of which A has been deprived by the impact is 48, the difference
between 72 and 24. On the other hand, B loses by the impact the
quantity 30 in the direction B C, which is equivalent to receiving
30 in the direction A C. But it also acquires a quantity 18 in
the direction A C, which, added to the former 30, gives a total
of 48 received by B in the direction A C. Thus the same quantity
of motion which A loses in the direction A C, is received by B in
the same direction. The law of “action and reaction” is, therefore,
fulfilled.
This result may in like manner be generalised. Retaining the former
symbols, the moving forces of A and B before impact will be A × _a_ and
B × _b_ and their forces after impact will be A × _x_ and B × _x_. The
force lost by A will therefore be A × _a_ - A × _x_. The mass B will
have lost all the force B × _b_ which it had in its former direction,
and will have received the force B × _x_ in the opposite direction.
Therefore the actual force imparted to B by the collision will be B
× _b_ + B × _x_. But since the force lost by A must be equal to that
imparted to B, we shall have
A × _a_ - A × _x_ = B × _b_ + B × _x_
and therefore
(A + B) × _x_ = A × _a_ - B × _b_
and if the common velocity after impact be required, we have
_x_ = (A × _a_ - B × _b_)/(A + B)
As a general rule, therefore, to find the common velocity after impact.
Multiply the weights by the previous velocities and take their sum if
the bodies move in the same direction, and their difference if they
move in opposite directions, and divide the one or the other by the sum
of their weights. The greatest will be the velocity after impact.
(67.) The examples of the equality of action and reaction in the
collision of bodies may be exhibited experimentally by a very simple
apparatus. Let A, _fig. 6._, and B be two balls of soft clay, or
any other substance which is inelastic, or nearly so, and let these
be suspended from C by equal strings, so that they may be in contact;
and let a graduated arc, of which the centre is C, be placed so that
the balls may oscillate over it. One of the balls being moved from its
place of rest along the arc, and allowed to descend upon the other
through a certain number of degrees, will strike the other with a
velocity corresponding to that number of degrees, and both balls will
then move together with a velocity which may be estimated by the number
of degrees of the arc through which they rise.
(68.) In all these cases in which we have explained the law of “action
and reaction,” the transfer of motion from one body to the other has
been made by impact or collision. The phenomenon has been selected only
because it is the most ordinary way in which bodies are seen to affect
each other. The law is, however, universal, and will be fulfilled
in whatever manner the bodies may affect each other. Thus A may be
connected with B by a flexible string, which, at the commencement of
A’s motion, is slack. Until the string becomes stretched, that is,
until A’s distance from B becomes equal to the length of the string,
A will continue to have all the motion first impressed upon it. But
when the string is stretched, a part of that motion is transferred to
B, which is then drawn after A; and whatever motion B in this way
receives, A must lose. All that has been observed of the effect of
motion transferred by impact will be equally applicable in this case.
Again, if B, _fig. 4._, be a magnet moving in the direction
B C with a certain quantity of motion, and while it is so moving a
mass of iron be placed at rest at A, the attraction of the magnet will
draw the iron after it towards C, and will thus communicate to the iron
a certain quantity of motion in the direction of C. All the motion thus
communicated to the iron A must be lost by the magnet B.
If the magnet and the iron were both placed quiescent at B and A, the
attraction of the magnet would cause the iron to move from A towards B;
but the magnet in this case not having any motion, cannot be literally
said to _transfer_ a motion to the iron. At the moment, however, when
the iron begins to move from A towards B, the magnet will be observed
to begin also to move from B towards A; and if the velocities of the
two bodies be expressed by numbers, and respectively multiplied by the
numbers expressing their masses, the quantities of motion thus obtained
will be found to be exactly equal. We have already explained why a
quantity of motion received in the direction B A, is equivalent
to the same quantity lost in the direction A B. Hence it appears,
that the magnet in receiving as much motion in the direction B A,
as it gives in the direction A B, suffers an effect which is
equivalent to losing as much motion directed towards C as it has
communicated to the iron in the same direction.
