the fall of bodies?
_Mrs. B._ Undoubtedly. I have given you the most familiar illustrations
in order to render the explanation clear; but since you seek for more
scientific examples, you may say that cohesion is the force which binds
the particles of bodies together, and heat that which drives them
asunder.
The motion of a body acted upon by a single force, is always in a
straight line, and in the direction in which it received the impulse.
_Caroline._ That is very natural; for as the body is inert, and can move
only because it is impelled, it will move only in the direction in which
it is impelled. The degree of quickness with which it moves, must, I
suppose, also depend upon the degree of force with which it is impelled.
_Mrs. B._ Yes; the rate at which a body moves, or the shortness of the
time which it takes to move from one place to another, is called its
velocity; and it is one of the laws of motion, that the velocity of the
moving body is proportional to the force by which it is put in motion.
We must distinguish between absolute and relative velocity.
The velocity of a body is called _absolute_, if we consider the motion
of the body in space, without any reference to that of other bodies.
When, for instance, a horse goes fifty miles in ten hours, his velocity
is five miles an hour.
The velocity of a body is termed _relative_, when compared with that of
another body which is itself in motion. For instance, if one man walks
at the rate of a mile an hour, and another at the rate of two miles an
hour, the relative velocity of the latter is double that of the former;
but the absolute velocity of the one is one mile, and that of the other
two miles an hour.
_Emily._ Let me see if I understand it--The relative velocity of a body
is the degree of rapidity of its motion compared with that of another
body; thus if one ship sail three times as far as another ship in the
same space of time, the velocity of the former is equal to three times
that of the latter.
_Mrs. B._ The general rule may be expressed thus: the velocity of a
body is measured by the space over which it moves, divided by the time
which it employs in that motion: thus if you travel one hundred miles in
twenty hours, what is your velocity in each hour?
_Emily._ I must divide the space, which is one hundred miles, by the
time, which is twenty hours, and the answer will be five miles an hour.
Then, Mrs. B., may we not reverse this rule, and say that the time is
equal to the space divided by the velocity; since the space, one hundred
miles, divided by the velocity, five miles per hour, gives twenty hours
for the time?
_Mrs. B._ Certainly; and we may say also that the space is equal to the
velocity multiplied by the time. Can you tell me, Caroline, how many
miles you will have travelled, if your velocity is three miles an hour,
and you travel six hours?
_Caroline._ Eighteen miles; for the product of 3 multiplied by 6, is 18.
_Mrs. B._ I suppose that you understand what is meant by the terms
_uniform_, _accelerated_ and _retarded_ motion.
_Emily._ I conceive uniform motion to be that of a body whose motion is
regular, and at an equal rate throughout; for instance a horse that goes
an equal number of miles every hour. But the hand of a watch is a much
better example, as its motion is so regular as to indicate the time.
_Mrs. B._ You have a right idea of uniform motion; but it would be more
correctly expressed by saying, that the motion of a body is uniform when
it passes over equal spaces in equal times. Uniform motion is produced
by a force having acted on a body once and having ceased to act; as, for
instance, the stroke of a bat on a ball.
_Caroline._ But the motion of a ball is not uniform; its velocity
gradually diminishes till it falls to the ground.
_Mrs. B._ Recollect that the ball is inert, and has no more power to
stop, than to put itself in motion; if it falls, therefore, it must be
stopped by some force superior to that by which it was projected, and
which destroys its motion.
_Caroline._ And it is no doubt the force of gravity which counteracts
and destroys that of projection; but if there were no such power as
gravity, would the ball never stop?
_Mrs. B._ If neither gravity nor any other force, such as the resistance
of the air, opposed its motion, the ball, or even a stone thrown by the
hand, would proceed onwards in a right line, and with a uniform velocity
for ever.
_Caroline._ You astonish me! I thought that it was impossible to
produce perpetual motion?
_Mrs. B._ Perpetual motion cannot be produced by art, because gravity
ultimately destroys all motion that human power can produce.
_Emily._ But independently of the force of gravity, I cannot conceive
that the little motion I am capable of giving to a stone would put it in
motion for ever.
