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.