-
what more quickly, the friction does not last long
enough to pass all the motion to the second ; and
this one cannot acquire the same velocity, but will
move more slowly, and the next draught still more
slowly ; and so the column will be upset. Finally,
if the lowest man be rapidly struck with a thin but
heavy body — for example, the back of a dinner knife
— ^it will be seen to fly away, while the column re-
mains undisturbed, and merely falls vertically. Here
again we see that the bodies forming the pile cannot
move themselves ; if they are at rest, and if no force
acts upon them, they will remain at rest. This ex-
periment succeeds best if the knife be placed on the
table so as to move in a perfectly horizontal direc-
tion. With a little practice, a piece may thus be
struck even from the middle of a column without
upsetting it.

(3.) Lay a card on the top of a wine-glass, and
place a coin upon the card. In obedience to the
first law of motion the coin will remain on the card
for ever, or until some force acts on it. If the card
be now smartly struck by the finger, it will be seen

THE FIRST LAW OP MOTION.

Fig. 17.— Inertia.

to fly ofi*, while the coin will drop into the glass.
The friction between the card and the coin was not
suflicient to overcome the inertia of the latter; when
the card was removed, the coin, being left without

any support, fell into the
glass. If the coin were
closely-watched, it would
be found to move a little,
but not enough to cause
it to fall beyond the glass.
Instead of one glass and
one coin, we might em-
ploy two glasses placed
side by side, covered with
a long card, and having a coin over each glass.
This arrangement, however, demands a little more
skill to insure success. Instead of using the finger,
the card may be struck by a spring, as in Fig. 17.

(4.) Instances of the action of the law of inertia
frequently come under our notice in everyday life.
When people are sitting in a train, and the train
suddenly moves forward, the bodies of the people,
tending to remain at rest, are thrown against the
back of the carriage.

(5.) Again: if a man is sitting loosely upon a
horse, and the horse suddenly starts forward, the
man falls ofi^ backward. The man tumbles off in
accordance with the first law of motion; for in
order to cause him to change his previous state of
rest, and move along with the horse, force must be
applied to his body. Now, this force can only be
applied at those points at which he is in contact

THE FIRST LAW OP MOTION. 57

with the horse, so that if he is sitting loosely he
will fall backwards; but if the man grasps the
horse firmly with his knees, he becomes, as it were,
part of the horse; the muscular force of the horse is
transmitted to his body, and he moves safely along
with the animal.

(6.) When a man is standing on the stern of a
boat, and the boat is suddenly pushed off, the man's
feet partake of the motion of the boat, but his head
and body, tending to remain at rest, lag behind, and
he may fall into the water.

(7.) If an open vessel, a^ a cup or a basin, con-
taining some leaden shot or some pease,
be moved suddenly and quickly down-
ward, the shot or the pease will be found
to linger behind, and to fall into the yiq. is.
cup again after it is brought to rest, inertia of Pease.
The muscular force of the arm causes the vessel to
move, but does not affect the shot. The pellets keep
their state of rest, and lag behind until gravity brings
them down into the pan. (Fig. 18.)

53. Practical Applications of Inertia. — In modern
rifles there is a groove cut in the barrel in a spiral
manner, designed to give the bullet a twisting motion.
If the bullet does not fit tightly, this rotation will
not occur, so that it is a point of great importance
to get the bullet to fit the barrel perfectly. In the
Enfield rifle the bullet is made with a hollow base,
and in this hole a wooden plug is loosely placed.
When the powder suddenly explodes, the plug is
forced forward, and before the bullet has had time
to take up the same velocity, the plug forces out

5B THE riBST LAW OP HOTION.

the lower end of the bullet, thus making it fit tightly
into the groove of the barrel.

The inertia of water is taken advantf^ of in the
well-known arrangement for supplying railway en-
gines with water while the train is running. The
Irish mail runs from Chester to Holyhead, a dis-
tance of 84<| miles, in two hours; and the tender
(the waggon behind the engine on which are the
coab) picks up about 1,000 gallons of water from a
long trough, 18 inches wide and 6 inches deep,
which is laid between the rails for a length of 441
yards near to Conway, A
scoop 10 inches wide dips
2 inches into the water, and
is connected with a pipe
leading up into the tender.
(Fig. 19.) As the engine
rushes along, the mouth of
the scoop sUces off a layer
of water; and before the
liquid has had time to ac-
quire the velocity of the
, l'°; ^?;~^?'"/°^- , , the trwn, it slides up the

a, b, Wnter-Ttoogh ; d, Wheel "' „ „ ^

Tender; «. Scoop; /, c, Tmk In- feW feet of pipe leading to

the tender, and rushes into
the tank as if it were being discharged from a
most powerful force-pump. What really happens is
exactly the contrary of what appeai-s to happen.
The water is at rest, but the inclined plane formed
by the scoop and pipe is pushed underneath it, with
a velocity of some forty miles per hour, and the water
is lifted into the tender by reason of its inertia.

YIIL— INERTIA OF MATTER IN MOTION.

M. The First Law of Motion m applied to moTlng Bodies — 66. Friction is the grwi
Destroyer of Motion— 66. Inertia of Matter in Motion— 67. Examples of the
Inertia of Matter In Motion — 68. Familiar Illnstrations of the Inertia of
Matter in Motion— 60. Practical Applications of Inertia of Matter in
Motion.

54. The First Law of Motion as applied to moving
Bodies. — The second statement in the first law of
motion is this, that when a body is not acted on
by any force, " if it be in motion it will continue to
move in a straight line with uniform velocity." In
other words, if a body is in motion it will for ever
continue to move in exactly the same manner unless
some external force interferes. A moving stone,
for example, has no power in itself to go either
faster or slower, or to stop ; neither can it turn to
the right hand nor to the left.

This statement seems quite contrary to our every-
day experience. All the bodies which we see in
motion on the globe tend, sooner or later, to cease
moving. We may make a clock to go for a week,
or a month, or even a year, and we admire the
steady motion of its wheels; but we know that
sooner or later it will stop. All moving things that
we see come to rest at last. But why is this ?
Why do they not move on for ever ? It is because

60 INERTIA OF MATTER IN MOTION.

we can never place a body where it can be perfectly
free. Wherever we may put the body, it will
always be acted upon by some force. Do what we
can, we can never get rid of friction. The moving
body is sure to ^oucA . something else, and wherever
it does touch there will be friction ; and by friction
alone, though unaided by any other force, the body
will be brought to rest. The more we can lessen
the friction the longer will the body move, and the
nearer we shall be to realizing the fact that motion
is as natural as rest. It is only in the heavens that
we see bodies in motion without friction ; and the
motions of these bodies — the sun, moon, and stars
— seem perpetual. For thousands of years our
Earth has been whirling round the sun at the
same rate as it does now, 'and as it will probably
continue to do for thousands of years more. As-
tronomers can foretell with certainty whereabouts
in the sky any of the heavenly bodies — ^the sun,
the moon, or the planets — will be next week, next
year, or at any more distant time. This is a proof
of Newton's law, that " if a body be in motion it will
continue to move in a straight line, and with uni-
form velocity, so long as it is not acted on by any
external force."

55. Friction is the great Destroyer of Motion. — We
have seen that if a body be stationary, it will re-
main at rest until it is acted on by some force which
can set it in motion. Suppose that a ball is placed
on a table : in virtue of its inertia it will tend to
remain where it has been placed ; but let some force
act on it — muscular force, for example. Then we

INERTIA OF MATTER IN MOTION. 61

shall see the ball move ; but having rolled a certain
distance it comes to rest. Why is this ? It is be-
cause both the table and the ball are more or less
rough, and there is a certain amount of rubbing or
friction between the table and the ball ; it is this
friction that sooner or later brings the ball to rest.
If such is the case, then if the ball is started with an
dqual force, on a rougher table it should come to rest
sooner, and on a smoother table it should continue in
motion for a longer time. If we try these experi-
ments this will be found perfectly true. Thus we
see that it is friction that causes the motion of a roll-
ing ball to cease ; and the less the friction the longer
will the motion continue. We know that if a stone
be thrown along a road it soon comes to rest ; but
if thrown along smooth ice it will travel very much
further. From this it seems clear that could we get
a perfectly smooth horizontal surface, and a perfectly
smooth stone, the motion of the stone would continue
for ever. Moreover, the stone would move with
uniform velocity and in a true line ; for there is no
reason why it should become either faster or slower,
or why the stone should go off to the right hand or
to the left. On the Earth, however, we cannot get
perfectly smooth surfaces, or perfectly smooth
bodies; and hence all bodies in motion, however
great their velocity may be for a time, soon come to
rest, their motion being gradually destroyed by fric-
tion, if by no other cause.

56. Inertia of Matter in Motion. — It now appears
clear that when the. motion of a body is changed or is
destroyed, it is on account of the action of some out-

62 INERTIA OF HATTER IN MOTION.

side force. Moving bodies have no power in them-
selves by which they can bring themselves to rest or
cause themselves to move faster or slower. This
fact is spoken of as " the inertia ofTnatter in motionJ*
Just as we saw that matter at rest was dead or
inert, in that it could not put itself into motion, so
matter in motion is inert, for it has the same want
of power — ^it cannot either slacken or increase its
speed — it cannot bring itsdf to rest. Thus we
speak of the inertia of matter in motion, as well as
of the inertia of matter at rest.

57. Examples of the Inertia of Matter in Motion. —
(1.) If a little pan or cup containing pease be jerked
upward and then suddenly stopped, the pease will be
found to fly out of the pan. At first they move
with the same velocity as the pan, but when the
pan is suddenly stopped the pease seek to continue
their motion, and so leave the pan. They would
continue moving upward in straight lines for ever
were it not for the friction against the air, and the
action of gravity, which at last cause them to stop
and to return to the Earth. (See Fig. 18.)

(2.) Fig. 20 shows a piece of apparatus by means
of which we may illustrate the law of inertia. It
consists of a little circular table of wood, a, which
can be made to revolve very rapidly by turning the
wheel, by the two being connected by an endless
band : to a screw in the centre of this wooden disc or
table the brass rod, c, having a brass ball at the end,
is fastened so that it can move freely while the table,
a, is at rest. Make a chalk mark on the edge of a,
and place the ball, c, over it. When the handle of

INERTIA OF MATTER IN MOTION. 63

the wheel, 6, is turned smartly, the table, a, moves
also ; but the ball, c, being at rest, tries to remain at
rest, and lags behind, until at length the friction
compels it to move with the table. Then, after
a time, when table and ball are both spinning round
with the same velocity, let the table, a, be suddenly
stopped by means of the hand, and it will be found
that in virtue of its inertia the ball, c, will continue
moving, until, after perhaps half a dozen revolu-
tions, its motion is destroyed by its friction against
the table, and it also comes to rest.

Fio. 20.— WhirUng Table.

(3.) A top when spun on the ground speedily
comes to rest, principally on account of the friction
of its sides against the air and of the point of the
peg against the ground. If the friction be made
less in any way, the motion will continue for a
longer time. For example, if the top be spun a
second time upon a smooth surface, as the inside
of a watch-gla^s, it will be found to continue spin-
ning for perhaps five or ten minutes ; and if it is
spim under a bell-jar which has been exhausted by
means of an air-pump, the top will continue to spin,
perhaps, for an hour or more.

58. Familiar Illnstrations of the Inertia of Matter in
Motion. — The people sitting in a railway carriage

64 INERTIA OF MATTER IN MOTION.

liave of course the same velocity as the train.
When a swiftly-moving train is suddenly brought
to a standstill the people in it are thrown forward,
since their bodies try to retain the motion they had
in common with the train. This is the reason why
such terrible injuries are often sustained by the
passengers of trains which come into collision.

When a vehicle approaches a sharp turn in a
road, a cautious driver always slackens his pace for
fear of an accident. When the horse is driven
round a sharp comer at too high a speed, the
carriage, in virtue of its inertia, tends to proceed
in the same straight line in which it was previ-
ously moving. The consequence is that it over-
turns, and the passengers are thrown out in the same
direction.

Persons who incautiously alight from a train in
motion frequently sustain severe falls. The reason
is, that a man's whole body when in the train par-
takes of its motion; but when he jumps out, his
feet are stopped by touching the ground, while his
body endeavours to move forward with the old
velocity. The consequence is that he is thrown down
in the direction in which he was travelling.

59. Practical Applications of Inertia of Matter in
Motion. — (1.) Every time we use a hammer we take
advantage of the force of inertia. If the hammer
be merely laid on the nail no effect is obtained ; but
when the hammer is made to move quickly through
the air, on reaching the head of the nail it tends to
continue in its state of motion, and in so doing drives
the nail into the wood.

INERTIA OF MATTER IN MOTION. 65

(2.) When we wish to fasten the head tightly on
a hammer, we knock the opposite end of the handle
smartly on the ground, and after one or two blows
the head is found to be firmly fixed. The reason
is, that the hammer-head endeavours to continue in
motion after the handle has stopped, and so fixes
itself firml J on the handle.

(3.) The pile-engine, a machine for the purpose of
driving large pieces of tim-
ber {or piles) into the ground,
depends for its utility on the
inertia of matter in motion.
A heavy piece of iron is
raised by means of a chain
to a height of several feet,
and is then suddenly allowed
to fall on the head of the
pile. The iron weight, in
endeavouring to continue in
motion with the velocity it
has gained in falling, forces ^^^ ^^ -puo-Eimine
the pile into the earth.

