sphere or globe the centre of gravity is the centre of the sphere. For a cube the centre of gravity is at the point which we may call the centre of the figure, namely the point where straight lines joining opposite corners meet. The same rule gives the centre of gravity of a body shaped like a brick, which in geometry is called a rectangular parallelepiped, or more briefly a right solid. The centre of gravity of a cylinder is midway between the centres of the circular ends. 172. We have hitherto spoken of the centre of gravity of bodies, but we may also speak of the centre of gravity of plane figures, although strictly these are not bodies inasmuch as they have no thickness. Thus we may say that the centre of gravity of a circle is at its centre. This is a short expression of which the full meaning may be easily supplied. Suppose we have a circle cut out of very thin metal; then we may fix our attention on either of the two faces, and, speaking roughly, we may say that the centre of gravity of the body is at the centre. The exact truth is that each circular face has its own centre, and that the centre of gravity of the body is midway between the two geometrical centres. This is strictly true whatever be the thickness of the metal; in fact the body is really a cylinder, and the centre of gravity is found by the rule of Art. 171. In like manner we may speak of the centre of gravity of a triangle, and the following is the rule for determining its position. Draw a straight line from an angle of the triangle to the middle point of the opposite side; the centre of gravity of the triangle is in this straight line. Draw a straight line from another angle to the middle point of the opposite side; the centre of gravity is also in this straight line. It is therefore at the point of intersection of the two straight lines. It can be proved by geometry, and verified by measurement, that the distance of this point from any angle of the triangle is twice the distance from the middle point of the opposite side. The interpretation of the phrase centre of gravity of a triangle is like that we have given respecting a circle. Suppose a triangle to be cut out of metal or wood; if the material is very thin we may take practically for the centre of gravity of the body the point on either face determined by the preceding rule. But if we wish to be quite exact we may suppose two points found, one on each face, by the preceding rule, and the centre of gravity of the body is midway between the two points. In like manner we may understand what is meant by the centre of gravity of any plane figure. As another example we may say that the centre of gravity of a straight line is at its middle point. We mean that if we take a straight slender rod which is of the same thickness throughout, as for instance a straight piece of wire, then the centre of gravity of the body may be said to be practically at its middle point If we wish to be quite exact we must observe that the rod is really a cylinder, and the centre of gravity is found by the rule of Art. 171. 173. The centre of gravity of a cone or pyramid is found by the following rule: join the vertex with the centre of gravity of the base, and measure off three quarters of the length of this straight line from the vertex; the point so obtained is the centre of gravity. 174. The centre of gravity of a body may be at a point where no particle of the body is situated. For example, suppose we have a spherical shell, everywhere of the same thickness, which may be called a hollow sphere; then the centre of gravity of it will be at the centre of the sphere. Also the centre of gravity of a ring is at the centre of the ring. Likewise for a wooden bowl, or for a drum, the centre of gravity will fall at some point of a certain straight line which may be called the axis of the body, but will not be coincident with any particle of the body. In fact this will be the case for innumerable bodies which we see around us. Take for instance a chair; it may by chance happen that the weights of the different parts are so adjusted as to bring the centre of gravity to some point of the seat: but probably this will not be the case, and the centre of gravity may very likely be below the seat. 175. The reader must notice that whether the centre of gravity of a body does or does not coincide with some particle of it, what we have stated in Art. 169 holds, namely that we may for most purposes suppose that the T. P. 5 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 base 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 |