In the same manner, if the body B had any property in virtue of which
it might _repel_ A, it would itself be repelled with the same quantity
of motion. In a word, whatever be the manner in which the bodies may
affect each other, whether by collision, traction, attraction, or
repulsion, or by whatever other name the phenomenon may be designated,
still it is an inevitable consequence, that any motion, in a given
direction, which one of the bodies may receive, must be accompanied by
a loss of motion in the same direction, and to the same amount, by
the other body, or the acquisition of as much motion in the contrary
direction; or, finally, by a loss in the same direction, and an
acquisition of motion in the contrary direction, the combined amount of
which is equal to the motion received by the former.
(69.) From the principle, that the force of a body in motion depends on
the mass and the velocity, it follows, that any body, however small,
may be made to move with the same force as any other body, however
great, by giving to the smaller body a velocity which bears to that of
the greater the same proportion as the mass of the greater bears to the
mass of the smaller. Thus a feather, ten thousand of which would have
the same weight as a cannon-ball, would move with the same force if
it had ten thousand times the velocity; and in such a case, these two
bodies encountering in opposite directions, would mutually destroy each
other’s motion.
(70.) The consequences of the property of inertia, which have been
explained in the present and preceding chapters, have been given by
Newton, in his PRINCIPIA, and, after him, in most English treatises on
mechanics, under the form of three propositions, which are called the
“laws of motion.” They are as follow:--
I.
“Every body must persevere in its state of rest, or of uniform motion
in a straight line, unless it be compelled to change that state by
forces impressed upon it.”
II.
“Every change of motion must be proportional to the impressed force,
and must be in the direction of that straight line in which the force
is impressed.”
III.
“Action must always be equal and contrary to reaction; or the actions
of two bodies upon each other must be equal, and directed towards
contrary sides.”
When _inertia_ and _force_ are defined, the first law becomes an
identical proposition. The second law cannot be rendered perfectly
intelligible until the student has read the chapter on the composition
and resolution of forces, for, in fact, it is intended as an expression
of the whole body of results in that chapter. The third law has
been explained in the present chapter, as far as it can be rendered
intelligible in the present stage of our progress.
We have noticed these formularies more from a respect for the
authorities by which they have been proposed and adopted, than from any
persuasion of their utility. Their full import cannot be comprehended
until nearly the whole of elementary mechanics has been acquired, and
then all such summaries become useless.
* * * * *
(71.) The consequences deduced from the consideration of the quality
of inertia in this chapter, will account for many effects which fall
under our notice daily, and with which we have become so familiar, that
they have almost ceased to excite curiosity. One of the facts of which
we have most frequent practical illustration is, that the quantity of
motion or _moving force_, as it is sometimes called, is estimated by
the velocity of the motion, and the weight or mass of the thing moved
conjointly.
If the same force impel two balls, one of one pound weight, and the
other of two pounds, it follows, since the balls can neither give force
to themselves, nor resist that which is impressed upon them, that they
will move with the same force. But the lighter ball will move with
twice the speed of the heavier. The impressed force which is manifested
by giving velocity to a double mass in the one, is engaged in giving a
double velocity to the other.
If a cannon-ball were forty times the weight of a musket-ball, but the
musket-ball moved with forty times the velocity of the cannon-ball,
both would strike any obstacle with the same force, and would overcome
the same resistance; for the one would acquire from its velocity as
much force as the other derives from its weight.
A very small velocity may be accompanied by enormous force, if the mass
which is moved with that velocity be proportionally great. A large
ship, floating near the pier wall, may approach it with so small a
velocity as to be scarcely perceptible, and yet the force will be so
great as to crush a small boat.
A grain of shot flung from the hand, and striking the person, will
occasion no pain, and indeed will scarcely be felt, while a block of
stone having the same velocity would occasion death.
