10.8:
Moment of Inertia and Rotational Kinetic Energy
Consider a rigid body made up of many infinitesimal particles in rotational motion. Its kinetic energy can be related to its rotational speed.
Using the moment of inertia notation, the rotational kinetic energy of the body is observed to be proportional to the moment of inertia and the square of the angular velocity.
It is similar to how the kinetic energy of a rigid body in translational motion is proportional to its mass and the square of the linear velocity.
A body with a greater moment of inertia will gain more kinetic energy when it is sped up by the same angular speed. Hence, more work would need to be done to change its angular speed. Thus, the moment of inertia gives a quantitative measure of rotational inertia.
10.8:
Moment of Inertia and Rotational Kinetic Energy
The rotational kinetic energy of a body is equal to half the square of its angular speed and the moment of inertia.
This relationship between the rotational kinetic energy of a body and its angular speed implies that for the same angular speed, the rotational kinetic energy is greater if its moment of inertia is greater. Thus, more work needs to be done on the body to change its rotational kinetic energy and rotate it at a specific angular speed. Hence, the moment of inertia quantifies the body's rotational inertia.
In translational motion, the mass of a body is a measure of its inertia. Likewise, in rotational motion, the moment of inertia gives a quantitative measure of rotational inertia. It can be shown that the angular momentum of an object is proportional to the moment of inertia and the angular velocity, similar to linear momentum.
The moment of inertia of a body is expressed as the sum of the mass of the constituents weighted by the square of their perpendicular distances from the rotational axis. Hence, it is the second moment of mass, a measure of inertia, which justifies its name.
This text is adapted from Openstax, University Physics Volume 1, Section 10.4: Moment of Inertia and Rotational Kinetic Energy.