Newton's Second Law for Rotating Rigid Bodies

The general motion of a rigid body can be completely described by the linear motion of its center of mass and its rotation about the center of mass.

Newton's Second Law therefore has two parts:

Linear Motion

$$\vec{F}_{{\rm net}} = M \vec{a}_{{\rm cm}}= {d \vec{p}_{{\rm tot}}\over dt}$$

where $\vec{F}_{{\rm net}}=\vec{F}_1+\vec{F}_2+...$ is the net force on the object, M is the total mass, $\vec{a}_{{\rm cm}}$ is th acceleration of the center of mass of the object, and $\vec{p}_{{\rm tot}}=M\vec{v}_{{\rm cm}}$ is its total linear momentum.

Rotational Motion

$$\vec{\tau}_{{\rm net}}= I \vec{\alpha} = {d\vec{L}\over dt}$$

where $\tau_{{\rm net}}= \vec{r}_1\times \vec{F}_1 +\vec{r}_2\times \vec{F}_2+...$ is the net torque on the object, I is its moment of inertia about its center of mass, and $\vec{L}$ is its angular moment about the center of mass.

Note

• When applying Newton's laws to solve for the motion of a system of objects, one must apply the rotational version of the Second law to each extended object in the system that is free to rotate.
• In the absence of net external torque the total angular momentum of any system of objects (about any axis) is conserved.