Question

A solid sphere of radius R has moment of inertia \(I\) about its diameter. What will be moment of inertia of a shell of same mass and same radius about its diameter?

• A. \( \frac{3}{5}I \)
• B. \( \frac{5}{3}I \)
• C. \( \frac{2}{3}I \)
• D. \( \frac{2}{5}I \)

Hint:

Remember standard results: solid sphere \( \frac{2}{5}MR^2 \), shell \( \frac{2}{3}MR^2 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The moment of inertia depends on how mass is distributed relative to the axis of rotation. In a shell, all mass is at the surface (further from the axis), while in a solid sphere, mass is distributed throughout.
: Key Formula or Approach:
1. \( I_{\text{solid sphere}} = \frac{2}{5} MR^2 \).
2. \( I_{\text{spherical shell}} = \frac{2}{3} MR^2 \).
Step 2: Detailed Explanation:
We are given that the solid sphere has moment of inertia \( I \).
\[ I = \frac{2}{5} MR^2 \implies MR^2 = \frac{5}{2} I \]
The moment of inertia of a spherical shell (\( I' \)) with the same mass \( M \) and radius \( R \) is:
\[ I' = \frac{2}{3} MR^2 \]
Substitute the value of \( MR^2 \) from the first equation:
\[ I' = \frac{2}{3} \times \left( \frac{5}{2} I \right) \]
\[ I' = \frac{10}{6} I = \frac{5}{3} I \]
Step 3: Final Answer:
The moment of inertia of the shell is \( 5/3 \text{ I} \).