familiar properties which it has in reference to detached bodies of
inconsiderable magnitude, placed on or near the surface of the earth,
yet it possesses properties of a much more general and not less
important nature. One of the most remarkable of these is, that the
centre of gravity of any number of separate bodies is never affected
by the mutual attraction, impact, or other influence which the bodies
may transmit from one to another. This is a necessary consequence of
the equality of action and reaction explained in Chapter IV. For if A
and B, _fig. 67._, attract each other, and change their places
to A′ and B′, the space a a′ will have to _b b′_ the same
proportion as B has to A, and therefore by what has just been proved
(171) the same proportion as _a_ C has to _b_ C. It follows, that the
remainders _a′_ C and _b′_ C will be in the proportion of B to A, and
that C will continue to be the centre of gravity of the bodies after
they have approached by their mutual attraction.
Suppose, for example, that A and B were 12lbs. and 8lbs. respectively,
and that _a b_ were 40 feet. The point C must (171) divide
_a b_ into two parts, in the proportion of 8 to 12, or of 2 to
3. Hence it is obvious that _a_ C will be 16 feet, and _b_ C 24 feet.
Now, suppose that A and B attract each other, and that A approaches
B through two feet. Then B must approach A through three feet. Their
distances from C will now be 14 feet and 21 feet, which, being in the
proportion of B to A, the point C will still be their centre of gravity.
Hence it follows, that if a system of bodies, placed at rest, be
permitted to obey their mutual attractions, although the bodies will
thereby be severally moved, yet their common centre of gravity must
remain quiescent.
(175.) When one of two bodies is moving in a straight line, the other
being at rest, their common centre of gravity must move in a parallel
straight line. Let A and B, _fig. 69._, be the centres of gravity
of the bodies, and let A move from A to _a_, B remaining at rest.
Draw the lines A B and _a_ B. In every position which the body B
assumes during its motion, the centre of gravity C divides the line
joining them into parts A C, B C, which are in the proportion
of the mass B to the mass A. Now, suppose any number of lines drawn
from B to the line A _a_; a parallel C _c_ to A _a_ through C divides
all these lines in the same proportion; and therefore, while the body A
moves from A to _a_, the common centre of gravity moves from C to _c_.
If both the bodies A and B moved uniformly in straight lines, the
centre of gravity would have a motion compounded (74) of the two
motions with which it would be affected, if each moved while the other
remained at rest. In the same manner, if there were three bodies, each
moving uniformly in a straight line, their common centre of gravity
would have a motion compounded of that motion which it would have if
one remained at rest while the other two moved, and that which the
motion of the first would give it if the last two remained at rest; and
in the same manner it may be proved, that when any number of bodies
move each in a straight line, their common centre of gravity will have
a motion compounded of the motions which it receives from the bodies
severally.
It may happen that the several motions which the centre of gravity
receives from the bodies of the system will neutralise each other; and
this does, in fact, take place for such motions as are the consequences
of the mutual action of the bodies upon one another.
(176.) If a system of bodies be not under the immediate influence of
any forces, and their mutual attraction be conceived to be suspended,
they must severally be either at rest or in uniform rectilinear
motion in virtue of their inertia. Hence, their common centre of
gravity must also be either at rest or in uniform rectilinear motion.
Now, if we suppose their mutual attractions to take effect, the
state of their common centre of gravity will not be changed, but the
bodies will severally receive motions compounded of their previous
uniform rectilinear motions and those which result from their mutual
attractions. The combined effects will cause each body to revolve in
an orbit round the common centre of gravity, or will precipitate it
towards that point. But still that point will maintain its former state
undisturbed.
This constitutes one of the general laws of mechanical science, and is
of great importance in physical astronomy. It is known by the title
“the conservation of the motion of the centre of gravity.”
(177.) The solar system is an instance of the class of phenomena to
which we have just referred. All the motions of the bodies which
compose it can be traced to certain uniform rectilinear motions,
received from some former impulse, or from a force whose action has
been suspended, and those motions which necessarily result from the
principle of gravitation. But we shall not here insist further on this
subject, which more properly belongs to another department of the
science.
(178.) If a solid body suffer an impact in the direction of a line
passing through its centre of gravity, all the particles of the body
will be driven forward with the same velocity in lines parallel to the
direction of the impact, and the whole force of the motion will be
equal to that of the impact. The common velocity of the parts of the
body will in this case be determined by the principles explained in
Chapter IV. The impelling force being equally distributed among all the
parts, the velocity will be found by dividing the numerical value of
that force by the number expressing the mass.