If a body in motion strike a body at rest, the striking body must
sustain as great a shock from the collision as if it had been at rest,
and struck by the other body with the same force. For the loss of force
which it sustains in the one direction, is an effect of the same kind
as if, being at rest, it had received as much force in the opposite
direction. If a man, walking rapidly or running, encounters another
standing still, he suffers as much from the collision as the man
against whom he strikes.
If a leaden bullet be discharged against a plank of hard wood, it will
be found that the round shape of the ball is destroyed, and that it
has itself suffered a force by the impact, which is equivalent to the
effect which it produces upon the plank.
When two bodies moving in opposite directions meet, each body sustains
as great a shock as if, being at rest, it had been struck by the other
body with the united forces of the two. Thus, if two equal balls,
moving at the rate of ten feet in a second, meet, each will be struck
with the same force as if, being at rest, the other had moved against
it at the rate of twenty feet in a second. In this case one part of the
shock sustained arises from the loss of force in one direction, and
another from the reception of force in the opposite direction.
For this reason, two persons walking in opposite directions receive
from their encounter a more violent shock than might be expected. If
they be of nearly equal weight, and one be walking at the rate of three
and the other four miles an hour, each sustains the same shock as if he
had been at rest, and struck by the other running at the rate of seven
miles an hour.
This principle accounts for the destructive effects arising from ships
running foul of each other at sea. If two ships of 500 tons burden
encounter each other, sailing at ten knots an hour, each sustains the
shock which, being at rest, it would receive from a vessel of 1000 tons
burden sailing ten knots an hour.
It is a mistake to suppose, that when a large and small body encounter,
the small body suffers a greater shock than the large one. The shock
which they sustain must be the same; but the large body may be better
able to bear it.
When the fist of a pugilist strikes the body of his antagonist, it
sustains as great a shock as it gives; but the fist being more fitted
to endure the blow, the injury and pain are inflicted on his opponent.
This is not the case, however, when fist meets fist. Then the parts
in collision are equally sensitive and vulnerable, and the effect is
aggravated by both having approached each other with great force. The
effect of the blow is the same as if one fist, being held at rest, were
struck by the other with the combined force of both.
CHAP. V.
THE COMPOSITION AND RESOLUTION OF FORCE.
(72.) Motion and pressure are terms too familiar to need explanation.
It may be observed, generally, that definitions in the first rudiments
of a science are seldom, if ever, comprehended. The force of words is
learned by their application; and it is not until a definition becomes
useless, that we are taught the meaning of the terms in which it is
expressed. Moreover, we are perhaps justified in saying, that in the
mathematical sciences the fundamental notions are of so uncompounded a
character, that definitions, when developed and enlarged upon, often
draw us into metaphysical subtleties and distinctions, which, whatever
be their merit or importance, would be here altogether misplaced. We
shall, therefore, at once take it for granted, that the words _motion_
and _pressure_ express phenomena or effects which are the subjects
of constant experience and hourly observation; and if the scientific
use of these words be more precise than their general and popular
application, that precision will soon be learned by their frequent use
in the present treatise.
(73.) FORCE is the name given in mechanics to whatever produces motion
or pressure. This word is also often used to express the motion or
pressure itself; and when the cause of the motion or pressure is not
known, this is the only correct use of the word. Thus, when a piece of
iron moves toward a magnet, it is usual to say that the cause of the
motion is “the attraction of the magnet;” but in effect we are ignorant
of the _cause_ of this phenomenon; and the name _attraction_ would
be better applied to the effect of which we have experience. In like
manner the _attraction_ and _repulsion_ of electrified bodies should be
understood, not as names for unknown causes, but as words expressing
observed appearances or effects.
When a certain phraseology has, however, gotten into general use, it
is neither easy nor convenient to supersede it. We shall, therefore,
be compelled, in speaking of motion and pressure, to use the language
of causation; but must advise the student that it is effects and not
causes which will be expressed.
