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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

X-Y Plane Motion Equations 1” = C

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.

X-Y Plane Motion Equations 2” = C

Angular momentum can be expressed as

X-Y Plane Motion Equations 3” = C

So, the conclusion is HG=  wIG. It means

X-Y Plane Motion Equations 4” = C

Now, from all the equations, it is clear that

X-Y Plane Motion Equations 5” = C

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.