Question

Which of the following has the highest moment of inertia?

• A. A disc about any of its diameters
• B. A ring about any of its diameters
• C. A solid sphere about any of its diameters
• D. A hollow sphere about any of its diameters

Answer 1

The correct option is D

A hollow sphere about any of its diameters

Explanation:

1. The moment of inertia of an object is a measure of how much torque is required to rotate the object about its axis with a certain angular acceleration.
2. The moment of inertia of an object depends on its geometric shape and is proportional to the mass and the square of the radiation of gyration.
3. The moment of inertia of a disc about its diameter is $\frac{1}{4}M{R}^2$, where, $M$ is its mass and $R$ is the radius of the disc.
4. The moment of inertia of a ring about its diameter is $\frac{1}{2}M{R}^2$, where, $M$ is its mass and $R$ is the radius of the ring.
5. The moment of inertia of a solid sphere about its diameter is $\frac{2}{5}M{R}^2$, where, $M$ is its mass and $R$ is the radius of the sphere.
6. The moment of inertia of a hollow sphere about its diameter is $\frac{2}{3}M{R}^2$, where, $M$ is its mass and $R$ is the radius of the sphere.

Since, $\frac{2}{3}>\frac{1}{2}>\frac{2}{5}>\frac{1}{4}$, hence the moment of inertia of the hollow sphere will be greatest if they all have the same mass and radii.

Hence, option D is the correct option.