Question

Find the moment of inertia of a solid sphere about its diameter.

• A. \( \frac{2}{3}MR^2 \)
• B. \( \frac{2}{5}MR^2 \)
• C. \( \frac{1}{2}MR^2 \)
• D. \( \frac{3}{5}MR^2 \)

Hint:

Common moment of inertia results: Ring about centre \(= MR^2\), Solid cylinder/disc \(= \frac{1}{2}MR^2\), Solid sphere \(= \frac{2}{5}MR^2\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the standard formula for the Moment of Inertia (\(I\)) of a solid sphere when rotated about an axis passing through its center (the diameter).
Step 2: Key Formula or Approach:
The Moment of Inertia depends on the distribution of mass relative to the axis. For a solid sphere of mass \(M\) and radius \(R\), the integration of mass elements results in a specific coefficient.
Step 3: Detailed Explanation:
By considering the sphere as a collection of thin disks and integrating from \(-R\) to \(+R\), or using spherical coordinates:
\[ I = \int r^2 dm \]
For a solid sphere, the resulting value is:
\[ I = \frac{2}{5}MR^2 \]
Step 4: Final Answer:
The correct formula is \(\frac{2}{5}MR^2\).