Question

The change in moment of inertia of a solid sphere of mass \( M \), radius \( R \), for a small change in temperature \( \Delta t \) is:

• A. \(\frac{2}{5} MR^2 \alpha \Delta t\)
• B. \(\frac{4}{5} MR^2 \alpha \Delta t\)
• C. \(\frac{7}{5} MR^2 \alpha \Delta t\)
• D. \(\frac{3}{5} MR^2 \alpha \Delta t\)

Hint:

The change in moment of inertia with temperature is related to the change in the radius of the body due to the linear expansion of materials.

Answer 1

The Correct Option is B

The moment of inertia \( I \) of a solid sphere is given by: \[ I = \frac{2}{5} MR^2 \] When the temperature changes by \( \Delta t \), the radius of the sphere expands. The moment of inertia changes as the sphere expands, and the change in the moment of inertia can be calculated using the coefficient of linear expansion \( \alpha \). The change in the moment of inertia is given by: \[ \Delta I = \frac{4}{5} MR^2 \alpha \Delta t \] Thus, the change in moment of inertia is proportional to the coefficient of linear expansion, the mass, the radius squared, and the change in temperature.