Question

A uniform solid sphere has a moment of inertia I about an axis tangent to its surface. What is the moment of inertia of this sphere about an axis through its center?

Answer 1

Concepts and reason

The concepts used to solve this problem are moment of inertia of the body about the center of mass of a solid sphere and parallel-axis theorem.

Use the parallel-axis theorem to find the moment of inertia of the body about the support.

Use the moment of inertia of the body about the support and the moment of inertia of the body about the center of mass to find the moment of inertia of the sphere about an axis through its center in terms of .

Fundamentals

The expression for the parallel-axis theorem is as follows:

Here, is the moment of inertia of the body about the support, is the mass of the body, is the distance of the center of the body from the support, and is the moment of inertia of the body about the center of mass.

The expression for the moment of inertia of the body about the center of mass of a solid sphere is as follows:

Here, the radius of the sphere is .

The expression for the parallel-axis theorem is as follows:

Substitute for and for .

The expression for the moment of inertia of the body about the center of mass of a solid sphere is as follows:

For the sphere:

From the above diagram:

Substitute for and for .

Ans:

The moment of inertia of the sphere about an axis through its center is .