CENTRE OF GRAVITY.
[Illustration: Fig. 30.]
[Illustration: Fig. 31. Fig. 32.]
95. =Centre of Gravity Illustrated.=--If you balance a ruler on
your finger, as in Fig. 30, it is balanced because there is just
as much weight on one side as on the other. Now just over your
finger, in the middle of the ruler, there is a point that we call
the centre of gravity; or, in other words, the centre of the weight
of the ruler. This point is indicated in the figure. There is as
much of the weight of the ruler on the one side of this point as on
the other, and also as much above it as below it. If your finger
should be a little to the one side or the other of this point, the
ruler would not be balanced, and would fall. When balanced it does
not fall, simply because this central point is supported by being
directly over the end of the finger. The weight of the ruler, then,
may be considered practically as all being in that point, for it is
there that is exerted all the downward pressure of the ruler as it
is balanced. So, also, when the ruler is balanced on the finger,
as represented in Fig. 31 (p. 68), this same centre of gravity is
directly over the point of the finger, and is therefore supported.
If it be to the one side or the other, as in Fig. 32, it is not
supported, and the ruler therefore falls. You see, then, that if
a body be balanced, the centre of gravity is directly _over_ the
point of support. If, on the other hand, a body is suspended, the
centre of gravity is directly _under_ the point of support.
[Illustration: Fig. 33.]
96. =Definition.=--If a plumb-line from the centre of gravity could
be extended into the earth it would go directly to its centre. The
body may be considered as making all its pressure from its centre
of gravity in that direction, in obedience to the attraction of
gravitation. The best definition, then, that we can give of the
centre of gravity is, that it is _that point in a body from which
its pressure as a whole toward the centre of the earth proceeds_.
It is that point, therefore, the support of which insures the
support of the whole body. And in speaking of the weight of a body,
or its downward pressure, we may consider all the matter composing
it as collected or concentrated in that point. The body, therefore,
can be balanced in any position in which this point is supported,
as shown in Figs. 30 and 31. And when a body is suspended, it is at
rest only when the centre of gravity is directly under the point of
support. Thus, if you have a circular plate suspended at E, Fig.
33, it will not be at rest when it is moved to the one side or
the other, as represented by the dotted lines, but only when the
centre of gravity, _c_, is directly under the point E.
[Illustration: Fig. 34. Fig. 35.]
97. =How to Find the Centre of Gravity of a Body.=--If you take
a piece of board, and suspend it at A, Fig. 34, and suspend a
plumb-line from the same point, the centre must be somewhere in
that line. But exactly at what point it is you do not know. How
will you ascertain this? Mark the line A B on the board, and
suspend the board by another point, as in Fig. 35. As the centre
of gravity must be somewhere in the plumb-line as it now hangs, of
course it is at O, where the two lines cross.
[Illustration: Fig. 36. Fig. 37.]
98. =Scales and Steelyards.=--When two bodies are connected by a
rod or bar, the centre of gravity of the whole is somewhere in
the connection. If the two bodies be equal in weight, as in Fig.
36, the centre of gravity is exactly in the middle of the rod, as
marked. But if the bodies are unequal, as in Fig. 37, the centre
of gravity is nearer to the larger body than to the smaller. In
balancing a body in one scale with weights in another, you have a
case parallel to that of Fig. 36. The centre of gravity of the
body weighed, the weights, and the scales, as a whole, is midway
between the scales, at the point of support. In the steelyard you
have the heavy body to be weighed nearer the centre of gravity than
the small weight is on the long arm, and so the case is parallel
with that of Fig. 37.
[Illustration: Fig. 38.]
99. =The Centre of Gravity of a Body not Always in the Body
Itself.=--The centre of gravity of a hollow ball of uniform
thickness is not in the substance of the ball, but it is in the
centre of the space in the ball, for the line of the ball's
downward pressure would be from that point. If the ball had a
frame-work in it, as represented in Fig. 38, the centre of gravity
would obviously be at A, the centre of this frame-work. But if
there were no frame-work, and perpendicular lines were supposed to
be drawn from different points of suspension, C, B, D, and E, these
would intersect at the point A, showing that this is the centre
of gravity, according to the rule for finding it given in § 97.
