This formula includes the entire theory of centrifugal force.
But it can be shown that the number expressed by _c_ is 1·226, and
consequently
C = (1·226 × W × R)/T^2.
It is often more convenient to use the number of revolutions made in
a given time than the time of one revolution. Let N then express the
number of revolutions, or fraction of a revolution, made in one second,
and we shall have
T = 1/N.
Therefore
C = 1·226 × W × R × N^2.
(140.) The consideration of centrifugal force proves, that if a body
be observed to move in a curvilinear path, some efficient cause must
exist which prevents it from flying off, and which compels it to
revolve round the centre. If the body be connected with the centre by
a thread, cord, or rod, then the effect of the centrifugal force is
to give tension to the thread, cord, or rod. If an unyielding curved
surface be placed on the convex side of the path, then the force will
produce pressure on this surface. But if a body is observed to move
in a curve without any visible material connection with its centre,
and without any obstruction on the convex side of its path to resist
its retreat, as is the case with the motions of the planets round
the sun, and the satellites round the planets, it is usual to assign
the cause to the attraction of the body which occupies the centre:
in the present instance the sun is that body, and it is customary to
say that the _attraction_ of the sun, neutralising the effects of the
centrifugal force of the planets, _retains them_ in their orbits. We
have elsewhere animadverted on the inaccurate and unphilosophical style
of this phraseology, in which terms are admitted which intimate not
only an unknown cause, but assign its seat, and intimate something of
its nature. All that we are entitled to declare in this case is, that
a motion is continually impressed upon the planet; that this motion is
directed towards the sun; that it counteracts the centrifugal force;
but from whence this motion proceeds, whether it be a virtue resident
in the sun, or a property of the medium or space in which both sun and
planets are placed, or whatever other influence may be its proximate
cause, we are altogether ignorant.
* * * * *
(141.) Numerous examples of the effects of centrifugal force may be
produced.
If a stone or other weight be placed in a sling, which is whirled round
by the hand in a direction perpendicular to the ground, the stone will
not fall out of the sling, even when it is at the top of its circuit,
and, consequently, has no support beneath it. The centrifugal force, in
this case, acting from the hand, which is the centre of rotation, is
greater than the weight of the body, and therefore prevents its fall.
In like manner, a glass of water may be whirled so rapidly that even
when the mouth of the glass is presented downwards, the water will
still be retained in it by the centrifugal force.
If a bucket of water be suspended by a number of threads, and these
threads be twisted by turning round the bucket many times in the same
direction, on allowing the cords to untwist, the bucket will be whirled
rapidly round, and the water will be observed to rise on its sides and
sink at its centre, owing to the centrifugal force with which it is
driven from the centre. This effect might be carried so far, that all
the water would flow over and leave the bucket nearly empty.
(142.) A carriage, or horseman, or pedestrian, passing a corner moves
in a curve, and suffers a centrifugal force, which increases with
the velocity, and which impresses on the body a force directed from
the corner. An animal causes its weight to resist this force, by
voluntarily inclining its body towards the corner. In this case, let
A B, _fig. 30._, be the body; C D is the direction of
the weight perpendicular to the ground, and C F is the direction
of the centrifugal force parallel to the ground and _from_ the corner.
The body A B is inclined to the corner, so that the diagonal force
(74), which is mechanically equivalent to the weight and centrifugal
force, shall be in the direction C A, and shall therefore produce
the pressure of the feet upon the ground.
As the velocity is increased, the centrifugal force is also increased,
and therefore a greater inclination of the body is necessary to resist
it. We accordingly find that the more rapidly a corner is turned, the
more the animal inclines his body towards it.
A carriage, however, not having voluntary motion, cannot make this
compensation for the disturbing force which is called into existence
by the gradual change of direction of the motion; consequently it
will, under certain circumstances, be overturned, falling of course
outwards, or _from_ the corner. If A B be the carriage, and
C, _fig. 31._, the place at which the weight is principally
collected, this point C will be under the influence of two forces: the
weight, which may be represented by the perpendicular C D, and the
centrifugal force, which will be represented by a line C F, which
shall have the same proportion to C D as the centrifugal force
has to the weight. Now the combined effect of these two forces will be
the same as the effect of a single force, represented by C G.
Thus, the pressure of the carriage on the road is brought nearer to the
outer wheel B. If the centrifugal force bear the same proportion to the
weight as C F (or D B), _fig. 32._, bears to C D,
the whole pressure is thrown upon the wheel B.
If the centrifugal force bear to the weight a greater proportion than
D B has to C D, then the line C F, which represents it,
_fig. 33._, will be greater than D B. The diagonal C G,
which represents the combined effects of the weight and centrifugal
force, will in this case pass outside the wheel B, and therefore this
resultant will be unresisted. To perceive how far it will tend to
overturn the carriage, let the force C G be resolved into two, one
in the direction of C B, and the other C K, perpendicular to
C B. The former C B will be resisted by the road, but the
latter C K will tend to lift the carriage over the external wheel.