_Mrs. B._ The quantity of motion you communicate to the stone would not
influence its duration; if you threw it with little force it would move
slowly, for its velocity you must remember, will be proportional to the
force with which it is projected; but if there is nothing to obstruct
its passage, it will continue to move with the same velocity, and in the
same direction as when you first projected it.
_Caroline._ This appears to me quite incomprehensible; we do not meet
with a single instance of it in nature.
_Mrs. B._ I beg your pardon. When you come to study the motion of the
celestial bodies, you will find that _nature_ abounds with examples of
perpetual motion; and that it conduces as much to the harmony of the
system of the universe, as the prevalence of it on the surface of the
earth, would to the destruction of all our comforts. The wisdom of
Providence has therefore ordained insurmountable obstacles to perpetual
motion here below; and though these obstacles often compel us to contend
with great difficulties, yet these appear necessary to that order,
regularity and repose, so essential to the preservation of all the
various beings of which this world is composed.
Now can you tell me what is _retarded motion_?
_Caroline._ Retarded motion is that of a body which moves every moment
slower and slower: thus when I am tired with walking fast, I slacken my
pace; or when a stone is thrown upwards, its velocity is gradually
diminished by the power of gravity.
_Mrs. B._ Retarded motion is produced by some force acting upon the body
in a direction opposite to that which first put it in motion: you who
are an animated being, endowed with power and will, may slacken your
pace, or stop to rest when you are tired; but inert matter is incapable
of any feeling of fatigue, can never slacken its pace, and never stop,
unless retarded or arrested in its course by some opposing force; and as
it is the laws of inert bodies of which mechanical philosophy treats, I
prefer your illustration of the stone retarded in its ascent. Now
Emily, it is your turn; what is _accelerated motion_?
_Emily._ Accelerated motion, I suppose, takes place when the velocity of
a body is increased; if you had not objected to our giving such active
bodies as ourselves as examples, I should say that my motion is
accelerated if I change my pace from walking to running. I cannot think
of any instance of accelerated motion in inanimate bodies; all motion of
inert matter seems to be retarded by gravity.
_Mrs. B._ Not in all cases; for the power of gravitation sometimes
produces accelerated motion; for instance, a stone falling from a
height, moves with a regularly accelerated motion.
_Emily._ True; because the nearer it approaches the earth, the more it
is attracted by it.
_Mrs. B._ You have mistaken the cause of its accelerated motion; for
though it is true that the force of gravity increases as a body
approaches the earth, the difference is so trifling at any small
distance from its surface, as not to be perceptible.
Accelerated motion is produced when the force which put a body in
motion, continues to act upon it during its motion, so that its velocity
is continually increased. When a stone falls from a height, the impulse
which it receives from gravitation in the first instant of its fall,
would be sufficient to bring it to the ground with a uniform velocity:
for, as we have observed, a body having been once acted upon by a force,
will continue to move with a uniform velocity; but the stone is not
acted upon by gravity merely at the first instant of its fall; this
power continues to impel it during the whole time of its descent, and it
is this continued impulse which accelerates its motion.
_Emily._ I do not quite understand that.
_Mrs. B._ Let us suppose that the instant after you have let a stone
fall from a high tower, the force of gravity were annihilated; the body
would nevertheless continue to move downwards, for it would have
received a first impulse from gravity; and a body once put in motion
will not stop unless it meets with some obstacle to impede its course;
in this case its velocity would be uniform, for though there would be no
obstacle to obstruct its descent, there would be no force to accelerate
it.
_Emily._ That is very clear.
_Mrs. B._ Then you have only to add the power of gravity constantly
acting on the stone during its descent, and it will not be difficult to
understand that its motion will become accelerated, since the gravity
which acts on the stone at the very first instant of its descent, will
continue in force every instant, till it reaches the ground. Let us
suppose that the impulse given by gravity to the stone during the first
instant of its descent, be equal to one; the next instant we shall find
that an additional impulse gives the stone an additional velocity, equal
to one; so that the accumulated velocity is now equal to two; the
following instant another impulse increases the velocity to three, and
so on till the stone reaches the ground.