(74.) If two forces act upon the same point of a body in different
directions, a single force may be assigned, which, acting on that
point, will produce the same result as the united effects of the other
two.
Let P, _fig. 7._, be the point on which the two forces act, and
let their directions be P A and P B. From the point P, upon
the line P A, take a length P _a_, consisting of as many inches
as there are ounces in the force P A; and, in like manner, take P
_b_, in the direction P B, consisting of as many inches as there
are ounces in the force P B. Through _a_ draw a line parallel to
P B, and through _b_ draw a line parallel to P A, and suppose
that these lines meet at _c_. Then draw P C. A single force,
acting in the direction P C, and consisting of as many ounces as
the line P c consists of inches, will produce upon the point P
the same effect as the two forces P A and P B produce acting
together.
(75.) The figure P _a c b_ is called in GEOMETRY a
_parallelogram_; the lines P _a_, P _b_, are called its _sides_, and
the line P _c_ is called its _diagonal_. Thus the method of finding an
equivalent for two forces, which we have just explained, is generally
called “the parallelogram of forces,” and is usually expressed thus:
“If two forces be represented in quantity and direction by the sides of
a parallelogram, an equivalent force will be represented in quantity
and direction by its diagonal.”
(76.) A single force, which is thus mechanically equivalent to two or
more other forces, is called their _resultant_, and relatively to it
they are called its _components_. In any mechanical investigation,
when the resultant is used for the components, which it always may
be, the process is called “the composition of force.” It is, however,
frequently expedient to substitute for a single force two or more
forces, to which it is mechanically equivalent, or of which it is the
resultant. This process is called “the resolution of force.”
(77.) To verify experimentally the theorem of the parallelogram
of forces is not difficult. Let two small wheels, M N,
_fig. 8._, with grooves in their edges to receive a thread, be
attached to an upright board, or to a wall. Let a thread be passed over
them, having weights A and B, hooked upon loops at its extremities.
From any part P of the thread between the wheels let a weight C be
suspended: it will draw the thread downwards, so as to form an angle
M P N, and the apparatus will settle itself at rest in some
determinate position. In this state it is evident that since the weight
C, acting in the direction P C, balances the weights A and B,
acting in the directions P M and P N, these two forces must
be mechanically equivalent to a force equal to the weight C, and acting
directly upwards from P. The weight C is therefore the quantity of the
resultant of the forces P M and P N; and the direction of the
resultant is that of a line drawn directly upwards from P.
To ascertain how far this is consistent with the theorem of “the
parallelogram of forces,” let a line P O be drawn upon the upright
board to which the wheels are attached, from the point P upward, in the
direction of the thread C P. Also, let lines be drawn upon the
board immediately under the threads P M and P N. From the
point P, on the line P O, take as many inches as there are ounces
in the weight C. Let the part of P O thus measured be P _c_, and
from _c_ draw _c a_ parallel to P N, and _c b_ parallel
to P M. If the sides P _a_ and P _b_ of the parallelogram thus
formed be measured, it will be found that P _a_ will consist of as many
inches as there are ounces in the weight A, and P _b_ of as many inches
as there are ounces in the weight B.
In this illustration, _ounces_ and _inches_ have been used as the
subdivisions of _weight_ and _length_. It is scarcely necessary to
state, that any other measures of these quantities would serve as well,
only observing that the same denominations must be preserved in all
parts of the same investigation.
(78.) Among the philosophical apparatus of the University of London,
is a very simple and convenient instrument which I constructed for
the experimental illustration of this important theorem. The wheels
M N are attached to the tops of two tall stands, the heights
of which may be varied at pleasure by an adjusting screw. A jointed
parallelogram, A B C D, _fig. 9._, is formed, whose
sides are divided into inches, and the joints at A and B are moveable,
so as to vary the lengths of the sides at pleasure. The joint C is
fixed at the extremity of a ruler, also divided into inches, while
the opposite joint A is attached to a brass loop, which surrounds the
diagonal ruler loosely, so as to slide freely along it. An adjusting
screw is provided in this loop so as to clamp it in any required
position.
