CENTER OF GRAVITY is that point in a body or system of bodies rigidly connected, upon which the body or system acted upon only by the force of gravity, will balance itself in all positions. Though the action of gravity enters this definition, many of the j properties of the point are independent of that force, and might be enunciated and proved without conceiving it to exist. By sonic, accordingly, the point has been called the center of magnitude, and by others, the center of parallel forces. Such a point exists in every body and system, and only one such point. Every body may be supposed to be made up of a multitude of minute particles connected by cohesion, and so far as its balance under gravity is concerned, each of these may be supposed to be removed, and its place occupied by a force proportioned to its weight. Instead of the body, on these suppositions, we should then have a system of parallel forces, the lines from the various particles to the earth's center being regarded as parallel. But a system of parallel forces (see PARALLEL FORCES) has a single resultant acting through a fixed point, whose position is independent of the position in space of the points of application of the com ponent forces, provided their relative positions in the system continue unchanged. This C Q A IT c ast -13 All,)1111111110466..
Fig. 1. Fig. 2.
point is the C. of G.; and if it be supported, it is clear that the body will balance itself upon it in all positions. The same reasoning obviously applies to any system of bodies rigidly connected. It is usual to demonstrate this and the general rule for finding the C. of G. by proving it first in the case of two heavy particles forming a body or system, and then extending the proof to the case of any number of particles. Let P and (see fig. 1) be two heavy particles. Join P and Q, and divide the line PQ in C, so that weight of P : weight of Q : : CQ : CP. Then C will be the C. of G. of P and Q. Draw AC13 horizontal, and PM, QN vertical, meeting AB in M and N. Then if P and Q represent the weights of Pand Q, we have P : Q : : CQ : CP. But CQ : CP : : CN • CM by similar triangles. Therefore P : Q : : CN : CM, and P.CINI = Q.CN. P and Q, therefore are balanced about C. See BALANCE and LEVER. This is true in all positions of P and Q, for no assumption was made as to their positions. C, therefore, is their center of gravity. Also, we may conceive P and Q to be removed (see PARALLEL FORCES), and in their stead a particle at C. equal to them taken together in weight. If now, the system contained three, it is clear how we should proceed to find its center of gravity; having found the C. of G. of two, we should consider the system as formed of two—viz., the equivalent of the first two at their C. of G., and the third, when the case
would fall under that already treated; and so on, extending the rule to a system con taining any number of particles. Apart from this rule, however, it is possible, in the case of most regular homogeneous bodies, to fix upon their centers of gravity from gen eral considerations. The C. of G. of a straight line, for instance, must clearly be in its middle point. So the C. of G. of a uniform homogeneous cylinder must be in the mid dle point of its axis. It must be in the axis, for the cylinder clearly is equally balanced about its axis. It must also be somewhere in its middle circular section, for it will bal ance itself on a knife-edge under that section. It must, therefore, be in the point where that section cuts the axis, or in the middle of the axis, The C. of G. of a uniform material plane triangle may be found from similar considerations. The triangle ABC (see fig. 2) may be supposed to be made up of uniform material lines parallel to its base AB; each of these will balance upon its middle point. The whole triangle, therefore, will balance upon the line CD, which bisects the base AB and all lines parallel to it. In the same way, the triangle will balance upon the line AE, bisecting BC. But if a figure balances itself upon a line, its C. of G. must lie in that line. The C. of G. of the trian gle is therefore in CD, and also in CB. It must therefore be at g where these lines inter sect, g being. the only point they have in common. Now, by geometry, we know that g divides CD, so that Cg = CD. Hence the rule for finding the C. of G. of a triangle: Draw a line from the vertex, bisecting the base, and measure off Cg, two thirds of the line. g is the center of gravity. By a similar method, the C. of G. of a great number of figures may be determined.
The above method applies only where the figure of the body is regular, and its mass homogeneous. But many bodies, besides being irregular, are formed by the agglomeration of particles of different specific gravities. Of these, the C. of G. can be found only by experiment, though not always satisfactorily. Let the body be suspended by a string, and allowed to find its position of equilibrium. The equilibrium being due to the ten sion of the string counterbalancing gravity, it follows that the tension is in the same line with that on which gravity acts on the body. But the tension acts on the line of the string, which therefore passes through the center of gravity. Mark its direction through the body. Suspending it then by another point, we should ascertain a second line in which lies the center of gravity. The C. of G., then, must be where these lines inter sect.—For the effect on the stability of bodies of the position of the C. of G., see STA BILITY.