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.
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.
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.
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.