(4.) Fly-wheels are large heavy wheels attached
to steam-engines, or to other machines that are re-
quired to work smoothly and regularly. When a
fly-wheel is set in rapid motion its inertia is so great
as to compel all the moving parts of the machine
to maintain a nearly uniform speed.

(5.) The inertia of water in motion is usefully
employed in obtaining a pure water supply for Man-
chestra. The water is obtained from the moorland
lying between Manchester and Sheffield, and is

INERTIA OP MATTER IN MOTION.

RESCRVOIK FOR CLEAK WATER.

TO RESERVOrR PORTORBIO WATER.

Fio. 22.— Inertia of Water in Motion.

sparkling and clear in dry weather, but becomes dis-
coloured by the peat after rain. The question is

how to prevent
the muddy from
mixing with the
pure water. Fig.
22 shows the ar-
rangement adopt-
ed for a small
stream, which
flows over a ledge,
having an opening
at A. When the
weather is dry and
the supply of
water small, the stream flows but slowly, drains
through the opening, and falls into the clear-water
reservoir : when the stream is swollen by rain,
and is therefore muddy, the inertia of the water
causes it to leap across the gap, and to pass away
in another direction.

(6.) In the corn warehouses at Liverpool the
grain is earned on a plain flat band, eighteen inches
broad, made of canvas or india-rubber. The band
runs on rollers, and is caused to move round and
round by means of a steam-engine. The first
law of motion is here applied very ingeniously to
divert the grain from one path into another during
its passage. At the point where the change of path
occurs the carrying band is bent a little upwards
(Fig. 23). The result is, that as the stream of grain
retains the velocity which is given to it by the

INERTIA OP MATTER IN MOTION. 6

band, it is carried forward in a jet over the top of
the pulley, B, just as if it were a stream of water.

6HO0TOFCRAIN

TRAVCLLINC BAND

Fio. 23.— Inertia of Matter in Motion.

The spout, c, diverts the com into a new channel,
and may pass it on to another travelling band for
transport in a new direction if necessary.

IX.— FRICTION

GO. Caiue of Friction— 61. Disadvantages of Friction~62. Advantages of Fric-
tion— 63. Kinds of Friction— 64. Friction produces Heat.

60. Cause of Friction. — By "friction" is meant the
rubbing together of two surfaces. The surface of
every body has upon it a certain number of little pro-
jections and little hollows, and no amount of polish-
ing, although it may remove the greater number of
these irregularities, can render the surface perfectly
smooth. The inequalities may be very small, too
small to be seen by the naked eye, but they are
always present, and may be rendered visible by the
aid of the microscope. When any two surfaces are
placed in contact, some of the little projections or
roughnesses on the one catch in the hollows in the
other, and thus the two bodies are held together.
When the little projections and the little hollows are
of the same shape, as they are in two pieces of the
same substance, then they fit closely into one another
like the teeth of two similar saws, and are more diffi-
cult to separate than they would be if the projections
were of different shape to the hollows. Hence it is
found that the friction between pieces of the same
material is greater than between pieces of different

FRICTION. 69

materials. Thus axles of steel are generally made
to revolve on brass or gun-metal, and in watches the
steel axles of the wheels move in little cups of a very
hard stone called agate. When the hollows are
filled up with some smooth substance, bodies will
slide over one another with greater ease ; therefore
those surfaces in machines which rub one against
the other are often greased or oiled to make the
friction less. Of late years black-lead has been
employed for the same purpose.

61. Disadvantages of Friction. — A large part of the
power used in driving machinery is always lost ;
that is, it cannot be used in doing the work we
desire the machine to accomplish. Wherever two
parts of the machine rub together there will always
be friction, and some part of the power applied to
the machine will be used up in overcoming this fric-

' tion. This sometimes amounts to as much as one-
fourth of the power applied. Again, nearly all the
labour expended every year, by men, horses, railway
engines, etc., in carrying bodies from place to place,
is used in overcoming friction. When we see a
horse toiling along a level road with a heavy load
in a cart, we know that a child could do the same
amount of work were it not for the friction of the
wheels against the road and of the axles against
the wheels.

62. Advantages of Friction. — Though friction is so
great a hindrance to all work, we should be much
worse off without any friction at all. Without friction
the erection of houses would be impossible; the slight-
est disturbance would cause them to fall to pieces. We

FRICTION.

ourselves could not move a step, for it is the friction
between our feet and the earth which renders walk-
ing so easy to us. We all know how difficult it is
to walk upon ice, for the friction between our feet
and the ice is very small; but were that friction
absent altogether, walking would be quite impossible.
Nor could we hold things in our hands : the least
force would make them slip through our fingers. It
is friction, too, that enables a screw to hold together
two pieces of wood. Again, when we drive a wedgfe
into a block of wood, we rely on friction to keep the
wedge in its place. When we weave the short fibres
of cotton or wool into long threads, it is the friction
between the fibres which makes them hold together.
What we generally require is a certain amount of
friction, but not too much. Our roa^s must not be
too rough and stony, for then the friction against
wheels and feet would be too much ; but, on the other
hand, their surface must not be smooth as glass or
as ice, for then the friction would be too little, and
we should all slip or slide about.

63. Kinds of Friction. — Friction is usually said to

be of two kinds
— ^namely, slid-
ing friction and
rolling friction.
When one sur-
face slides over
another the fric-
tion is called sliding fmction (Fig. 24). Thus sliding
friction is produced when a log of wood is dragged
endways along a road, or when a rope is pulled

Fig. 24.— Sliding Friction.

FRicyriON. 71

through the hands. Rolling friction is produced
when a body turns round and round, moving onward
at the same time, as when a wheel rolls over the
ground. Rolling friction is very much less than
sliding friction, for the inequalities on the touching
surfaces are not dragged but lifted out from one
another. Thus, in an experiment which was actually
tried, a roughly-chiselled block of stone weighing
1080 lbs. was made to slide over a stone surface by
a force of 758 lbs. The stone was next placed on
a wooden sledge, and then a force of 606 lbs. was
sufficient to make the loaded sledge slide over a
wooden floor. When the wooden surfaces in contact
were smeared with tallow the force necessary to
draw the stone was reduced to 182 lbs. Finally,
when the stone was placed on wooden rollers three
feet in diameter, the force necessary to move it was
reduced to 28 lbs. only.

In this experiment we have an illustration of
what is generally done in every-day life. When-
ever we can, we substitute rolling friction for sliding
friction. Thus, except when the ground is covered
with snow or ice, we always use wheeled carriages
to convey materials from one place to another. A
labourer puts rollers xmder heavy blocks of stone, in
order to shift them more easily from place to place ;
and we mount our chairs and tables on castors that
they may be easily moved about.

At other times, however, we find it convenient to
change the rolling friction into sliding friction.
Thus when a laden waggon is moving down a hill,
the drag is placed under one wheel in order that

72 FRICTION.

the extra friction so produced may check too rapid
motion. In other vehicles a brake is applied to the
rim of the wheel for the same purpose. The brake
generally consists of a block of wood, which can be
caused to rub against the circumference of the wheel
when it is desired to slacken speed.

64. Friction produces Heat. — Any one who has
watched a heavy waggon with the drag under one
wheel coming down a stony hill, cannot have failed
to see the sparks fly, and the ground smoke after
the passage of the drag across it. This indicates
the fact that friction is a powerful agent in the
production of heat; indeed, the old method of
obtaining a light by the aid of flint and steel was
only a means of utilizing the heat resulting from the
friction of the steel against the flint. Similarly the
Indians of North America can obtain a light by rub-
bing one piece of stick upon another. The friction of
a railway-carriage wheel on the metal rails is very
small, but the metals are always found to be heated
after the passage, of a train ; and the sparks and
flames which often come from the brake when it is
necessary to suddenly stop a train give us an idea
of the heat that can be generated by friction.

X.— MASS AND MOMENTUM.

65. Definition of Matter— 66. Bodies, Particles, and Molecules— 67. Volume of
a Body— 68. How the Masses of Bodies are compared— 69. Momentum—
70. Examples of Momentum.

65. Definition of Matter. — Matter may be defined to
be "that which affects our senses." Thus, if we
take an apple in our hand, the sense of touch tells
us that it is round and smooth. By the aid of the
sense of sight we perceive its red colour. Our nose
informs us of its pleasant smell ; and should we place
a portion of it in our mouth we learn at once that
it has an agreeable taste. Since the apple affects
our senses, we say it is a piece of matter. It is not
necessary that all the senses should be affected. A
stone, for example, would probably not affect the sense
of taste ; the air does not affect the sense of sight ;
and we cannot smell a piece of iron ; but inasmuch
as we detect these substances by the aid of one or
more pf the senses, we call them all Tnatter, Again :
we have learned that the force of gravity attracts
all things towards the centre of the Earth, and that
the force with which any substance presses down-
ward is called its weight. It is clear, therefore, that
all matter must have weight. Further, we may say
that matter exerts force. Thus the Earth, which is

74 MASS AND MOMENTUM.

a large ball composed of various kinds of matter,
exerts the force called Gravity ; a magnet made of
the matter called steel exerts the force of Magnetism ;
and a stick of sealing-wax rubbed with flannel exerts
the force of Electricity. Not only does matter
exert force, but force always acts on matter. We
cannot recognize force, indeed, except when it is
acting upon matter. Thus we should know nothing
about the magnetism in the magnet did we not see
its action in attracting pieces of iron ; nor about the
force of gravity, yere it not that we recognize its
effects on bodies placed near the Earth. The force
itself is entirely beyond our observation. We can
only form ideas about it by studying its action upon
matter. We may sum up our knowledge by saying
that matter is that which affects our senses, which
has weight, which exerts force and is acted upon
by force.

66. Bodies, Particles, and Molecules. — A portion of
matter large enough to be handled is called a hody.
Thus a stone, a tree, or an orange, might be called a
body. But any body may be broken up into many
smaller parts, as a piece of sugar would be by crush-
ing it or by grinding it to a powder. The portions
so obtained, too small to be handled, but large enough
to be seen, are generally called particles. Now take
the sugar particles, and drop them slowly into
water. They will gradually disappear, dissolving,
as we say, in the water. What has really happened
is that the particles of sugar have been broken up
into still smaller pieces by the water — into pieces
too small to be seen. These are the smallest pieces

MASa AND MOMENTUM.

of sugar that can be obtained, and are called mole-
aides. Thus a particle may be said to be built up
of molecules, and a body to be made of particles.

67. Volume of a Body. — All bodies take up a cer-
tain amount of space or room ; in other words, they
all have a certain size. The amount of space taken
up by any body is called its volumCy and is measured
in cubic feet or cubic inches. Thus a piece of coal of
the shape of a brick, and measuring three inches
every way, would have a volume 3x3x3 = 27 cubic
inches.

68. How the Masses of Bodies are compared. — The
quantity of matter which any body contains is
called its Tnass. The quantity of matter in two
cubic inches of iron is twice that contained in one
cubic inch ; therefore the mass of the former is
twice that of the latter. But the mass of a body
is not always in proportion to its volume. A piece of
wood of twenty cubic
inches would have less
matter in it, and there-
fore less mass, than one
cubic inch of platinum.
How, then, are we to
estimate the masses of
diflferent bodies? We
have learned that grav-
ity attracts bodies in
proportion to their mass ; ^^^ 2^

therefore when any two Bails of Cork and Lead; equal in mass,
t -,. 1 • • 1 aoeqaal in volume.

bodies contammg each

the same quantity of matter are placed in the oppo-

76 MASS AND MOMENTUM.

site scale-pans of a true balance they will be equally
attracted by the Earth, and will balance each other
whatever their size may be (Fig. 25). But the strength
of the Earth's attraction for any body is called its
weight ; therefore mass is usually measured by
weight. The unit of mass, or standard of weight,
in this country, is defined by Act of Parliament
to be a piece of platinum marked "P. S., 1844,
1 lb. ; " which is kept in London, and copies of which
are preserved in various parts of the kingdom, so
that if the original pound were destroyed the others
would remain.

It is necessary to understand clearly the differ-
ence between the terms mass and weight Mass,
or the quantity of matter in a body, is something
which cannot change, whether the body be at the
equator or at the pole, on the Earth or on the Moon ;
but weight, which is used only as the measure of
mass, does vary. Thus a body weighing 200 ounces
at the equator would weigh 201 ounces at the pole
(tested in a spring balance), for the weight of a
body increases by about ttujj in passing from the
equator to the pole ; yet the mass of the body would
be exactly the same at each place. The mass of a
body is invariably the same under all circumstances,
but the weight may vary. The weight of a body
is used only as the measure of its mass, and must
not be confounded with mass itself.

69. Momentum. — If a cricket-ball and a cannon-
ball of the same size be moving with the same
velocity, it will be a much harder task to stop the
latter than the former. The cricket-ball may be

MASS AND MOMENTUM. 77

stopped with the hand, but it will be found im-
possible to stop the cannon-ball in the same way.
Why is this ? The answer is plain — that in the
cannon-ball there is more force than in the cricket-
ball, just as there is more heat in two gallons of
water than in one gallon, though both may be of
the same temperature.

