CENTRE OF GRAVITY.
_That point about which all the parts of a body do, in any situation,
exactly balance each other._
The discovery of this fact is due to Archimedes, and it is a point in
every solid body (whatever the form may be) in which the _forces_ of
_gravity_ may be considered as _united_. In our globe, which is a
sphere, or rather an oblate spheroid, the centre of gravity will be the
centre. Thus, if a plummet be suspended on the surface of the earth, it
points directly to the centre of gravity, and, consequently, two
plummet-lines suspended side by side cannot, strictly speaking, be
parallel to each other.
If it were possible to bore or dig a gallery through the whole substance
of the earth from pole to pole, and then to allow a stone or the fabled
Mahomet's coffin to fall through it, the momentum--_i.e._, the force of
the moving body, would carry it beyond the centre of gravity. This
force, however, being exhausted, there would be a retrograde movement,
and after many oscillations it would gradually come to rest, and then,
unsupported by anything material, it would be suspended by the force of
gravitation, and now enter into and take part in the general attracting
force; and being equally attracted on every side, the stone or coffin
must be totally without weight. _Momentum_ is prettily illustrated by a
series of inclined planes [Page 33] cut in mahogany, with a grooved
channel at the top, in imitation of the famous Russian ice mountains:
and if a marble is allowed to run down the [Page 34] first incline, the
momentum will carry it up the second, from which it will again descend
and pass up and down the third and last miniature mountain.
[Illustration: Fig. 40. F. The centre. A B C D E. Plummet-lines, all
pointing to the centre, and therefore diverging from each other.]
[Illustration: Fig. 41. P P P. Inclined planes, gradually decreasing in
height, cut out of inch mahogany, with a groove at the top to carry an
ordinary marble. B B B. Different positions of the marble, which starts
from B A.]
In a sphere of uniform density, the centre of gravity is easily
discovered, but not so in an irregular _mass_; and here, perhaps, an
explanation of terms may not be altogether unacceptable.
_Mass_, is a term applied to solids, such as a mass of lead or stone.
_Bulk_, to liquids, such as a bulk of water or oil.
_Volume_, to gases, such as a volume of air or oxygen.
[Illustration: Fig. 42. A B D, The three points of suspension. C, The
point of intersection, and, therefore, the centre of gravity. P, The
line of plummet.]
To find the centre of gravity of any mass, as, for example, an ordinary
school-slate, we must first of all suspend it from any part of the
frame; then allow a plumb-line to drop from the point of suspension, and
mark its direction on the slate. Again, suspend the slate at various
other points, always marking the line of direction of the plummet, and
at the point where the lines intersect each other, there will be the
centre of gravity.
[Illustration: Fig. 43.]
If the slate be now placed (as shown in Fig. 43) on a blunt wooden point
at the spot where the lines cross each other, it will be found to
balance exactly, and this place is called the _centre of gravity_, being
the point with which all other particles of the body would move with
parallel and equable motion during its fall. The equilibrium of bodies
is therefore much affected by the position of the centre of gravity.
Thus, if we cut out an elliptical figure from a board one inch in
thickness, and rest it on a flat surface by one of its edges (as at No.
1, fig. 44), this point of contact is called the point of support, and
the centre of gravity is immediately above it.
In this case, the body is in a state of secure equilibrium, for any
motion on either side will cause the centre of gravity to ascend in
these directions, and an oscillation will ensue. But if we place it upon
the smaller end, as shown at No. 2 (fig. 44), the position will be one
of [Page 35] equilibrium, but not stable or secure; although the centre of
gravity is directly above the point of support, the slightest touch will
displace the oval and cause its overthrow. The famous story of Columbus
and the egg suggests a capital illustration of this fact; and there are
two modes in which the egg may be poised on either of the ends.
[Illustration: Fig. 44. The point of support. C, The centre of gravity.]
