Repeating the analysis of Section 2.6, we can sum Equation (8.1) over all mass elements to obtain
Here, is the total mass, the position vector of the center of mass [see Equation (2.27)], and the total external force. It can be seen that the center of mass of a rigid body moves under the action of the external forces like a point particle whose mass is identical with that of the body.Again repeating the analysis of Section 2.6, we can sum Equation (8.1) over all mass elements to obtain
Here, is the total angular momentum of the body (about the origin), and the total external torque (about the origin). The preceding equation is only valid if the internal forces are central in nature. However, this is not a particularly onerous constraint. Equation (8.3) describes how the angular momentum of a rigid body evolves in time under the action of the external torques.In the following, we shall only consider the rotational motion of rigid bodies, because their translational motion is similar to that of point particles [see Equation (8.2)] and, therefore, is fairly straightforward in nature.