In studying bodies in motion we soon learn that
we have to pay attention to two things — the masa
of the body in motion, and the velocity with which
it is moving. If a piece of matter weighing one
pound were moving at the rate of one foot per
second, it would possess a certain quantity of
motion ; if it were moving at the rate of ten feet
per second, it would have ten times the quantity :
now if a body weighing ten pounds were moving
at the rate of one foot per second, it also would
have ten times the quantity of motion of the
body weighing one pound and moving at the same
rate. The quantity of motion possessed by a mov-
ing body is measured by multiplying the weight of
the body expressed in pounds by its velocity mea-
sured in feet per second. The number so obtained
expresses the Tnomentum of the body. The unit
or standard of momentum is the quantity of mo-
tion possessed by a body weighing one pound and
moving at the rate of one foot per second.

70. Examples of Momentum. — (1.) A body weighing
one hundredweight, and moving at the rate of nine
feet per second, possesses double the momentum of
a body weighing half a hundredweight and moving
with the same velocity ; for the quantity of motion

78 MASS AND MOMENTUM.

in the former will be 112x9 = 1,008 units; while
that of the latter will be 56x9 = 504 units.

(2.) The momentum of a cannon-ball weighing 64
lbs., and moving with a velocity of 1,500 feet per
second, is 64 x 1,500 = 96,000 units.

(3.) The momentum of a train weighing 200,000
lbs. (or about 90 tons) and travelling 30 miles per
hour (that is, 44 feet per second) is 200,000 x 44 =
8,800,000 units. From this we can understand
the terrible results which follow when two heavy
trains dash into each other.

(4.) Two icebergs (weighing thousands of tons
each), though moving but slowly, can crush the
strongest iron-clad between them as we crush an
egg-shell between our fingers : their great mass
makes up for their small velocity.

(5.) It is often noticed that small, light boys,
dodge much better at football or other games than
big, heavy lads. The reason is that their momentum
is less, and it therefore requires less force to change
their course. Hares, too, often escape from the
hounds by doubling ; that is, by turning quickly on
one side and running in a new direction. The
hound, with his heavier body and greater speed,
has much greater momentum than the hare ; and,
when the hare has doubled, this momentum takes
the hound many yards onward in a straight line
before he can alter his course and chase the hare
anew.

XL— COMPOSITION AND RESOLUTION OF

FORCES.

71. Bepresentation of Forces— 72. Forces acting in the same Direction—
73. Forces acting in opposite Directions — 74. Forces acting in parallel Lines—
75. Forces acting at an Angle with one another — 76. Resolution of Forces.

71. Bepresentation of Forces. — Force has been de-
fined to mean " Thxit which "(noves or tmes to move a
body, or which changes or tries to change the motion
of a body'' In talking about forces it is frequently-
found convenient to represent them by lines drawn
upon paper, forming what is called a diagram. But
before this can be done it is necessary that we
should know three things about the force we are
dealing with : — (1.) The point of application of the
force ; (2.) The direction of the force ; (3.) The mag-
nitude of the force.

(1.) The point of application of the force, — All
bodies, as we have seen, are made up of particles.
By the point of application of any force we mean
the position of that particle of the body on which
the force acts. This particle we may represent on
a piece of paper or on a slate by a dot, and this dot
will then indicate the point of application.

(2.) The direction of the force. — When a force is
acting on a body, it moves or tends to move the

80 COMPOSITION AND RESOLUTION OF FORCES.

particle on which it acts in a certain direction.
The line along which the particle moves, or tends
to move, is called the direction of the force. If a
line be drawn from the dot which represents the
point of application of the force, in the direction in
which the particle is caused to move, it will repre-
sent the direction in which the force acts.

(3.) The magnitude of the force, — We measure a
force by saying how many pounds weight it can
support. Thus the force which the muscles of a
man's arm can exert may be measured by the weight
he can lift. If a weight be suspended from an
india-rubber or a steel spring, it is evident that the
weight will stretch the spring, until the weight
pulls the spring down and the spring pulls the
weight up with equal force ; hence the number
of pounds in the weight is the measure of the
force the spring is exerting. Now, in our diagram,
if we fix: on a line of a certain length and agree that
it shall represent a force of one pound, we can repre-
sent the magnitude of any
, , force hy taking as many

of these lengths as there
<C ' ' • are pounds in the force.

^ Let us suppose, for ex-

^'®- ^- ample, that a force of three

c, The Standard line, half an inch ^ j • i* ji j. xi.

in length, representing a force of one PO^UdS IS applied at the

pound A, The Point of Application, point A (Fig. 26) tending

AB, Line one inch and a half long, rep- *^ .

resenting a force of three pounds act- tO make A mOVC in the

ing on the point a, in the direction ab. ^i^^^y^^ ^ 3 ^ standard

line (c) of a certain length (say half an inch) is
taken to represent a force of one pound. A dot

COMPOSITION AND RESOLUTION OF FORCES. 81

placed at A will represent the point of application
of the force; the line AB bearing the arrow-head
represents its direction; and if AB be made three
times the length of C (that is, one inch and a half),
it will indicate the Toagnitude of the force. Thus
the line A B completely represents the force in
question.

72. Forces acting in the same Direction. — Take a
spring balance and hang on the hook a weight of
two pounds. Here we have a force of two pounds
acting downward, and to represent it we must draw a
line, BD (Pig. 27),
from the point of ' ^

application, b, and

twice the length • ' ^ ' )>

of A, which rep-

' „ ^ Fig. 27.

resents a torce a, standard line (^ inch) representing a force of one

of one nOUnd P^^^^* bp, Line one inch long, representing a force

1^ ' of two pounds. DC, Line one inch long, representing

Now, place on a second force, also of two pounds, bo, Line two

, ^ . 1 ■, inches long, representing a force of four pounds, act-

the hook a second ing on the point B in the direction bc. This line

wmahf also of representstheresultantof the two forces bd, do.

two pounds; this force will act in the same direc-
tion as the first, and we can represent it by con-
tinuing the line BD to c, and by making DC equal
to twice A. The whole line BC now represents the
two forces, and from it we learn that they are
together equal to one force of four poimds act-
ing in the same direction (bc). We can test the
accuracy of this statement by putting a four-pound
weight in place of the two two-pound weights on the
spring balance, and we perceive that the pointer
marks the same place on the scale as it did before.

(765) 6

82 COMPOSITION AND RESOLUTION OF FORCES.

Here, then, we have one force equal to two others
and producing the same result. This single force is
called the resultant of the other two forces, and its
magnitude is found by adding the former forces (its
components, as they are called) together. For
example, three men pulling at a rope in the same
direction, one with a force of ten pounds, another
with a force of fifteen pounds, and a third with a force
of twenty-five pounds, would, by their united efforts,
produce on the rope a strain of fifty pounds, which
might be called the resultant of the three forces.

73. Forces acting in opposite Directions. — The in-
dividual forces, however, though acting in the same

^ B straight line, may be act-

^^^^^^^^^^^^^^^^^^^ ing in opposite directions.

Let us examine this case.
In Fig. 28 we have repre-
sented two equal weights,
A and B, to which strings are
attached. These strings,
j,j^ after passing over pulleys.

Equal Forces acting in opposite are fastened in a knot (c).

directions. rni • i l • nii i

ihis knot IS pulled by two
equal and opposite forces. Let us mark off'cD, CE
to indicate the forces. Since there is no reason
why c should move to one side more than to the
other, it remains at rest. Two forces counteracting
each other in this way are said to be in equi-
libriwm.; they must be equal and opposite. If the
forces are made unequal by placing an additional
pound on one of the hooks, say on B, the knot will
no longer remain at rest. It will then move in

-«, — .-

COMPOSITION AND RESOLUTION OP FORCES. 83

the direction of the greater force. If we have a
force of one pound acting in the direction CE and
a force of two pounds in the direction CD, they
might be replaced, and exactly the same effect pro-
duced, by a force of one pound acting in the direc-
tion CD. This last-named would be the resultant of
the two other forces ; and, since the original forces
are acting in opposite directions, the magnitude of
their resultant is found by subtracting the lesser
force from the greater. Thus, in Fig. 28, if A were
five pounds and B three pounds, the resultant would
be a force of two pounds acting along ce.

74. Forces acting in parallel Lines. — Forces that
act in the same direction
but not in the same straight
line are often called parallel
forces, since they act in par-
allel lines. They may be rep-
resented as before by lines of
proper length drawn in the
direction in which the forces
act. Thus let ab (Fig. 29) ^

represent a wooden rod ; let R

a force (ap) of two pounds ^ /^I'T' .a u .^.

^, ' , ."*• A B, Wooden Rod, acted on by the

act at A in the direction ap, two parallel forces a p (two pounda)

1 /» i* j.-i_ J and BQ (three pounds). The single

and a lorce oi three pounds force or (five pounds) would pro-
(bq; act at b m tne airec ^^^^ ^^ ^^^^^. ,t jg therefore

tion BQ, parallel to A p. These oaHed their resultant. In this dia-

„ •n-Lx xi- gram a force of one pound is repre-

tWO lOrCeS will be together sented by a Une one-quarter of an

equal in their effect to one i^<'»^ ^^ i^^^*^-

force of five pounds (cr) acting at c in the direction CR.

This force R is the resultant of the other two forces.

COMPOSITION AND RESOLUTION OP FORCES.

and its magnitude is found by adding the force
p to the force Q. The following experiment will
help to make this clear : — In Fig. 30, AB is a bar
of wood supported by cords passing over pulleys at
c and D, and having scale-pans attached to the cords.
In the scale-pan p place a one-pound weight, and in
the pan w a three-pound weight, and hang a four-
pound weight on AB at /. The three weights will be
found to balance one another on being left free.
That is, the weights P and w, of one and three
pounds respectively, which tend to pull the bar up-

Fia. 80.

ward, are balanced by a single weight of four
pounds acting downward from /. Instead, there-
fore, of the two forces P and w, we might have a
single force of four pounds acting upward from /.
This last force would be the resultant of the other
two forces, and would be equal to their sum.

But the forces, though parallel, may act in oppo-
site directions. Thus in Fig. 30 the forces P and
w actually act upward on the rod, while / acts down-

COMPOSITION AND RESOLUTION OF FORCES.

ward ; and these three forces are in equilibrium. But
add one pound more to /and the rod will move down-
ward. In that case we should have two forces of
three pounds and one pound respectively acting up-
ward, and a single force of five pounds acting down-
ward. The resultant force would be found by sub-
tracting the sum of the two smaller forces from the
greater force, and it would act in the direction of
the greater. Thus the resultant of the three forces
in this last example would be a force of one pound
acting downward.

75. Forces acting at an Angle with one another. —
We have an example of forces acting at an angle with
one another when two boys at the same moment
strike a ball, A (Fig. 31), one
urging it in the direction AB
and the other trying to send
it in the direction AC. It is
easy to see that the ball will
then move along a line some-
where between the two lines
AB and AC — along the line
AD, for instance. But to

understand the exact direC- same moment by two equal forces,

one actmg in the direction a b, and
tlOn which the body would the other in the direction a c, the
. 1 -I II 1 •J. 'xi. ball will move in the direction A D.

take and the velocity with

which it would move requires a knowledge of the
parallelogram of forces and the parallelogram of
velocities. These we shall not explain at present,
but they will be considered in the third part (Third
Year's Course) of this book.

76. Resolntion of Forces. — We have learned that

D c

Fio. 81.
If the ball, a, is acted on at the

86 COMPOSITION AND RESOLUTION OP FORCES.

several forces may be combined ; and we are able
to find the single force, or resultant, which is equal
to several forces acting either (1) in the same
direction or (2) in opposite directions. But it is
also possible to resolve a single force into two or
more other forces, whose combined effect will be
equal to that of the one original force : this process is
called the resolution of forces. For example, we can
imagine a man pulling a truck with a force equal,
say, to one hundred pounds : if we now replace the
man by two boys of equal strength, who are able
together to pull the truck with a force exactly equal
to that of the man, we shall have replaced the one
force of one hundred pounds by two forces of fifty
pounds each. We might then say that we had re-
solved the one great force into two other smaller
forces.

XII.— THE SECOND LAW OF MOTION.

77. Efifect produced by a Force acting on a Body in Motion— 78. Forces acting
in the same Direction — 79. The Velocity of a falling Body is in accordance
with the Second Law of Motion— 80. Forces acting in opposite Directions.

77. Effect produced by a Force acting on a Body in
Motion. — ^We have already learned something about
Newton's first law of motion, and we must now con-
sider the second law which he discovered. In his
first law Newton describes what would happen if
no force acted upon a body. In the second law of
motion a force is supposed to have acted on a body,
and to have set it in motion ; and while this motion
continues, a second force is supposed to act on the body.
Now, what effect will this second force produce ?

We have the answer to this question in Newton's
second law of motion, which says that " When a
force acta upon a body in motion, the change of mo-
tion is the same in magnitude and direction as
if the force acted upon the body at rest"

78. Forces acting in the same Direction. — There
are three cases to be considered under this law:
First, that in which the forces cause motion in the
same direction ; secondly, that in which the forces
act in opposite directions ; and, thirdly, that in
which the forces act at an angle with one another.

