22 PRINCIPLES OF ELEMENTARY DYNAMICS
fixed in the body at which the weight of the body is applied. Thus, it is assumed that every body has a center of gravity coincident with the point at which the resultant of the gravitational forces acting on the body would be applied if they were parallel.
If a body or system of bodies be conceived to be divided into particles of equal mass, then that point, the distance of which from any given plane is equal to the average distance from that plane of all the constituent particles, is termed the center of mass, or center of inertia, of the body or system of bodies.
If we represent by m,, m2, etc., the masses of the particles composing a body of total mass M, and by x,, x2, etc., the respective distances of these particles from any given plane, then it can be shown that the distance of the center of mass of the body from the
given plane is X = m,xl + m2x2 + etc. 29 M
The center of mass has the following properties:
(a) The center of mass is coincident with the centroid.
(b) The weight of a body acts approximately at its center of mass.
(c) If the resultant of all the forces acting on a rigid body is a single force the line of action of which passes through the center of mass of the body, the motion of the body is without angular acceleration.
(d) The center of mass of a body is so situated that the linear motion of the body will not be changed if the total mass were concentrated at this point and all the forces acting on the body were transferred to this point without change of magnitude or direction.
(e) The motion of the center of mass of any material system is not affected by the internal forces between the parts of the system, but only by external forces.
(f) No material system can of itself, without the action of external forces, change the motion of its center of mass.
(g) An unconstrained body, when acted upon by a system of forces equivalent to a couple, will rotate about its center of mass with constant angular acceleration.
§2. Simple Harmonic Motion
18. Simple Harmonic Motion of Translation. - When a body moves with a reciprocating motion which has at every instant a linear acceleration directed toward the center of the path and that