**Polar Moment of Inertia**

If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis *k* perpendicular to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the *polar moment of inertia*. The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.

If a system of n particles, P_{i}, i=1,...,n, are assembled into a rigid body, then the momentum of the system can be written in terms of position and velocity relative to a reference point **R**,

where ω is the angular velocity of the system.

For planar movement the angular velocity vector is directed along the unit vector *k* which is perpendicular to the plane of movement. Introduce the unit vectors **e**_{i} from the reference point **R** to a point **r**_{i}, and the unit vector **t**_{i}=*k*x**e**_{i} so

This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.

Read more about this topic: Moment Of Inertia

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