So, also, the centre of gravity of an empty box, or an empty ship,
would be an imaginary point in the space inside. In a hoop it is
the centre of the hoop's circle.
[Illustration: Fig. 39.]
100. =The Centre of Gravity Seeks the Lowest Point.=--The centre of
gravity always takes the lowest place which the support of the body
will allow. In a suspended body, therefore, it is always directly
under the point of suspension. To get to the one side or the other
of this position it must rise. This the attraction of gravity
forbids, and if by any force it is made to rise, this attraction at
once brings it back. This is manifest in the case of a suspended
ball, Fig. 39. If the ball be moved to _b_, it will, on being let
go, return to its first position, simply because its centre of
gravity, in obedience to the earth's attraction, seeks the lowest
place possible. From inertia (§ 49) it moves beyond this point,
and continues to vibrate back and forth for some time; but when
its motion is stopped, it hangs perpendicularly; that is, in such
a way that its centre of gravity shall have the lowest possible
position. I add a few other illustrations of the same point. When
a rocking-horse is at rest, its centre of gravity is directly
over the point at which it touches the floor, for thus it has its
lowest possible place. If it be rocked, the centre of gravity is
moved to a higher point, and for this reason it rocks back again.
The same is seen in the swing, the cradle, the rocking-chair, etc.
Most interesting illustrations of the same thing are found in the
Laggan, or Loggan Stones as they are called, several of which
are seen on the rugged parts of the British coast. An immense
rock, which has been loosened by some convulsion, rests with a
slightly-rounded base on another rock which is flat, and it is so
nicely balanced that one person alone can produce a perceptible
rocking motion in it. I saw, many years ago, a large rock near
Salem, Massachusetts, thus situated. There is one also in Great
Barrington, Massachusetts.
[Illustration: Fig. 40. Fig. 41.]
[Illustration: Fig. 42.]
[Illustration: Fig. 43.]
101. =Farther Illustrations.=--It is because the centre of gravity
always seeks the lowest place that an egg lies upon its side. When
on its side, the centre of gravity is at its lowest point, as is
manifest from comparing Fig. 40 with Fig. 41 (p. 71). Children
often have a toy, called a witch, which illustrates the same thing
in another way. It is a piece of light substance, as pith, with a
shot fastened in one end. It always stands up on its loaded end,
and can not be made to lie down on its side, because the centre of
gravity would not then be at the lowest point. There is an amusing
Chinese toy of the same kind. It is a figure of a fat old woman,
Fig. 42, loaded with lead at the bottom, so that its centre of
gravity is at _a_. If the figure be thrust over to one side, as
shown by the dotted lines, the centre of gravity is raised, and the
upright position is at once resumed. If the toy were not loaded, it
would lie in the position represented in Fig. 43, just as the egg
lies on its side.
[Illustration: Fig. 44.]
[Illustration: Fig. 45. Fig. 46.]
[Illustration: Fig. 47.]
102. =Curious Experiments.=--You can not hang a pail of water on a
stick laid upon a table, as represented in Fig. 44, for the centre
of gravity is not supported. But if you place another stick a as a
brace, in the manner represented in Fig. 45 (p. 73), so as to push
the pail under the table, it will hang securely, because the centre
of gravity is now under the point of suspension. The explanation of
the following experiment is the same: Run a large needle through
a cork; fasten to the cork a fork, and you can suspend the whole
on the edge of a table, as seen in Fig. 46. Here the centre of
gravity is directly under the point of suspension, which is at the
point of the needle. The same can be said of the very common toy
represented in Fig. 47. The horse, made of very light material,
stands securely, because the centre of gravity of the whole is in
the heavy ball, which is under the point of suspension. If the
horse be made to rock back and forth, the centre of gravity in the
ball moves in a curved line, as in the case of a ball suspended by
a string (Fig. 39). It is at its lowest place only when the horse
is at rest. The hanging of a cane with a hook-shaped handle on the
edge of a table is to be explained in the same way.