_Caroline._ Now I understand it; the effects of preceding impulses
continue, whilst gravity constantly adds new ones, and thus the velocity
is perpetually increased.
_Mrs. B._ Yes; it has been ascertained, both by experiment, and
calculations which it would be too difficult for us to enter into, that
heavy bodies near the surface of the earth, descending from a height by
the force of gravity, fall sixteen feet the first second of time, three
times that distance in the next, five times in the third second, seven
times in the fourth, and so on, regularly increasing their velocities in
the proportion of the odd numbers 1, 3, 5, 7, 9, &c. according to the
number of seconds during which the body has been falling.
_Emily._ If you throw a stone perpendicularly upwards, is it not the
same length of time in ascending, that it is in descending?
_Mrs. B._ Exactly; in ascending, the velocity is diminished by the force
of gravity; in descending, it is accelerated by it.
_Caroline._ I should then imagine that it would fall, quicker than it
rose?
_Mrs. B._ You must recollect that the force with which it is projected,
must be taken into the account; and that this force is overcome and
destroyed by gravity, before the body begins to fall.
_Caroline._ But the force of projection given to a stone in throwing it
upwards, cannot always be equal to the force of gravity in bringing it
down again; for the force of gravity is always the same, whilst the
degree of impulse given to the stone is optional; I may throw it up
gently, or with violence.
_Mrs. B._ If you throw it gently, it will not rise high; perhaps only
sixteen feet, in which case it will fall in one second of time. Now it
is proved by experiment, that an impulse requisite to project a body
sixteen feet upwards, will make it ascend that height in one second;
here then the times of the ascent and descent are equal. But supposing
it be required to throw a stone twice that height, the force must be
proportionally greater.
You see then, that the impulse of projection in throwing a body upwards,
is always equal to the action of the force of gravity during its
descent; and that whether the body rises to a greater or less distance,
these two forces balance each other.
I must now explain to you what is meant by the _momentum_ of bodies. It
is the force, or power, with which a body in motion, strikes against
another body. The momentum of a body is the product of its _quantity of
matter_, multiplied by its _quantity of motion_; in other words, its
weight multiplied by its velocity.
_Caroline._ The quicker a body moves, the greater, no doubt, must be the
force which it would strike against another body.
_Emily._ Therefore a light body may have a greater momentum than a
heavier one, provided its velocity be sufficiently increased; for
instance, the momentum of an arrow shot from a bow, must be greater than
that of a stone thrown by the hand.
_Caroline._ We know also by experience, that the heavier a body is, the
greater is its force; it is not therefore difficult to understand, that
the whole power, or momentum of a body, must be composed of these two
properties, its weight and its velocity: but I do not understand why
they should be _multiplied_, the one by the other; I should have
supposed that the quantity of matter, should have been _added_ to the
quantity of motion?
_Mrs. B._ It is found by experiment, that if the weight of a body is
represented by the number 3, and its velocity also by 3, its momentum
will be represented by 9, not by 6, as would be the case, were these
figures added, instead of being multiplied together.
_Emily._ I think that I now understand the reason of this; if the
quantity of matter is increased three-fold, it must require three times
the force to move it with the same velocity; and then if we wish to give
it three times the velocity, it will again require three times the force
to produce that effect, which is three times three, or nine; which
number therefore, would represent the momentum.
_Caroline._ I am not quite sure that I fully comprehend what is
intended, when weight, and velocity, are represented by numbers alone; I
am so used to measure space by yards and miles, and weight by pounds and
ounces, that I still want to associate them together in my mind.
_Mrs. B._ This difficulty will be of very short duration: you have only
to be careful, that when you represent weights and velocities by
numbers, the denominations or values of the weights and spaces, must not
be changed. Thus, if we estimate the weight of one body in ounces, the
weight of others with which it is compared, must be estimated in ounces,
and not in pounds; and in like manner, in comparing velocities, we must
throughout, preserve the same standards both of space and of time; as
for instance, the number of feet in one second, or of miles in one hour.
