To understand the equation of a plane motion of a rigid body, consider a rigid body moving in X-Y plane. So, here is the diagram to make the things convenient. The center of mass G has the acceleration ‘ a’. The angular velocity
The angular momentum of mass center G for the particle m can be represented as a scalar equation and, finally, it is clear through this equation that the angular acceleration is same as the rate of change of angular velocity with time.
Angular momentum can be expressed as
So, the conclusion is HG= wIG. It means
Now, from all the equations, it is clear that
The term I, is a constant and measure the inertia of rotation. IG is the mass moment of inertia about G or the mass center. It is also known as the moment of inertia of mass M through Z axis. Now, through the above explanation you can easily get that how to explain x-y Plane Motion Equations.