The one usually attributed to the great discoverer, is that of scraping
or slightly breaking away a little of the shell, so as to flatten one of
the ends, thus--
[Illustration: Fig. 45. A Represents the egg in its natural state, and,
therefore, in unstable equilibrium; B, another egg, with the surface, S,
flattened, by which the centre of gravity is lowered, and if not
disturbed beyond the extent of the point of support the equilibrium is
stable.]
The most philosophical mode of making the egg stand on its end and
without disturbing the exterior shell is to alter the position of the
yolk, which has a greater density than the white, and is situated about
the centre. If the egg is now shaken so as to break the membrane
enclosing the yolk, and thus allow it to sink to the bottom of the
smaller end, the centre of gravity is lowered; there is a greater
proportion of weight [Page 36] concentrated in the small end, and the
egg stands erect, as depicted at fig. 46.
It is this variable position of the centre of gravity in ivory balls
(one part of which may be more dense than another) that so frequently
annoys even the best billiard-players; and on this account a ball will
deviate from the line in which it is impelled, not from any fault of the
player, but in consequence of the ivory ball being of unequal density,
and, therefore, not having the centre corresponding with the centre of
gravity. A good billiard-player should, therefore, always try the ball
before he engages to play for any large sum.
[Illustration: Fig. 46.--No. 1. Section of egg. C. Centre of gravity. Y.
The yolk. W. The white.
No. 2. C. Centre of gravity, much lowered. Y. The yolk at the bottom of
the egg.]
The toy called the "tombola" reminds us of the egg-experiment, as there
is usually a lump of lead inserted in the lower part of the hemisphere,
and when the toy is pushed down it rapidly assumes the upright position
because the centre of gravity is not in the lowest place to which it can
descend; the latter position being only attained when the figure is
upright.
[Illustration: Fig. 47.--No. 1. C. Centre of gravity in the lowest
place, figure upright.
No. 2. C. Centre of gravity raised as the figure is inclined on either
side, but falling again into the lowest place as the figure gradually
comes to rest.]
There is a popular paradox in mechanics--viz., "a body having a tendency
to fall by its own weight, may be prevented from falling by adding to it
a weight on the same side on which it tends to fall," and the paradox is
demonstrated by another well-known child's toy as depicted in the next
cut.
[Page 37]
[Illustration: Fig. 48. The line of direction falling beyond the base;
the bent wire and lead weight throwing the centre of gravity under the
table and near the leaden weight; the hind legs become the point of
support, and the toy is perfectly balanced.]
[Illustration: Fig. 49.--No. 1. Sword balanced on handle: the arc from C
to D is very small, and if the centre, C, falls out of the line of
direction it is not easily restored to the upright position.
No 2. Sword balanced on the point: the arc from C to D much larger, and
therefore the sword is more easily balanced.]
After what has been explained regarding the improvement of the stability
of the egg by lowering the situation of the centre of gravity, it may at
first appear singular that a stick loaded with a weight at its upper
extremity can be balanced perpendicularly with greater ease and
precision than when the weight is lower down and nearer the hand; and
that a sword can be balanced best when the hilt is uppermost; [Page 38]
but this is easily explained when it is understood that with the handle
downwards a much smaller arc is described as it falls than when
reversed, so that in the former case the balancer has not time to
re-adjust the centre, whilst in the latter position the arc described is
so large that before the sword falls the centre of gravity may be
restored within the line of direction of the base.
[Illustration: Fig. 50.--No. 1. The two pieces of mahogany, carved to
represent a man and a boy, one being 10 and the other 5 inches long,
attached to board by hinges at H H.]
[Illustration: Fig. 51.--No. 2. The board pushed forward, striking
against a nail, when the short piece falls first, and the long one
second.]
For the same reason, a child tripping against a stone will fall quickly;
whereas, a man can recover himself; this fact can be very nicely shown
by fixing two square pieces of mahogany of different lengths, by hinges
on a flat base or board, then if the board be pushed rapidly forward and
struck against a lead weight or a nail put in the [Page 39] table, the
short piece is seen to fall first and the long one afterwards; the
difference of time occupied in the fall of each piece of wood (which may
be carved to represent the human figure) being clearly denoted by the
sounds produced as they strike the board.