88 THE SECOND LAW OP MOTION.

Suppose that a boy strikes a ball with his bat, so
as to make it travel with a velocity of ten feet per
second, and that, while it is going at this speed,
another boy strikes the ball in the same direction
just as hard as the first one did. Its velocity will
then be twenty feet per second. Here the second
force, acting on the body in motion, has evidently
produced its full effect ; or, in other words, has pro-
duced a velocity the same " in magnitude and direc-
tion as if the force had acted on the body at rest."

Another illustration may be found in the motion
of a boat on a running stream. Suppose that in
still water the rower can propel his boat with a
velocity of five miles per hour, and that he is about
to row in a stream which runs at the rate of four miles
per hour. When he rows in the direction in which
the current is flowing, the force of his muscles and
the current of the stream each produces its full eflfect,
and the boat travels at the rate of 5+4 = 9 miles
per hour. This latter velocity may be termed the
resultant velocity ; and the process of combining the
two velocities to produce this resultant tnay be called
the composition of velocities, just as in the preceding
chapter we had the composition of forces and the
production of a resultant force.

79. The Velocity of a falling Body is in accordance
with the Second Law of Motion. — Another illustration
of the second law of motion will be found in the
consideration of the velocity of falling bodies. Sup-
pose that the force of gravity acts for one second
upon a body. During this second the body will fall
a distance of sixteen feet, and will have acquired a

THE SECOND LAW OF MOTION. 89

velocity of thirty-two feet per second. If gravity
then suddenly ceased to act, the body would fall
thirty-two feet during each succeeding second. Grav-
ity, however, does not cease, but acts during the
next second just as it did during the first. During
the second second, therefore, the body will fall
through thirty-two feet in virtue of the velocity it
had at the end of the first second, and through an
additional sixteen feet in virtue of the continued
action of gravity ; making a total distance traversed
during the second second of forty-eight feet. At the
end of the second second it will have a velocity of
sixty-four feet per second, which would carry it
through sixty-four feet in the third second ; but the
force of gravity urges the body through sixteen feet
in this third second just as in the first, causing the
total distance traversed in the third second to be
64 + 16 = 80 feet. Hence we see that a force act-
ing on a body in motion will produce its full effect,
apart from any motion the body may already have; in
other words, it will produce exactly the same effect
as if it acted on the body at rest. The rule, then, for
finding the resultant velocity of a body urged onward
by more than one force in the same direction, will
be to add together the velocities that the forces
would produce if they were to act separately on the
body at rest.

80. Forces acting in opposite Directions. — The second
case is that in which, when a body is moving with
a certain force (which has produced a given veloc-
ity) in one direction, another force acts on it from
the opposite direction.

90 THE SECOND LAW OF MOTION.

A boatman rowing against the stream is a good
example of this. If he pulls at the rate of five miles
per hour, while the stream runs at the rate of four
miles per hour, we may suppose that his strength
produces its full effect in moving the boat five miles
up the stream, but during the hour spent in doing
this, the current carries him four miles in the oppo-
site direction; and hence at the end of the hour
he will be found only one mile above his starting-
place. The rule for finding the resultant of veloci-
ties acting in opposite directions is, subtract the
smaller velocity from the greater. The motion will
take place in the direction of the greater, with the
velocity of the difference.

Again : suppose two balls of clay of the same
mass to be rolling toward each other, one with a
velocity of five feet per second, and the other with a
velocity of twenty feet per second. When they meet,
the velocity of the first will neutralize, as it were,
five feet per second of the velocity of the other ball,
which will be left with a velocity of fifteen feet per
second. This will be divided equally between the
two balls, which will consequently roll away each
with a velocity of seven and a half feet per second,
in the direction in which the swifter ball was mov-
ing when they met.

In the case of bodies thrown upward we have
another instance of a force acting on a body in
motion, in a contrary direction to that in which the
body is moving. By muscular force we can cause
a ball to rise through the air ; but all the time that
it is rising, another force — the force of gravity —

THE SECOND LAW OP MOTION. 91

is pulling it down. The muscular force only acts
on the ball for a moment ; but the force of gravity
is a constant, never-ceasing force, and it continues
to act on the ball all the time it is rising. The
upward motion of the ball is consequently soon
stopped ; and as soon as this happens (for the force
of gravity is then opposed by no other force), the
ball is drawn to the Earth again.

In all our movements on the Earth we illustrate
the second law of motion, since, in addition to the
motion of our bodies caused by our own muscular
force, we partake of the motions of the Earth. Thus,
when we jump, we fall again on the same spot of
ground. People have not always remembered this.
A man once proposed to cross the Atlantic Ocean by
going up in a balloon and waiting in the air till
America came under him I He forgot that his bal-
loon, with the air and all things near the Earth,
would move on in exactly the same way as the
Earth itself.

XIII.— THE THIRD LAW OF MOTION.

81. Action and Beaction— 82. Newton's Third Law of Motion— 83. Illastrations

of the Third Law of Motion.

81. Action and Reaction. — If we take a strong
spring balance and place a weight on the hook,
gravity, acting on the iron w^eight, causes it to
stretch the spring and move downward. But when
the weight has moved a certain distance downward,
it comes to rest. The cause of this stoppage is, that
in addition to the action of gravity pulling down-
ward, we have the opposite action of the spring
pulling upward ; and when the weight comes to
rest, these two forces are in equilibrium. Instead
of using the words "opposite action," we may say
reaction, which means the same thing (Latin re, back
or opposite) ; and we can express what happens by
saying that when the weight is in equilibrium it is
because the action of gravity and the reaction of
the spring are equal and opposite.

82. Newton's Third Law of Motion. — Newton's third
law of motion states, that " To every action there is
an equal and contrary reaction ;" so that what we
found to be true in the case of the spring and the
weight, we shall find to hold good in all cases. Let

THE THIRD LAW OF MOTION. 93

US take another illustration. If a piece of iron be
suspended by a thread, and a magnet be brought near
it, the iron will be attracted toward the magnet.
If the magnet be now suspended, the iron will be
found also to attract it. The action of the magnet
on the iron and the reaction of the iron on the mag-
net are equal. If the iron be placed in a scale-
pan and balanced by a one-pound weight, we will
say, placed in the opposite scale, it will require, per-
haps, a second pound weight to balance it when the
magnet is brought underneath the scale-pan con-
taining the iron. If the magnet and the piece of
iron are then made to change places, exactly the same
extra weight will be found necessary to restore equi-
librium when the iron is brought below the scale-pan
containing the magnet as when the magnet was
brought below the balanced iron. Action and reaction,
therefore, are equal and opposite in this example also.

83. Illustrations of the Tliird Law of Motion.— (1.) If
a boy press with his two hands against a wall, the
wall will be found to press with equal force against
him ; and if he suddenly increase his pressure, the
reaction may become so great as to force him back-
ward away from the wall.

(2.) So, too, if a boy wishes to break a piece of
cord, he may pull at one end, and may get another
boy to pull equally hard in the opposite direction.
But he would have had the same power for break-
ing the cord if he had tied one end to the wall, and
himself had pulled as before at the opposite end ;
for the reaction of the wall would have done the
work of the second boy, and would have pulled in

94 THE THIRD LAW OF MOTION.

the opposite direction with a force equal to his
own.

(3.) When a ball is held in the hand, the force
of gravity exerted by the Earth draws it downward ;
but does the ball pull the Earth upward ? Yes ;
and when the ball is set free, the Earth is drawn
upward by the ball, just as the ball is drawn down-
ward by the Earth. But how is the motion to be
measured in each case ? It must be measured by
the momentum (or quantity of motion) of the two
bodies. When they meet, their momenta will be
equal. As the mass of the Earth is so much greater
than the mass of the ball, the velocity of the Earth
will be proportionally less than the velocity of
the ball. The ball, if it has been falling down-
ward for one second, will be moving at the rate of
thirty-two feet per second ; but the upward velocity
of the Earth will be so small as to be inappreciable.
From this we see that one body cannot attract an-
other without being itself attracted by that other
body; in other words, we cannot have action without
reaction ; the one always accompanies the other, and
they are always equal and opposite.

(4.) The attraction of the Earth and the Moon is
mutual ; the Earth attracts the Moon and the Moon
attracts the Earth ; but as the Earth is much the
larger and heavier body, the motion of the Moon is
affected far more than the motion of the Earth. By
reason of the Earth's greater mass, the Moon is
caused to circle round the Earth. And just as a boy
pulling at a man's coat draws the coat away from
the man by his reaction, though he may be compelled

THE THIRD LAW OP MOTION. 95

to move after the man, so the reacting force of the
Moon draws towards it the waters of the ocean —
the loose jacket of the Earth, as it were — and pro-
duces the phenomenon of the tides.

(5.) When a gun is fired, we have not only the
action of the powder in forcing the bullet out of the
muzzle, but we also have an equal and contrary re-
action in the recoil (or " kick," as it is sometimes
called) of the gun against the shoulder of the per-
son who is firing it. If we suppose the mass of the
gun to be one hundred times that of the bullet, then
the velocity with which the gun is forced back
against the shoulder will be only y^ of that im-
parted to the bullet.

(6.) Many more instances of the third law of mo-
tion might easily be found, but we shall only men-
tion two others. A sky-rocket when fired shoots
high into the air, from the reaction of the force with
which the exploding gas, rushing out at the lower
end of the rocket, pushes against the air.

(7.) When a man jumps out of a boat, the action
of his feet sends him on to the bank ; but there is
an equal reaction in the opposite direction, and the
boat is seen to move away from the shore.

XIV.— WORK, AND HOW TO MEASURE IT.

84k Definition of the Term "Work"— 85. Cases in which no Work is done—
88. Measurement of Work— 87. What is meant by "One Horse-Power"?—
88. Labour and Time.

84. Definition of the Term " Work." — The word work
is one which we use daily, and to which we assign
various meanings. By work a carpenter means
making doors and windows, and so on ; a bricklayer
calls laying bricks and making mortar, work ; while
by work a clerk would mean the writing of letters
and the making up of accounts. In short, any occu-
pation which causes mental or bodily fatigue is com-
monly called work. In Mechanics, however, the
word has only one meaning — namely, that ** work is
the production of motion against resistance"

Let us consider this definition. Suppose a four-
pound weight to be on the floor with a stool beside
it ; if the weight be raised from the floor and placed
upon the stool, we shall readily admit that work has
been done. For in virtue of its inertia the weight
will remain on the floor until some force acts upon it.
We may apply muscular force to the task of raising
the weight ; but we shall not succeed unless the mus-
cular force is sufiiciently strong to overcome the re-
sistance offered by gravity to the raising of the

WORK, AND HOW TO MEASURE IT. 97

weight. Since we cause the weight to move in
spite of that resistance, we do mechanical work.

85. Gases in wMcli no Work is done. — By keeping
the above definition closely in view, we shall find
that there are many cases in which, at first sight,
work appears to be done, while in reality none is
performed. Thus, when the weight has been placed
on the stool, it would appear as though the stool
were doing work in maintaining it in its place
against the attraction of the Earth. But since there
is no motion, since the weight retains its position,
moving neither upward nor downward, no work is
being done. In like manner, a man who stands
still with a weight on his shoulder is doing no
work.

Now let a string be fastened to the weight, and
let it be dragged along the top of the stool. We
feel a certain amount of resistance to the movement
of the iron weight, caused, not by the attraction of
gravity, for that is balanced by the reaction of the
stool, but by the friction of the rough iron upon the
rough wood. If the iron were polished, and if the
top of the stool were covered with glass, the friction
would be less, and in dragging the weight along we
should do less work, as there would be less resistance
to the motion of the weight. Now suppose that the
weight and the stool-top were perfectly smooth,
there would be no friction, and however much we
moved the weight about we could do no work, as
there would be no resistance to be overcome. In
pulling a cart along a level road, friction is the only
force which the horse has to overcome. When he

(765) 7

98 WORK, AND HOW TO MEASURE IT.

comes to a hill, however, then the horse has> in
addition, the force of gravity to contend with, and
the work done in pulling the cart must be greater.

86. Measurement of Work. — When we wish to
make a measurement of any kind, it is always neces-
sary first to fix on some standard or unit of what
we want to measure. Hence we must fix on a unit
of work. To do this it will be necessary to con-
sider both the motion of the body on which work
is done, and the resistance that has to be overcome
in order to move it. The first (the motion) can be
clearly defined by stating how many feet the body
has been moved; and the second (the resistance)
will be most easily expressed by comparing it with
the resistance to be overcome when a body is raised
from the Earth, which resistance we generally esti-
mate in pounds. This resistance is, of course, due to
the force of gravity; and since this force is constant
and always in action, it aflfords the best means of
measuring work. The unit or standard of work
generally adopted is the work that is done in raising
a weight of one pound through a vertical height of
one foot. This unit of work is called the foot-pound.
If a weight of two pounds be raised one foot, twice
as much work will be done as when one pound was
raised one foot. Again : if a weight of one pound
be raised two feet, it will take twice as much power
as is required to raise one pound through one foot.
How many units of work will be required to raise
five pounds to a height of three feet ? To raise five
pounds through one foot . requires a force of five
foot-pounds, and this must be repeated three times

WORK, AND HOW TO MEASURE IT. 99

before the weight arrives at the required height;
hence for the whole operation fifteen foot-pounds
of work will be necessary. The rule to find the
amount of work required to be done in any case
is therefore seen to be — Multiply the weight (in
poimds) hy the vertical distance through which it
is raised (in feet). Here is an illustration of this
rule : — How many units of work will be expended
in lifting two hundredweight to a height of fifty
feet ?