If the velocity and the curvature of the course be continued for a
sufficient time to enable this force C K to elevate the weight,
so that the line of direction shall fall on B, the carriage will be
overthrown.
It is evident from what has been now stated, that the chances of
overthrow under these circumstances depend on the proportion of
B D to C D, or what is to the same purpose, of the distance
between the wheels to the height of the principal seat of the load.
It will be shown in the next chapter, that there is a certain point,
called the centre of gravity, at which the entire weight of the vehicle
and its load may be conceived to be concentrated. This is the point
which in the present investigation we have marked C. The security of
the carriage, therefore, depends on the greatness of the distance
between the wheels and the smallness of the elevation of the centre of
gravity above the road; for either or both of these circumstances will
increase the proportion of B D to C D.
(143.) In the equestrian feat exhibited in the ring at the
amphitheatre, when the horse moves round with the performer standing on
the saddle, both the horse and rider incline continually towards the
centre of the ring, and the inclination increases with the velocity of
the motion: by this inclination their weights counteract the effect
of the centrifugal force, exactly as in the case already mentioned
(142.)
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
(144.) If a body be allowed to fall by its weight down a convex
surface, such as A B, _fig. 34._, it would continue upon the
surface until it arrive at B but for the effect of the centrifugal
force: this, giving it a motion from the centre of the curve, will
cause it to quit the curve at a certain point C, which can be easily
found by mathematical computation.
(145.) The most remarkable and important manifestation of centrifugal
force is observed in the effects produced by the rotation of the earth
upon its axis. Let the circle in _fig. 35._ represent a section
of the earth, A B being the axis on which it revolves. This
rotation causes the matter which composes the mass of the earth to
revolve in circles round the different points of the axis as centres
at the various distances at which the component parts of this mass
are placed. As they all revolve with the same angular velocity, they
will be affected by centrifugal forces, which will be greater or less
in proportion as their distances from the centre are greater or less.
Consequently the parts of the earth which are situated about the
equator, D, will be more strongly affected by centrifugal force than
those about the poles, A B. The effect of this difference has been
that the component matter about the equator has actually been driven
farther from the centre than that about the poles, so that the figure
of the earth has swelled out at the sides, and appears proportionally
depressed at the top and bottom, resembling the shape of an orange. An
exaggerated representation of this figure is given in _fig. 36._;
the real difference between the distances of the poles and equator
from the centre being too small to be perceptible in a diagram. The
exact proportion of C A to C D has never yet been certainly
ascertained. Some observations make C D exceed C A by 1/277,
and others by only 1/333. The latter, however, seems the more probable.
It may be considered to be included between these limits.
The same cause operates more powerfully in other planets which revolve
more rapidly on their axes. Jupiter and Saturn have forms which are
considerably more elliptical.
(146.) The centrifugal force of the earth’s rotation also affects
detached bodies on its surface. If such bodies were not held upon the
surface by the earth’s attraction, they would be immediately flung
off by the whirling motion in which they participate. The centrifugal
force, however, really diminishes the effects of the earth’s attraction
on those bodies, or, what is the same, diminishes their weights. If
the earth did not revolve on its axis, the weight of bodies in all
places equally distant from the centre would be the same; but this is
not so when the bodies, as they do, move round with the earth. They
acquire from the centrifugal force a tendency to fly from the axis,
which increases with their distance from that axis, and is therefore
greater the nearer they are to the equator, and less as they approach
the pole. But there is another reason why the centrifugal force is more
efficient, in the opposition which it gives to gravity near the equator
than near the poles. This force does not act from the centre of the
earth, but is directed from the earth’s axis. It is, therefore, not
directly opposed to gravity, except on the equator itself. On leaving
the equator, and proceeding towards the poles, it is less and less
opposed to gravity, as will be plain on inspecting _fig. 35._,
where the lines P C all represent the direction of gravity, and
the lines P F represent the direction of the centrifugal force.
Since, then, as we proceed from the equator towards the poles, not only
the amount of the centrifugal force is continually diminished, but also
it acts less and less in opposition to gravity, it follows that the
weights of bodies are most diminished by it at the equator, and less so
towards the poles.
Since bodies are commonly weighed by balancing them against other
bodies of known weight, it may be asked, how the phenomena we have been
just describing can be ascertained as a matter of fact? for whatever
be the body against which it may be balanced, that body must suffer
just as much diminution of weight as every other, and consequently, all
being diminished in the same proportion, the balance will be preserved
though the weights be changed.
To render this effect observable, it will be necessary to compare the
effects of gravity with some phenomenon which is not affected by the
centrifugal force of the earth’s rotation, and which will be the same
at every part of the earth. The means of accomplishing this will be
explained in a subsequent chapter.