Boat-accidents frequently arise in consequence of ignorance on the
subject of the centre of gravity, and when persons are alarmed whilst
sitting in a boat, they generally rise suddenly, raise the centre of
gravity, which falling, by the oscillation of the frail bark, outside
the line of direction of the base, cannot be restored, and the boat is
upset; if the boat were fixed by the keel, raising the centre of gravity
would be of little consequence, but as the boat is perfectly free to
move and roll to one side or the other, the elevation of the centre of
gravity is fatal, and it operates just as the removal of the lead would
do, if changed from the base to the head of the "tombola" toy.
A very striking experiment, exhibiting the danger of rising in a boat,
may be shown by the following model, as depicted at Nos. 1 and 2, figs.
52 and 53.
[Illustration: Fig. 52.--No. 1. Sections of a toy-boat floating in
water. B B B. Three brass wires placed at regular distances and screwed
into the bottom of the boat, with cuts or slits at the top so that when
the leaden bullets, L L L, which are perforated and slide upon them like
beads, are raised to the top, they are retained by the brass cuts
springing out; when the bullets are at the bottom of the lines they
represent persons sitting in a boat, as shown in the lower cuts, and the
centre of gravity will be within the vessel.]
We thus perceive that the stability of a body placed on a base depends
upon the position of the line of direction and the height of the centre
of gravity.
Security results when the line of direction falls within the base.
Instability when just at the edge. Incapability of standing when falling
without the base.
[Page 40]
[Illustration: Fig. 53.--No. 2. The leaden bullets raised to the top now
show the result of persons suddenly rising, when the boat immediately
turns over, and either sinks or floats on the surface with the keel
upwards.]
[Illustration: Fig. 54. F. Board cut and painted to represent the
leaning-tower of Pisa. G. The centre of gravity and plummet line
suspended from it. H. The hinge which attaches it to the base board. I.
The string, sufficiently long to unwind and allow the plummet to hang
outside the base, so that, when cut, the model falls in the direction of
the arrow.]
The leaning-tower of Pisa is one hundred and eighty-two feet in height,
and is swayed thirteen and a half feet from the perpendicular, but yet
remains perfectly firm and secure, as the line of direction falls
considerably within the base. If it was of a greater altitude it could
no longer stand, because the centre of gravity would be so elevated that
the line of direction would fall outside the base. This fact may be
illustrated by taking a board several feet in length, and having cut
[Page 41] it out to represent the architecture of the leaning-tower of
Pisa, it may then be painted in distemper, and fixed at the right angle
with a hinge to another board representing the ground, whilst a
plumb-line may be dropped from the centre of gravity; and it may be
shown that as long as the plummet falls within the base, the tower is
safe; but directly the model tower is brought a little further forward
by a wedge so that the plummet hangs outside, then, on removing the
support, which may be a piece of string to be cut at the right moment,
the model falls, and the fact is at once comprehended.
The leaning-towers of Bologna are likewise celebrated for their great
inclination; so also (in England) is the hanging-tower, or, more
correctly, the massive wall which has formed part of a tower at
Bridgenorth, Salop; it deviates from the perpendicular, but the centre
of gravity and the line of direction fall within the base, and it
remains secure; indeed, so little fears are entertained of its tumbling
down, that a stable has been erected beneath it.
[Illustration: Fig. 55.--No. 1. Two billiard-cues arranged for the
experiment and fixed to a board: the ball is rolling _up_.
No. 2. Sections showing that the centre of gravity, C, is higher at A
than at B, which represents the thick end of the cues; it therefore, in
effect, rolls down hill.]
One of the most curious paradoxes is displayed in the ascent of a
billiard-ball from the thin to the thick ends of two billiard-cues
placed at an angle, as in our drawing above; here the centre of gravity
is raised at starting, and the ball moves in consequence of its actually
_falling_ from the high to the low level.
