weight of the body is collected at this point. Thus, taking the example of the chair just given, if we hang the chair up by a string attached anywhere to it, the line of direction of the string when the chair is at rest will always pass through a certain point which, although not coincident with any particle of the body, has a fixed position with respect to the body: thus in whatever way we hang up the chair the position which it takes is the same as if all the weight were collected at that certain point. Another mode of bringing the nature of the centre of gravity before the mind is sometimes given: suppose this point to be connected with various parts of the body by strong rods without weight, then let the point be supported and the body, allowed to turn round the point in any way; it will be found that the body will remain at rest in any position in which it may be left. If the supposition of strong rods without weight appears difficult or extravagant to any reader, we may take another which will answer our purpose as well. Suppose the weights of these strong rods to be so adjusted that the centre of gravity of the whole of them shall just fall at the same point as the centre of gravity of the body: then, as before, the body will remain at rest in any position in which it may be left. XI. PROPERTIES OF THE CENTRE OF 176. One of the most important facts relating to the centre of gravity is thus stated: When a body is placed on a horizontal plane it will stand or fall according as the vertical straight line through its centre of gravity passes within or without the base. Let G be the centre of gravity of the body. Let the vertical line through G cut the horizontal plane on which the body stands at H. Let any horizontal straight line be -drawn through H, and let AB be that portion of it which is within the base of the body. First suppose H to be between A and B. No motion can take place round A. For the weight of the body acts vertically downwards at G, and therefore any motion of turning round A which this weight might produce would tend to make G move in the direction GK; and such motion is prevented by the resistance of the horizontal plane. Similarly no motion can take place round B. Next suppose H to be on AB produced through B. Then, as before, no motion can take place round A. But motion will take place round B; for the weight of the body would tend to make G move round in the direction GK, and there is nothing to prevent this. The body then would fall over round B. 177. In order to understand the preceding proposition we must pay careful attention to the meaning of the word tase there used. It may happen that the portions of surface common to the body and the ground on which it is placed form one undivided area, and then the base is this area; for instance, when a brick is placed on the ground the base is the area of the face of the brick which is in contact with the ground. Or it may happen that the portions of surface common to the body and the ground form various separate areas; this is the case with a chair, where there are four separate areas corresponding to the four feet. Here we may suppose a string stretched round the four feet close to the ground, so as to include the four separate areas; then the figure bounded by the string is what we mean by the base of the chair. 178. If the vertical straight line drawn through the centre of gravity passes within the base the body will stand, but if the vertical passes extremely near the boundary of the base the body will not stand very securely; for then a small push or shake may bring the vertical beyond the boundary of the base, and the body will tumble over. Suppose, for example, that one leg of a chair is broken off; then the base of the chair is reduced to the figure formed by stretching a string round the other three legs close to the ground. The vertical through the centre of gravity of the chair may pass within the base, and so the chair stand on three legs, but the vertical will be extremely near to that portion of the string which passes diagonally from front to back, and thus the chair falls over very easily in the direction of the absent leg. An experiment may be easily tried, which is the same in principle, without waiting until accident supplies a damaged chair. Take a common chair and put three pieces of wood of the same thickness under three of the legs; it will most likely be found impossible to keep the fourth leg off the ground, if it be one of the back legs: but if the weight of the back of the chair is considerable the centre of gravity will be decidedly nearer to the two back legs than to the two front legs, and it will be possible by putting the pieces of wood under two back legs and one front leg to keep the fourth leg off the ground. 179. It is easily seen by a little reflection on the diagram of Art. 176 that if the base remains unchanged, the lower the centre of gravity of a body is the more securely the body stands. If the centre of gravity in the left-hand case instead of being at G were between G and H, the body would have to be turned through a large angle about A or B before the vertical through the centre of gravity would pass beyond the base. Thus if a waggon is loaded with stones or coals the centre of gravity of the whole is about half way between the top and the bottom of the load; and if the waggon is by any accident tilted up a little to the right hand or to the left hand, still it does not fall over. But suppose that instead of stones or coals the waggon is loaded with an equal weight of hay; then the hay is piled up to a great height, and the centre of gravity comes to a point much above its former position: thus the same amount of angular tilting as before may bring the vertical through the centre of gravity outside the base, and so lead to an upset. Similarly if persons in a small boat stand up the centre of gravity of the system is raised so high that a very little disturbance of the boat may overturn it. 180. The fact that in some positions a body may stand more securely than in others is well illustrated by the case of a body shaped like a brick. It stands most securely when one of its two largest faces is placed on the ground; the base is then greater, and the centre of gravity is lower than for any other position. The body stands least securely when one of its two smallest faces is placed on the ground; the base is then smaller, and the centre of gravity is higher than for any other position in which the body can be made to rest. 181. An example on this subject which is frequently given in popular works requires a little notice. There is a famous tower at Pisa which is called the Leaning Tower of Pisa because it is very much out of the perpendicular. It is sometimes said that the tower remains safe in that position because the vertical through the centre of gravity falls within the base; but this is rather a misleading remark. For the tower is not placed on the ground but thrust into the ground, by reason of the foundation; and the main thing to be regarded is whether the parts of the mass are fastened strongly enough together by mortar and other means. If they are the tower will remain in its position even if the vertical through the centre of gravity falls without the base. It is well known that a stick thrust a little way into the ground will stop where it is placed whether in an oblique or an upright position. So also trees may be seen so much bent from the perpendicular that the centre of gravity must be beyond the base; but they are held fast by the roots in the ground. 182. A body at rest when acted on by forces is said to be in equilibrium; sce Art. 143. Now there are different kinds of equilibrium. Suppose that a body in equilibrium is slightly disturbed by some new force, and then left to the action of the old forces. The body may move back towards its original position, or it may move farther away from it; in the former case the equilibrium is said to be stable and in the latter case unstable. 183. A very good example of stable equilibrium is furnished by a weight hanging from a fixed point by a string. If we draw the weight a little aside from its position of rest, and then leave it, the weight moves backwards towards its original position. Again, suppose we have a hole bored through a body, so that the body can turn round a rod passed through this hole, and held fast in a horizontal position. As in the former case the body is in stable equilibrium when the centre of gravity is vertically below the rod. The body may however be turned round and placed so as to have its centre of gravity vertically abore the rod; and then also it will be in equilibrium. But the equilibrium is now unstable; for if the body be moved a little way from this position and left to itself, it does not fall back towards its original position, but further away from it. If the hole happens to pass through the centre of gravity of the body, then the body rests in any position in which it may be placed; if it is disturbed and then left to itself it neither goes back to the former position nor further away from it: the equilibrium in this case is said to be neutral. This case is comparatively rare in practice; equilibrium is in general either stable or unstable. A common grindstone furnishes an example of neutral equilibrium. A cone standing on its own base is in stable equilibrium; it might be balanced on its point and then would be in unstable equilibrium; it might rest on its slant side, and then the equilibrium would be neutral. 184. We may say then that in general a body is in stable equilibrium when it rests with the centre of gravity in its lowest possible position. For the weight of a body is a force tending downwards; and if the centre of gravity is originally in its lowest possible position, any slight displacement of it must bring it to a higher point, and then the weight of the body will bring the centre of gravity down again, that is turn the body back towards its original position. In like manner when the body rests with the centre of gravity in its highest possible position the equili |