## Center of Mass and Its Motion

The center of mass of a body B, living or nonliving, is defined by the following equation:

2m1 is the total mass in B, rc is the position vector for the center of mass of B, mi is the mass of the ith element in B, and r' is its position vector. These entities are shown in Fig. 3.1. Note that the center of mass is also commonly known as the center of gravity.

Let us now perceive center of mass as if it were a particle in space. In reality, the center of mass may not correspond to any point of the object B. The position of center of mass may be occupied by different particles of B at different times during motion. The time rate of change of position vector rc (derivative of rc with respect to time t) is equal to the velocity of the center of mass, which we denote by vc. The acceleration of the center of mass ac is the time derivative of vc:

Using Eqn. 3.8b, the linear momentum L of a system of particles can be written as

Substituting Eqn. 3.8c into Eqn. 3.6, we obtain an equation governing the motion of center of mass of an object:

According to this equation, the net force acting on a system of particles is equal to the mass of the system times the acceleration of the center of mass. For a sphere of uniform mass density, the center of mass is positioned at the center of the sphere. As in the case of an L-shaped body, the center of mass may lie outside the body. The tables at the end of the text provide information about the position of the center of mass for solid bodies of a variety of geometric shapes as well as human body configurations (see Appendix 2).

Example 3.1. The Center of Mass of a Human Body as Represented by Two Rods. Consider a body consisting of two slender rods ab and bc of length L and mass m that are connected with a pin at b. The bar ab is tilted 45° from the horizontal axis. Determine the center of mass of B.

Solution: We draw a reference frame whose origin O is at the pin b connecting the two rods (Fig. 3.2). In a uniform slender rod, the center of mass occupies the midpoint along the axis of the rod. Thus, the centers of mass of the two rods ab and bc are located at the following positions:

x1 = —(L/2) cos 45°; x2 = (L/2) sin 45°; x3 = 0 for the rod ab x1 = (L/2), x2 = 0, x3 = 0 for the rod bc a