10.7:
Moment of Inertia
When a rigid body is in linear motion, every point moves with the same velocity. However, when it is in rotational motion, different points on the body have different velocities and therefore different kinetic energies.
If the i-th particle, placed at a perpendicular distance ri from the axis of rotation, has a tangential velocity of vi, its kinetic energy is calculated by replacing vi with the product of angular speed and ri.
The total kinetic energy of the rigid body is the sum of the individual kinetic energies of the constituent particles. Since angular speed is the same for each particle, it can be pulled out of the summation.
On comparing this equation with the kinetic energy equation for translational motion, a new rotational variable is defined. This quantity is called the moment of inertia and has units of kilogram meters squared.
For a single particle rotating about a fixed axis, the moment of inertia is the product of its mass and the square of its distance from the axis of rotation.
10.7:
Moment of Inertia
The comparability between linear and angular velocities, linear and angular accelerations, and the kinematic equations of translational and rotational motion can be extended to the concept of inertia.
If a rigid body is rotating about an axis but is not in translational motion, its translational kinetic energy is zero. However, since each particle undergoes rotational motion, it possesses non-zero velocity and kinetic energy. Thus, the kinetic energy of the rigid body, which is the sum of the kinetic energy of its constituents, is non-zero. The rotational kinetic energy of a rigid body is given as half the square of the angular speed times the moment of inertia.
Rigid bodies and systems of particles with more mass concentrated at a greater distance from the axis of rotation have greater moments of inertia than bodies and systems of the same mass but concentrated near the axis of rotation. For instance, a hollow cylinder has more rotational inertia than a solid cylinder of the same mass when rotating about an axis through the center.
Although defined while keeping rigid bodies in mind, the moment of inertia also applies to single particles. It helps treat objects rotating with respect to inertial frames of reference as single particles with their total mass concentrated at the center of mass.
This text is adapted from Openstax, University Physics Volume 1, Section 10.4: Moment of Inertia and Rotational Kinetic Energy.