Much of the stability of a body depends on the height through which the
centre of gravity must be elevated before the body can be overthrown.
The greater this height, the greater will be the immovability of the
mass. One of the grandest examples of this fact is shown in the ancient
Pyramids; and whilst gigantic palaces, with vast columns, [Page 42] and
all the solid grandeur belonging to Egyptian architecture, have
succumbed to time and lie more or less prostrate upon the earth, the
Pyramids, in their simple form and solidity, remain almost as they were
built, and it will be noticed, in the accompanying sketch, how
difficult, if not impossible, it would be to attempt to overthrow bodily
one of these great monuments of ancient times.
[Illustration: Fig. 56. C. Centre of gravity, which must be raised to D
before it can be overthrown.]
[Illustration: Fig. 57. No. 1. The centre of gravity is near the ground,
and falls within the wheels. No. 2. The centre of gravity is much
elevated, and the line of direction is outside the wheels.]
The principles already explained are directly applicable to the
construction or secure loading of vehicles; and in proportion as the
centre of gravity is elevated above the point of support (that is, the
wheels), so is the insecurity of the carriage increased, and the
contrary takes place if the centre of gravity is lowered. Again, if a
waggon be loaded [Page 43] with a very heavy substance which does not
occupy much space, such as iron, lead, or copper, or bricks, it will be
in much less danger of an overthrow than if it carries an equal weight
of a lighter body, such as pockets of hops, or bags of wool or bales of
rags.
In the one instance, the centre of gravity is near the ground, and falls
well within the base, as at No. 1, fig. 57. In the other, the centre of
gravity is considerably elevated above the ground, and having met with
an obstruction which has raised one side higher than the other, the line
of direction has fallen outside the wheels, and the waggon is
overturning as at No. 2.
The various postures of the human body may be regarded as so many
experiments upon the position of the centre of gravity which we are
every moment unconsciously performing.
To maintain an erect position, a man must so place his body as to cause
the line of direction of his weight to fall within the base formed by
his feet.
[Illustration: Fig. 58.]
The more the toes are turned outwards, the more contracted will be the
base, and the body will be more liable to fall backwards or forwards;
and the closer the feet are drawn together, the more likely is the body
to fall on either side. The acrobats, and so-called "India-Rubber
Brothers", dancing dogs, &c., unconsciously acquire the habit of
accurately balancing themselves in all kinds of strange positions; but
as these accomplishments are not to be recommended to young people, some
other marvels (such as balancing a pail of water on a stick laid upon a
table) may be adduced, as illustrated in fig. 59.
Let A B represent an ordinary table, upon which place a broomstick, C D,
so that one-half shall lay upon the table and the other extend from
[Page 44] it; place over the stick the handle of an empty pail (which
may possibly require to be elongated for the experiment) so that the
handle touches or falls into a notch at H; and in order to bring the
pail well under the table, another stick is placed in the notch E, and
is arranged in the line G F E, one end resting at G and the other at E.
Having made these preparations, the pail may now be filled with water;
and although it appears to be a most marvellous result, to see the pail
apparently balanced on the end of a stick which may easily tilt up, the
principles already explained will enable the observer to understand that
the centre of gravity of the pail falls within the line of direction
shown by the dotted line; and it amounts in effect to nothing more than
carrying a pail on the centre of a stick, one end of which is supported
at E, and the other through the medium of the table, A B.
[Illustration: Fig. 59.]
This illustration may be modified by using a heavy weight, rope, and
stick, as shown in our sketch below.
[Illustration: Fig. 60.]
