be able to find the position of the centre of gravity of any plate of
uniform thickness and density which is bounded by straight edges, as
will be shown hereafter. (173.)
(155.) The centre of gravity is not always included within the volume
of the body, that is, it is not enclosed by its surfaces. Numerous
examples of this can be produced. If a piece of wire be bent into any
form, the centre of gravity will rarely be in the wire. Suppose it be
brought to the form of a ring. In that case, the centre of gravity of
the wire will be the centre of the circle, a point not forming any
part of the wire itself: nevertheless this point may be proved to have
the characteristic property of the centre of gravity; for if the ring
be suspended by any point, the centre of the ring must always settle
itself under the point of suspension. If this centre could be supposed
to be connected with the ring by very fine threads, whose weight would
be insignificant, and which might be united by a knot or otherwise at
the centre, the ring would be balanced upon a point placed under the
knot.
In like manner, if the wire be formed into an ellipse, or any other
curve similarly arranged round a centre point, that point will be its
centre of gravity.
(156.) To find the centre of gravity experimentally, the method
described in (149, 150) may be used. In this case two points of
suspension will be sufficient to determine it; for the directions of
the suspending cord being continued through the body, will cross each
other at the centre of gravity. These directions may also be found
by placing the body on a sharp point, and adjusting it so as to be
balanced upon it. In this case a line drawn through the body directly
upwards from the point will pass through the centre of gravity, and
therefore two such lines must cross at that point.
(157.) If the body have two flat parallel surfaces like sheet metal,
stiff paper, card, board, &c., the centre of gravity may be found by
balancing the body in two positions on an horizontal straight edge.
The point where the lines marked by the edge cross each other will
be immediately under the centre of gravity. This may be verified by
showing that the body will be balanced on a point thus placed, or that
if it be suspended, the point thus determined will always come under
the point of suspension.
The position of the centre of gravity of such bodies may also be found
by placing the body on an horizontal table having a straight edge.
The body being moved beyond the edge until it is in that position in
which the slightest disturbance will cause it to fall, the centre of
gravity will then be immediately over the edge. This being done in two
positions, the centre of gravity will be determined as before.
(158.) It has been already stated, that when the body is perfectly
free, the centre of gravity must necessarily move downwards, in a
direction perpendicular to an horizontal plane. When the body is not
free, the circumstances which restrain it generally permit the centre
of gravity to move in certain directions, but obstruct its motion in
others. Thus if a body be suspended from a fixed point by a flexible
cord, the centre of gravity is free to move in every direction except
those which would carry it farther from the point of suspension than
the length of the cord. Hence if we conceive a globe or sphere to
surround the point of suspension on every side to a distance equal to
that of the centre of gravity from the point of suspension, when the
cord is fully stretched, the centre of gravity will be at liberty to
move in every direction within this sphere.
There are an infinite variety of circumstances under which the motion
of a body may be restrained, and in which a most important and useful
class of mechanical problems originate. Before we notice others,
we shall, however, examine that which has just been described more
particularly.
Let P, _fig. 44._, be the point of suspension, and C the centre
of gravity, and suppose the body so placed that C shall be within the
sphere already described. The cord will therefore be slackened, and in
this state the body will be free. The centre of gravity will therefore
descend in the perpendicular direction until the cord becomes fully
extended; the tension will then prevent its further motion in the
perpendicular direction. The downward force must now be considered as
the diagonal of a parallelogram, and equivalent to two forces C D
and C E, in the directions of the sides, as already explained in
(149). The force C D will bring the centre of gravity into the
direction P F, perpendicularly under the point of suspension.
Since the force of gravity acts continually on C in its approach to
P F, it will move towards that line with accelerated speed, and
when it has arrived there it will have acquired a force to which no
obstruction is immediately opposed, and consequently by its inertia
it retains this force, and moves beyond P F on the other side.
But when the point C gets into the line P F, it is in the lowest
possible position; for it is at the lowest point of the sphere which
limits its motion. When it passes to the other side of P F, it
must therefore begin to ascend, and the force of gravity, which, in the
former case, accelerated its descent, will now for the same reason, and
with equal energy, oppose its ascent. This will be easily understood.
Let C′ be any point which it may have attained in ascending;
C′ G′, the force of gravity, is now equivalent to C′ D′ and
C′ E′. The latter as before produces tension; but the former
C′ D′ is in a direction immediately opposed to the motion, and
therefore retards it. This retardation will continue until all the
motion acquired by the body in its descent from the first position
has been destroyed, and then it will begin to return to P F, and
so it will continue to vibrate from the one side to the other until
the friction on the point P, and the resistance of the air, gradually
deprive it of its motion, and bring it to a state of rest in the
direction P F.
But for the effects of friction and atmospheric resistance, the body
would continue for ever to oscillate equally from side to side of the
line P F.