_Caroline._ I now understand it perfectly, and think that I shall never
forget a thing which you have rendered so clear.
_Mrs. B._ I recommend it to you to be very careful to remember the
definition of the momentum of bodies, as it is one of the most important
points in mechanics: you will find that it is from opposing velocity, to
quantity of matter, that machines derive their powers.
The _reaction_ of bodies, is the next law of motion which I must explain
to you. When a body in motion strikes against another body, it meets
with resistance from it; the resistance of the body at rest will be
equal to the blow struck by the body in motion; or to express myself in
philosophical language, _action_ and _reaction_ will be equal, and in
opposite directions.
_Caroline._ Do you mean to say, that the action of the body which
strikes, is returned with equal force by the body which receives the
blow?
_Mrs. B._ Exactly.
_Caroline._ But if a man strike another on the face with his fist, he
surely does not receive as much pain by the reaction, as he inflicts by
the blow?
_Mrs. B._ No; but this is simply owing to the knuckles, having much less
feeling than the face.
Here are two ivory balls suspended by threads, (plate 1. fig. 3.) draw
one of them, A, a little on one side,--now let it go;--it strikes, you
see, against the other ball B, and drives it off, to a distance equal to
that through which the first ball fell; but the motion of A is stopped;
because when it struck B, it received in return a blow equal to that it
gave, and its motion was consequently destroyed.
_Emily._ I should have supposed, that the motion of the ball A was
destroyed, because it had communicated all its motion to B.
_Mrs. B._ It is perfectly true, that when one body strikes against
another, the quantity of motion communicated to the second body, is lost
by the first; but this loss proceeds from the reaction of the body which
is struck.
Here are six ivory balls hanging in a row, (fig. 4.) draw the first out
of the perpendicular, and let it fall against the second. You see none
of the balls except the last, appear to move, this flies off as far as
the first ball fell; can you explain this?
_Caroline._ I believe so. When the first ball struck the second, it
received a blow in return, which destroyed its motion; the second ball,
though it did not appear to move, must have struck against the third;
the reaction of which set it at rest; the action of the third ball must
have been destroyed by the reaction of the fourth, and so on till motion
was communicated to the last ball, which, not being reacted upon, flies
off.
_Mrs. B._ Very well explained. Observe, that it is only when bodies are
elastic, as these ivory balls are, and when their masses are equal, that
the stroke returned is equal to the stroke given, and that the striking
body loses all its motion. I will show you the difference with these two
balls of clay, (fig. 5.) which are not elastic; when you raise one of
these, D, out of the perpendicular, and let it fall against the other,
E, the reaction of the latter, on account of its not being elastic, is
not sufficient to destroy the motion of the former; only part of the
motion of D will be communicated to E, and the two balls will move on
together to _d_ and _e_, which is not so great a distance as that
through which D fell.
Observe how useful reaction is in nature. Birds in flying strike the air
with their wings, and it is the reaction of the air, which enables them
to rise, or advance forwards; reaction being always in a contrary
direction to action.
_Caroline._ I thought that birds might be lighter than the air, when
their wings were expanded, and were by that means enabled to fly.
_Mrs. B._ When their wings are spread, this does not alter their weight,
but they are better supported by the air, as they cover a greater extent
of surface; yet they are still much too heavy to remain in that
situation, without continually flapping their wings, as you may have
noticed when birds hover over their nests: the force with which their
wings strike against the air, must equal the weight of their bodies, in
order that the reaction of the air, may be able to support that weight;
the bird will then remain stationary. If the stroke of the wings is
greater than is required merely to support the bird, the reaction of the
air will make it rise; if it be less, it will gently descend; and you
may have observed the lark, sometimes remaining with its wings extended,
but motionless; in this state it drops quietly into its nest.
_Caroline._ This is indeed a beautiful effect of the law of reaction!
But if flying is merely a mechanical operation, Mrs. B., why should we
not construct wings, adapted to the size of our bodies, fasten them to
our shoulders, move them with our arms, and soar into the air?