The number of pounds is 112x2 = 224, the
number of feet =50 ; therefore the number of foot-
pounds = 224x50 = 11,200.

87. What is meant by "One Horse -Power"? — In
estimating the amount of work done by machines,
the number of foot-pounds often becomes inconveni-
ently large. Hence a measure of work larger than
a foot-pound has been established, just as we find it
convenient to use the mile as a measure of length
in addition to the yard. This larger measure of
work is called " one horse-power." But since the
power of horses varies considerably, it is necessary
to state exactly how much work we understand by
a single horse-power. It was James Watt, the in-
ventor of the steam-engine, who introduced this
standard of work, and he defined one horse-power
to mean 33,000 foot-pounds of work done in one
minute of time. This is probably beyond the power
of most horses ; but it has passed into general use,
and is always understood when the "horse-power"
of an engine is mentioned. A machine, then, of
eight horse-power would be one capable of perform-

100 WORK, AND HOW TO MEASURE IT.

ing 8x33,000 = 264,000 foot-pounds of work in
one minute. The words "horse power" are often
represented by the letters " H. P.," so that an engine
of " six H. P." means one of six horse-power.

It will be seen that the idea of time is introduced
into our definition of horse-power, while it was
expressly left out when treating of foot-pounds.
This is important, for a child could do the amount
of work known as a horse-power if allowed time
enough. Thus a boy could easily lift a weight
of thirty-three pounds to the height of one foot,
thus doing thirty-three foot-pounds of work ; and
if this were done a thousand times, 33,000 foot-
pounds of work would be accomplished. But this
would probably take the boy a day or more ;
while in order to obtain one horse-power of work
the 33,000 foot-pounds of work must be performed
in one minute.

88. Labour and Time. — Many observations have
been made as to the amount of work that can be
done by men and animals, and as to the way in
which it is performed. Thus the greater part of the
labour of walking appears to consist in raising the
body a small distance at each step; and a great
part of the exertion in throwing up earth with a
spade is due to the fact that part of the digger's
body has to be raised each time a spadeful of earth
is thrown out. Thus the amount of useful work
done may be much less than we should be led to
expect if we considered only the fatigue of the per-
son who does it. It is found, also, that when a man
works so that he can do the greatest amount of

WORK, AND HOW TO MEASURE IT. 101

work in a day, keeping on day by day for a long
time, he must not work too hard nor too long. If
he works too hard, he will soon break down ; if he
goes more slowly and tries to make up for it by
working longer, he will not accomplish so much in
the long run as one who works at a fair medium
pace.

XV.— ENERGY.

89. Definition of Energy— 00. Measure of Energy— 91. Forms of Energy.

89. Definition of Energy. — In ordinaiy language, a
man is said to have great energy when he is capable
of overcoming great obstacles, or of getting through
a large amount of work. Thus a blacksmith who
shoes two horses while his neighbour shoes one is
said to have twice the energy of the other man. In
this respect, too, we may compare the energies of
men, horses, and machines respectively, measuring
the energy of each by the work accomplished.
Thus a man and a horse may be employed sepa-
rately to raise coal from a mine. The horse will
raise, perhaps, ten times as much as the man in
the same time, and will then be said to possess ten
times the energy of the man. Again, a steam-
engine may raise a ten times greater weight of coal
than the horse could in an equal period of time, and
will, therefore, have ten times as much energy as
the horse, or one hundred times as much ais the man.

From this it will be seen that by energy is meant
" the 'power of doing work!' Work has been already
defined to be the " production of motion against
resistance." The resistance may be of any kind:

ENERGY. 103

but in all cases where a body is moved against some
resistance work is done, and the power which over-
comes the resistance is called energy. Thus, if a
bullet from a gun pierces the leaves of a book, the
force which the moving bullet possesses will be
called energy. If this bullet can pierce three hun-
dred pages while another bullet can pierce only one
hundred, the former will be said to have three times
the energy of the latter body.

90. Measure of Energy. — We have already learned
that to measure the magnitude of any force, we
must consider how many units of work it is capable
of performing. Thus a force that could raise nine
pounds to the height of eight feet might be spoken
of as doing 9x8 = 72 foot-pounds of work. Energy,
being the power of doing work, will be estimated in
the same way. Thus if two machines are working
side by side, and one does twenty foot-pounds of
work while the other does ten, the former will have
double the energy of the latter.

When we know (1) the velocity of a moving body
and (2) its weight, we can easily find (3) how many
foot-pounds of work the body is capable of perform-
ing; and this is the true measure of its energy.
Thus, if two forces act on two bodies of the same
mass (say one pound each), causing them to move,
the one with a velocity of thirty-two feet per
second, and the other with a velocity of sixty-four
feet per second, we might at first be inclined to
think that the energy of the latter body was only
double that of the former; but in fact it would
be much more. For if the two bodies were thrown

104 ENERGT.

upwards with the velocities of thirty-two feet and
sixty-four feet per second respectively, they would
rise — the former sixteen feet, but the latter sixty-
four feet. In other words, the former would do
sixteen foot-pounds of work, and the latter sixty-
four foot-pounds (if each body weighed one pound).
If, then, in two bodies of equal mass one has twice
the velocity of the other, it will have four times the
energy. The energy, in fact, increases according to
the square of the velocity. The energy of a iliov-
ing body can only be measured by multiplying its
weight by the height through which it would have
to fall in order to acquire the velocity which it
actually has.

91. Forms of Energy. — The various forces of nature
may be considered as so many forms of energy, or
sources of power. By considering them under this
common name of energy, we shall be able to see
more clearly how closely related to one another
these forces are.

We will now examine each force as a form of
energy : —

(1.) Gravitation is one of the most apparent of
the forms of energy. All falling bodies owe their
energy or power of overcoming resistance to gravita-
tion. This energy is employed in many ways:
mills are driven by the energy of falling water;
clocks by that of falling weights ; and so on.

(2.) Cohesion is the attraction of the molecules of
a body for one another. When we try to bend, or
twist, or lengthen a rod of iron, the resistance we
experience is due to this form of energy.

ENERGY. 105

(3.) Chemical attraction is a form of energy
of vast importance. When coal is burned, the
carbon of which it is composed joins with the
oxygen from the air, and forms a new substance
called carbonic acid gas. The force which causes
this to happen, and which afterwards holds the
molecules of the carbon and the oxygen fast bound
together, is known as chemical attraction.

(4.) Heat is a form of energy that is used in
almost all processes of manufacture. Steam-engines
owe all their energy to the heat produced by the
burning of coal in their furnaces.

(5.) Magnetism is the form of energy which
causes the needle of the mariner's compass always
to point to the north, and thus to guide him across
the sea ; and by which all magnets are able to attract
pieces of iron. Our Earth possesses some of this
kind of energy, and it is the action of the Earth's
magnetism on the needle that causes the latter to
point north and south.

(6.) Electricity is energy of a very similar
nature to magnetism. We are indebted to this
form of energy for the electric telegraph and for the
dazzling electric light ; and we may, perhaps, some
day use electrical energy to drive our machinery, in
the same way as we use steam-engines now.

(7.) Light. This is caused by an exceedingly
rapid motion of the molecules of luminous bodies,
transmitted by an extremely thin fluid called ether,
which pervades all space. Light is undoubtedly a
form of energy. It is the energy by the aid of
which plants live and grow, and thus prepare food

106 ENERGY.

for men and animals. In photography it is the
energy of the rays of light which produces the pic-
tures.

(8.) Muscular energy, or the power possessed
by the muscles of living animals, is that form of
energy which enables them to move and do work.
As it is only possessed by living things, it is fre-
quently called " vital energy," or vital force.

(9.) Mechanical energy is a convenient name
for the energy that a moving body possesses. Thus
the energy of an arrow flying through the air is
called mechanical energy.

XYL— POTENTIAL ENERGY.

92. Stonga of Energj— B3. Eiamplea ol
Soorce of Koergj—W. Polenti

92. StorftKQ of EnergT.- — To show how energy may
he stored up or accuimilated in a body, let us take
a tripod stand (Fig. 32) supporting a pulley at a
height of about nine feet from the ground, and
pass over the ptilley a rope
lifteen feet long, bearing at
one end a weight (A) of four-
teen pounds, and at the other
a weight (b) of twenty-eight
pounds. When the whole is
left free, the heavy weight
will, of course, be upon the
ground, and the lighter one
will hang about three feet
from the ground. Now, let
the lighter weight, A, be 3
raised by means of the rope
to the height of the pulley, PoJt'^Sag,.

c. To do this we must exett

a certain amount of muscular force ; enough, in
fact, to raise fourteen pounds through a height

108 POTENTIAL ENERGY.

of six feet — that is 14x6=84 foot-pounds.
This energy is now stored up in the weight a,
and by setting that weight free, we can get the
energy back again, and use it to perform work.
Since this stored-up energy is capable of doing
work, it is called potential energy (from the Latin
patens, powerful). Now, set a free, and it will
at once commence to fall ; but when it has fallen
six feet, and is still three feet from the ground,
the rope is pulled tight, and it can now fall further
only by raising the weight B. This the small
weight A is able to do by using the store of energy
accumulated in it. We know that eighty-four foot-
pounds of energy were stored up in A ; and could
this all be used in raising B, the latter would rise
three feet; for, in rising this distance, 3 x 28 = 84
foot-pounds of work (that is, a's whole store) are
performed. As a matter of fact, B would not rise
quite so high as three feet, for part of A's energy is
wasted in overcoming the friction of the pulley and
in bending the rope.

93. Examples of Potential Energy. — Energy thus
stored up in a body may be used in various ways.

(1.) It is this accumulated or potential energy
which enables us to make such good use of the
hammer. The hammer merely laid on the head of
a nail would have little or no effect on it. We
first raise the hammer to a height of one or two
feet, thus storing up potential energy in it; and
then by suddenly bringing it down we expend this
energy (and some of our muscular force also) in
driving the nail into the wood. The harder the

POTENTIAL ENERGY. 109

blow we desire to strike, the heavier do we make
the hammer-head and the higher do we raise it, in
order to store up in it a greater amount of potential
energy.

(2.) The pile-engine is a hammer on a large scale.
Piles are long pieces of timber, sharpened at one
end and driven firmly into the ground, in order to
bear great weights. The pile-engine consists of a
tripod frame bearing a pulley, over which a rope
passes to a heavy iron block called the "monkey,"
which represents the hammer-head. (See Fig. 21.)
This is raised as high as the pulley will allow, and
thus gains potential energy. This energy the monkey
gives out when released, by falling on the top of the
pile and driving it into the ground. Suppose
the monkey to weigh three hundredweight (336
pounds), and to be raised 1 5 feet ; it will accumu-
late 336 X 15=5040 foot-pounds of energy, which
represents the force of the blow given to the pile to
drive it further into the ground.

(3.) When a stone is thrown into the air, it rises
for a certain length of time, storing up energy as it
goes, until, on reaching its highest point, it is for a
moment at rest. If it were caught at that moment
and suspended in the air, it might remain at rest for
any length of time ; but the energy which has been
stored up in the stone would not be lost ; it would
be retained as potential energy, which would at once
become active if the stone were again set free. Had
the stone remained lying on the ground, it would have
had none of this potential energy, for it would have
had no advantage of position over the other things

110 POTENTIAL ENERGY.

around it. Its potential energy depends on its poai-
Hon, or height above the ground, and potential
energy is therefore frequently spoken of as " energy
of position." A great stone perched high on a hiU-
side has, in virtue of its position, a store of energy.
This stone may be held in its place by a small stone
in front of it, and, thus supported, may remain at
rest for ages, until some chance dislodges the small
stone. Then the large stone goes thundering down
the hill, giving up in its descent the energy of
position that has been stored up in it for so long a
time.

(4.) A brick on a house-top has energy of posi-
tion,, which was accumulated in it when the labourer
carried it up, and which it will retain undimin-
ished until the house is pulled down, and the energy
is employed to bring it back to the ground again.

In all these cases the advantage the body has is
one of position with respect to the Earth and to the
force of gravity. Work is expended in raising the
body from the Earth ; and this work is stored up,
as it were, in giving the body such a position that
gravity, by causing it to fall, can restore the exact
amount of energy expended in raising it.

94. The Sun as a Source of Energy. — The Sun is con-
stantly at work laying up a store of energy for us ;
and we as constantly take advantage of his labour,
by using this potential energy to perform various
kinds of work.

