Polar Moment of Inertia For Planar Dynamics
The mass properties of a rigid body that is constrained to move parallel to a plane are defined by its center of mass R=(x, y) in this plane, and its polar moment of inertia IR around an axis through R that is perpendicular to the plane. The parallel axis theorem provides a convenient relationship between the moment of inertia IS around an arbitrary point S and the moment of inertia IR about the center of mass R.
Recall that the center of mass R has the property
where r is integrated over the volume V of the body. The polar moment of inertia of a body undergoing planar movement can be computed relative to any reference point S,
where S is constant and r is integrated over the volume V.
In order to obtain the moment of inertia IS in terms of the moment of inertia IR, introduce the vector d from S to the center of mass R,
The first term is the moment of inertia IR, the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector d. Thus,
which is known as the parallel axis theorem.
Read more about this topic: Parallel Axes Rule
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