_Mrs. B._ Such an experiment has been repeatedly attempted, but never
with success; and it is now considered as totally impracticable. The
muscular power of birds, is incomparably greater in proportion to their
weight, than that of man; were we therefore furnished with wings
sufficiently large to enable us to fly, we should not have strength to
put them in motion.
In swimming, a similar action is produced on the water, to that on the
air, in flying; in rowing, also, you strike the water with the oars, in
a direction opposite to that in which the boat is required to move, and
it is the reaction of the water on the oars which drives the boat along.
_Emily._ You said, that it was in elastic bodies only, that the whole
motion of one body, would be communicated to another; pray what bodies
are elastic, besides the air?
_Mrs. B._ In speaking of the air, I think we defined elasticity to be a
property, by means of which bodies that are compressed, return to their
former state. If I bend this cane, as soon as I leave it at liberty, it
recovers its former position; if I press my finger upon your arm, as
soon as I remove it, the flesh, by virtue of its elasticity, rises and
destroys the impression I made. Of all bodies, the air is the most
eminent for this property, and it has thence obtained the name of an
elastic fluid. Hard bodies are in the next degree elastic; if two ivory,
or hardened steel balls are struck together, the parts at which they
touch, will be flattened; but their elasticity will make them
instantaneously resume their former shape.
_Caroline._ But when two ivory balls strike against each other, as they
constantly do on a billiard table, no mark or impression is made by the
stroke.
_Mrs. B._ I beg your pardon; you cannot, it is true, perceive any mark,
because their elasticity instantly destroys all trace of it.
Soft bodies, which easily retain impressions, such as clay, wax, tallow,
butter, &c. have very little elasticity; but of all descriptions of
bodies, liquids are the least elastic.
_Emily._ If sealing-wax were elastic, instead of retaining the
impression of a seal, it would resume a smooth surface, as soon as the
weight of the seal was removed. But pray what is it that produces the
elasticity of bodies?
_Mrs. B._ There is great diversity of opinion upon that point, and I
cannot pretend to decide which approaches nearest to the truth.
Elasticity implies susceptibility of compression, and the susceptibility
of compression depends upon the porosity of bodies; for were there no
pores or spaces between the particles of matter of which a body is
composed, it could not be compressed.
_Caroline._ That is to say, that if the particles of bodies were as
close together as possible, they could not be squeezed closer.
_Emily._ Bodies then, whose particles are most distant from each other,
must be most susceptible of compression, and consequently most elastic;
and this you say is the case with air, which is perhaps the least dense
of all bodies?
_Mrs. B._ You will not in general find this rule hold good; for liquids
have scarcely any elasticity, whilst hard bodies are eminent for this
property, though the latter are certainly of much greater density than
the former; elasticity implies, therefore, not only a susceptibility of
compression, but depends upon the power possessed by the body, of
resuming its former state after compression, in consequence of the
peculiar arrangement of its particles.
_Caroline._ But surely there can be no pores in ivory and metals, Mrs.
B.; how then can they be susceptible of compression?
_Mrs. B._ The pores of such bodies are invisible to the naked eye, but
you must not thence conclude that they have none; it is, on the
contrary, well ascertained that gold, one of the most dense of all
bodies, is extremely porous; and that these pores are sufficiently large
to admit water when strongly compressed, to pass through them. This was
shown by a celebrated experiment made many years ago at Florence.
_Emily._ If water can pass through gold, there must certainly be pores
or interstices which afford it a passage; and if gold is so porous, what
must other bodies be, which are so much less dense than gold!
_Mrs. B._ The chief difference in this respect, is I believe, that the
pores in some bodies are larger than in others; in cork, sponge and
bread, they form considerable cavities; in wood and stone, when not
polished, they are generally perceptible to the naked eye; whilst in
ivory, metals, and all varnished and polished bodies, they cannot be
discerned. To give you an idea of the extreme porosity of bodies, sir
Isaac Newton conjectured that if the earth were so compressed as to be
absolutely without pores, its dimensions might possibly not be more than
a cubic inch.