CHAP. IX.
THE CENTRE OF GRAVITY.
(147.) By the earth’s attraction, all the particles which compose
the mass of a body are solicited by equal forces in parallel
directions downwards. If these component particles were placed in mere
juxtaposition, without any mechanical connection, the force impressed
on any one of them could in nowise affect the others, and the mass
would, in such a case, be contemplated as an aggregation of small
particles of matter, each urged by an independent force. But the bodies
which are the subjects of investigation in mechanical science are not
found in this state. Solid bodies are coherent masses, the particles
of which are firmly bound together, so that any force which affects
one, being modified according to circumstances, will be transmitted
through the whole body. Liquids accommodate themselves to the shape of
the surfaces on which they rest, and forces affecting any one part are
transmitted to others, in a manner depending on the peculiar properties
of this class of bodies.
As all bodies, which are subjects of mechanical enquiry, on the surface
of the earth, must be continually influenced by terrestrial gravity,
it is desirable to obtain some easy and summary method of estimating
the effect of this force. To consider it, as is unavoidable in the
first instance, the combined action of an infinite number of equal and
parallel forces soliciting the elementary molecules downwards, would
be attended with manifest inconvenience. An infinite number of forces,
and an infinite subdivision of the mass, would form parts of every
mechanical problem.
To overcome this difficulty, and to obtain all the ease and simplicity
which can be desired in elementary investigations, it is only necessary
to determine some force, whose single effect shall be equivalent to the
combined effects of the gravitation of all the molecules of the body.
If this can be accomplished, that single force might be introduced into
all problems to represent the whole effect of the earth’s attraction,
and no regard need be had to any particles of the body, except that on
which this force acts.
(148.) To discover such a force, if it exist, we shall first enquire
what properties must necessarily characterise it. Let A B,
_fig. 37._, be a solid body placed near the surface of the
earth. Its particles are all solicited downwards, in the directions
represented by the arrows. Now, if there be any single force equivalent
to these combined effects, two properties may be at once assigned to
it: 1. It must be presented downwards, in the common direction of
those forces to which it is mechanically equivalent; and, 2. it must
be equal in intensity to their sum, or, what is the same, to the force
with which the whole mass would descend. We shall then suppose it to
have this intensity, and to have the direction of the arrow D E.
Now, if the single force, in the direction D E, be equivalent
to all the separate attractions which affect the particles, we may
suppose all these attractions removed, and the body A B influenced
only by a single attraction, acting in the direction D E. This
being admitted, it follows that if the body be placed upon a prop,
immediately under the direction of the line D E, or be suspended
from a fixed point immediately above its direction, it will remain
motionless. For the whole attracting force in the direction D E
will, in the one case, press the body on the prop, and, in the other
case, will give tension to the cord, rod, or whatever other means of
suspension be used.
(149.) But suppose the body were suspended from some point P, not in
the direction of the line D E. Let P C be the direction of
the thread by which the body is suspended. Its whole weight, according
to the supposition which we have adopted, must then act in the
direction C E. Taking C F to represent the weight; it may be
considered as mechanically equivalent to two forces (74), C I and
C H. Of these C H, acting directly from the point P, merely
produces pressure upon it, and gives tension to the cord P C; but
C I, acting at right angles to C P, produces motion round P
as a centre, and in the direction C I, towards a vertical line
P G, drawn through the point P. If the body A B had been on
the other side of the line P G, it would have moved in like manner
towards it, and therefore in the direction contrary to its present
motion.
Hence we must infer, that when the body is suspended from a fixed
point, it cannot remain at rest, if that fixed point be not placed in
the direction of the line D E; and, on the other hand, that if
the fixed point _be_ in the direction of that line, it cannot move. A
practical test is thus suggested, by which the line D E may be at
once discovered. Let a thread be attached to any point of the body, and
let it be suspended by this thread from a hook or other fixed point.
The direction of the thread, when the body becomes quiescent, will
be that of a single force equivalent to the gravitation of all the
component parts of the mass.
(150.) An enquiry is here suggested: does the direction of the
equivalent force thus determined depend on the position of the body
with respect to the surface of the earth, and how is the direction
of the equivalent force affected by a change in that position? This
question may be at once solved if the body be suspended by different
points, and the directions which the suspending thread takes in each
case relatively to the figure and dimensions of the body examined.
The body being suspended in this manner from any point, let a small
hole be bored through it, in the exact direction of the thread, so that
if the thread were continued below the point where it is attached to
the body, it would pass through this hole. The body being successively
suspended by several different points on its surface, let as many small
holes be bored through it in the same manner. If the body be then cut
through, so as to discover the directions which the several holes have
taken, they will be all found to cross each other at one point within
the body; or the same fact may be discovered thus: a thin wire, which
nearly fills the holes being passed through any one of them, it will be
found to intercept the passage of a similar wire through any other.
