Question

What is the moment of inertia of a solid sphere of density \( \rho \) and radius \( R \) about its diameter?

• A. \( \frac{105}{176} R^5 \rho \)
• B. \( \frac{105}{176} R^2 \rho \)
• C. \( \frac{176}{105} R^2 \rho \)
• D. \( \frac{176}{105} R^5 \rho \)

Hint:

The moment of inertia of a solid sphere about its diameter is proportional to the square of the radius and the density.

Answer 1

The Correct Option is C

Step 1: Formula for moment of inertia of a solid sphere.
The moment of inertia of a solid sphere about its diameter is given by: \[ I = \frac{2}{5} M R^2 \] where \( M \) is the mass of the sphere and \( R \) is its radius.

Step 2: Substitute mass.
The mass of the sphere is related to its density \( \rho \) by \( M = \rho \frac{4}{3} \pi R^3 \). So, substituting this in the formula for \( I \): \[ I = \frac{2}{5} \left( \rho \frac{4}{3} \pi R^3 \right) R^2 = \frac{176}{105} R^2 \rho \] Thus, the correct answer is (3).