Before we dismiss this subject it is advisable to explain a term
referring to a very useful truth, called the centre of percussion; a
knowledge of which, gained instinctively or otherwise, enables the
workman to wield his tools with increased power, and gives greater force
to the cut of the swordsman, so that, with some physical strength, he
may perform the feat of cutting a sheep in half, cleaving a bar of lead,
or [Page 45] neatly dividing, _à la Saladin_, in ancient Saracen
fashion, a silk handkerchief floating in the air. There is a feat,
however, which does not require any very great strength, but is
sufficiently startling to excite much surprise and some inquiry--viz.,
the one of cutting in half a broomstick supported at the ends on
tumblers of water without spilling the water or cracking or otherwise
damaging the glass supports.
[Illustration: Fig. 61.]
These and other feats are partly explained by reference to time: the
force is so quickly applied and expended on the centre of the stick that
it is not communicated to the supports; just as a bullet from a pistol
may be sent through a pane of glass without shattering the whole square,
but making a clean hole through it, or a candle may be sent through a
plank, or a cannon-ball pass through a half opened door without causing
it to move on its hinges. But the success of the several feats depends
in a great measure on the attention that is paid to the delivery of the
blows at the _centre of percussion_ of the weapon; this is a point in a
moving body where the percussion is the greatest, and about which the
impetus or force of all parts is balanced on every side. It may be
better understood by reference to our drawing below. Applying this
principle to a model sword made of wood, cut in half in the centre of
the blade, and then united with an elbow-joint, the handle being fixed
to a board by a wire passed through it and the two upright pieces of
wood, the fact is at once apparent, and is well shown in Nos. 1, 2, 3,
fig. 62.
[Page 46]
[Illustration: Fig. 62. No. 1, is the wooden sword, with an elbow-joint
at C. No. 2. Sword attached to board at K, and being allowed to fall
from any angle shown by dotted-line, it strikes the block, W, outside
the centre of percussion, P, and as there is unequal motion in the parts
of the sword it bends down (or, as it were, breaks) at the elbow-joint,
C.
No. 3 displays the same model; but here the blow has fallen on the
block, W, precisely at the centre of percussion of the sword, P, and the
elbow-joint remains perfectly firm.]
When a blow is not delivered with a stick or sword at the centre of
percussion, a peculiar jar, or what is familiarly spoken of as a
_stinging_ sensation, is apparent in the hand; and the cause of this
disagreeable result is further elucidated by fig. 63, in which the post,
A, corresponds with the handle of the sword.
[Page 47]
[Illustration: Fig. 63. A. The post to which a rope is attached. B and C
are two horses running round in a circle, and it is plain that B will
not move so quick as C, and that the latter will have the greatest
moving force; consequently, if the rope was suddenly checked by striking
against an object at the centre of gravity, the horse C would proceed
faster than B, and would impart to B a backward motion, and thus make a
great strain on the rope at A. But if the obstacle were placed so as to
be struck at a certain point nearer C, viz., at or about the little
star, the tendency of each horse to move on would balance and neutralize
the other, so that there would be no strain at A. The little star
indicates the _centre of percussion_.]
All military men, and especially those young gentlemen who are intended
for the army, should bear in mind this important truth during their
sword-practice; and with one of Mr. Wilkinson's swords, made only of the
very best steel, they may conquer in a chance combat which might
otherwise have proved fatal to them. To Mr. Wilkinson, of Pall Mall, the
eminent sword-cutler, is due the great merit of improving the quality of
the steel employed in the manufacture of officers' swords; and with one
of his weapons, the author has repeatedly thrust through an iron plate
about one-eighth of an inch in thickness without injuring the point, and
has also bent one nearly double without fracturing it, the perfect
elasticity of the steel bringing the sword straight again. These, and
other severe tests applied to Wilkinson's swords, show that there is no
reason why an officer should not possess a weapon that will bear
comparison with, nay, surpass, the far-famed _Toledo_ weapon, instead of
submitting to mere army-tailor swords, which are often little better
than hoops of beer barrels; and, in dire combat with Hindoo or Mussulman
fanatics' Tulwah, may show too late the folly of the owner.
[Illustration: Fig. 64.]
[Page 48]