If any number of impacts be given simultaneously to different points
of a body, a certain complex motion will generally ensue. The mass
will have a relative motion round the centre of gravity as if it were
fixed, while that point will move forward uniformly in a straight line,
carrying the body with it. The relative motion of the mass round the
centre of gravity may be found by considering the centre of gravity
as a fixed point, round which the mass is free to move, and then
determining the motion which the applied forces would produce. This
motion being supposed to continue uninterrupted, let all the forces be
imagined to be applied in their proper directions and quantities to the
centre of gravity. By the principles for the composition of force they
will be mechanically equivalent to a single force through that point.
In the direction of this single force the centre of gravity will move
and have the same velocity as if the whole mass were there concentrated
and received the impelling forces.
(179.) These general properties, which are entirely independent of
gravity, render the “centre of gravity” an inadequate title for this
important point. Some physical writers have, consequently, called it
the “centre of inertia.” The “centre of gravity,” however, is the name
by which it is still generally designated.
CHAP. X.
THE MECHANICAL PROPERTIES OF AN AXIS.
(180.) When a body has a motion of rotation, the line round which it
revolves is called an _axis_. Every point of the body must in this
case move in a circle, whose centre lies in the axis, and whose radius
is the distance of the point from the axis. Sometimes while the body
revolves, the axis itself is moveable, and not unfrequently in a state
of actual motion. The motions of the earth and planets, or that of
a common spinning-top, are examples of this. The cases, however, which
will be considered in the present chapter, are chiefly those in which
the axis is immovable, or at least where its motion has no relation to
the phenomena under investigation. Instances of this are so frequent
and obvious, that it seems scarcely necessary to particularise them.
Wheel-work of every description, the moving parts of watches and
clocks, turning lathes, mill-work, doors and lids on hinges, are all
obvious examples. In tools or other instruments which work on joints or
pivots, such as scissors, shears, pincers, although the joint or pivot
be not absolutely fixed, it is to be considered so in reference to the
mechanical effect.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
In some cases, as in most of the wheels of watches and clocks,
fly-wheels and chucks of the turning lathe, and the arms of wind-mills,
the body turns continually in the same direction, and each of its
points traverses a complete circle during every revolution of the
body round its axis. In other instances the motion is alternate or
reciprocating, its direction being at intervals reversed. Such is
the case in pendulums of clocks, balance-wheels of chronometers, the
treddle of the lathe, doors and lids on hinges, scissors, shears,
pincers, &c. When the alternation is constant and regular, it is called
_oscillation_ or _vibration_, as in pendulums and balance-wheels.
(181.) To explain the properties of an axis of rotation it will be
necessary to consider the different kinds of forces to the action of
which a body moveable on such an axis may be submitted, to show how
this action depends on their several quantities and directions, to
distinguish the cases in which the forces neutralise each other and
mutually equilibrate from those in which motion ensues, to determine
the effect which the axis suffers, and, in the cases where motion is
produced, to estimate the effects of those centrifugal forces (137.)
which are created by the mass of the body whirling round the axis.
Forces in general have been distinguished by the duration of their
action into instantaneous and continued forces. The effect of an
instantaneous force is produced in an infinitely short time. If
the body which sustains such an action be previously quiescent and
free, it will move with a uniform velocity in the direction of the
impressed force. (93.) If, on the other hand, the body be not free,
but so restrained that the impulse cannot put it in motion, then the
fixed points or lines which resist the motion sustain a corresponding
shock at the moment of the impulse. This effect, which is called
_percussion_, is, like the force which causes it, instantaneous.
A continued force produces a continued effect. If the body be free and
previously quiescent, this effect is a continual increase of velocity.
If the body be so restrained that the applied force cannot put it in
motion, the effect is a continued pressure on the points or lines which
sustain it. (94.)
It may happen, however, that although the body be not absolutely free
to move in obedience to the force applied to it, yet still it may not
be altogether so restrained as to resist the effect of that force and
remain at rest. If the point at which a force is applied be free to
move in a certain direction not coinciding with that of the applied
force, that force will be resolved into two elements; one of which is
in the direction in which the point is free to move, and the other at
right angles to that direction. The point will move in obedience to
the former element, and the latter will produce percussion or pressure
on the points or lines which restrain the body. In fact, in such cases
the resistance offered by the circumstances which confine the motion
of the body modifies the motion which it receives, and as every change
of motion must be the consequence of a force applied (44.), the fixed
points or lines which offer the resistance must suffer a corresponding
effect.
It may happen that the forces impressed on the body, whether they
be continued or instantaneous, are such as, were it free, would
communicate to it a motion which the circumstances which restrain it
do not forbid it to receive. In such a case the fixed points or lines
which restrain the body sustain no force, and the phenomena will be
the same in all respects as if these points or lines were not fixed.
It will be easy to apply these general reflections to the case in which
a solid body is moveable on a fixed axis. Such a body is susceptible
of no motion except one of rotation on that axis. If it be submitted
to the action of instantaneous forces, one or other of the following
effects must ensue. 1. The axis may resist the forces, and prevent any
motion. 2. The axis may modify the effect of the forces sustaining a
corresponding percussion, and the body receiving a motion of rotation.
