Here is a collection of notes on the planar dynamics of rigid bodies. When we say planar dynamics, we are referring to the dynamics of bodies that can only rotate about a single axis (perpendicular to the “plane”). The main relationships are summarized below, followed by rigorous derivations.
Moment equation for a rigid body (derivation here).
Parallel Axis Theorem (derivation here).
Kinetic energy of a rigid body (derivation coming).
Angular momentum of a rigid body (derivation coming).
Here is a set of notes deriving the rotational equations of motion for the planar dynamics of a rigid body. As usual, we start with Newton’s Second Law. We use rigid body kinematics. Then we write the results in terms of moment of inertia.
If you know the moment of inertia about the center of mass, then the parallel axis theorem allows you to easily compute the moment of inertia about a different, parallel axis. These notes provide a derivation of the equation that transforms one moment of inertia into the other.