Inertia Tensor - Scalar Moment of Inertia of A Simple Pendulum

Scalar Moment of Inertia of A Simple Pendulum

Moment of inertia can be obtained by considering the movement of a mass at the end of a lightweight rod forming a simple pendulum, which can be studied using Newton's second law of motion. The weight of the mass is a force that accelerates it around the pivot point.

This weight also generates a torque T on the pendulum around the pivot point and the acceleration of the mass a = rα is defined by the angular acceleration α of the pendulum, therefore

where r is the length of the pendulum. The quantity I = mr2 is the moment of inertia of the pendulum mass around the pivot point.

In the same way, the kinetic energy of the pendulum mass is defined by its velocity v = rω using the angular velocity ω of the pendulum to yield

The angular momentum of the pendulum mass is given by

This shows that the quantity I = mr2 plays the same role for rotational movement, as mass does for translational movement. The moment of inertia of an arbitrarily shaped body is the sum of the values mr2 for all of the elements of mass in the body.