This singular fact teaches us, what indeed can be proved by
mathematical reasoning without experiment, that there is _one_ point in
every body through which the single force, which is equivalent to the
gravitation of all its particles, must pass, in whatever position the
body be placed. This point is called _the centre of gravity_.
(151.) In whatever situation a body may be placed, the centre of
gravity will have a tendency to descend in the direction of a
line perpendicular to the horizon, and which is called the _line
of direction_ of the weight. If the body be altogether free and
unrestricted by any resistance or impediment, the centre of gravity
will actually descend in this direction, and all the other points of
the body will move with the same velocity in parallel directions,
so that during its fall the position of the parts of the body, with
respect to the ground, will be unaltered. But if the body, as is most
usual, be subject to some resistance or restraint, it will either
remain unmoved, its weight being expended in exciting pressure on the
restraining points or surfaces, or it will move in a direction and
with a velocity depending on the circumstances which restrain it.
In order to determine these effects, to predict the pressure produced
by the weight if the body be quiescent, or the mixed effects of motion
and pressure, if it be not so, it is necessary in all cases to be
able to assign the place of the centre of gravity. When the magnitude
and figure of the body, and the density of the matter which occupies
its dimensions, are known, the place of the centre of gravity can be
determined with the greatest precision by mathematical calculation.
The process by which this is accomplished, however, is not of a
sufficiently elementary nature to be properly introduced into this
treatise. To render it intelligible would require the aid of some
of the most advanced analytical principles; and even to express the
position of the point in question, except in very particular instances,
would be impossible, without the aid of peculiar symbols.
(152.) There are certain particular forms of body in which, when they
are uniformly dense, the place of the centre of gravity can be easily
assigned, and proved by reasoning, which is generally intelligible;
but in all cases whatever, this point may be easily determined by
experiment.
(153.) If a body uniformly dense have such a shape that a point may be
found on either side of which in all directions around it the materials
of the body are similarly distributed, that point will obviously be
the centre of gravity. For if it be supported, the gravitation of the
particles on one side drawing them downwards, is resisted by an effect
of exactly the same kind and of equal amount on the opposite side, and
so the body remains balanced on the point.
The most remarkable body of this kind is a globe, the centre of which
is evidently its centre of gravity.
A figure, such as _fig. 38._, called an _oblate spheroid_, has its
centre of gravity at its centre, C. Such is the figure of the earth.
The same may be observed of the elliptical solid, _fig. 39._,
which is called a prolate spheroid.
A cube, and some other regular solids, bounded by plane surfaces, have
a point within them, such as above described, and which is, therefore,
their centre of gravity. Such are _fig. 40._
A straight wand of uniform thickness has its centre of gravity at the
centre of its length; and a cylindrical body has its centre of gravity
in its centre, at the middle of its length or axis. Such is the point
C, _fig. 41._
A flat plate of any uniform substance, and which has in every part
an equal thickness, has its centre of gravity at the middle of its
thickness, and under a point of its surface, which is to be determined
by its shape. If it be circular or elliptical, this point is its
centre. If it have any regular form, bounded by straight edges, it is
that point which is equally distant from its several angles, as C in
_fig. 42._
(154.) There are some cases in which, although the place of the centre
of gravity is not so obvious as in the examples just given, still
it may be discovered without any mathematical process, which is not
easily understood. Suppose A B C, _fig. 43._, to be
a flat triangular plate of uniform thickness and density. Let it be
imagined to be divided into narrow bars, by lines parallel to the side
A C, as represented in the figure. Draw B D from the angle
B to the middle point D of the side A C. It is not difficult to
perceive, that B D will divide equally all the bars into which the
triangle is conceived to be divided. Now if the flat triangular plate
A B C be placed in a horizontal position on a straight edge
coinciding with the line B D, it will be balanced: for the bars
parallel to A C will be severally balanced by the edge immediately
under their middle point; since that middle point is the centre of
gravity of each bar. Since, then, the triangle is balanced on the edge,
the centre of gravity must be somewhere immediately over it, and must,
therefore, be within the plate at some point under the line B D.
The same reasoning will prove that the centre of gravity of the plate
is under the line A E, drawn from the angle A to the middle
point E of the side B C. To perceive this, it is only necessary
to consider the triangle divided into bars parallel to B C,
and thence to show that it will be balanced on an edge placed under
A E. Since then the centre of gravity of the plate is under the
line B D, and also under A E, it must be under the point G,
at which these lines cross each other; and it is accordingly at a depth
beneath G, equal to half the thickness of the plate.
This may be experimentally verified by taking a piece of tin or card,
and cutting it into a triangular form. The point G being found by
drawing B D and A E, which divide two sides equally, it will
be balanced if placed upon the point of a pin at G.