We are now ready to write down the general equations of motion for rigid bodies in terms of for the center of mass and for the rotation of the body about its center of mass.
As discussed above, it is useful to decompose the motion of a rigid body into
The linear motion is governed by Newton's second law , where is the total mass, is the velocity of the center-of-mass, and is the sum of all external forces on the rigid body. (Equivalently, is the sum of the radial force components pointing toward or away from the center of mass.) Since this is so straightforward, essentially no harder than dealing with a point mass, we will not consider it further.
The angular motion is governed the rotational version of Newton's second law introduced in §B.4.19:
The driving torque is given by the resultant moment of the external forces, using Eq. (B.27) for each external force to obtain its contribution to the total moment. In other words, the external moments (tangential forces times moment arms) sum up for the net torque just like the radial force components summed to produce the net driving force on the center of mass.