Consider the Sun shining on the water of the
ocean, and changing some of it into vapour. This
vapour rises into the air and forms clouds, and the

POTENTIAL ENEROT. Ill

clouds are drifted by the wind over the land. Here
they are condensed, and the water falls as rain on
the hilltops, and, running over the surface of the
groimd, collects in little streamlets. The streamlets
unite to form streams, and these go rushing down
the hillsides, the potential energy bestowed on the
water by the Sun being gradually lost as the river ■
flows downward. -Wheels are frequently placed in
the course of the streams, and the energy of the
streams is used to turn them. Fig. 33 represents
one of these water - wheels
(called an overshot- wheel, be-
cause the water flows over the
top, to distinguish it from those
in which the water flows un-
derneath, and which are called '
undershot). The water is car-
ried along a trough to the top
of the wheel, and flows into the buckets arranged
round its circumference. One side of the wheel is
thus made heavier than the other side, and the
water, continually falling, causes the wheel to revolve.
An axle from the wheel leads to a mill containing
machinery whereby the energy once stored up in the
water is used, perhaps, to grind our com. In the
famous Falls of Niagara there is an enormous and
unfailing supply of energy, and it is now proposed
to use it for driving machinery. When our coal
runs short, it is probable that many other waterfalls
will also be used as sources of energy.

95. Potential Energy of elastic Bodiea — ^Use is some-
times made o£ the property of elasticity in order to

112 POTENTIAL ENERGY.

store up potential energy. A bow and arrow is an
instance of this. When we wish to shoot with the
bow, we first bend it, thereby using our muscular
force to overcome the force of cohesion, which endeav-
ours to hold the molecules of the bow in their places,
and which also strives to make them return to their
old positions when they are moved out of them. In
these molecules, now removed from their places, and
tugging at one another in their attempt to regain
their old positions, we have a store of potential energy,
which is set free all at once by liberating the bow-
string, and is used to propel the arrow through the
air. In a watch-spring we have another instance of
energy stored by the aid of elasticity — the watch-
spring slowly giving out in a day the potential
energy that was accumulated in it during the few
seconds occupied in winding up the watch.

XY 11. —KINETIC ENERGY.

96. Energy of Matter In Motion— 97. Examples of Kinetic Energy— 98. Kinetic
I Enei^ varies as the Mass of a Body— 99. Kinetic Energy varies as the

Square of the Velocity— 100. Kinetic Energy and Momentum.

96. Energy of Matter in Motion. — We have ex-
plained that the word energy means the power of
doing work, and that potential energy is the name
for power stored up in a body and ready to be used.
By kinetic energy we mean energy that is actually
being used. The word " kinetic" comes from the Greek
word kineOy I move ; and thus kinetic energy means
" the energy of a body that is in motion."

We know that a cannon-ball moving rapidly
through the air possesses a great amount of energy,
or power of doing work ; and we see this energy
expended in overcoming, first, the resistance of the
air ; and, secondly, the far greater resistance of the
target. While this energy was stored up in the
gunpowder, it was potential energy ; now that it is
being put forth by the moving ball, we call it kinetic
energy. A hammer, poised high in the air by the
hand of a workman, has a store of energy in virtue
of its position above the nail he intends to strike ;
but unless he makes it fall through the air on the
head of the nail, no work will be done. This energy

(766) 8

114 KINETIC ENERGY.

which the hammer has while descending, and while
actually driving in the nail, is an example of what
we mean by kinetic energy. Whenever we see work
being done, we may be sure that it is kinetic energy
that is engaged in doing it, whatever may have been
the source from which the kinetic energy was
derived.

97. Examples of Kinetic Energy. — (1.) A cricket-ball,
as it lies on the ground by the bowler's foot, has
neither potential nor kinetic energy ; but let a man
take it up and throw it at the wickets, then it has
kinetic energy, and we see the effect of the work
done by the energy of the ball in the falling and
perhaps broken stumps. Similarly, when the bats-
man strikes the ball, he must overcome the kinetic
energy imparted to it by the bowler, and give it
sufficient energy to carry it in another direction
across the field, before he can score a run.

(2.) A stream of water running down-hill has
kinetic energy, which carries it onward in its course.
If we want to change its course, we must overcome
the kinetic energy it has in the old direction, and
force it into a new one. We may make use, too, of
the kinetic energy of the running water, and cause it
to turn mill-stones or other machinery, by placing
in its course a water-wheel, which will transfer, as
it were, a part of the kinetic energy of the run-
ning water to the machinery to be moved. Now,
build a dam across the stream above the mill. The
water no longer flows, and the mill does no work.
What has become of the energy of the mill-stream ?
It is now being stored up as potential energy in the

KINETIC ENERGY. 115

water behind the dam. Soon, however, the water
will rise above the dam, and the potential energy
that has been accumulating will be converted into
kinetic energy, as we shall see in the downward
rush of the water and the sudden turning of the
mill-wheel.

98. Kinetic Energy varies as the Mass of a Body. —
If a stone weighing one pound and another weigh-
ing two pounds fall from the same height, they will
each have the same velocity on reaching the ground ;
but the latter will have twice the energy of the
former, because it is twice as heavy. A hammer
weighing ten pounds will strike twice as hard a
blow as one weighing five pounds, when both are
moved with the same velocity. The energy of a
moving body, then, depends partly on the mass of
the body, or on the quantity of matter it contains.
Other things being equal, the body that has the
greater mass has the greater energy. It i^ because
the mass of an ironclad ship-of-war is so enormous,
that it is able to cut another vessel completely in
two, although it may not be moving faster than a
boy can run. Its energy depends largely on its
mass.

99. Kinetic Energy varies as the Square of the Ve-
locity. — But when the mass remains unchanged, the
energy is found to vary with the velocity. Thus
a bullet shot out of a gun with a velocity of one
hundred feet per second may perhaps pierce a plank
three inches thick. If it has a velocity of two
hundred feet per second it will pierce, not two, but
four such planks ; and with a velocity of three

116 KINETIC ENERGY.

hundred feet per second it will pierce nine planks.
Thus, by doubling the velocity we have increased
the energy fourfold, and by trebling the velocity we
make the energy nine times greater. In other
words, the energy increases as the square of the
velocity.

If a cannon-ball be shot out of a cannon with a
velocity of one thousand feet per second, it will
travel a certain distance ; if, however, its velocity
be doubled, it will travel four times the distance (not
twice) ; and if the velocity be quadrupled, it will
travel, not four times as far, but four times squared ;
that is, sixteen times the distance. If the ball be
sent upward from the Earth, instead of parallel to
it, the same result will be obtained. Thus a ball
thrown upward with a velocity of thirty-two feet
per second rises sixteen feet. Now double the
velocity at starting, making it sixty-four feet per
second, and the height reached is quadrupled ; that
is, the ball rises sixty-four feet. Next, treble the
starting velocity, making it sixty-four feet per
second, and the height reached will be nine times
as great ; that is, one hundred and forty-four feet.

100. Kinetic Energy and Momentum. — When a force
acts on a body, its effect may be measured either by
the quantity of motion, or momentum, imparted to
the body ; or by the power of doing work, or energy,
given to the body.

For example: let a ball weighing two pounds
fall for one second under the action of gravity
alone ; at the end of that time its velocity will be 32
feet per second. Its momentum will therefore be

KINETIC ENERGY. 117

32x2 = 64 units. As it has fallen only 16 feet,
however, its energy will be 16x2«32 foot-pounds.
Now, let the mass of the ball be doubled ; that is,
let it now weigh four jxtunds. If it fall for the
same time, both its momentun^ and its energy will
be doubled. The former will be 32x4 = 128 units,
and the latter 16 x 4 = 64 foot-pounds. But now,
while the mass still remains at two polmds, let
the time of falling be increased to two seconds.
The velocity will be doubled ; that is, it will be 64
feet per second : but the total distance fallen through
will be four times as great; that is, 64 feet. The
momentum in this case will be 64x2 = 128 units,
and the energy 64x2 = 128 foot-pounds. At the
end of the first second the number representing
the momentum was double that representing the
energy, but at the end of the second second they
are equal. In other words, by doubling the veloc-
ity we have doubled the momentum or quantity of
motion, but we have increased the energy or power
of doing work fourfold.

The great point to be borne in mind in consider-
ing momentum and energy is, that the units em-
ployed in measuring each must be clearly understood
and kept quite separate. The unit of Tnomentvmi
is the quantity of motion in a mass of one pound
moving with a velocity of one foot per second ; the
unit of BTiergy is the power required to raise a mass
of one pound to the height of one foot. The
momentum and the energy must be calculated sepa-
rately, and we must not confound the energy of a
moving body witlj the morrientumy or vice versa.

XVIIL— INDESTRUCTIBILITY OF ENERGY.

101. Energy may be transferred but cankiot be destroyed— 102. The Energy of
the Pendulum— 103. Loss of Energy by Friction— 104. Energy in a Ball
thrown upward- 105. What becomes of the Energy of falling Bodies?-
106. Energy can be changed from one Form into another— 107. Heat
changed into Chemical Force— 108. The Energy of the Sun— 109. The
Potential Energy of Coal.

101. Energy may be transferred but cannot be de-
stroyed. — It has long been known that matter
cannot be destroyed ; it may be changed or altered,
but it can never be got rid of altogether. It is,
however, only during the last thirty years or so
that this idea has been extended to energy. But
we now know that energy cannot be destroyed ;
and bearing this in mind, many facts can be ex-
plained which previously had been matters of wonder
even to scientific men.

We have learned that energy is of two kinds : —
(1.) Potential energy, or the energy a body has
in virtue of its position ; and (2.) Kinetic energy,
or the energy of matter in motion. The actual
energy a body has may be either kinetic or poten-
tial, or it may be partly one and partly the other.
Now it has been discovered that energy can be
changed from the one kind into the other in the
same body, but that it cann/)t be destroyed. Energy

INDESTBUCTIBILITY OF ENERGY. 119

may even be passed on from one body to another,
but it will always remain energy of some kind or
other.

102. The Energy of the Pendulum. — Perhaps the
pendulum affords as good an example of this fact
as we can get. When it hangs vertically at rest
(PM, Fig. 34) it has no kinetic energy; and if we
suppose the cord to
be too strong ever to
break, it has no poten-
tial energy either,
since the bob, M, can
fall no lower.

In order to start -
the pendulum we pull
it to one side, but in
so doing we raise it d m c

vertically through the ^ ,, ^^?*^* , ♦!, i, *v

•^ ^ ^ Pendalum arranged to swing through the

distance c6 above its arc an 6. when swinging (or vibrating), its

f* , .. 1 . potential energy is greatest when the bob is

IirSt resting-place at M. at a or at 6 ; as the bob passes m, its energy is

To raise the pendulum J^^^^'^'^iy-
in. this way, we must use a certain amount of
force ; this force is now stored up in the weight
at 6 as so much potential energy. Now release
the pendulum : its potential energy is gradually
changed into kinetic energy, which carries it
downward to the point M. When at this point it
has no longer any potential energy, but kinetic
energy only; still the velocity which it now pos-
sesses will exactly represent the potential energy
lost, and will carry the pendulum onward to the
point a, as high on that side as it was on the other

120 INDESTRUCTIBILITY OF ENERGY.

side at 6. When the bob is at a, it is for a moment
still, and has no kinetic energy, but potential only ;
just as much potential energy, in fact, as it had
when at b. And now the bob will fall again, grad-
ually exchanging its potential energy for kinetic, till,
when it reaches M, its energy is again all kinetic ;
then rising towards b it will lose kinetic energy
and gain potential as it mounts higher and higher,
until the change is complete and it is momentarily
at rest again at 6.

103. Loss of Energy by Friction. — In every swing
of the pendulum we thus have a change of energy
from potential to kinetic, and baxjk again from
kinetic to potential. If this change is complete
and no energy is destroyed, the pendulum should
swing on for ever. But it presently stops. Why
is this ? The reason lies in the fact that in every
vibration of the pendulum a certain quantity of air
must be pushed aside. Force is required for this,
and also to overcome the friction of the rod of the
pendulum on the support from which it hangs.
Could this friction be done away with, the pendu-
lum would go on vibrating for ever. But what be-
comes of the energy used in overcoming the friction?
Is it completely lost ? No ; it is changed into heat
Could we collect the heat generated in this way, we
should have exactly the same amount of energy as
the pendulum has lost, but in another form. The
energy has only been changed from one form into
another.

104. Energy in a Ball thrown upward. — When a ball
is thrown into the air a certain amount of muscular

INDESTRUCTIBILITY OF ENERGY. 121

force is exerted. This force is expended in causing
the ball to rise in opposition to the force of gravity ;
and the amount of muscular energy lost by the per-
son throwing is exactly equal to the energy gained
by the ball. In risiDg, however, the kinetic energy
of the ball is gradually changed into potential energy,
and at a certain point this change is complete. At
this point the ball is for a moment perfectly still.
Now the reverse change begins to take place. The
potential energy is gradually changed into kinetic,
and the ball again begins to move; but this time
in the opposite direction — namely, downward. At
the moment when the ball touches the ground the
change is again complete: the kinetic energy at
that moment is at its greatest, but on striking the
ground it is completely lost. What has become
of it?

105. Wliat becomes of the Energy of flailing Bodies?
— When the ball strikes the ground it comes to
rest, having apparently no energy, kinetic or
potential. What has become of its kinetic energy ?
It is dianged into heat ; and the heat developed is
exactly equal to the kinetic energy lost by the
ball. Could the heat so produced be collected and
used in the right way, it would raise the ball again
to the height from which it fell. As a matter of
fact the heat is carried away by the air and by the
ground, so that it cannot usually be recognized. But
when a stone is thrown violently upon a hard road,
the heat produced is often rendered visible in the
sparks which fly around. When a blacksmith
hammers a piece of iron, the iron speedily becomes

122 INDESTBUCTIBILITT OF ENERGY.

hot ; the heat is in reality only the muscular force
of the blacksmith changed into another form— first
into the visible energy of the falling hammer, and
then into that motion of the molecules of the nail
which we call heat.