(159.) The phenomenon just developed is only an example of an extensive
class. Whenever the circumstances which restrain the body are of such
a nature that the centre of gravity is prevented from descending below
a certain level, but not, on the other hand, restrained from rising
above it, the body will remain at rest if the centre of gravity be
placed at the lowest limit of its level; any disturbance will cause
it to oscillate around this state, and it cannot return to a state of
rest until friction or some other cause have deprived it of the motion
communicated by the disturbing force.
(160.) Under the circumstances which we have just described, the body
could not maintain itself in a state of rest in any position except
that in which the centre of gravity is, at the lowest point of the
space in which it is free to move. This, however, is not always the
case. Suppose it were suspended by an inflexible rod instead of a
flexible string; the centre of gravity would then not only be prevented
from receding from the point of suspension, but also from approaching
it; in fact, it would be always kept at the same distance from it.
Thus, instead of being capable of moving anywhere within the sphere,
it is now capable of moving on its surface only. The reasoning used in
the last case may also be applied here, to prove that when the centre
of gravity is on either side of the perpendicular P F, it will
fall towards P F and oscillate, and that if it be placed in the
line P F, it will remain in equilibrium. But in this case there is
another position, in which the centre of gravity may be placed so as to
produce equilibrium. If it be placed at the highest point of the sphere
in which it moves, the whole force acting on it will then be directed
on the point of suspension, perpendicularly downwards, and will be
entirely expended in producing pressure on that point; consequently,
the body will in this case be in equilibrium. But this state of
equilibrium is of a character very different from that in which the
centre of gravity was at the lowest part of the sphere. In the present
case any displacement, however slight, of the centre of gravity, will
carry it to a lower level, and the force of gravity will then prevent
its return to its former state, and will impel it downwards until it
attain the lowest point of the sphere, and round that point it will
oscillate.
(161.) The two states of equilibrium which have been just noticed, are
called stable and instable equilibrium. The character of the former is,
that any disturbance of the state produces oscillation about it; but
any disturbance of the latter state produces a total overthrow, and
finally causes oscillation around the state of stable equilibrium.
Let A B, _fig. 45._, be an elliptical board resting on
its edge on an horizontal plane. In the position here represented,
the extremity P of the lesser axis being the point of support, the
board is in stable equilibrium; for any motion on either side must
cause the centre of gravity C to ascend in the directions C O,
and oscillation will ensue. If, however, it rest upon the smaller
end, as in _fig. 46._, the position would still be a state of
equilibrium, because the centre of gravity is directly above the point
of support; but it would be instable equilibrium, because the slightest
displacement of the centre of gravity would cause it to descend.
Thus an egg or a lemon may be balanced on the end, but the least
disturbance will overthrow it. On the contrary, it will easily rest on
the side, and any disturbance will produce oscillation.
(162.) When the circumstances under which the body is placed allow the
centre of gravity to move only in an horizontal line, the body is in a
state which may be called _neutral equilibrium_. The slightest force
will move the centre of gravity, but will neither produce oscillation
nor overthrow the body, as in the last two cases.
An example of this state is furnished by a cylinder placed upon an
horizontal plane. As the cylinder is rolled upon the plane, the centre
of gravity C, _fig. 47._, moves in a line parallel to the plane
A B, and distant from it by the radius of the cylinder. The body
will thus rest indifferently in any position, because the line of
direction always falls upon a point P at which the body rests upon the
plane.
If the plane were inclined, as in _fig. 48._, a body might be
so shaped, that while it would roll the centre of gravity would move
horizontally. In this case the body would rest indifferently on any
part of the plane, as if it were horizontal, provided the friction be
sufficient to prevent the body from sliding down the plane.
If the centre of gravity of a cylinder happen not to coincide with
its centre by reason of the want of uniformity in the materials of
which it is composed, it will not be in a state of neutral equilibrium
on an horizontal plane, as in _fig. 47._ In this case let G,
_fig. 49._, be the centre of gravity. In the position here
represented, where the centre of gravity is immediately _below_ the
centre C, the state will be stable equilibrium, because a motion
on either side would cause the centre of gravity to ascend; but in
_fig. 50._, where G is immediately above C, the state is instable
equilibrium, because a motion on either side would cause G to descend,
and the body would turn into the position _fig. 49._
(163.) A cylinder of this kind will, under certain circumstances, roll
up an inclined plane. Let A B, _fig. 51._, be the inclined
plane, and let the cylinder be so placed that the line of direction
from G shall be _above_ the point P at which the cylinder rests upon
the plane. The whole weight of the body acting in the direction
G D will obviously cause the cylinder to roll towards A, provided
the friction be sufficient to prevent sliding; but although the
cylinder in this case ascends, the centre of gravity G really descends.
