Moment of Inertia:
When we try to set a body rotating about an axis (by applying to it a moment of a force) the body tends to resist this change in its state of rest (i.e., it developed a reaction) on account of its rotational inertia. This rotational inertia on account of which the reaction is developed is called the moment of inertia of the body about the axis of rotation. Consider a body AB of mass M which rotates about the axis XY. Imagine the body cut up into particles of masses m1, m2, …. Drop perpendiculars from m1, m2, …. on the line XY. Then the sum of the products m1r12, m2r22 ….. for the whole body is called its Moment of Inertia, I, r about the axis XY.
k is called the radius of gyration of the body. It is the distance from the axis of rotation, at which the whole mass of the body may be supported to be concentrated; the moment of inertia of the body remains the same as for the actual distribution of mass in the body.
The moment of inertia of a body depends upon:
- The mass of the body.
- The position of the axis of rotation.
- The way the mass is distributed within the body with respect to the axis.
Moment of inertia plays the same part in rotating bodies that mass plays when bodies move in straight lines. Thus,
(1) Force = mass x acceleration, ∴ Torque = moment of inertia x angular acceleration
(2) Kinetic Energy = (1/2) mass x (velocity)2 = (1/2) I x ω2, where ω = angular velocity