[Illustration: Fig. 48. Fig. 49. Fig. 50.]
[Illustration: Fig. 51.]
[Illustration: Fig. 52.]
103. =Stability of Bodies.=--The firmness with which a body
stands depends upon two circumstances--the height of its centre
of gravity, and the extent of its base. The lower the centre of
gravity, and the broader the base, the firmer does the body stand.
A cube, represented in Fig. 48, is more stable, that is, less
easily turned over, than a body shaped as Fig. 49, because it has
a larger base. The contrast is still greater between Figs. 48 and
50. The reason of the stability of a body with a broad base is
found in the fact, that in turning it over the centre of gravity
must be raised more than in turning over one of a narrower base.
The curved lines indicate the paths of the centres of gravity as
the bodies are turned over. In the case of a perfectly round ball,
the base is a mere point, and therefore the least touch turns it
over. Its centre of gravity does not rise at all, but moves in a
horizontal line, as shown in Fig. 51. The pyramid is the firmest
structure in the world, because it possesses in the highest degree
the two elements--a broad base, and a low position of the centre of
gravity. On both these accounts the centre of gravity must ascend
considerably when the body is turned over, as seen in Fig. 52.
[Illustration: Fig. 53.]
[Illustration: Fig. 54.]
104. =Bodies not Upright Unstable.=--When a body does not stand
upright, its stability is diminished simply because only a portion
of the base is concerned in its support. In Fig. 53 the base is
broad, but the body is so far from being upright that the centre
of gravity bears upon the very extremity of the base on one side,
as the perpendicular line from it indicates. The least jostle will
turn it over, because the centre of gravity need not ascend the
least when this is done. You see, then, that the less upright a
body is, the less of the base is of service in its support, because
the farther is the line of direction of the downward pressure of
the centre of gravity from the centre of the base. The famous tower
of Pisa, Fig. 54, one hundred and thirty feet high, overhangs its
base fifteen feet. It was undoubtedly built intentionally in this
way to excite wonder and surprise, for what would otherwise have
been a very unsafe structure is rendered stable and safe by the
arrangement of its materials. Its lower portion is built of very
dense rock, the middle of brick, and the upper of a very light
porous stone. In this way the centre of gravity of the whole
structure is made to have a very low position.
[Illustration: Fig. 55.]
105. =Familiar Illustrations.=--You see now the explanation of
the fact which common experience teaches every one, that the
taller is a body, and the narrower its base, the more easily is
it overturned. This is exemplified in the two loads, Fig. 55. The
base is the space included by the wheels. The centre of gravity
is so high in the tall load that a perpendicular line drawn from
it falls outside of the base if the cart come upon a considerable
lateral inclination of the road. But the smaller load, under the
same circumstances, is perfectly secure from overturning. A high
carriage is more easily overturned than a low one, for the same
reason. A stage, if loaded on its top, is very unsafe on a rough
road. Stability is given to articles of furniture by making their
bases broad and heavy, as you see in tables supported by a central
pillar, candlesticks, lamps, etc. The tall chairs in which children
sit at table would be very insecure if their legs were not widely
separated at the bottom, thus widening the base of support. In
the ladder, so commonly used now in picking fruit, a broad base
is furnished between the foot of the ladder and the two standards
which are spread out to sustain its top.
[Illustration: Fig. 56. Fig. 57.]
106. =Support of the Centre of Gravity in Animals.=--The base of
support which quadrupeds have, viz., the space included between
their four feet, is quite large; and this is one reason that they
walk so soon after birth. A child does well that can walk at the
end of ten or twelve months, for the supporting base is quite small
compared with that of a quadruped. It consists of the feet and the
space between them. It requires skill, therefore, in the child to
manage the centre of gravity in standing and walking, and this is
gradually acquired. If one should grow up without ever standing on
his feet, he would find, as the infant does, that some training
is necessary to enable him to do it. It is on account of the
smallness of the base furnished by the feet that the statue of a
man is always made with a large base or pedestal. Although we exert
considerable skill in walking, by no means so much is requisite
as the Chinese ladies must put in exercise with their small feet.