When G is so placed that the line of direction G D shall fall on
the point P, the cylinder will be in equilibrium, because its weight
acts upon the point on which it rests. There are two cases represented
in _fig. 52._ and _fig. 53._, in which G takes this position.
_Fig. 52._ represents the state of stable, and _fig. 53._ of
instable equilibrium.
(164.) When a body is placed upon a base, its stability depends upon
the position of the line of direction and the height of the centre of
gravity above the base. If the line of direction fall within the base,
the body will stand firm; if it fall on the edge of the base, it will
be in a state in which the slightest force will overthrow it on that
side at which the line of direction falls; and if the line of direction
fall without the base, the body must turn over that edge which is
nearest to the line of direction.
In _fig. 54._ and _fig. 55._, the line of direction G P
falls within the base, and it is obvious that the body will stand firm;
for any attempt to turn it over either edge would cause the centre of
gravity to ascend. But in _fig. 56._ the line of direction falls
upon the edge, and if the body be turned over, the centre of gravity
immediately commences to descend. Until it be turned over, however, the
centre of gravity is supported by the edge.
In _fig. 57._ the line of direction falls outside the base, the
centre of gravity has a tendency to descend from G towards A, and the
body will accordingly fall in that direction.
(165.) When the line of direction falls within the base, bodies will
always stand firm, but not with the same degree of stability. In
general, the stability depends on the height through which the centre
of gravity must be elevated before the body can be overthrown. The
greater this height is, the greater in the same proportion will be the
stability.
Let B A C, _fig. 58._, be a pyramid, the centre of
gravity being at G. To turn this over the edge B, the centre of
gravity; must be carried over the arch G E, and must therefore
be raised through the height H E. If, however, the pyramid were
taller relatively to its base, as in _fig. 59._, the height
H E would be proportionally less; and if the base were very small
in reference to the height, as in _fig. 60._, the height H E
would be very small, and a slight force would throw it over the edge B.
It is obvious that the same observations may be applied to all figures
whatever, the conclusions just deduced depending only on the distance
of the line of direction from the edge of the base, and the height of
the centre of gravity above it.
(166.) Hence we may perceive the principle on which the stability of
loaded carriages depends. When the load is placed at a considerable
elevation above the wheels, the centre of gravity is elevated, and the
carriage becomes proportionally insecure. In coaches for the conveyance
of passengers, the luggage is therefore sometimes placed below the body
of the coach; light parcels of large bulk may be placed on the top with
impunity.
When the centre of gravity of a carriage is much elevated, there is
considerable danger of overthrow, if a corner be turned sharply and
with a rapid pace; for the centrifugal force then acting on the centre
of gravity will easily raise it through the small height which is
necessary to turn the carriage over the external wheels (142).
(167.) The same waggon will have greater stability when loaded with
a heavy substance which occupies a small space, such as metal, than
when it carries the same weight of a lighter substance, such as hay;
because the centre of gravity in the latter case will be much more
elevated.
[Illustration: _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
If a large table be placed upon a single leg in its centre, it will
be impracticable to make it stand firm; but if the pillar on which it
rests terminate in a tripod, it will have the same stability as if it
had three legs attached to the points directly over the places where
the feet of the tripod rest.
(168.) When a solid body is supported by more points than one, it is
not necessary for its stability that the line of direction should fall
on one of those points. If there be only two points of support, the
line of direction must fall between them. The body is in this case
supported as effectually as if it rested on an edge coinciding with a
straight line drawn from one point of support to the other. If there
be three points of support, which are not ranged in the same straight
line, the body will be supported in the same manner as it would be by
a base coinciding with the triangle formed by straight lines joining
the three points of support. In the same manner, whatever be the
number of points on which the body may rest, its virtual base will be
found by supposing straight lines drawn, joining the several points
successively. When the line of direction falls within this base, the
body will always stand firm, and otherwise not. The degree of stability
is determined in the same manner as if the base were a continued
surface.
(169.) Necessity and experience teach an animal to adapt its postures
and motions to the position of the centre of gravity of his body. When
a man stands, the line of direction of his weight must fall within the
base formed by his feet. If A B, C D, _fig. 61._, be the
feet, this base is the space A B D C. It is evident,
that the more his toes are turned outwards, the more contracted the
base will be in the direction E F, and the more liable he will be
to fall backwards or forwards. Also, the closer his feet are together,
the more contracted the base will be in the direction G H, and the
more liable he will be to fall towards either side.
When a man walks, the legs are alternately lifted from the ground, and
the centre of gravity is either unsupported or thrown from the one
side to the other. The body is also thrown a little forward, in order
that the tendency of the centre of gravity to fall in the direction of
the toes may assist the muscular action in propelling the body. This
forward inclination of the body increases with the speed of the motion.