106. Energy can be changed from one Form into
another. — In all cases where energy seems to be
destroyed, it will be found to be merely changed in
form. We may not at first recognize the energy
that has been produced, but careful attention will
generally make this clear.

Thus, when a piece of sealing-wax is rubbed with
cat-skin, the muscular force of the person rubbing is
changed partly into the force of heat and partly into
the force of electricity. We recognize the heat, for
both the cat-skin and the sealing-wax feel warm.
The electricity, however, may pass unnoticed unless
we make some special arrangement in order to render
its effects visible. A very simple means of doing
this is to place some small pieces of paper on a
table and to bring the rubbed wax near them.
(Fig. 35.) They will be seen to jump up from

the table to the wax, and

'"•" 'l'o'"***S^^5^^>v sometimes to fly away

o°oVr3'o%i!^§i^^/^ again. The electric energy

. . S*^,A!!.J^L 2hL.^^ which causes the pieces of

Fig. 35.— Attraction of light bodies paper to do this is derived

by electrmed Sealing-wax ^^^^ ^^^ mUSCular Cneigy

used in rubbing the sealing-wax.

When we place certain metals, as strips of zinc
and copper, into a mixture of sulphuric acid and
water, the chemical force begins to act. If we now

INDraTRUCTIBILlTY OF ENEROV. 123

connect these strips in the manner shown in Fig. 36,
we shall obtain a current of electricity which may
be able to produce a dazzling light. Here we have
the chemical force changing first into the electrical
force, and then into the forces of light and heat.

Fia. 36.— Tha Electric Ligbt u pioduced bf CbemicBl Action.

107. Heat changed into chemical Force. — When a
match is rubbed on a rough surface the kinetic
energy of the moving match is changed into heat.
We choose a rough surface on which to rub the
match, for experience has taught us that a rough
surface will change the kinetic energy into heat
more rapidly than a smooth one, since the greater
the friction the greater is the amount of heat pro-
duced. The heat in its turn produces chemical

124 INDESTRUCTIBILITY OP ENERGY.

energy, which is the form of energy that causes
the phosphorus on the end of the match to join
with the oxygen in the air, thus making the match
burn. The chemical force continues to act as long
as there is any of the match left to be burned ; and
during all that time is itself constantly being
changed, partly into heat, which can be felt by
placing the hand near the flame, and partly into
light, which illumines the room. The heat pro-
duced from the chemical energy can be used to
generate more chemical energy ; as when we bring
the match close to the wick of a candle, which
almost immediately bursts into flame, showing us in
this way that chemical energy is again at work as
it was when the match was burning.

Many more illustrations of the change of one kind
of energy into another might be given. In fact, we
can change any one force into any other force pro-
vided we set about doing it in the proper way. The
examples that have been given will be suiBcient to
point out the great lesson, that energy rfiay he
altered from one form to another, but can never he
destroyed. This principle is known as the con-
servation of energy.

108. The Energy of the Sun. — The Sun is constantly
supplying us with stores of force. By the aid of
its light and heat plants are able to take into their
leaves carbonic acid gas from the atmosphere, and
to split up this gaseous substance into the carbon
and oxygen of which it is composed. The carbon
is retained by the plant to form its wood, but the
oxygen is returned to the atmosphere. A plant

INDESTRUCTIBILITY OF ENERGY. 125

cannot grow in a very cold place or in the dark ;
it needs the two forces of light and heat to enable
it to grow. In decomposing the carbonic acid gas,
the plant is making use of the Sun's energy (in
the forms of light and heat) to accumulate a store of
potential energy ; for the separated atoms of the
carbon and oxygen are ready at any time to join
together again.

109. The Potential Energy of CoaL — In ages long
ago countless numbers of plants lived and died,
and their wood has been buried deep within the
Earth for thousands and thousands of years, be-
coming at last changed into coal, which consists
mainly of the carbon of the plant. At last the coal
is dug up and used, it may be, to feed the furnace-
fire of an engine. The coal burns; but what do we
mean by burning ? We mean that the carbon of the
coal, joining with oxygen in the air, forms carbonic
acid gas and produces light and heat. We are, in
fact, bringing together again the very atoms of
carbon and of oxygen which the plant, aided by the
sunlight, separated so many ages ago ; and in doing
so we are changing the potential energy then stored
up, into kinetic energy, which we may recognize in
the moving steam-engine and in the work it accom-
plishes. It is said that the famous engineer George
Stephenson once asked a companion what it was
that really moved a railway train. " The engine" —
" the coal" — his friend guessed. " No," said Stephen-
son ; " it is * bottled sunlight ' ! " And in the main
he was right. It was the Sun's energy in the form
of light and heat which had enabled the plants

126 INDESTRUCTIBILITY OF ENERGY.

whose remains form coal, to obtain carbon from the
air in past ages and to form it into wood. We use
this carbon to produce heat once more in the engine-
furnace, and the heat is then converted into the
kinetic energy of the moving train.

XIX.— THE NATURE OF HEAT.

110. What is a " Theory " ?— 111. The Material Theory of Heat-112. The Mechanical
Theory of Heat— 113. Expansion of Bodies by Heat— 114. Change of State
effected by Heat— 115. Transmission of Heat by Conduction— 116. Trans-
mission of Heat by Convection— 117. Transmission of Heat by Badiation.

110. What is a "Theory "7 — We must now consider
more closely the nature of the force known as heat
The effects produced by heat come under our notice
every dfty. Indeed they are so common that they
excite no surprise, although most people would be
puzzled if they were asked to eocplain some of
the simplest phenomena of . heat. Why, for in-
stance, do some substances feel hot and others feel
cold ? Why do two bodies become heated by being
rubbed one against the other ? This we shall now
endeavour to make clear. But before we do so it
will be well to explain a word that we shall have
frequent occasion to use — the word "theory." A
theory is an explanation of a natural occurrence
which we find convenient to use when we cannot be
absolutely certain what the cause of the occurrence
is. Thus, we may never be able to see with our eyes
what the cause of heat is, but we may have an idea
about it, and the explanation that our idea enables
us to give of the nature of heat is called a theory.

128 THE NATURE OF HEAT.

111. The Material Theory of Heat — Two theories
have been put forward as to the nature of heat.
The one is known as the material theory, and the
other as the mechanical theory of heat. The mate-
rial theory is the older of the two, and it supposed
"heat" to be a very thin fluid or gas, much lighter even
than hydrogen, which is the lightest substance we
know on the Earth. It was supposed that when a
body contained a large quantity of this heat fluid —
caloric, as it was called — the body felt hot, and that
when it contained but little it was cold. Heating
a body was supposed to be simply putting more
caloric into it; and the cooling of any body was
thought to be the result of taking caloric away
from it.

The chief objections to this theory are, firstly,
that if heat is a substance it ought to weigh some"
thing, for all matter has weight. It is certain, how-
ever, that a body weighs no more when hot than
when cold. Secondly, no body can contain more
than a certain definite quantity of any substance ;
but it is possible to produce an unlimited quantity
of heat out of any two bodies by simply rubbing
them together. It is pretty certain, therefore, that
the material theory of heat cannot be true, and that
there is no such substance as caloric.

112. The Mechanical Theory of Heat. — The more
modern theory of heat is called the mechanical
theory, for it does not consider heat to be a sub-
stance, but a kind of rnotion. We have learned
that bodies are composed of very small pieces called
molecules. Now heat is thought to be a motion of

THE NATURE Of HEAT.

129

the molecules of bodies. When the motion of the
molecules is rapid, the heat is great ; and when a
body becomes cold, it is because the molecules move
more slowly. We do not know exactly in what
way the molecules move, for they are so small that
we cannot see them, but it is convenient to think
of them as swinging to and fro like little pendulums
rapidly vibrating.

This mechanical theory of heat is much preferable
to the material theory. It explains all the facts
connected with heat in a clear and satisfactory way ;
and when a theory can do that, it is generally
accepted as being the true one.

113. Expansion of Bodies by Heat. — By the aid of
this theory we can clearly understand why bodies
expand when they are
heated. Let us take a
brass ball (Fig. 37) which
just passes through a
brass ring when both ring
and ball are cold. Now
heat the brass ball, and
again try to pass it
through the ring. It
will not go through, for
it has grown larger by
being heated. When the
ball was cold its mol-
ecules were moving to and
fro with a certain velocity, but when the ball was
heated the molecules moved more rapidly and through
longer distances. They consequently wanted more

(765) 9

Pig. 87.
Gravesande's Ball and Ring.

130 THE NATURE OF HEAT.

room, and jostled against one another, pushing one
another further apart, and thus causing the ball, as
a whole, to take up more room, or, in other words,
to expand.

When the ball is cooled, the opposite change
takes place. The molecules vibrate less widely and
less rapidly, and thus lie closer together, causing the
ball, as a whole, to contract.

114. Change of State effected by Heat. — The mol-
ecules of a body are held together by the force of
cohesion, while heat, as we have seen, causes them to
move further apart. The more the molecules are
separated one from another the weaker their cohesion
becomes, for the force of cohesion can act only over very
small distances. Thus the greater the heat the less is
the cohesion. If we cdntinue the heating above a
certain point, the force of cohesion will be so far
overcome that it cannot keep the molecules in their
fixed places, but allows them to roll freely over one
another. The body is then changed from the solid
into the liquid state.

If the heat be still further increased, the mol-
ecules will be driven further and further asunder,
and will move about so violently that at last cohe-
sion will be unable to restrain them any longer, and
they will fly off from the surface of the liquid into
the air. The heat has now changed the liquid into
the form of a gas. By cooling a gas — that is, by
reducing the motion of its molecules — we can
bring it into the liquid state. Then, by still further
cooling the liquid, the vibration of the molecules is
still more checked, the force of cohesion obtains more

THE NATURE OP HEAT.

131

hold on them, and the liquid again becomes a
solid.

115. Transmission of Heat by Conduction. — When an
iron poker is thrust into the fire, the point in the
fire soon becomes heated. Before long, however, the
knob at the end farthest from the fire also grows hot.
The heat must therefore have been conveyed in some
way along the poker from the point in the fire to the
knob outside. On the principle that heat is due to
the vibration of the molecules of a body, we can
readily explain the passage of heat along the poker.
The heat of the fire
throws the molecules of
the iron point intoa state
of vibration. These
molecules pass on the
motion to the next
molecules, and so the
vibrations travel along
the poker until the
molecules of the knob
are last of all set in
motion. This i& called
the passage of heat by
conduction.

116. Transmission of
Heat by Convection. —
Liquids and gases are,
as a rule, heated in a
very different way to that just described. When a
kettleful of cold water is put on the fire, the water
next the bottom of the kettle is the first to become

FlQ. 88.
Transmission of Heat by Convection.

132 THE NATURE OF HEAT.

hot. The heat drives the molecules of this bottom
layer of water further apart, so that it becomes less
dense, and consequently lighter. The warm water
therefore rises to the top, and a colder layer takes
its place, to be heated and to rise in its turn. This
is called the passage of heat by convection. Where
hot-water pipes are used, the air of a room is also
heated by convection. The molecules of air touch-
ing the pipes become heated by their contact with
the hot surface of the pipes. They rise, and fresh
molecules take their place. This goes on until every
molecule of air in the room has in turn been in
contact with, and been heated by, the hot pipes.

117. Transmission of Heat by RadiatioxL — But how
is heat transmitted from the Sun to the Earth ?
The air only extends about two hundred miles
above the Earth's surface, and what is there between
the Earth and the Sun — which is ninety -three
millions of miles away — to carry the heat from the
latter to the former ? We believe that all the space
between the- Earth and the Sun and stars is filled
with a substance called ether. We cannot prove
the existence of this ether, but we are obliged to
conceive of such a substance as existing. The Sun
is immensely hot, much hotter than molten iron.
The vibrations of the molecules of the hot matter
of which the Sun is composed produce waves in the
ether surrounding the Sun, and these waves of heat
spread out in every direction, just as waves do when
we throw a stone into the middle of a pond. These
heat-waves travel all the w^ay from the Sun to the
Earth ; and when they strike on the Earth, the

THE NATURE OF HEAT. 133

motion of. the ether is changed into motion of the
molecules of the matter on which the heat-waves
fall. This is called the passage of heat by radia-
tion. All hot bodies radiate or shoot out heat in
this way. When we hold our hands in front of
a bright fire, it is the heat-waves which the fire is
raying out, or radiating, that fall on our skin and
produce a sensation of warmth.

We see, then, that there are three ways in which
heat can be transmitted : — 1. By conduction ; 2. By
convection ; and 3. By radiation. -

XX.— HEAT AS A FORM OF ENERGY.

118. Heat produced by Friction— 119. How to reduce the Heat produced by
Friction— ISO. Heat produced by Mechanical Energy— 121. The Mechanical
Equivalent of Heat — 122. Conversion of Heat into Mechanical Energy—
123. The Steam-Engine.

