John Henry Pepper

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.

Fig. 40. Fig. 40.

f. The centre. a b c d e. Plummet-lines, all pointing to the centre, and therefore diverging from 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 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 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.

Fig. 41. 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.

Fig. 42. 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.

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.

Fig. 43. Fig. 43.

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 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.

Fig. 44. 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—

Fig. 45. 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 concentrated in the small end, and the egg stands erect, as depicted at fig. 46.

Fig. 46. 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.

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.

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.

Fig. 47. 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.

Fig. 48. 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.

Fig. 49. 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; 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.

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 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.

Fig. 50. 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.

Fig. 51. Fig. 51.

No. 2. The board pushed forward, striking against a nail, when the short piece falls first, and the long one second.

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, maybe shown by the following model, as depicted at Nos. 1 and 2, figs. 52 and 53.

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.

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 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.

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, 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.

c. Centre of gravity, which must be raised to d before it can be overthrown.

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 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.

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 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.

This illustration may be modified by using a heavy weight, rope, and stick, as shown in our sketch below.

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 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.

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.

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.

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.