3. The forces applied may be such as would cause the body to spin round
the axis even were it not fixed, in which case the body will receive a
motion of rotation, but the axis will suffer no percussion.
What has been just observed of the effect of instantaneous forces is
likewise applicable to continued ones. 1. The axis may entirely resist
the effect of such forces, in which case it will suffer a pressure
which may be estimated by the rules for the composition of force. 2.
It may modify the effect of the applied forces, in which case it must
also sustain a pressure, and the body must receive a motion of rotation
which is subject to constant variation, owing to the incessant action
of the forces. 3. The forces may be such as would communicate to the
body the same rotatory motion if the axis were not fixed. In this case
the forces will produce no pressure on the axis.
The impressed forces are not the only causes which affect the axis of
a body during the phenomenon of rotation. This species of motion calls
into action other forces depending on the inertia of the mass, which
produce effects upon the axis, and which play a prominent part in the
theory of rotation. While the body revolves on its axis, the component
particles of its mass move in circles, the centres of which are placed
in the axis. The radius of the circle in which each particle moves is
the line drawn from that particle perpendicular to the axis. It has
been already proved that a particle of matter, moving round a centre,
is attended with a centrifugal force proportionate to the radius of the
circle in which it moves and to the square of its angular velocity.
When a solid body revolves on its axis, all its parts are whirled round
together, each performing a complete revolution in the same time. The
angular velocity is consequently the same for all, and the difference
of the centrifugal forces of different particles must entirely depend
upon their distances from the axis. The tendency of each particle to
fly from the axis, arising from the centrifugal force, is resisted by
the cohesion of the parts of the mass, and in general this tendency is
expended in exciting a pressure or strain upon the axis. It ought to
be recollected, however, that this pressure or strain is altogether
different from that already mentioned, and produced by the forces which
give motion to the body. The latter depends entirely upon the quantity
and directions of the applied forces in relation to the axis: the
former depends on the figure and density of the body, and the velocity
of its motion.
These very complex effects render a simple and elementary exposition
of the mechanical properties of a fixed axis a matter of considerable
difficulty. Indeed, the complete mathematical development of this
theory long eluded the skill of the most acute geometers, and it was
only at a comparatively late period that it yielded to the searching
analysis of modern science.
(182.) To commence with the most simple case, we shall consider the
body as submitted to the action of a single force. The effect of this
force will vary according to the relation of its direction to that of
the axis. There are two ways in which a body may be conceived to be
moveable around an axis. 1. By having pivots at two points which rest
in sockets, so that when the body is moved it must revolve round the
right line joining the pivots as an axis. 2. A thin cylindrical rod may
pass through the body, on which it may turn in the same manner as a
wheel upon its axle.
If the force be applied to the body in the direction of the axis, it
is evident that no motion can ensue, and the effect produced will be a
pressure on that pivot towards which the force is directed. If in this
case the body revolved on a cylindrical rod, the tendency of the force
would be to make it slide along the rod without revolving round it.
Let us next suppose the force to be applied not in the direction of
the axis itself, but parallel to it. Let A B, _fig. 70._,
be the axis, and let C D be the direction of the force applied.
The pivots being supposed to be at A and B, draw A G and B F
perpendicular to A B. The force C D will be equivalent to
three forces, one acting from B towards A, equal in quantity to the
force C D. This force will evidently produce a corresponding
pressure on the pivot A. The other two forces will act in the
directions A G and B F, and will have respectively to the
force C D the same proportion as A E has to A B. Such
will be the mechanical effect of a force C D parallel to the axis.
And as these effects are all directed on the pivots, no motion can
ensue.
If the body revolve on a cylindrical rod, the forces A G and
B F would produce a strain upon the axis, while the third force
in the direction B A would have a tendency to make the body slide
along it.
(183.) If the force applied to the body be directed upon the axis,
and at right angles to it, no motion can be produced. In this case,
if the body be supported by pivots at A and B, the force K L,
perpendicular to the line A B, will be distributed between the
pivots, producing a pressure on each proportional to its distance from
the other. The pressure on A having to the pressure on B the same
proportion as L B has to L A.
If the force K H be directed obliquely to the axis, it will be
equivalent to two forces (76.), one K L perpendicular to the axis,
and the other K M parallel to it. The effect of each of these may
be investigated as in the preceding cases.
In all these observations the body has been supposed to be submitted
to the action of one force only. If several forces act upon it, the
direction of each of them crossing the axis either perpendicularly
or obliquely, or taking the direction of the axis or any parallel
direction, their effects may be similarly investigated. In the same
manner we may determine the effects of any number of forces whose
combined results are mechanically equivalent to forces which either
intersect the axis or are parallel to it.
