Question

The moment of inertia of a solid sphere (radius \( R \) and mass \( M \)) about the axis which is at a distance of \( \frac{R}{2} \) from the center is:

• A. \( \frac{3}{20}MR^2 \)
• B. \( \frac{1}{2}MR^2 \)
• C. \( \frac{13}{20}MR^2 \)
• D. \( \frac{9}{20}MR^2 \)

Hint:

Always use the parallel axis theorem for axes not passing through the center: \( I = I_{\text{center}} + Md^2 \).

Answer 1

The Correct Option is C

Step 1: Recall the moment of inertia of a solid sphere about its diameter.
For a solid sphere about its central axis, \[ I_{\text{center}} = \frac{2}{5}MR^2. \]
Step 2: Use the parallel axis theorem.
If the new axis is at a distance \( d = \frac{R}{2} \) from the center, then \[ I = I_{\text{center}} + Md^2 = \frac{2}{5}MR^2 + M\left(\frac{R}{2}\right)^2. \]
Step 3: Simplify.
\[ I = \frac{2}{5}MR^2 + \frac{1}{4}MR^2 = \frac{8 + 5}{20}MR^2 = \frac{13}{20}MR^2. \]
Step 4: Final Answer.
Hence, the moment of inertia about the axis at a distance \( \frac{R}{2} \) from the center is \( \frac{13}{20}MR^2 \).