But for the flexibility of the knee-joint the labour of walking would
be much greater than it is; for the centre of gravity would be more
elevated by each step. The line of motion of the centre of gravity in
walking is represented by _fig. 62._, and deviates but little
from a regular horizontal line, so that the elevation of the centre
of gravity is subject to very slight variation. But if there were no
knee-joint, as when a man has wooden legs, the centre of gravity would
move as in _fig. 63._, so that at each step the weight of the body
would be lifted through a considerable height, and therefore the labour
of walking would be much increased.
If a man stand on one leg, the line of direction of his weight must
fall within the space on which his foot treads. The smallness of this
space, compared with the height of the centre of gravity, accounts for
the difficulty of this feat.
The position of the centre of gravity of the body changes with the
posture and position of the limbs. If the arm be extended from one
side, the centre of gravity is brought nearer to that side than it was
when the arm hung perpendicularly. When dancers, standing on one leg,
extend the other at right angles to it, they must incline the body in
the direction opposite to that in which the leg is extended, in order
to bring the centre of gravity over the foot which supports them.
When a porter carries a load, his position must be regulated by the
centre of gravity of his body and the load taken together. If he bore
the load on his back, the line of direction would pass beyond his
heels, and he would fall backwards. To bring the centre of gravity
over his feet he accordingly leans forward, _fig. 64._
If a nurse carry a child in her arms, she leans back for a like reason.
When a load is carried on the head, the bearer stands upright, that the
centre of gravity may be over his feet.
In ascending a hill, we appear to incline forward; and in descending,
to lean backward, but in truth, we are standing upright with respect to
a level plane. This is necessary to keep the line of direction between
the feet, as is evident from _fig. 65._
A person sitting on a chair which has no back cannot rise from it
without either stooping forward to bring the centre of gravity over
the feet, or drawing back the feet to bring them under the centre of
gravity.
A quadruped never raises both feet on the same side simultaneously,
for the centre of gravity would then be unsupported. Let
A B C D, _fig. 66._, be the feet. The base on
which it stands is A B C D, and the centre of gravity
is nearly over the point O, where the diagonals cross each other. The
legs A and C being raised together, the centre of gravity is supported
by the legs B and D, since it falls between them; and when B and D
are raised it is, in like manner, supported by the feet A and C. The
centre of gravity, however, is often unsupported for a moment; for the
leg B is raised from the ground before A comes to it, as is plain from
observing the track of a horse’s feet, the mark of A being upon or
before that of B. In the more rapid paces of all animals the centre of
gravity is at intervals unsupported.
The feats of rope-dancers are experiments on the management of the
centre of gravity. The evolutions of the performer are found to be
facilitated by holding in his hand a heavy pole. His security in
this case depends, not on the centre of gravity of his body, but on
that of his body and the pole taken together. This point is near the
centre of the pole, so that, in fact, he may be said to hold in his
hands the point on the position of which the facility of his feats
depends. Without the aid of the pole the centre of gravity would be
within the trunk of the body, and its position could not be adapted to
circumstances with the same ease and rapidity.
(170.) The centre of gravity of a mass of fluid is that point which
would have the properties which have been proved to belong to the
centre of gravity of a solid, if the fluid were solidified without
changing in any respect the quantity or arrangement of its parts. This
point also possesses other properties, in reference to fluids, which
will be investigated in HYDROSTATICS and PNEUMATICS.
(171.) The centre of gravity of two bodies separated from one another,
is that point which would possess the properties ascribed to the centre
of gravity, if the two bodies were united by an inflexible line, the
weight of which might be neglected. To find this point mathematically
is a very simple problem. Let A and B, _fig. 67._, be the two
bodies, and _a_ and _b_ their centres of gravity. Draw the right line
_a b_, and divide it at C, in such a manner that _a_ C shall have
the same proportion to _b_ C as the mass of the body B has to the mass
of the body A.
This may easily be verified experimentally. Let A and B be two bodies,
whose weight is considerable, in comparison with that of the rod
_a b_, which joins them. Let a fine silken string, with its
ends attached to them, be hung upon a pin; and on the same pin let a
plumb-line be suspended. In whatever position the bodies may be hung,
it will be observed that the plumb-line will cross the rod _a b_
at the same point, and that point will divide the line _a b_ into
parts _a_ C and _b_ C, which are in the proportion of the mass of B to
the mass of A.
(172.) The centre of gravity of three separate bodies is defined in the
same manner as that of two, and may be found by first determining the
centre of gravity of two; and then supposing their masses concentrated
at that point, so as to form one body, and finding the centre of
gravity of that and the third.
In the same manner the centre of gravity of any number of bodies may be
determined.
(173.) If a plate of uniform thickness be bounded by straight edges,
its centre of gravity may be found by dividing it into triangles by
diagonal lines, as in _fig. 68._, and having determined by (154)
the centres of gravity of the several triangles, the centre of gravity
of the whole plate will be their common centre of gravity, found as
above.