Mass Moment of Inertia

The mass moment of inertia is a measure of a rigid object's resistance to rotational motion.

For an elemental mass of an object, the differential moment of inertia equals the product of the differential mass and the square of the distance from the object's rotational axis.

Integrating this expression over the entire mass distribution yields the object's moment of inertia.

For a constant density, the density term is factored out of the integral, and the integral reduces to a geometrical factor.

The moment of inertia for a solid sphere about its diameter can be estimated by considering a thin elemental disc.

The differential moment of inertia can be expressed as half the product of the differential mass and the square of its radius.

Integrating the expression over the sphere's diameter gives the moment of inertia for the sphere.

Expressing the disc radius in terms of the sphere's radius and adjusting the limits of integration, the integral is simplified.

So, the sphere's moment of inertia depends on its mass and radius.