(184.) If any force be applied whose direction lies in a plane oblique
to the axis, it can always be resolved into two elements (76.), one of
which is parallel to the axis, and the other in a plane perpendicular
to it. The effect of the former has been already determined, and
therefore we shall at present confine our attention to the latter.
Suppose the axis to be perpendicular to the paper, and to pass through
the point G, _fig. 71._ and let A B C be a section of
the body. It will be convenient to consider the section vertical and
the axis horizontal, omitting, however, any notice of the effect of the
weight of the body.
Let a weight W be suspended by a cord Q W from any point Q. This
weight will evidently have a tendency to turn the body round in the
direction A B C. Let another cord be attached to any other
point P, and, being carried over a wheel R, let a dish S be attached to
it, and let fine sand be poured into this dish until the tendency of
S to turn the body round the axis in the direction of C B A
balances the opposite tendency of W. Let the weights of W and S be
then exactly ascertained, and also let the distances G I and
G H of the cords from the axis be exactly measured. It will be
found that, if the number of ounces in the weight S be multiplied by
the number of inches in G H, and also the number of ounces in W
by the number of inches in G I, equal products will be obtained.
This experiment may be varied by varying the position of the wheel R,
and thereby changing the direction of the string P R, in which
cases it will be always found necessary to vary the weight of S in
such a manner, that when the number of ounces in it is multiplied by
the number of inches in the distance of the string from the axis, the
product obtained shall be equal to that of the weight W by the distance
G I. We have here used ounces and inches as the measures of weight
and distance; but it is obvious that any other measures would be
equally applicable.
From what has been just stated it follows, that the energy of the
weight of S to move the body on its axis, does not depend alone upon
the actual amount of that weight, but also upon the distance of the
string from the axis. If, while the position of the string remains
unaltered, the weight of S be increased or diminished, the resisting
weight W must be increased or diminished in the same proportion. But
if, while the weight of S remains unaltered, the distance of the string
P R from the axis G be increased or diminished, it will be found
necessary to increase or diminish the resisting weight W in exactly the
same proportion. It therefore appears that the increase or diminution
of the distance of the direction of a force from the axis has the
same effect upon its power to give rotation as a similar increase
or diminution of the force itself. The power of a force to produce
rotation is, therefore, accurately estimated, not by the force alone,
but by the product found by multiplying the force by the distance of
its direction from the axis. It is frequently necessary in mechanical
science to refer to this power of a force, and, accordingly, the
product just mentioned has received a particular denomination. It is
called the _moment_ of the force round the axis.
(185.) The distance of the direction of a force from the axis is
sometimes called the _leverage_ of the force. The _moment_ of a force
is therefore found by multiplying the force by its leverage, and the
energy of a given force to turn a body round an axis is proportional to
the leverage of that force.
From all that has been observed it may easily be inferred that, if
several forces affect a body moveable on an axis, having tendencies
to turn it in different directions, they will mutually neutralise
each other and produce equilibrium, if the sum of the moments of those
forces which tend to turn the body in one direction be equal to the
sum of the moments of those which tend to turn it in the opposite
direction. Thus, if the forces A, B, C, ... tend to turn the body from
right to left, and the distances of their directions from the axis be
_a_, _b_, _c_, ... and the forces A′, B′, C′, ... tend to move it from
left to right, and the distances of their directions from the axis be
_a′_, _b′_, _c′_, ...; then these forces will produce equilibrium,
if the products found by multiplying the ounces in A, B, C, ...
respectively by the inches in _a_, _b_, _c_, ... when added together
be equal to the products found by multiplying the ounces in A′, B′,
C′, ... by the inches in _a′_, _b′_, _c′_, ... respectively when added
together. But if either of these sets of products when added together
exceed the other, the corresponding set of forces will prevail, and the
body will revolve on its axis.
(186.) When a body receives an impulse in a direction perpendicular to
the axis, but not crossing it, a uniform rotatory motion is produced.
The velocity of this motion depends on the force of the impulse, the
distance of the direction of the impulse from the axis, and the manner
in which the mass of the body is distributed round the axis. It is to
be considered that the whole force of the impulse is shared amongst the
various parts of the mass, and is transmitted to them from the point
where the impulse is applied by reason of the cohesion and tenacity
of the parts, and the impossibility of one part yielding to a force
without carrying all the other parts with it. The force applied acts
upon those particles nearer to the axis than its own direction under
advantageous circumstances; for, according to what has been already
explained, their power to resist the effect of the applied force is
small in the same proportion with their distance. On the other hand,
the applied force acts upon particles of the mass, at a greater
distance than its own direction, under circumstances proportionably
disadvantageous; for their resistance to the applied force is great in
proportion to their distances from the axis.