Still more skill is exercised by one who has two wooden legs, or
one who walks on stilts. The base made by the feet can be varied
much by their position. If the toes be turned out and the heels
brought near to each other, the base will not be as large as when
the feet are straight forward and far apart, as is manifest in
Figs. 56 and 57. It is for this reason that the child, in his first
essays at standing and walking, instinctively manages his feet as
in Fig. 56.
[Illustration: Fig. 58.]
[Illustration: Fig. 59.]
[Illustration: Fig. 60. Fig. 61.]
107. =Motions of the Centre of Gravity in Walking.=--In walking,
the centre of gravity is alternately brought over one foot and the
other, and so moves in a waving line. This is very manifest as you
see people before you going down the aisle out of a church. When
two are walking together, if they keep step the two waving lines
of their centres of gravity run parallel, as in Fig. 58, and they
walk easily; but if they do not keep step these lines run as in
Fig. 59, and the movement is both awkward and embarrassing. The
line of movement of the centre of gravity is always slightly waving
_upward_ also, as seen in Fig. 60 (p. 78). In the case of a man
with wooden legs the line would not be gently waving, but somewhat
angular, as represented in Fig. 61.
[Illustration: Fig. 62. Fig. 63.]
[Illustration: Fig. 64.]
[Illustration: Fig. 65.]
108. =The Centre of Gravity and Attitudes.=--The object of various
attitudes assumed under different circumstances is to keep the
centre of gravity over the base of support. A man with a load on
his back would not assume the position of Fig. 63, but that of Fig.
62, so that the centre of gravity of his load may be directly over
his feet. So a man carrying any thing in front leans backward, as
in Fig. 64. In ascending a hill a man appears to lean forward, and
in descending to lean backward; but in fact he is in both cases
upright in reference to the plain on which the hill stands, as may
be seen in Fig. 65. A perpendicular line drawn from his centre
of gravity strikes the ground midway between the feet, that is,
in the middle of the base, and if prolonged would go straight to
the centre of the earth. When one rises from a chair he draws his
feet backward, and then bends his body forward to bring the centre
of gravity over the feet. Unless this is done, it is impossible
to rise, at least deliberately, as you will find by trying the
experiment. A man standing with his heels close to a wall can not
stoop forward and pick up any thing, for the wall prevents him
from moving any part of his body backward, and therefore when he
stoops forward, the centre of gravity being brought far in advance
of the base, he loses his balance and falls. A man who did not
understand this undertook to stoop in this way to pick up a purse
containing twenty guineas, which he was to have if he succeeded,
the forfeiture in case of failure being ten guineas. Of course his
centre of gravity made him lose his wager.
109. =Rope-Dancers, Tops, etc.=--Great skill is exhibited by the
rope-dancer in supporting the centre of gravity. Similar skill is
seen in feats of balancing, as, for example, in balancing a long
stick upright on the finger. In these cases the centre of gravity
is very little of the time directly over the point of support. It
is kept in constant motion nearly but not quite over this point,
this unstable equilibrium, as it is called, being vastly less
difficult to maintain than stable equilibrium; that is, keeping
the balance in one unvarying position. It is the motion of the top
that makes it to stand upright upon its point--a very beautiful
example of unstable equilibrium. The centre of gravity revolves
around a perpendicular line, at exceedingly little distance from it
at first, but greater and greater as its motion becomes less rapid,
till at length the centre of gravity gets so far from this line
that the top falls. For a similar reason an intoxicated man may not
be able to keep himself up if he undertake to stand still, and yet
may do so if he keep moving. As in the case of the top, his centre
of gravity must be kept in motion, or he must fall.