In making the experiment, the sides A B and A D, C B
and C D are adjusted by the joints B and A to the same number
of inches respectively as there are ounces in the weights A and B,
_fig. 8._ Then the diagonal A C is adjusted by the loop and
screw at A, to as many inches as there are ounces in the weight C.
This done, the point A is placed behind P, _fig. 8._, and the
parallelogram is held upright, so that the diagonal A C shall be
in the direction of the vertical thread P C. The sides A B
and A D will then be found to take the direction of the threads
P M and P N. By changing the weights and the lengths of the
diagonal and sides of the parallelogram, the experiment may be easily
varied at pleasure.
(79.) In the examples of the composition of forces which we have here
given, the effects of the forces are the production of pressures, or,
to speak more correctly, the theorem which we have illustrated, is “the
composition of pressures.” For the point P is supposed to be at rest,
and to be drawn or pressed in the directions P M and P N.
In the definition which has been given of the word force, it is
declared to include motions as well as pressures. In fact, if motion be
resisted, the effect is converted into pressure. The same cause acting
upon a body, will either produce motion or pressure, according as the
body is free or restrained. If the body be free, motion ensues; if
restrained, pressure, or both these effects together. It is therefore
consistent with analogy to expect that the same theorems which regulate
pressures, will also be applicable to motions; and we find accordingly
a most exact correspondence.
(80.) If a body have a motion in the direction A B, and at the
point P it receive another motion, such as would carry it in the
direction P C, _fig. 10._, were it previously quiescent at
P, it is required to determine the direction which the body will take,
and the speed with which it will move, under these circumstances.
Let the velocity with which the body is moving from A to B be such,
that it would move through a certain space, suppose P N, in one
second of time, and let the velocity of the motion impressed upon it
at P be such, that if it had no previous motion it would move from P
to M in one second. From the point M draw a line parallel to P B,
and from N draw a line parallel to P C, and suppose these lines to
meet at some point, as O. Then draw the line P O. In consequence
of the two motions, which are at the same time impressed upon the body
at P, it will move in the straight line from P to O.
Thus the two motions, which are expressed in quantity and direction
by the sides of a parallelogram, will, when given to the same body,
produce a single motion, expressed in quantity and direction by its
diagonal; a theorem which is to motions exactly what the former was to
pressures.
There are various methods of illustrating experimentally the
composition of motion. An ivory ball, being placed upon a perfectly
level square table, at one of the corners, and receiving two equal
impulses, in the directions of the sides of the table, will move along
the diagonal. Apparatus for this experiment differ from each other only
in the way of communicating the impulses to the ball.
(81.) As two motions simultaneously communicated to a body are
equivalent to a single motion in an intermediate direction, so
also a single motion may be mechanically replaced, by two motions
in directions expressed by the sides of any parallelogram, whose
diagonal represents the single motion. This process is “the resolution
of motion,” and gives considerable clearness and facility to many
mechanical investigations.
(82.) It is frequently necessary to express the portion of a given
force, which acts in some given direction different from the
immediate direction of the force itself. Thus, if a force act from
A, _fig. 11._, in the direction A C, we may require to
estimate what part of that force acts in the direction A B. If the
force be a pressure, take as many inches A P from A, on the line
A C, as there are ounces in the force, and from P draw P M
perpendicular to A B; then the part of the force which acts along
A B will be as many ounces as there are inches in A M. The
force A B is mechanically equivalent to two forces, expressed by
the sides A M and A N of the parallelogram; but A N,
being perpendicular to A B, can have no effect on a body at A,
in the direction of A B, and therefore the effective part of the
force A P in the direction A B is expressed by A M.