118. Heat produced by Friction. — ^Almost every one
is in the habit, at some time or other, of rubbing his
hands together in order to warm them. This is an
excellent instance of heat being produced by fric-
tion. The human skin is not perfectly smooth, and
when one hand is rubbed against the other, the
rough places catch one against the other, and the
motion of the hands is more or less stopped.

What becomes of this arrested motion ? We have
learned that it is converted into heat. Motion of
the two hands, as a whole, is changed into motion of
the molecules of the skin, and this molecular motion
produces in us the sensation of heat. As an experi-
ment, let us fix a bright metal button in a cork, so
that we can grasp it easily, and rub the button
smartly on a piece of wood. The button will soon
become so hot as to burn the skin, to set fire to a
piece of phosphorus, ot even to a lucifer match.
The friction produced when a brake is applied
sharply to the wheel of a waggon or an engine is

HEAT AS A FOBU OF ENEROT. 135

often 80 great as to produce a stream of sparks.
In striking a match we have another example of
the heat produced by friction. All common matches
have a little phosphorus on them, and the heat pro-
duced hy the friction of the match on the rough hox
is suflScient to cause the phosphorus to catch fire.

Pio. 39— Indian mode of obUlnlng ■ light by tha friction of on« piec« ol nood

Many Indians obtain fire by the friction of a
hard, dry, pointed piece of wood against another
piece. Shooting stars or meteors are masses of

136 HEAT AS A FORM OF ENERGY.

stone or iron travelling through the air with such a
great velocity (foYty to sixty miles a second) that
their friction (igainst the air sets them on fire.
Very often friction produces heat where it is not
wanted. In this way the axles of railway carriages
sometimes become so heated as to set fire to the
wood-work of the train; and by friction a work-
man's tools are often made too hot to be handled.

119. How to reduce the Heat produced by Friction. —
In such cases the remedy is to apply oil, or grease,
or blacklead, or some other lubricant which will fill

^up the rough places and make the one surface glide
more smoothly over the other. The friction will
then be less, and less heat will consequently.be
produced.

120. Heat produced by Mechanical Energy. — By
mechanical energy we mean the energy possessed
by a body which is moving as a whole. It may be
said that this is the same as kinetic energy, but the
latter term includes molecular motion as well as
motion of the body as a whole. Thus a red-hot
ball hanging by a chain has kinetic energy, because
its molecules are in motion ; but it has no mechani-
cal energy, for as a ball it is at rest. By means of
our muscular force, and the force of gravity, we can
set a hammer-head in motion, and impart to it much
mechanical energy. By a few smart blows with a
hammer an iron nail may be made so hot as to ignite
a match, and it is said that a clever blacksmith can
so hammer a nail as to make it red-hot. In that
case the mechanical energy possessed by the moving
hammer is converted into the molecular energy we

HEAT AS A FORM OP ENERGY. 137

call heat. If we pick up a bullet that has just
been fired at, and has struck, an iron target, we
shall find the lead so hot that we shall quickly drop
it. When the bullet was moving through the air
it had great mechanical energy. When it struck the
target its energy was not lost, but it was converted
into the kind of molecular energy ce^Ued heat.

121. The Mechanical Equivalent of Heat. — We can
now explain what becomes of the energy of falling
bodies. When a stone, or a ball, or a meteor strikes
the ground, its mechanical energy may appear to be
lost, but it is really changed into heat -energy.
Both the moving body and the spot on which it
struck will be a little warmer after the collision
than before. If we require to know how Tnuch
mechanical energy is equal to a given amount of
heat, the best answer is to state from what height a
pound-weight would have to fall in order to raise
one pound of water through one degree Fahrenheit.
A great many careful experiments, more especially
those made by Dr. Joule of Manchester, have shown
that this height is seven hundred and* seventy- two
feet. If we could take a one-pound weight up in a
balloon to a height of seven hundred and seventy-
two feet, and let it fall from thence into a basin
containing one pound of water at, say, sixty degrees
Fahrenheit, and if all the heat produced by the
. blow went to warm the water, then its temperature
after the weight had fallen into it would be exactly
sixty-one degrees Fahrenheit. But a weight of one
pound falling from a height of seven hundred and
seventy-two feet does seven hundred and seventy-

138 HEAT AB A FOBH Of ENEROT.

hoo foot-povmds of work. This amount is known
as the mechanical equivalent of heat.

122. OoiiTeraion of Heat isto Mechanical Energ;. —
Suppose that we place some water, at a temperature
of, say, 60°, in a kettle on the fire, and keep a
thermometer in the water. The temperature of the
water steadily rises until it reaches 212°, but at this

^.—.. point it remains constant.

.f^/^^!-.~- Yet heat from the fire is

; t"^ ■^,-'-, continually passing into the

\^-~{''S water. What becomes of this

'^^''' I'^t'' Is it lost? Certainly

t:il: not ; it is spent in driving

the molecules of water asun-
der — in overcoming their
cohesion — so that the liquid
changes into a gas. It is
found that water - gas or
steam takes up one thou-
sand seven hundred times
as much room as the water
from which it was derived,
i One pint of water would
produce one thousand seven
pji^ ^ hundred pints of steam.

ConTsraioii of liquid wBter Into Heat is the force which has

ol (hs waUr an drlTan unnder by produced motion of the

ti»i»c«(>fh«t. molecules of water, driving

them further apart, so that they take up a great
deal more room. This is an example of the con-
version of heat into mechanical energy.

123. The Steam -Engiiie. — It is the expansion of

HEAT AS A FORM OP ENERGY. 139

water when it is changed from the liquid to the
gaseous state that does work in the machine called
the steam-engine. The water in the boiler is heated
by a fire in the furnace. As the water turns into
steam it expands and pushes up a piston. After it
has done this, the steam is condensedy and the piston
falls down again. This up and down motion of the
piston is changed by a suitable contrivance into the
circular motion of the wheels. Nothing can be
clearer than that in a steam-engine we put in heat
and take out mechanical energy. Unfortunately we
are not able to obtain in this or in any other machine
the full ^mechanical equivalent of the heat we em-
ploy. A great deal of heat is wasted in warming
the different parts of the machine, and much escapes
up the chimney. It is certain that the best con-
structed engines do not give us the mechanical
equivalent of more than one-tenth of the heat pro-
duced by the coal burned in their furnaces.

APPEIJJ'DIX.

-•♦-

QUESTIONS AND EXERCISES.

1. Name some property by which matter can be distinguished from
force,

2. If there is no forcer that can directly affect our senses, how do
we know that any forces exist ? ' *

3. How many kinds of motion are there? Name them. Which
kind is recognized by the sense of sight ?

II.

4. Name all the forces you know. What one force is very different
in its mode of action from aU the others ?

5. Describe an experiment by which you could produce some of the
force called electricity.

6. What do you know about the muscular force ?

III.

7. Write out the first law of gravitation. Where could you place
a body so that it would not be acted upon by this force ?

8. Write out the second law of gravitation. Find the squares of
the numbers 1, 32, and 1000.

9. Compare the attraction between two balls, each weighing four
pounds, and hanging two feet apart, with that between two balls, each
weighing one pound, and hanging one foot apart.

IV.

10. Explain the words up and down. When a boy in New Zealand
throws a ball " up," in what direction is he throwing it compared with
a ball thrown " up " by a boy in England ?

APPENDIX. 141

11. What is meant by the centre of gravity of a body? Where-
abouts in the body is the centre of gravity of a cricket-ball, a brick, a
biscuit, and a slate ?

12. Name the three kinds of equilibrium, and give an illustration of each.

13. What did Galileo discover about falling bodies ? How and where
did he experiment upon them?

14. Describe two experiments by which you can prove that very
light bodies fall with the same velocity as heavy ones.

15. Explain the meaning of the word uniform. A train passes a
station (A) at twelve o'clock, with a velocity of thirty miles an hour ; ,
twenty minutes later this train passes a station (B) ten miles off : has
its velocity been uniform, or variable ? Give a reason for your answer.

VI.

16. State any facts you have noticed which prove that falling bodies
move with variable velocity.

17. Explain the words accelei*aied, retarded, and vdocUy.

18. A stone dropped into a well is heard to strike tlie water in four
seconds; find (1) the distance from the water to the surface, and
(2) the velocity of the stone at the moment it reached the water.

VII.

19. Are the houses which form our streets at rest, or in motion ? If
they are in motion, how is it that we cannot see them move ?

20. Write out the first law of motion. What is another name for
this law ?

21. Describe any two experiments you would perform to prove that
inertia is a universal property of matter.

VIII.

22. Explain how it happens that people frequently fall when they
step out of a moving train or carriage.

23. Why is it more difficult for a boy to set a loaded truck in motion
than for him to keep it in motion after it has once been started ?

24. Describe any operations in which the inertia of matter in motion
is taken advantage of and utilized.

IX.

25. What effect has friction on a moving body? How can you
reduce the friction of any two surfaces ?

142 APPENDIX.

26. Why does a ball roll to a greater distance on ice than on grass
(supposing it to be thrown with the same force in each case) ? How
far would a ball rpU on a perfectly smooth horizontal surface ?

27. Point out the advantages and the disadvantages of friction.

28. Explain the words matter, molecule, partide, and body,

29. Which will strike the harder blow — a ten pound hammer moving
with a velocity of fifteen feet per second, or a twenty-five pound
hammer moving six feet per second ?

30. State exactly what ia meant by the word Ttuus. How could we
prove that a cubic inch of platinum has twenty- two times the mass of
a cubic inch of water ?

XI.

31. Three policemen, who can lift separately weights of 290, 310,
and 336 pounds, engage in a " tug of war " with three soldiers, whose
strength is equal to 300, 306, and 230 pounds respectively. Find which
side will win, and draw a diagram representing the six forces in action.

32. Find the resultant of two forces of six pounds and eight pounds
respectively, actix^ parallel to each other on a given point : (1) when
the forces act in the same direction, (2) when they act in opposite
directions.

33. Three forces, of six, eight, and ten pounds, act on a given point.
What is the greatest resultant they can have, and what is the least ?

xn.

34. Write out the second law of motioi), and give three examples
of its action.

35. Why is it impossible for a soldier in a moving train to hit any
object at which he may fire, provided that he aims straight at it ?

36. Show how to find the resultant of forces acting in opposite direc-
tions. Give an example.

XIII.

37. Write out the third law of motion, and give one illustra-
tion of it.

38. A man sitting in a boat attempts to make the boat go back-
wards by pushing the stern with his oar. Why does he fail to move
the boat ?

39. When a man-of-war is chasing another ship the sailors object to
fire the bow-guns. Have they any reason for this ? What would be
the effect of firing cannon from the stem of the ship ?

APPENDIX. 143

XIV.

40. A man, weighing 144 lbs., carries a wheel, weighing 50 lbs., to
the top of a hill 1000 feet high ; how much work has he done ? Another
man, of the same weight, rolls the wheel down again ; in doing this
how much work has this second man performed ?

41. What is the imit by which work is measiured ? How much water
(ten pounds to the gallon) could a steam-engine of ten horse-power
raise in ten minutes from the bottom of a weU one hundred feet deep ?

XV.

42. What kind of energy does each of the following bodies possess ?—
(1) gunpowder, (2) a lion, (3) a river, (4) a rubbed piece of sealing-wax,
(5) a fire, (6) the wind, (7) a lump of cpal.

43. What do you mean by energy t How can the energy of a steam-
engine be measured ?

XVI.

44. Explain the word potential, and give four examples of bodies
possessing potential energy.

45. Point out the advantage of the Sun to the Earth as a source of
energy.

46. How could you impart some potential energy to each of the
following bodies ? — (1) a stone, (2) a ca^e, (3) a piece of india-rubber,
(4) a clock-weight.

xvn.

47. What is the name of the kind of energy possessed by a body in
motion ? In the case of a stone lodged on a house-top, what becomes
of the energy which the stone had while in motion ?

48. Which has the greater energy — a body weighing one pound, and
moving with a velocity of ten feet per second; or a body weighing two
pounds, and moving five feet per second ?

49. If a cannon-ball weighs one pound, and moves with a velocity
of one thousand feet per second, find the force of the blow which it
will strike in foot-pounds.

xvin.

50. If energy cannot be destroyed, what becomes of it, and where
does it go? For example, what becomes of the energy of a falling
stone?

51. Give at least three examples of the change of eneigy from one
form into another.

52. A boy weighing fifty pounds seats himself in a basket, and by

144 APPENDIX.

means of a rope passed over a pulley raises himself to a height of
twenty feet. How much energy has he expended in doing this, and
where did it come from ? When he lets himself down, what becomes of
the energy ?

XIX.

53. Explain these words : — Theory, mcUerial, mechanical, expansion,
transmission.

54. Why is it impossible that heat can be any kind of matter ? If it
is not matter, what is it ?

XX.

55. Describe any experiment by which- you can change mechanical
energy into heat.

56. What is the mechanical equivalent of heat ? From what height
would you have to drop a pound of water in order to raise its tempera-
ture ten degrees ? (You may suppose that all the heat resulting from
the fall is communicated to the water.)

57. Wliat is the source of the mechanical energy of a steam-engine?
If an engine can raise a million gallons of water per hoiu: from a
depth of one hundred feet, find the horse-power of the engine.

THE END.

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Full text of "Elementary Mechanics, Or First Lessons in Natural Philosophy: 2d Year's Course"

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