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Determination of Moment of Inertia & Radius of Gyration of Flywheel (Part – VI) 

If the moment of inertia has been calculated for rotations about the centroid of a rigid body, we can conveniently recalculate the moment of inertia for all parallel rotations as well, without having to resort to the formal definition. If the axis of rotation is displaced by a distance R from the centroid axis of rotation (e.g., spinning a disc about a point on its periphery, rather than through its center), the new moment of inertia equals:

where

M is the total mass of the rigid body, and

R is the distance of the axis of rotation from the centroid axis of rotation (as described above).

This theorem is also known as parallel axes rule or Steiner's theorem.

For a system with N point masses mi moving with speeds vi, the kinetic energy T always equals

For a rigid body rotating with angular speed ω, the speeds can be written

where again ri is the shortest distance from the point mass to the rotation axis. Therefore, the kinetic energy can be written

The final formula also holds for a continuous distribution of mass.

Compiled by Samar Das